Stochastics and Financial Mathematics

Master Thesis

Pricing of Contingent Convertible Bonds

Author: Supervisor:Mike Derksen dhr. prof. dr. P.J.C. Spreij

dhr. prof. dr. S.J.G. vanWijnbergen

Examination date:August 29, 2017

Korteweg-de Vries Institute forMathematics

Abstract

Contingent Convertible bonds (CoCos) are bonds designed to convert to equity whena bank is close to becoming insolvent. This conversion is typically triggered by anaccounting ratio falling below some threshold. However, in the existing literature on thepricing of CoCos no difference is made between accounting values and market values.In this thesis a model is proposed which attempts to fill this gap. In the proposedmodel, debt is valued under the assumption that the only information available is noisyaccounting information, which is only received at discrete moments in time. In thisway, it is possible to distinct between market values and book values in the valuationof CoCos. Another important contribution of the model is the inclusion of the MDAregulations concerning the payment of coupons.

Title: Pricing of Contingent Convertible BondsAuthor: Mike Derksen, mike.derksen@student.uva.nl, 11060808Supervisor: dhr. prof. dr. P.J.C. Spreij, dhr. prof. dr. S.J.G. van WijnbergenSecond Examiner: mw. dr. A. KhedherExamination date: August 29, 2017

Korteweg-de Vries Institute for MathematicsUniversity of AmsterdamScience Park 105-107, 1098 XG Amsterdamhttp://kdvi.uva.nl

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Contents

Introduction 5

1 Literature overview 9

2 A structural model involving jumps 152.1 The firm’s asset value process . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 The firm’s capital structure . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.1 Straight debt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.2 Contingent convertible bonds . . . . . . . . . . . . . . . . . . . . . 162.2.3 Default . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Valuation of the firm’s liabilities . . . . . . . . . . . . . . . . . . . . . . . 172.3.1 Valuation of the straight debt . . . . . . . . . . . . . . . . . . . . . 172.3.2 Valuation of the contingent convertibles . . . . . . . . . . . . . . . 19

2.4 Computing the transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 A model with imperfect accounting information 283.1 Description of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 Conditioning on one noisy accounting report . . . . . . . . . . . . . . . . 293.3 Survival probability and default intensity . . . . . . . . . . . . . . . . . . 33

3.3.1 Survival probability . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3.2 Default intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4 Conditioning on several noisy accounting reports . . . . . . . . . . . . . . 373.5 Valuation of CoCos with a principal write-down . . . . . . . . . . . . . . . 40

3.5.1 CoCos with a regulatory trigger . . . . . . . . . . . . . . . . . . . . 403.5.2 CoCos with only a book value trigger . . . . . . . . . . . . . . . . 49

3.6 Valuation of CoCos with a conversion into shares . . . . . . . . . . . . . . 54

4 Applications 624.1 Impact of model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.1.1 Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.1.2 Accounting noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.1.3 The conversion trigger . . . . . . . . . . . . . . . . . . . . . . . . . 664.1.4 The number of shares received at conversion . . . . . . . . . . . . 66

4.2 Impact CoCos on capital structure . . . . . . . . . . . . . . . . . . . . . . 674.2.1 Replacing debt with CoCos . . . . . . . . . . . . . . . . . . . . . . 674.2.2 Replacing equity with CoCos . . . . . . . . . . . . . . . . . . . . . 684.2.3 Risk taking incentives . . . . . . . . . . . . . . . . . . . . . . . . . 684.2.4 Investment incentives . . . . . . . . . . . . . . . . . . . . . . . . . 70

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4.3 The role of the MDA trigger . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Popular summary 75

Bibliography 76

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Introduction

A Contingent Convertible bond (CoCo) is a special type of bond, which is designed toabsorb losses when the capital of the issuing bank becomes too low. When this happens,the bond converts into equity or is (partially) written down. In both cases, the debtof the issuing bank is reduced and its equity is raised. The conversion of the bond istriggered by a specified trigger event, for example the capital ratio of the bank fallingbelow a certain threshold. CoCos were first issued in 2009, when Lloyds Banking groupoffered some of its debt holders the possibility to exchange their bonds for bonds whichpossibly would convert into shares. After the financial crisis in 2007 it was realizedthat stronger regulation of the banking sector was necessary, this lead to the Basel IIIframework. In this new framework, contingent convertible instruments were included asa part of Capital, Additional Tier 1 Capital to be specific, which lead to an increasingpopularity of CoCo bonds.

Contingent convertible bonds

A Contingent Convertible bond is a bond which converts into equity or is (partially)written down at the conversion date. This means that the design of a CoCo contract isspecified by two main characteristics:

• The trigger event: when does conversion happen?

• The conversion mechanism: what does happen at conversion?

Trigger event

The trigger event specifies at which moment the conversion takes place. As in [9], wecan distinguish three types of trigger events; an accounting trigger, a market trigger anda regulatory trigger.In case of an accounting trigger, the conversion is triggered by an accounting ratio, e.g.the Common Equity Tier 1 Ratio (defined as the fraction of common equity and riskweighted assets) falling below a certain barrier. This type of trigger is typical in practice,it is however criticised by the academic world. By Flannery [11], it is argued that a bookvalue will only be triggered long after the damage has already be done, because bookvalues are not up-to-date at any moment. Therefore, the market trigger is proposed inthe literature. In case of a market trigger, the conversion would happen if a marketvalue, e.g. the share price of the issuing bank, falls below a certain threshold. A marketprice would better reflect the current situation of the issuing bank, because a marketprice is a forward looking parameter; it reflects the market’s opinion on the future of the

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bank. However, the market trigger also has its shortcomings. Sundaresan and Wang [18]and Glasserman and Nouri [13] point out that a market trigger could lead to a multipleequilibria problem for the pricing of a CoCo if the terms of conversion are beneficialto CoCo holders. In this case, a market trigger could also encourage CoCo holders toshort-sell shares of the issuing bank, to profit from a conversion, which could lead toa “death spiral”. Although the market trigger is preferred by a substantial part of theacademic world, there exist no CoCos with a market trigger in practice.

A third type of trigger is the regulatory trigger, which allows the regulator to callfor a conversion. In reality, CoCos often have a trigger which is a combination of anaccounting trigger and a regulatory trigger.

Conversion mechanism

The conversion mechanism specifies what happens at the moment of conversion, thereare two possibilities: a (partial) principal write-down or a conversion into shares. Incase of a (partial) principal write-down mechanism, the principal of the CoCo bond is(partially) written down at the moment of conversion, to strengthen the capital positionof the issuing bank. In case of a conversion into shares, the principal of the CoCo bondis converted into an amount of shares. Of course, it needs to be specified how manyshares a CoCo holder receives at conversion. This conversion rule can be designed intwo different ways. One possibility is that the CoCo holder receives a fixed number ofshares for every dollar of principal. Another option is a variable number of shares, inthis case the CoCo holder would “buy” a number of shares against a conversion price.This conversion price can be set as the market price of shares, which possibly leads toan infinitely large dilution of the existing shareholders. A way to avoid this, is to placea cap on the conversion price.

Pricing of CoCos

Due to the hybrid nature of Contingent Convertible bonds, a lot of different pricingmodels have been proposed in the literature. Following [19], these can be roughly groupedinto three categories: structural models (see e.g. [1], [6],[12], [17]), equity derivativemodels (see e.g. [8], [9]) and credit derivative models or reduced form models (see e.g.[9]). For a comprehensive overview of the existing literature on the pricing of CoCos,see Chapter 1. As pointed out in this literature overview, the above mentioned modelsdiffer in applications and complexity (jumps/no jumps, constant/stochastic risk freerate, finite maturities/perpetuities, fixed/variable shares at conversion, etc.). However,they all have in common that a conversion trigger based on market values is used, whilein reality all CoCos have a trigger based on the regulatory capital ratio, a book value.

Maximum Distributable Amount

As Contingent Convertible bonds qualify as a form of capital in the Basel III regulations,they are also affected by the concept of the Maximum Distributable Amount (MDA),

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which requires regulators to stop earnings distributions when the bank’s capital becomestoo low. An example of these earnings distributions are dividends. However, as CoCosqualify as a form of capital, the coupons on CoCos are also affected by the MDA. Thismeans that when the bank’s capital falls below some threshold, higher than the CoCo’sconversion trigger, the payment of coupons is stopped until the bank’s capital is againabove the MDA trigger.

Contents of this thesis

This thesis is organized as follows. In the first chapter an overview of the existing lit-erature on the pricing of Contingent Convertibles is provided. In the second chaptera structural model, allowing for jumps in the asset process, is described. This is basi-cally the model proposed by Chen, Glasserman, Nouri and Pelger [7], for which all themathematical details left out in the paper are filled in. This pricing model is a very richmodel, which is still tractable due to the use of exponential distributions and leads toclosed form solutions. However, as all the models in the existing literature, it makes nodistinction between market values and book values in the valuation of CoCos.In Chapter 3 of this thesis a model is proposed which does make a distinction betweenmarket values and accounting values and in which also early cancelling of coupons, ascaused by the above described MDA regulations, is considered. The model developedin this chapter thus contributes in two different ways to the existing literature; it dis-tinguishes between market and book values of assets in the valuation of CoCos and itallows the coupons of CoCos to be already cancelled at a moment before the conver-sion date. The model is based on the model by Duffie and Lando [10], in which debtis valued under the assumption that the only information available is noisy accountinginformation which is received at selected moments in time. This setting is particularlyrelevant for the pricing of CoCos since, as pointed out above, conversion triggers are al-ways based on imperfect accounting ratios observed at discrete moments in time, ratherthan on continuously observable market prices. The first part of Chapter 3 is devoted toa comprehensive description of the model proposed by Duffie and Lando, including thederivation of all the relevant formulas and proofs left out in the original paper. Afterthis, in the sections 3.5 and 3.6, this thesis goes beyond the paper by Duffie and Lando,as explicit formulas and algorithms for the pricing of CoCos are provided. The settingis applied to the valuation of different kinds of CoCo bonds, namely CoCos with a (par-tial) principal write down and CoCos with a conversion into shares. Also a distinctionis made between CoCos with a regulatory trigger, for which conversion could happenat any moment in time, and CoCos that can only be triggered at one of the accountingdates. The model does not lead to closed form solutions, but the expressions for CoCoprices involve integrals that are computed using MCMC-methods.The last Chapter of this thesis is devoted to some applications of the model describedin Chapter 3. The impact of several model parameters on the price of a CoCo is exam-ined and the impact of the issuance of CoCos on the capital structure and on incentivesfor shareholders is investigated. Finally, the model is applied in an attempt to explain

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the big downward price jump that CoCos of Deutsche Bank suffered at the beginningof 2016 after the release of a profit warning. In this particular case the added valueof the proposed model becomes most clear as it allows for the announcement of a badaccounting report. Also, the sudden price drop seemed to be more out of fear for theMDA trigger than for the conversion trigger, as the conversion trigger was still far out ofreach. So the two most important contributions of the model, the notion of accountingreports and the inclusion of the MDA trigger, seem to be very relevant to this case.

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1 Literature overview

Due to the hybrid nature of Contingent Convertible bonds, a lot of different pricingmodels have been proposed in the literature. Following [19], these can be roughly groupedinto three categories: structural models (see e.g. [1], [6], [12], [17]), equity derivativemodels (see e.g. [8], [9]) and credit derivative models or reduced form models (see e.g.[9]).In a structural model one starts to describe the value of the assets of a firm by introducinga stochastic process. Also the liabilities are described and the capital is given by thedifference between the assets and the liabilities. Conversion of CoCos occurs if a specifiedtrigger event happens, for example if the market value of the firm’s assets [1,6] or thefirm’s capital ratio [17] falls below a predetermined threshold.

Albul, Jaffee and Tchistyi [1] consider a model in which the firm’s value of assets Atfollows a geometric Brownian motion process under the risk-neutral measure, given by

dAt = µAtdt+ σAtdWt.

Here µ and σ are constants and W is a standard Brownian motion. Furthermore, therisk free rate is assumed to be constant. The firm issues two types of debt; a straightbond and a CoCo bond, both with perpetual maturities. Both pay coupons continuouslyat a constant rate. The CoCo is assumed to convert at the first time the value of assetsdrops below some specified threshold ac, i.e. the conversion time is given by

τ(ac) = inf{t ≥ 0 : At ≤ ac}.

It is assumed that the CoCo fully converts into equity against market value of sharesat a specified conversion ratio λ (λ equal to 1 means the CoCo holders receive equity,valued at its market price, equal to the face value of the bond). Liquidation of the firmis also incorporated in the model, by assuming that the equity holders liquidate the firmwhen the value of assets falls below some optimal threshold, chosen by the shareholdersto maximize equity value. Furthermore, it is assumed that default cannot occur beforeconversion. Now the value of the various claims (including the CoCos) is given by therisk-neutral expectation of the discounted future cashflows regarding the claim. In caseof the CoCos this leads to a value at time t < τ(ac), given by

Vt = EQ

(∫ τ(ac)

te−r(s−t)ccds+ e−r(τ(ac)−t)λ

ccr

).

Here the first term represents the discounted coupon payments, paid at rate cc, untilconversion, while the second term accounts for the equity received by the CoCo holders

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at conversion. Note that the face value of the CoCo is given by cc/r, such that the CoCoholders receive λcc/r at conversion. Due to the tractable setting (geometric Brownianmotion, constant risk free rate), this leads to easy closed form solutions. For instancethe value of a CoCo at time t reads

Vt =ccr

(1−

(Atac

)−γ)+

(Atac

)−γλccr,

where

γ =1

σ2

((µ− σ2/2) +

√(µ− σ2/2)2 + 2rσ2

).

Chen, Glasserman, Nouri and Pelger [6] propose a more involved model in which themarket value of assets follows not only a geometric Brownian motion, but in whichthe asset value process also involves jumps, with a distinction between market-widejumps and firm-specific jumps. For tractability it is assumed that the log-values ofthe jump sizes have exponential distributions. Again, the risk free rate is taken as aconstant. The bank issues four kinds of debt: insured deposits, senior and subordinateddebt and contingent convertible bonds. All pay coupons (interest in case of the deposits)continuously at a constant rate. It is assumed that all debt has an exponential distributedmaturity, which could easily be replaced by perpetuities, tractability would be retainedin that case, by taking the rate parameter of the exponential distribution equal to zero.This leads to a setting in which the par value of outstanding debt remains constant forall types of debt. Again, conversion of CoCos into equity is triggered the first time thevalue of assets falls below some specified threshold. In contrast to the variable sharesfeature in [1], at conversion the CoCo holders receive a fixed number of shares for everydollar of principal. The model also involves a notion of default, endogenously. Similar tothe situation in [1], it is assumed that the shareholders declare the firm bankrupt whenthe value of assets falls below some optimal threshold, chosen by the equity holdersto maximize equity value. Again the firm’s liabilities are valued by discounting theirfuture cashflows and taking expectations under the risk neutral measure. In case of aCoCo, these future cashflows are the coupons paid until either maturity or conversion,the principal paid at maturity if the bond matures before conversion and the value ofequity received at conversion if conversion occurs before maturity. Due to the use ofexponential distributions (for both jumpsizes and maturities) this valuation leads toclosed form solutions. For a detailed description of a simplified version of this model,see chapter 2.

Pennacchi [17] also considers a jump-diffusion process for the dynamics of the marketvalue of assets, however only one type of jumps is considered, no distinction betweenmarketwide or firm-specific jumps is made. To be specific, the instanteneous rate ofreturn A∗t earned on the firm’s assets, which have time t value At, is modeled by thefollowing dynamics under the risk-neutral measure Q

dA∗tA∗t

= (rt − λtkt)dt+ σdWt + (Yqt− − 1)dqt.

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Here W is a standard Brownian motion under Q and q is a Poisson process with inten-sity λt and kt := EQ(Yqt− − 1) is the expected proportional jump in case of a Poissonevent. Furthermore, the risk free rate is not taken as a constant, but assumed to evolvestochastically in time, the dynamics given by a Cox-Ingersoll-Ross model. The firmagain issues deposits, straight bonds and CoCos. The deposits of the bank have instan-teneous maturities, meaning that they are very short-term sources for funding of thebank, while the straight debt and the CoCos are perpetual bonds. The market valueof deposits outstanding, denoted by Dt, follows a process, which relates the growth ofdeposits positively to the asset-to-deposit ratio xt = At/Dt, given by

dDt = g(xt − x)dt.

Here, x > 1 is the target asset-to-deposit ratio and g > 0 is a constant. This can beinterpreted as follows; when the asset-to deposit ratio lies above the target, the bank willissue more deposits, while if the asset-to-deposit ratio is below the target, the value ofoutstanding deposits will decrease. In this way, a mean-reversion is created in which theasset-to-deposit ratio will converge to the target. This setting is supported by empiricalevidence that banks have target capital ratios and that deposits increase when there isa capital surplus, while they decrease in case of a capital shortage. Furthermore, thedeposits pay out interest plus deposit premiums continuously at some stochastic timedependent rate, given by an insurance premium on top of the risk free rate. Failure of thebank is also incorporated in the model, occuring the first time the value of assets dropsbelow the deposit value. The bonds (either straight debt or CoCos) continuously payfixed or floating coupons and have maturity T . The CoCos are triggered if the marketvalue of the bank’s asset-to-deposit ratio xt = At/Dt falls below some specified thresholdx. It is assumed that the face value of the CoCo is fully converted into equity, followinga conversion mechanism specified by two parameters, p and α. The first parameter paccounts for the maximum proportion of the par value that CoCo holders can receivein the form of new shares and α for the maximum proportion of all shares that CoCoholders can receive. In this way, the model is suitable for both a variable number ofshares (as in [1]) and a fixed number of shares (as in [6]). Denote by τc the time ofconversion, i.e. τc = inf{t ≥ 0 : xt ≤ x}, and by B the face value of the CoCo, then thevalue of the contingent capital at conversion is given by

Vtc =

pB if pB < α(Aτc −Dτc)

ατc(Aτc −Dτc) if 0 < α(Aτc −Dτc) ≤ pB0 if Aτc −Dτc ≤ 0

(1.1)

It is also assumed that the CoCo pays coupons continuously at rate ct. Now the valueof the CoCo at time 0 is given by

V0 = EQ

(∫ T

0e−

∫ t0 rsdsv(t)dt

),

where v(t) denotes the cashflow per unit time paid at date t. This cashflow is ctB aslong no conversion has happened. If at time T no conversion has happened and also the

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bank did not fail, there is a final cashflow, the payment of the principal B. If tc ≤ Tthere is a cashflow at tc, with a value given by equation (1.1). Due to the richness ofthe model, no closed form solution is obtained, but a Monte Carlo simulation method isproposed to evaluate the above expectation.

All of the above models make use of a conversion trigger based on the market price of afirm’s equity. Because the market price of equity depends on the firm’s capital structureitself, which changes in case of conversion, questions rise about the internal consistencyof this type of trigger. This problem is analysed by Glasserman and Nouri [13]. Theyconsider a post-conversion firm, for which the contingent capital is already convertedprior to time zero and a no-conversion firm, which is not subject to a conversion at all.The question then is whether there exists a stock price process for the original firm thatis equal to the no-conversion stock price before the trigger event and equal to the post-conversion value afterwards. Such a stock price is then called an equilibrium stock price.A second question is whether such an equilibrium is unique or if there are multiple equi-libria. Reasonable conditions are stated under which existence and uniqueness hold. Forexistence it is required that the no-conversion price is higher than the post-conversionprice, when both are above the trigger. This condition can be interpreted to mean thatshareholders suffer from a conversion. For uniqueness it is required that the likelihoodthat the no-conversion price reaches the trigger is sufficiently large. The results hold ina setting in which the asset value process is a geometric Brownian motion. By a smallmodification of the precise conditions, it is shown that a unique equilibrium also existsif jumps are incorporated in the asset value process.All the above described models have a similar structure, but they differ in applica-tions and complexity (jumps/no jumps, constant/stochastic risk free rate, finite matu-rities/perpetuities, fixed/variable shares at conversion, etc.). However, they all have incommon that a conversion trigger based on market values is used, while in reality almostall CoCos have a trigger based on the regulatory capital ratio, a book value. An attemptto distinguish between market values and book values of assets is made by Glassermanand Nouri [12]. They consider a model in which the conversion of contingent capital ispartial and ongoing. That means, every time the capital ratio falls below a threshold,just enough conversion takes place to retain the capital ratio at the minimum level re-quired. This is in contrast with the situation in reality and the above models in whichconversion takes place all at once, the first time the trigger event occurs. The dynamicsof the book value of assets are modeled by a geometric Brownian motion

dVtVt

= (r − δ)dt+ σdWt,

where r is the risk-free rate and δ is a constant pay out rate. The key assumption madeto relate book values to market values, denoted by At, is that they largely agree onwhether a bank is solvent. So it is assumed that the market value of assets is greaterthan the total debt outstanding whenever the book value of assets is greater than thetotal debt outstanding, that is

At > Bt +D whenever Vt > Bt +D,

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where D denotes the book value of the straight debt (assumed to be constant) and Btdenotes the book value of contingent capital outstanding. Then a second geometricBrownian motion U is introduced

Ut = U0 exp(θut+ σuW′t),

where W ′ is a second Brownian motion and the instanteneous correlation between Wand W ′ is given by ρ. U can be roughly interpreted as a market-to-book ratio. Thatis, the difference of the market value of assets and total debt is equal to this geometricBrownian motion times the difference of the book value of assets and the total debtoutstanding:

At −Bt −D = Ut(Vt −Bt −D).

This method to relate book values and market values introduces two extra parameters;the volatility σu of the second geometric Brownian motion (the “book-to-market volatil-ity” ) and the instantaneous correlation ρ between the two Brownian motions. Theseparameters should be calibrated using market values of the banks’s debt and equityand book values from financial statements. This leads to a problem regarding the bookvalues; using the above approach, it is assumed that book values can be observed con-tinuously. In practice however, regulatory capital ratios are only calculated quarterly.

Brigo, Garcia and Pede [4] also consider a trigger event which is not based solely onmarket values, but is, as in reality, related to regulatory capital. They propose a modelin which the value of the firm is modeled by a geometric Brownian motion, where thevolatility is allowed to be time-dependent. After this, also a process for the regulatorycapital of the firm, denoted by ct is needed, because the CoCos are triggered when thisregulatory capital falls below some threshold. Instead of modeling the regulatory capitaldirectly, it is seen as an exogenous variable. A proxy for its value is then estimated by alinear regression, where it is assumed that the asset-to-equity ratio is the driver for theregulatory capital, that is

ci = α+ βXi + εi,

where ε is the residual term and Xi is the asset-to-equity ratio, defined by Xi = Ai/(Ai−Li), where At denotes the value of assets at time t and Lt the value of liabilities.

The above described models can all be categorized as structural models. In [9] twowhole different approaches to the pricing of CoCos are described. The first one is a creditderivatives model, which is set up from the point of view of a fixed income investor. Afixed income investor would compute how much yield is needed on top of the risk free rateto compensate for the possible loss in case the CoCo is triggered. The second proposedapproach is an equity derivatives model. In this model the problem of pricing a CoCois approached from the point of view of an equity derivatives specialist, who will see aCoCo as a long position in shares that are knocked in when the CoCo is triggered.For the first case, a reduced form approach is used. One describes the likeliness of atrigger event by a trigger intensity λ. A Black-Scholes setting is used to determine thevalue for the intensity. That is, the stock price St is assumed to follow a geometricBrownian motion

dSt = µStdt+ σStdWt.

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It is assumed that the CoCo converts when the stock price falls below a certain thresholdS∗. Due to the Black-Scholes setting there is a closed form solution for the probabilityp∗ that the threshold is reached during the lifetime of the contingent convertible. Thevalue for the trigger intensity can then be derived from this probability, by

λ = − log(1− p∗)T

.

After this, it is possible to compute the credit spread needed on top of the risk free rateto compensate for the possible loss in case of a trigger event. This credit spread is givenby the trigger intensity times the fraction of the face value that is lost at conversion,that is

cs = λ

(1− S∗

Cp

),

where Cp is the conversion price.In the equity derivatives approach, one tries to replicate the payoff of the CoCo by usingequity derivatives. The CoCo is seen as the following combination of equity derivatives

CoCo = Straight Bond + Knock-In Forwards− Binary Down-In Options.

Here the long position in knock-in forwards corresponds to the possible purchase of shares(against the conversion price) in case the trigger event occurs (that is, the forwards areknocked-in). The short position in the binary down-in options reflects the reduction ofcoupons after conversion. Now the values of the knock-in forwards and BDI options canbe computed in a Black-Scholes setting, which leads to a closed form solution for the priceof a CoCo. Using this approach, it is assumed that the CoCo investor receives forwardsat conversion. However, in reality the investor receives shares at conversion. This makesa major difference if the trigger event occurs a long time before the expiration dateof the CoCo. For example, shares would entitle the investor immediately to dividendsand voting rights, while this is not the case for a forward on those shares. Under thereasonable assumption that dividends will be low after the trigger event occurs, thisequity derivatives approach will be an acceptable model. Another drawback of themodels proposed in [9] is the fact that both models use a trigger driven by the stockprice of the company, which should in some way be linked to the actual accounting ratiotrigger. However, it is unclear how these two quantities should be linked. Furthermore,both of the approaches make use of a Black-Scholes setting, in which the stock priceprocess follows a geometric Brownian motion. However, CoCos come with a lot of fattail risk, which can not easily be handled in the Black-Scholes model, so other, betterfitting, processes should be considered to improve the models. This will come at thecost of replacing the closed form solutions for simulation based solutions. Corcuera, DeSpiegeleer, Ferreiro-Castilla, Kyprianou, Madan and Schoutens [8] work this out, nowthe equity derivative approach is not applied in a Black-Scholes setting, but the stockprice dynamics follow an exponential Levy process incorporating jumps and fat tails.

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2 A structural model involving jumps

In this chapter the structural model proposed by Chen et al [7], which is an updatedversion of [6] and contains a simplified version of the original model, is considered.

2.1 The firm’s asset value process

Consider a firm generating cash continuously at a rate {δt, t ≥ 0}. The dynamics of theincome flow are taken as a jump-diffusion process, given by

dδtδt−

= µdt+ σdWt + d

Nt∑i=1

(Yi − 1)

. (2.1)

Here δt− is the value of the income flow just before a possible jump at time t, µ, σare constants and W = {Wt : t ≥ 0} is a standard Brownian motion. Furthermore,N = {Nt : t ≥ 0} is a Poisson process with intensity λ, which drives the jumps withsizes given by {Yi : i = 1, 2, . . . }. Because only downward jumps are relevant concerninga trigger event, it is assumed that Yi < 1 for every i. For the sake of tractability, thejumpsizes are taken to be log-exponentially distributed, that is

Zi := − log(Yi) ∼ exp(η) for some η > 0.

Furthermore, it is assumed that the jump sizes {Yi : i = 1, 2, . . . }, W and N are allindependent of each other and that the risk free rate r is constant. Denote by Ft thesigma-algebra generated by δt, that is Ft = σ (δs, s ≤ t).Now the dynamics of the asset value process can be stated explicitly, following Kou [15].

Theorem 2.1. In a rational expectations framework, the equilibrium price of a claimon future income of the firm is given by the expected value of the discounted payoff ofthe claim under a risk-neutral measure Q. It follows that the value of the firm’s assetsVt at time t is given by

Vt = EQ

(∫ ∞t

e−r(s−t)δsds∣∣∣Ft) .

Furthermore, Vt/δt is a constant, denote it by δ, and the Q-dynamics of Vt are given by

dVtVt−

=

(r − δ +

λ

1 + η

)dt+ σdWt + d

(Nt∑i=1

(Yi − 1)

), (2.2)

15

where, under Q, W is a Brownian motion and N is a Poisson process with intensity λ.The distribution of the jump sizes Yi is identical to that of Yi, but now with a differentparameter η.

The proof of the theorem and explicit expressions for the new parameters can be foundin [15].

2.2 The firm’s capital structure

The assets of the firm are financed through two types of debt, straight debt and contin-gent convertible debt (CoCos), and also through equity. In this setting deposits are aspecial case of straight debt, in which the maturity of straight debt corresponds to thetime until depositors withdraw their money.

2.2.1 Straight debt

The straight debt issued by the firm is the most senior claim at default, i.e. at defaultthe holders of straight debt have the first claim at the firm’s assets. We assume thatthe firm continuously issues straight debt at rate p1. That is, the par value of the debtissued in the interval (t, t + dt) is given by p1dt. Furthermore it is assumed that thematurity of the debt is an exponentially distributed variable with rate parameter m.The assumptions that the maturity is exponential and that the issuance rate is constantlead to a setting in which the total par value of debt outstanding is constant, given by

P1 :=

∫ ∞t

∫ t

−∞p1me

−m(s−u)duds =p1

m

This follows from the fact that in the interval (t, t+ dt) debt of value p1dt is issued, butfor all s ≥ 0 a portion of m exp(−ms)ds of the total value p1dt matures in the interval(t+ s, t+ s+ ds) . The debt also pays a coupon continuously at rate c1 per unit of thepar value of debt. The coupon payments are tax-deductible, where the marginal tax rateis given by κ1, 0 ≤ κ1 < 1. It follows that the net value of coupon payments is given by(1− κ1)c1P1.

2.2.2 Contingent convertible bonds

For the issuance and maturity of CoCos the same approach as for the straight debt isused. That is, CoCos are issued continuously at rate p2 and their maturity is againexponentially distributed with rate parameter m. This leads, similar as above, to aconstant total par value of CoCos outstanding, denoted by P2. Furthermore, the CoCosalso pay a coupon continuously at rate c2. Note that, to capture the case that straightdebt and CoCos have perpetual maturities, it suffices to set m = 0, which implies T =∞.Conversion of the CoCos is triggered when the asset value of the firm falls below a specificthreshold vc. That is, conversion occurs at

τc = inf{t ≥ 0 : Vt ≤ vc}. (2.3)

16

In reality a CoCo converts if the capital ratio of the firm falls below some given threshold.This could be implemented in this setting by saying that the CoCos convert if

Vt − P1 − P2

Vt≤ ρ,

where ρ ∈ (0, 1). This is compatible with the setting in equation (2.3), by taking

vc =P1 + P2

1− ρ.

It should be noted that in this setting a market value of the capital ratio is used regardingthe trigger event, while in reality this is a book value.At conversion, the CoCo holder receives a fixed number of shares, denoted by ∆, forevery dollar of principal value, so a total of ∆P2 shares is provided to the CoCo holders.By normalizing the number of shares before conversion to 1, it follows that the CoCo

holders own a fraction∆P2

1 + ∆P2of the firm after conversion.

2.2.3 Default

The firm is declared bankrupt the first time the asset value falls at or below somethreshold vb. Thus, bankruptcy occurs at

τb = {t ≥ 0 : Vt ≤ vb}.

At bankruptcy a fraction (1 − α), 0 ≤ α ≤ 1 of the asset value of the firm is lost tobankruptcy costs. This means that at the moment of bankruptcy the asset value is givenby αVτb . We assume that conversion takes place before the firm defaults, i.e. vb ≤ vc.This is a natural assumption in the sense that CoCos are actually designed to preventthe firm from defaulting.

2.3 Valuation of the firm’s liabilities

The model described above leads to closed-form solutions for the value of both thestraight debt and the CoCos.

2.3.1 Valuation of the straight debt

The value at time t of straight debt with unit face value and time to maturity T is givenby

b(Vt, T ) = EQ

(e−rT1{τb>T+t}

∣∣∣Ft)+ EQ

(e−r(τb−t)1{τb≤T+t}

αVτbP1

∣∣∣Ft)+ EQ

(∫ τb∧(T+t)

tc1e−r(s−t)ds

∣∣∣Ft) (2.4)

17

In this valuation, different cases are considered. The first term on the right describesthe principal payment if no default occurs before the debt matures. The second termdenotes the payment at default when default occurs before maturity; at default the assetvalue is given by αVτb , which has to be divided among all the straight debt holders, so a

holder of straight debt with unit face value will receiveαVτbP1

. The last term represents thediscounted value of the coupon payments. From now on, we will take t = 0 to simplifynotation (the conditional expectations become ordinary expectations in this case). Itshould be kept in mind that the value of both straight debt and CoCos depends on thecurrent value of assets, denoted by V . Now, because the total par value of straight debtis P1 and the maturity satisfies T ∼ exp(m), the total market value of straight debt isgiven by

B(V ) = P1

∫ ∞0

b(V, T )me−mTdT.

By inserting equation (2.4) in the above it follows that

B(V ) = P1

∫ ∞0

EQ(e−rT1{τb>T}

)me−mTdT + P1

∫ ∞0

EQ

(e−rτb1{τb≤T}

αVτbP1

)me−mTdT

+ P1

∫ ∞0

EQ

(∫ τb∧T

0c1e−rsds

)me−mTdT. (2.5)

Here, the first integral on the right hand side of equation (2.5) is given by

P1

∫ ∞0

EQ(e−rT1{τb>T}

)me−mTdT = mP1EQ

∫ τb

0e−(r+m)TdT

=mP1

m+ rEQ

(1− e−(m+r)τb

).

Furthermore, the second integral is given by

P1

∫ ∞0

EQ

(e−rτb1{τb≤T}

αVτbP1

)me−mTdT = mEQ

(e−rτbαVτb

∫ ∞τb

e−mTdT

)= EQ

(αVτbe

−(m+r)τb).

And the last integral is given by

P1

∫ ∞0

EQ

(∫ τb∧T

0c1e−rsds

)me−mTdT = P1mc1EQ

(∫ τb

0

∫ T

0e−rse−mTdsdT

)+ P1mc1EQ

(∫ ∞τb

∫ τb

0e−rse−mTdsdT

)= P1mc1EQ

(∫ τb

0

1

r(1− e−rT )e−mTdT

)+ P1mc1EQ

(∫ ∞τb

1

r(1− e−rτb)e−mTdT

)

18

=P1c1

rEQ(1− e−mτb

)− P1mc1

r(m+ r)EQ

(1− e−(m+r)τb

)+P1c1

rEQ

(e−mτb − e−(m+r)τb

)=

c1P1

m+ rEQ

(1− e−(m+r)τb

).

Inserting these three expressions into equation (2.5) implies that the market value oftotal straight debt outstanding is given by

B(V ) =P1(m+ c1)

m+ rEQ

(1− e−(m+r)τb

)+ EQ

(αVτbe

−(m+r)τb). (2.6)

From the expression in equation (2.6) it follows that the key to valuation of the straightdebt is the joint Laplace transform of τb and the log-asset value log Vt. In Section 2.4,it will be shown that this transform has a closed form solution, which leads to a closedform solution for the value of straight debt.

2.3.2 Valuation of the contingent convertibles

Again we start by computing the market value of a CoCo with a unit face value andmaturity T , which is given by

d(V, T ) = EQ(e−rT1{τc>T}

)+ EQ

(∫ T∧τc

0c2e−rsds

)+

∆

∆P2 + 1EQ(e−rτcEPC(Vτc)1{τc<T}

).

(2.7)

Here EPC(v) is the value of the firm’s equity after conversion, at asset value v. Sothis value corresponds to a firm with a total par value of straight debt given by P1 andwith no CoCos. At conversion, all the CoCo investors together own a fraction ∆P2

∆P2+1

of the firm, so the holder of a CoCo with unit face value will obtain a fraction ∆∆P2+1

of the firm, this explains the last term in equation (2.7). The first term represents thepayment of the principal if the CoCo matures before conversion is triggered, while thesecond term accounts for the coupon payments until either maturity or conversion.In the same way as before, the total market value of CoCos is now given by

D(V ) = P2

∫ ∞0

d(V, T )me−mTdT.

Following exactly the same calculations as in the case for straight debt, it follows that

D(V ) =P2(m+ c2)

m+ rEQ

(1− e−(m+r)τc

)+ EQ

(∆P2

∆P2 + 1EPC(Vτc)e

−(m+r)τc

). (2.8)

Now the post-conversion value of equity at conversion EPC(Vτc) still needs to be com-puted. To this end, first the post-conversion firm value is computed and thereafter the

19

value of debt is substracted to obtain the equity value. Note that after conversion,straight debt is the only debt remaining, so it follows that

EPC(Vτc) = FPC(Vτc)−B(Vτc).

Now note that the firm value upon conversion is given by

FPC(Vτc) = Vτc + EQ

(∫ τb

τc

κ1c1P1e−r(s−τc)ds

∣∣∣Fτc)− EQ

(e−r(τb−τc)(1− α)Vτb

∣∣∣Fτc)= Vτc +

κ1c1P1

rEQ

(1− e−r(τb−τc)

∣∣∣Fτc)− EQ

(e−r(τb−τc)(1− α)Vτb

∣∣∣Fτc) .Here the first term is just the unleveraged firm value, i.e. the firm’s value if it wouldcarry no debt and would be entirely financed trough equity. The second term accountsfor the tax benefits, while the third term represents bankruptcy costs. Furthermore,note that the conversion does not affect the value of straight debt, such that equation(2.6) applies. By modifying this equation to the setting in which the present time is τc,it follows that

B(Vτc) =P1(m+ c1)

m+ rEQ

(1− e−(m+r)(τb−τc)

∣∣∣Fτc)+ EQ

(αVτbe

−(m+r)(τb−τc)∣∣∣Fτc) .

This leads to the following expression for the equity value upon conversion

EPC(Vτc) = Vτc +κ1c1P1

rEQ

(1− e−r(τb−τc)

∣∣∣Fτc)− EQ

(e−r(τb−τc)(1− α)Vτb

∣∣∣Fτc)− P1(m+ c1)

m+ rEQ

(1− e−(m+r)(τb−τc)

∣∣∣Fτc)− EQ

(αVτbe

−(m+r)(τb−τc)∣∣∣Fτc) .

(2.9)

So, as before, the valuation boils down to finding a formula for the joint Laplace trans-form of τc, τb and log Vt. This is considered in the next section.

2.4 Computing the transforms

In this section closed form solutions for the transforms needed in the valuation of thefirm’s liabilities are derived following the method proposed by Cai et al [5]. In thissection all dynamics, expressions and expectations considered are with respect to therisk-neutral measure Q. Recall from equation (2.2) that the dynamics of the asset valueprocess are given by

dVtVt−

=

(r − δ +

λ

1 + η

)dt+ σdWt + d

(Nt∑i=1

(Yi − 1)

).

This stochastic differential equation is solved by

Vt = V0 exp

((r − δ +

λ

1 + η− σ2/2

)t+ σWt

) Nt∏i=1

Yi.

20

Now denote Xt = log(Vt), µ =(r − δ + λ

1+η − σ2/2)

and recall that − log(Yi) = Zi ∼exp(η), it then follows that

Xt = X0 + µt+ σWt +

Nt∑i=1

(−Zi).

Also, we again denote Ft = σ(Vs : s ≤ t) = σ(Xs : s ≤ t). Now note that X is a Levyprocess with Levy exponent

ξ(s) :=1

tlogE (exp(sXt)) = µs+

1

2σ2s2 − λ s

η + s,

which satisfies the following lemma, due to Kou and Wang [16].

Lemma 2.1. The equation ξ(s) = a has three distinct real roots β,−γ1,−γ2, whereβ, γ1, γ2 > 0 and all these roots are different from η.

Proof. Note that ξ(s) is convex on the interval (−η,∞). Also ξ(0) = 0, lims↓−η ξ(s) =∞and lims→∞ ξ(s) = ∞. So it follows there exists a unique −γ1 ∈ (−η, 0) such thatξ(−γ1) = a and there exists a unique β ∈ (0,∞) such that ξ(β) = a. Furthermore itholds that lims↑−η ξ(s) = −∞ and lims→−∞ ξ(s) = ∞, so there must be at least 1 rootof ξ(s) = a on (−∞,−η). But (η + s)ξ(s) is a polynomial of order three, so there areat most three real roots of the equation. Hence there exists a unique −γ2 ∈ (−∞,−η)such that ξ(−γ2) = a.

Now denote τx = inf{t ≥ 0 : Xt ≤ x} for a constant x. Note that X can reach or crossthe barrier x in two ways; with or without a jump at τx. Let J0 denote the event thatthe barrier is reached without a jump at τx and J1 the event that the barrier x is crossedwith a jump at τx. We want to say something about the overshoot x−Xτx in the secondcase, so define the events F0 := {Xτx = x} ∩ J0, F1 := {Xτx < x + y} ∩ J1 for somenegative y. As mentioned above, to find solutions for the pricing of the liabilities, theonly quantities that still need to be evaluated are of the form

ui(x0) = E(e−aτx+θXτx1Fi

∣∣∣X0 = x0

), i = 0, 1, (2.10)

for constants a ≥ 0 and θ.To this end, we first need the following two lemmas, of which the first one is a modifiedversion of a result by Kou and Wang [16].

Lemma 2.2. The joint distribution of τx and the overshoot x − Xτx satisfies, for anyy > 0:

(i) P (τx ≤ t, x−Xτx ≥ y) = e−ηyP (τx ≤ t, x−Xτx > 0) .

(ii) P (x−Xτx ≥ y|J1) = e−ηy.

(iii) P(τx ≤ t, x−Xτx ≥ y|J1) = P(τx ≤ t|J1)P(x−Xτx ≥ y|J1).

21

Proof.

(i) Denote by T1, T2, . . . the arrival times of the poisson process N . Note that for y > 0,x−Xτx ≥ y can only happen with a jump at τx, hence τx is one of this arrival times.So we can write

P (τx ≤ t, x−Xτx ≥ y) =

∞∑n=1

P(Tn = τx ≤ t, x−XTn ≥ y).

Now writePn := P(Tn = τx ≤ t, x−XTn ≥ y)

and observe that

Pn = P(

min0≤s<Tn

Xs > x, (−XTn) ≥ y − x, Tn ≤ t)

= E(E(1{−XTn≥y−x}1{min0≤s<Tn Xs>x,Tn≤t}|Tn,FTn−

))= E

(P (−XTn ≥ y − x|Tn,FTn−) 1{min0≤s<Tn Xs>x,Tn≤t}

),

where FT− is defined as FT− = σ (F0 ∪ {As ∩ {s < T} : As ∈ Fs, s ≥ 0}) for a stop-ping time T .Furthermore note that

−XTn +X0 + µTn + σWTn −n−1∑i=1

Zi = Zn ∼ exp(η),

from which it follows that

P (−XTn ≥ y − x|Tn,FTn−) = exp

(−η

(y − x+X0 + µTn + σWTn −

n−1∑i=1

Zi

))= e−ηyP (−XTn > −x|Tn,FTn−) .

Hence

Pn = e−ηyE(P(−XTn ≥ −x|Tn,FT−

n

)1{min0≤s<Tn Xs>x,Tn≤t}

)= e−ηyP

(min

0≤s<TnXs > x, x−XTn > 0, Tn ≤ t

)= e−ηyP (x−XTn > 0, Tn = τx ≤ t) .

So we conclude that

P (τx ≤ t, x−Xτx ≥ y) =∞∑n=1

Pn

=

∞∑n=1

e−ηyP (x−XTn > 0, Tn = τx ≤ t)

= e−ηyP (τx ≤ t, x−Xτx > 0) ,

which proves the first part.

22

(ii) First note that J1 = {x − Xτx > 0}. Furthermore, by letting t → ∞ in (i) andnoting that τx <∞ on the set J1 by definition, we have

P (x−Xτx ≥ y) = e−ηyP (x−Xτx > 0) ,

which implies

P (x−Xτx ≥ y|J1) =P (x−Xτx ≥ y)

P (x−Xτx > 0)= e−ηy.

(iii) From (i) and (ii) it follows that

P(τx ≤ t, x−Xτx ≥ y|J1) =P(τx ≤ t, x−Xτx ≥ y)

P(x−Xτx > 0)

= e−ηyP(τx ≤ t, x−Xτx > 0)

P(x−Xτx > 0)

= e−ηyP(τx ≤ t|J1)

= P(τx ≤ t|J1)P(x−Xτx ≥ y|J1).

Lemma 2.3. For any a > 0 and l ∈ iR, it holds that

Mt := exp(−at+ lXt)− exp(lX0)− (ξ(l)− a)

∫ t

0exp(−as+ lXs)ds

is a zero-mean martingale with respect to (Ft)t≥0.

Proof. First note that for s < t

E(Mt|Fs) = Ms + E(

exp(−at+ lXt)− exp(−as+ lXs)− (ξ(l)− a)

∫ t

sexp(−au+ lXu)

∣∣∣Fs) .By definition of ξ it holds that EelXt = eξ(l)t and we know that X has independent andstationary increments, so it follows that

E(

(ξ(l)− a)

∫ t

se−au+lXudu

∣∣∣Fs) = (ξ(l)− a)elXs−asE(∫ t

se−a(u−s)+l(Xu−Xs)du

∣∣∣Fs)= (ξ(l)− a)elXs−as

∫ t

sE(e−a(u−s)+lXu−sdu

)= (ξ(l)− a)elXs−as

∫ t

seξ(l)(u−s)−a(u−s)du

= elXs−as(eξ(l)(t−s)−a(t−s) − 1

)= elXs−as

(EelXt−s−a(t−s) − 1

).

23

Also, it holds that

E (exp(−at+ lXt)− exp(−as+ lXs)|Fs) = elXs−asE(e−a(t−s)+l(Xt−Xs) − 1|Fs

)= elXs−as

(EelXt−s−a(t−s) − 1

).

Hence

E(

exp(−at+ lXt)− exp(−as+ lXs)− (ξ(l)− a)

∫ t

sexp(−au+ lXu)

∣∣∣Fs) = 0,

so we conclude that E(Mt|Fs) = Ms and that M has zero mean.

Now, define a matrix M by

M :=

(e−γ1x e−γ1x η

η−γ1e−γ2x e−γ2x η

η−γ2

)

and note that M is invertible, because the roots −γ1,−γ2 are not equal. Recall that thegoal of this section is to find explicit expressions for ui(x0), i = 0, 1, defined by equation(2.10). The matrix M is used to compute these expressions, as stated in the followingtheorem.

Theorem 2.2. Let a > 0 and consider the negative roots −γ1,−γ2 of the equationξ(s) = a. Let w(x0) := (exp(−γ1x0), exp(−γ2x0))> and define

D :=

(eθx 0

0 eθx ηθ+ηe

(θ+η)y

).

Then it holds that (u0(x0)

u1(x0)

)= DM−1w(x0).

Proof. First note that

E (exp(−aτx + θXτx)1F0 |X0 = x0) = eθxE (exp(−aτx)1J0 |X0 = x0) . (2.11)

Also, by lemma 2.2, (ii) and (iii), we see that conditional on J1, x−Xτx is exponentiallydistributed with rate parameter η and is independent of τx. This leads to

E (exp(−aτx + θXτx)1F1 |X0 = x0)

= eθxE(exp(−aτx + θ(Xτx − x))1J11{Xτx<x+y}|X0 = x0

)= eθxE

(E(exp(−aτx + θ(Xτx − x))1{Xτx<x+y}|J1

)1J1 |X0 = x0

)= eθxE(exp(−aτx)1J1 |X0 = x0)E

(exp(−θ(x−Xτx))1{x−Xτx>−y}|J1

)= eθxE (exp(−aτx)1J1 |X0 = x0)

η

θ + ηe(θ+η)y. (2.12)

24

So it is sufficient to find expressions for vi, defined by

vi(x0) := E (exp(−aτx)1Ji |X0 = x0) , i = 0, 1.

Now consider the martingale M from lemma 2.3. By the optional sampling theorem itfollows that E(Mτx |X0 = x0) = 0, that is

E (exp(−aτx + lXτx)|X0 = x0)− elx0 − (ξ(l)− a)E(∫ τx

0exp(−as+ lXs)ds|X0 = x0

)= 0.

(2.13)

From setting y = 0 in equation (2.12) it follows that the first term on the left hand sidecan be written as

E (exp(−aτx + lXτx)|X0 = x0) = E (exp(−aτx + lXτx)1J0 |X0 = x0)

+ E (exp(−aτx + lXτx)1J1 |X0 = x0)

= elxE(e−aτx1J0 |X0 = x0

)+ elxE

(e−aτx1J1 |X0 = x0

) η

l + η

Inserting this into equation (2.13) leads to

0 = elxE(e−aτx1J0 |X0 = x0

)+ elxE

(e−aτx1J1 |X0 = x0

) η

l + η− elx0

− (ξ(l)− a)E(∫ τx

0exp(−as+ lXs)ds|X0 = x0

). (2.14)

Now let h(l) denote the right hand side of equation (2.14), then h(l) = 0, for all l ∈ iR.Define H(l) = (l + η)h(l), then H is well-defined and analytic in C and H(l) = 0 for alll ∈ iR. Then, by the identity theorem of holomorphic functions, we have that H(l) = 0for all l ∈ C, which implies that h(l) = 0 for all l ∈ C\{−η}. Now we can choosel = −γj , j = 1, 2, which gives ξ(l)− a = 0. From h(l) = 0 then follows

e−γjx0 = e−γjxE(e−aτx1J0 |X0 = x0

)+ e−γjxE

(e−aτx1J1 |X0 = x0

) η

η − γj= e−γjxv0(x0) + e−γjx

η

η − γjv1(x0), j = 1, 2.

Which is equivalent to

w(x0) = M

(v0(x0)

v1(x0)

).

Now note that, by equations (2.11) and (2.12), it follows that(u0(x0)

u1(x0)

)= D

(v0(x0)

v1(x0)

)= DM−1w(x0),

which concludes the proof.

25

Note that in order to compute equation (2.8), one has to compute iterated expressionsof the form

E(e−a1τx1+θ1Xτx1E

(e−a2(τx2−τx1 )+θ2Xτx2 1Fi

∣∣∣Fτx1)) ,where τx1 ≤ τx2 .The remainder of this section shows how to modify the computations in the proof oftheorem 2.2 to evaluate the Fτx1 -conditonal expectation. First we want to compute theconditional expectations

ui(Xτx1) := E

(e−a2τx2+θ2Xτx2 1Fi

∣∣∣Fτx1) .Now denote, similarly as before:

D2 :=

(eθx2 0

0 eθx2 ηθ2+ηe

(θ2+η)y

), w2(Xτx1

) =

(exp(−γ(2)

1 Xτx1)

exp(−γ(2)2 Xτx1

)

)and

M2 :=

e−γ(2)1 x2 e−γ

(2)1 x2 η

η−γ(2)1

e−γ(2)2 x2 e−γ

(2)2 x2 η

η−γ(2)2

,

where −γ(2)j , j = 1, 2 are the roots of the equation ξ(s) = a2. Then in the same way as

in the proof of theorem 2.2 the computation of the ui boils down to finding expressionsfor

vi(Xτx1) := E

(exp(−a2τx2)1Ji |Fτx1

),

where (u0(Xτx1

)

u1(Xτx1)

)= D2

(v0(Xτx1

)

v1(Xτx1)

).

Now adapting equation (2.13) to the Fτx1 -conditional setting leads to

0 = E(exp(−a2τx2 + lXτx2

)|Fτx1)− exp(−a2τx1 + lXτx1

)

− (ξ(l)− a2)E

(∫ τx2

τx1

exp(−as+ lXs)ds|Fτx1

),

which implies that equation (2.14) modifies into

0 = elx2E(e−a2τx21J0 |Fτx1

)+ elx2E

(e−a2τx21J1 |Fτx1

) η

l + η− exp(−a2τx1 + lXτx1

)

− (ξ(l)− a2)E

(∫ τx2

τx1

exp(−as+ lXs)ds|Fτx1

).

Following the same arguments as in the proof of theorem 2.2, we have

w(Xτx1) = ea2τx1M2

(v0(Xτx1

)

v1(Xτx1)

),

26

so we conclude that (u0(Xτx1

)

u1(Xτx1)

)= e−a2τx1D2M

−12 w(Xτx1

).

HenceE(e−a2(τx2−τx1 )+θ2Xτx2 1F0

∣∣∣Fτx1) =(D2M

−12 w(Xτx1

))

1,

E(e−a2(τx2−τx1 )+θ2Xτx2 1F1

∣∣∣Fτx1) =(D2M

−12 w(Xτx1

))

2

and

E(e−a1τx1+θ1Xτx1E

(e−a2(τx2−τx1 )+θ2Xτx2 1F0

∣∣∣Fτx1)) = E(e−a1τx1+θ1Xτx1

(D2M

−12 w(Xτx1

))

1

),

(2.15)

E(e−a1τx1+θ1Xτx1E

(e−a2(τx2−τx1 )+θ2Xτx2 1F1

∣∣∣Fτx1)) = E(e−a1τx1+θ1Xτx1

(D2M

−12 w(Xτx1

))

2

).

(2.16)

Now note that(D2M

−12 w(Xτx1

))i, i = 1, 2, are linear combinations of the terms exp(−γ(2)

1 Xτx1),

exp(−γ(2)2 Xτx1

) such that the expectations in equations (2.15) and (2.16) are solved byapplying theorem 2.2.

27

3 A model with imperfect accountinginformation

In this chapter a structural model is considered in which debt is valued under the as-sumption that the only information available is noisy accounting information which isreceived at selected times. The setting is that of Duffie and Lando [10]. The first foursections are used to describe the model proposed by Duffie and Lando and to provide allthe formulas and proofs that are left out in the original paper. In the last two sectionsthe setting is applied to the valuation of different forms of Contingent Convertible bonds.

3.1 Description of the model

The value of assets of the firm, denoted by Vt, is modeled by a geometric Brownianmotion, that is

dVtVt

= µdt+ σdWt.

Define Zt = log Vt and m = µ− σ2/2, then we can write

Zt = Z0 +mt+ σWt.

The firm issues straight debt with a total value P1. The straight debt has a perpetualmaturity and pays coupons continuously at rate c1. Furthermore it is assumed that therisk free interest rate r is constant. As before, default occurs the first time the value ofassets falls below some trigger vb, which means that the firm defaults at τb, defined by

τb = inf{s ≥ 0 : Vt ≤ vb}.

As mentioned before, the bond investors do not have all the information about theasset value, instead they receive imperfect accounting information at times t1 < t2 <. . . (typically every three months). At every observation date there is an imperfect ac-counting report of the asset value available, denoted by Vt, where log Vt and log Vt areassumed to be joint normal. This means that we can write

Yt := log Vt = Zt + Ut,

where Ut is normally distributed and independent of Zt. The information available tobond investors is now described by the filtration Ht, where

Ht = σ({Yt1 , . . . Ytn ,1{τb≤s} : s ≤ t}),

28

for the largest n such that tn ≤ t. Here, the indicator is included to ensure that it is alsoobserved whether the firm is liquidated at t. The goal in the following section is now tofind an expression for the conditional distribution of Vt, given Ht. Firstly, we considerthe simple case that there is only one imperfect observation at time t = t1. This will beextended to multiple observations later on.

3.2 Conditioning on one noisy accounting report

In this section t is a fixed time at which the only noisy accounting value Yt is observed,also we state Z0 = z0, for some z0 ∈ R. The goal is to compute gt(·|Yt, τb > t), theconditional density of Zt given Yt and τb > t. To this end, we first need an expressionfor the probability ψ(z0, x, σ

√t) that min{Zs : s ≤ t} > 0, conditional on Z0 = z0 > 0

and Zt = x > 0. This expression is stated in the following lemma.

Lemma 3.1. The probability ψ(z0, x, σ√t) that min{Zs : s ≤ t} > 0, conditional on

Z0 = z0 > 0 and Zt = x > 0, is given by

ψ(z0, x, σ√t) = 1− exp

(−2z0x

σ2t

).

Proof. To prove this Lemma, we will rely on the following result by Harisson [14, Chapter1.8]. Denote by Xt a Brownian motion with drift µ, variance σ2 and X0 = 0. Further-more define Mt := max{Xs : 0 ≤ s ≤ t}. Then the joint distribution of Xt and Mt

satisfies

P (Xt ∈ dx,Mt ≤ y) =1

σ√t

exp

(µx

σ2− µ2t

2σ2

)(φ

(x

σ√t

)− φ

(x− 2y

σ√t

))dx, (3.1)

where φ denotes the standard normal density function. Now define Xt = −Zt + z0, thenXt = −mt− σWt, which is a Brownian motion with drift −m, variance σ2 and X0 = 0.Denote by fXt its density, which is normal with mean −mt and variance σ2t. Also itholds that

min{Zs : 0 ≤ s ≤ t} > 0⇔Mt = max{Xs : 0 ≤ s ≤ t} < z0.

Then by Bayes’ rule and equation (3.1) it follows that

ψ(z0, x, σ√t) = P(min{Zs : 0 ≤ s ≤ t} > 0|Zt = x)

= P(Mt < z0|Xt = z0 − x)

=P (Mt < z0, Xt ∈ d(z0 − x))

fXt(z0 − x)dx

=exp

(−m(z0−x)

σ2 − m2t2σ2

)(exp

(− (z0−x)2

2σ2t

)− exp

(− (z0+x)2

2σ2t

))exp

(− (z0−x+mt)2

2σ2t

)= 1− exp

(−(z0 + x)2

2σ2t+

(z0 − x)2

2σ2t

)

29

= 1− exp

(−2z0x

σ2t

).

Of course we can write the stopping time τb as

τb = inf{s ≥ 0 : Zs ≤ zb},

for zb = log vb. Next we can compute the density b(·|Yt) of Zt, for τb > t, conditional onYt. That is, b(·|Yt) satisfies

b(x|Yt)dx = P(τb > t, Zt ∈ dx|Yt), for x ≥ zb.

Recall that Yt = Ut+Zt and that Zt and Ut are independent. Furthermore, it holds that

τb > t⇔ min{Zs : 0 ≤ s ≤ t} > zb.

So by Bayes’ rule it follows that

P(τb > t, Zt ∈ dx|Yt) = P(τb > t|Zt ∈ dx, Yt)P(Zt ∈ dx|Yt)= ψ(z0 − zb, x− zb, σ

√t)fZt(x|Yt)dx

= ψ(z0 − zb, x− zb, σ√t)fYt(Yt|Zt = x)fZt(x)

fYt(Yt)dx

= ψ(z0 − zb, x− zb, σ√t)fUt(Yt − x)fZt(x)

fYt(Yt)dx,

which is equivalent to writing

b(x|Yt) =ψ(z0 − zb, x− zb, σ

√t)fUt(Yt − x)fZt(x)

fYt(Yt), (3.2)

where fUt , fZt and fYt denote the densities of Ut, Zt and Yt, respectively. These areall normal, with respective means ut = EUt,mt + z0 and mt + z0 + ut, and respectivevariances a2 = Var(Ut), σ

2t and a2 + σ2t. Note that the standard deviation a of Utdetermines how noisy the accounting reports are.Now we can move forward to the main result of this subsection, an expression for theconditional density g(·|Yt, τb > t) of Zt, given Yt and τb > t.

Theorem 3.1. The conditional density gt(·|Yt, τb > t) of Zt, given Yt and τb > t, isgiven by

gt(x|y, τb > t) =

√β0π e−J(y,x,z0)

(1− exp

(−2z0xσ2t

))exp

(β21

4β0− β3

)Φ(

β1√2β0

)− exp

(β22

4β0− β3

)Φ(− β2√

2β0

) , (3.3)

30

where y = y−zb−ut, x = x−zb, z0 = z0−zb, Φ denotes the standard normal distributionfunction,

J(y, x, z0) =(y − x)2

2a2+

(z0 +mt− x)2

2σ2t,

and

β0 =a2 + σ2t

2a2σ2t,

β1 =y

a2+z0 +mt

σ2t,

β2 = −β1 + 2z0

σ2t,

β3 =1

2

(y2

a2+

(z0 +mt)2

σ2t

).

Proof. First note that

P(τb > t|Yt) =

∫ ∞zb

b(z|Yt)dz,

and recall thatb(x|Yt)dx = P(τb > t, Zt ∈ dx|Yt).

Using Bayes’ rule and equation (3.2), we can compute

gt(x|y, τb > t) =b(x|Yt = y)∫∞

zbb(z|Yt = y)dz

=ψ(z0, x, σ

√t)fUt(y − x)fZt(x)∫∞

zbψ(z0, z − zb, σ

√t)fUt(y − z)fZt(z)dz

, (3.4)

where the numerator is given by

ψ(z0, x, σ√t)fUt(y − x)fZt(x) =

(1− exp

(−2z0x

σ2t

))√

2πa22πσ2texp

(−(y − x− ut)2

2a2− (x−mt− z0)2

2σ2t

)=

(1− exp

(−2z0x

σ2t

))√

2πa22πσ2texp

(−(y − x)2

2a2− (z0 +mt− x)2

2σ2t

)=

1√2πa22πσ2t

e−J(y,x,z0)

(1− exp

(−2z0x

σ2t

)). (3.5)

Furthermore, the denominator of equation (3.4) can be written as

∫ ∞zb

1− exp(−2z0(z−zb)

σ2t

)√

2πa22πσ2texp

(−(y − z − ut)2

2a2− (z −mt− z0)2

2σ2t

)dz = (I)− (II),

where

(I) =1√

2πa22πσ2t

∫ ∞zb

exp

(−(y − z − ut)2

2a2− (z −mt− z0)2

2σ2t

)dz

31

=1√

2πa22πσ2t

∫ ∞0

exp

(−(y − z)2

2a2− (z0 +mt− z)2

2σ2t

)dz

=1√

2πa22πσ2t

∫ ∞0

exp(−(β0z

2 − β1z + β3))

dz

=1√

2πa22πσ2texp

(β2

1

4β0− β3

)∫ ∞0

exp

(−β0

(z − β1

2β0

)2)

dz

=1√

2β0a22πσ2texp

(β2

1

4β0− β3

)∫ ∞−β1√2β0

1√2πe−u

2/2du

=1√

2π(a2 + σ2t)exp

(β2

1

4β0− β3

)Φ

(β1√2β0

),

and where

(II) =1√

2πa22πσ2t

∫ ∞zb

exp

(−2z0(z − zb)

σ2t− (y − z − ut)2

2a2− (z −mt− z0)2

2σ2t

)dz

=1√

2πa22πσ2t

∫ ∞0

exp

(−2z0z

σ2t− (y − z)2

2a2− (z0 +mt− z)2

2σ2t

)dz

=1√

2πa22πσ2t

∫ ∞0

exp(−(β0z

2 + β2z + β3))

dz

=1√

2πa22πσ2texp

(β2

2

4β0− β3

)∫ ∞0

exp

(−β0

(z +

β2

2β0

)2)

dz

=1√

2β0a22πσ2texp

(β2

2

4β0− β3

)∫ ∞β2√2β0

1√2πe−u

2/2du

=1√

2π(a2 + σ2t)exp

(β2

2

4β0− β3

)Φ

(− β2√

2β0

).

Hence the denominator of equation (3.4) is equal to

1√2π(a2 + σ2t)

(exp

(β2

1

4β0− β3

)Φ

(β1√2β0

)− exp

(β2

2

4β0− β3

)Φ

(− β2√

2β0

)).

(3.6)

Now it follows from equations (3.4), (3.5) and (3.6) that

gt(x|y, τb > t) =

1√2πa22πσ2t

e−J(y,x,z0)(1− exp

(−2z0x

σ2t

))1√

2π(a2+σ2t)

(exp

(β21

4β0− β3

)Φ(

β1√2β0

)− exp

(β22

4β0− β3

)Φ(− β2√

2β0

))=

√β0π e−J(y,x,z0)

(1− exp

(−2z0xσ2t

))exp

(β21

4β0− β3

)Φ(

β1√2β0

)− exp

(β22

4β0− β3

)Φ(− β2√

2β0

) ,which concludes the proof.

32

3.3 Survival probability and default intensity

3.3.1 Survival probability

Denote by p(t, s) the Ht-conditional probability of survival until time s > t, that is

p(t, s) = P(τb > s|Ht), for s > t.

To obtain an expression for the survival probability, first consider the probability π(t, x)that Z hits 0 before time t, starting from x > 0. This probability is given by the followinglemma.

Lemma 3.2. The probability π(t, x) that Z hits 0 before time t, starting from x > 0, isgiven by

π(t, x) = 1− Φ

(x+mt

σ√t

)+ e−2mx/σ2

Φ

(−x+mt

σ√t

).

Proof. To prove this, we will rely on the following result by Harrison [14, Chapter 1.8].For a Brownian motion X with drift µ and variance σ2, denote T (x) = inf{t ≥ 0 : Xt =x}. Then the probability that X did not hit x > 0, starting from 0, before time t isgiven by

P(T (x) > t) = Φ

(x− µtσ√t

)− e2µx/σ2

Φ

(−x− µtσ√t

).

Now the expression for π(t, x) follows directly from this result by noting that the prob-ability of hitting 0 before time t, starting from x > 0, with drift m, is equal to theprobability of hitting x > 0 before time t, starting from 0, with drift −m.

Stationarity of Z now implies that the Ht-conditional survival probability p(t, s) for timet < τb, can be written as

p(t, s) =

∫ ∞zb

(1− π(s− t, x− zb))gt(x|Yt, τb > t)dx. (3.7)

3.3.2 Default intensity

One of the advantages of the current setting, is the fact that it is compatible with areduced form approach. That is, it is possible to define a stochastic intensity for default.This is possible because τb is a totally inaccessible Ht-stopping time, which means thatfor any sequence ofHt-stopping times dominated by τb, the probability that the sequenceapproaches τb, is zero. First consider the following definition.

Definition 3.1. A progressively measurable process λ = (λt)t≥0, is called an intensityprocess for a stopping time τ , with respect to a filtration (Gt)t≥0, if it satisfies

∫ t0 λsds <

∞ a.s. for all t ≥ 0 and {1{τ≤t} −∫ t

0 λsds} is a Gt-martingale.

The intuitive meaning of such an intensity process is what one would expect from anintensity, that is

P (τ ∈ (t, t+ dt]|Gt) = λtdt.

This intuitive meaning is made rigorous in the following result, due to Aven [2].

33

Lemma 3.3. Let τ be a stopping time with respect to a filtration (Gt)t≥0. Define

Yn(s) =1

hnP(τ ≤ s+ hn|Gs)1τ>s,

where (hn)n∈N is a sequence decreasing to 0. For (λt)t≥0 and (γt)t≥0 nonnegative mea-surable processes, assume that

(i) For all t ≥ 0, limn→∞

Yn(t) = λt a.s.

(ii) For all t ≥ 0, for almost all ω, there exists an n0 = n0(t, ω) such that for all s ≤ t,n ≥ n0 it holds that

|Yn(s, ω)− λs(ω)| ≤ γs(ω).

(iii)

∫ t

0γsds <∞ a.s., for all t ≥ 0.

Then it follows that {1{τ≤t} −∫ t

0 λsds} is a Gt-martingale, i.e. the intensity process ofτ with respect to (Gt)t≥0 is given by

λt = limh↓0

1

hP(τ ≤ t+ h|Gt)1{τ>t}.

From the results in the previous section it follows that, for every pair (ω, t) such thatτb(ω) > t, the Ht-conditional distribution of Zt admits a continuously differentiable con-ditional density f(t, ·, ω), which is zero in zb and therefore has a derivative fx(t, x, ω) :=∂∂xf(t, x, ω) which is positive at zb. This can be seen as follows. For a time t beforethe first accounting report at time t1, assuming t < τb, the density of Zt can be writtendown explicity and does not depend on ω. As a completion to [10], we will now derivean explicit expression for this density. Denote this density by f(t, ·, z0), then it needs tosatisfy

P (Zt ∈ dx|τb > t) = f(t, x, z0)dx.

By Bayes’ rule we can write

P (Zt ∈ dx|τb > t) =P (Zt ∈ dx, τb > t)

P(τb > t).

The denominator of this expression is given by

P(τb > t) = 1− π(t, z0 − zb) = Φ

(z0 − zb +mt

σ√t

)− e−2m(z0−zb)/σ2

Φ

(zb − z0 +mt

σ√t

)and the numerator can be computed using the same method as in the proof of Lemma3.1. That is, denote Xt = −Zt+z0, which is a Brownian motion with drift −m, varianceσ2 and X0 = 0. Furthermore, denote Mt = max{Xs : 0 ≤ s ≤ t}. Then equation (3.1)implies that

P (Zt ∈ dx, τb > t) = P(Zt ∈ dx, inf

0≤s≤tZs > zb

)

34

= P (Xt ∈ d(z0 − x),Mt ≤ z0 − zb)

=1

σ√t

exp

(−m(z0 − x)

σ2− m2t

2σ2

)(φ

(z0 − xσ√t

)− φ

(−z0 − x+ 2zb

σ√t

))dx.

(3.8)

So we conclude that

f(t, x, z0) =1

σ√t

exp(−m(z0−x)

σ2 − m2t2σ2

)(φ(z0−xσ√t

)− φ

(−z0−x+2zb

σ√t

))Φ(z0−zb+mt

σ√t

)− e−2m(z0−zb)/σ2Φ

(zb−z0+mt

σ√t

) . (3.9)

This density satisfies f(t, zb, z0) = 0 and is differentiable with respect to x, with deriva-tive fx(t, ·, z0) that is bounded uniformly on [t, t1) for all t > 0. The Ht-conditionaldensity at the time t1 of the first accounting report is given by theorem 3.1, denoted bygt1(x|Yt1 , τb > t1). This density gt1(x|Yt1(ω), τb > t1) equals zero at x = zb and has abounded derivative with respect to x, for all ω. Now let s > 0 such that t1 < t1 + s < T ,then the density of Zt1+s is given by

f(t1 + s, x, ω) =

∫ ∞zb

f(s, x, u)gt1(u|Yt1(ω), τb(ω) > t1)du,

which implies

fx(t1 + s, x, ω) =

∫ ∞zb

fx(s, x, u)gt1(u|Yt1(ω), τb(ω) > t1)du.

So it follows that the Ht-conditional density f(t, x, ω) of Zt also equals zero at x = zband has a derivative with respect to x, uniformly bounded on [t, T ], for all t > 0. Becauseof this, we can define an intensity process for the default stopping time, with respect tothe noisy accounting reports filtration (Ht)t≥0.

Theorem 3.2. Define a process λ by

λt(ω) =1

2σ2fx(t, zb, ω)1{τb>t}(ω), for t > 0.

Then λ is an intensity process of τb with respect to (Ht)t≥0.

Proof. Without loss of generality we can assume that zb = 0, for which we will denoteτ0 = inf{t ≥ 0 : Zt = 0}. To prove the result, we will rely on lemma 3.3. That is, wehave to show that on the event {τ0 > t} it holds that

limh↓0

1

hP(τ0(ω) ≤ t+ h|Ht) =

1

2σ2fx(t, 0, ω)

and that the integrability conditions are met. Note that the intensity process can onlybe defined for t > 0, because at t = 0 we have perfect accounting information, whichimplies that τ0 is not totally inaccessible. So we can only prove the existence of an

35

intensity process for t > 0, this will be done by proving existence on compact intervals[t, T ], for all t > 0.Define Yn(t) = 1

hnP(τ0 ≤ t + hn|Ht)1{τ0>t}, for a sequence (hn)n∈N such that hn ↓ 0 as

n → ∞. Recall that by π(t, x) we denote the probability that Z hits 0 before time t,starting from x > 0. So we can write

limn→∞

Yn(t, ω) = limn→∞

1

hnP(τ0(ω) ≤ t+ hn|Ht)1{τ0>t}(ω)

= limn→∞

1

hn

∫(0,∞)

π(hn, x)f(t, x, ω)dx1{τ0>t}(ω)

= limn→∞

∫(0,∞)

π(hn, σ√hnz)

f(t, σ√hnz, ω)

σ√hnz

σ2zdz1{τ0>t}(ω),

where the last equation follows by the substitution z = xσ√hn

.

Now denote

G1(z, h) = π(h,√hz), G2(z, h, ω) =

f(t, σ√hz, ω)

σ√hz

σ2

and note that by lemma 3.2 we have

G1(z, h) = 1− Φ

(z +

m√h

σ

)+ e−2m

√hz/σΦ

(−z +

m√h

σ

).

Furthermore, because f(t, 0, ω) = 0, it holds that

limh↓0

G2(z, h, ω) = fx(t, 0, ω)σ2.

Then, assuming that the dominated convergence theorem can be applied, it follows that

limn→∞

Yn(t, ω) = limn→∞

∫(0,∞)

G1(z, hn)G2(z, hn, ω)zdz1{τ0>t}(ω)

=

∫(0,∞)

(1− Φ(z) + Φ(−z))fx(t, 0, ω)σ2zdz1{τ0>t}(ω)

= σ2fx(t, 0, ω)

∫(0,∞)

(1− Φ(z) + Φ(−z))zdz1{τ0>t}(ω).

Since ∫(0,∞)

(1− Φ(z) + Φ(−z))zdz =

∫(0,∞)

2Φ(−z)zdz

=[Φ(−z)z2

]∞0

+

∫(0,∞)

φ(−z)z2dz

= 0 +1

2

∫(−∞,∞)

φ(z)z2dz

=1

2,

36

we conclude that

limn→∞

Yn(t, ω) =1

2σ2fx(t, 0, ω)1{τ0>t}(ω) = λt(ω).

The justification of the use of dominated convergence is as follows. Because fx(t, 0, ω)is bounded, there exists a constant C > 0 such that∣∣∣∣∣f(t, σ

√hz, ω)

σ√hz

∣∣∣∣∣ ≤ C.Also, for h < 1 it holds that

|G1(z, h)| ≤ G(z) := 1− Φ

(z − |m|

σ

)+ e2|m|z/σΦ(−z).

So it follows that for all h < 1

|G1(z, h)G2(z, h)z| < Cσ2zG(z).

Now note that 1 − Φ(x) behaves like φ(x)/x as x → ∞, which implies that G(z) → 0exponentially fast as z →∞, such that Cσ2zG(z) provides an integrable upper bound,which justifies the application of the dominated convergence theorem.It now suffices to check the conditions of lemma 3.3, to conclude the proof. Becausefx(t, 0, ω) is uniformly bounded on [t, T ] it follows that∫ T

t|λs(ω)|ds <∞

and the above discussion implies that∫ T

t|Yn(s, ω)|ds <∞,

for almost all ω and all n such that hn < 1. So, if we choose n0 such that hn < 1whenever n > n0, it follows that for all n > n0

|Yn(s, ω)− λs(ω)| ≤ |Yn(s, ω)|+ |λs(ω)|, where

∫ T

t|Yn(s, ω)|+ |λs(ω)|ds <∞,

which shows that the conditions of lemma 3.3 are met. Hence λt defines an intensityprocess for τ0.

3.4 Conditioning on several noisy accounting reports

The results in the previous sections hold in a setting in which only one single noisyaccounting value Yt is observed at time t. In this section, this is extended to a setting

37

in which noisy accounting reports Yi := Yti arrive at successive dates t1 < t2 < . . . ,typically every three months in practice. Of course, it is reasonable that there existssome correlation between the accounting noise U1, U2, . . . . To be more specific, followingDuffie and Lando [10], it is assumed that Yi = Zi + Ui, where

Ui = κUi−1 + εi,

for some fixed κ ∈ R and independent and indentically distributed ε1, ε2, . . . , which havea normal distribution with mean µε ∈ R and variance σε > 0, and are independent of Z.Consider the following notation for the relevant random vectors and its realisations:

Z(n) = (Z1, Z2, . . . , Zn) and its realisation z(n) = (z1, z2, . . . , zn),

Y (n) = (Y1, Y2, . . . , Yn) and its realisation y(n) = (y1, y2, . . . , yn),

U (n) = Y (n) − Z(n) and its realisation u(n) = y(n) − z(n).

Consistent with the previous notation, denote by bn(·|Y (n)) the conditional density ofZ(n) for τb > tn, conditional on Y (n), that is

bn(z(n)|Y (n))dz(n) = P(Z(n) ∈ dz(n), τb > tn|Y (n)).

Note that (Zn)n∈N and (Un)n∈N are Markov processes and denote by pZ(zn|zn−1) andpU (un|un−1) their respective transition densities for realisations z(n), u(n). Furthermore,denote by pY (yn|y(n−1)) the conditional density of Yn given Y (n−1) = y(n−1).By Bayes’ rule it follows that

bn(z(n)|y(n))dz(n) =P(τb > tn, Z

(n) ∈ dz(n), Y (n) ∈ dy(n))

P(Y (n) ∈ dy(n)).

Now note that{τb > tn, Z

(n) ∈ dz(n), Y (n) ∈ dy(n)} = A ∩B,

whereA = {τb > tn, Zn ∈ dzn, Yn ∈ dyn},

B = {τb > tn−1, Z(n−1) ∈ dz(n−1), Y (n−1) ∈ dy(n−1)}.

This implies that

bn(z(n)|y(n))dz(n) =P(A|B)P(B)

pY (yn|y(n−1))dynP(Y (n−1) ∈ dy(n−1))

=P(A|B)bn−1(z(n−1)|y(n−1))

pY (yn|y(n−1))dyndz(n−1),

where

P(A|B) = P(τb > tn, Zn ∈ dzn, Yn ∈ dyn|τb > tn−1, Z(n−1) ∈ dz(n−1), Y (n−1) ∈ dy(n−1))

= P(τb > tn|Zn ∈ dzn, Yn ∈ dyn, τb > tn−1, Z(n−1) ∈ dz(n−1), Y (n−1) ∈ dy(n−1))

38

× P(Yn ∈ dyn|Zn ∈ dzn, τb > tn−1, Z(n−1) ∈ dz(n−1), Y (n−1) ∈ dy(n−1))

× P(Zn ∈ dzn|τb > tn−1, Z(n−1) ∈ dz(n−1), Y (n−1) ∈ dy(n−1))

= ψ(zn−1 − zb, zn − zb, σ√tn − tn−1)

× P((Un + zn) ∈ dyn|Zn ∈ dzn, τb > tn−1, Z(n−1) ∈ dz(n−1), Y (n−1) ∈ dy(n−1))

× P(Zn ∈ dzn|τb > tn−1, Z(n−1) ∈ dz(n−1), Y (n−1) ∈ dy(n−1))

= ψ(zn−1 − zb, zn − zb, σ√tn − tn−1)pU (yn − zn|yn−1 − zn−1)pZ(zn|zn−1)dyndzn.

So we obtain the following expression for bn(z(n)|y(n)):

ψ(zn−1 − zb, zn − zb, σ√tn − tn−1)pZ(zn|zn−1)pU (yn − zn|yn−1 − zn−1)bn−1(z(n−1)|y(n−1))

pY (yn|y(n−1)).

(3.10)

Similarly as in the case with only one noisy accounting observation, treated in section3.2, it now follows that the conditional density gtn(·|Y (n), τb > tn) of Z(n) is given by

gtn(z(n)|y(n), τb > tn) =bn(z(n)|y(n))∫

Anbn(z(n)|y(n))dz(n)

, (3.11)

whereAn = {z = (z1, . . . , zn) ∈ Rn : z1 ≥ zb, . . . , zn ≥ zb}.

It should be noted that there is no explicit expression for the integral in the denominatorof equation (3.11), such that it needs to be evaluated using numerical integration. Nowthe marginal conditional density of Zn at time tn is given by

gtn(zn|y(n), τb > tn) =

∫An−1

gtn(z(n)|y(n), τb > tn)dz(n−1). (3.12)

For τb > tn, the Htn-conditional survival probability is then given by

p(tn, s) =

∫ ∞zb

(1− π(s− tn, zn − zb))gtn(zn|Y (n), τb > tn)dzn for s > tn, (3.13)

where, π(t, x) denotes again the probability that Z hits 0 before time t, starting fromx > 0, as computed in Lemma 3.2. Also the notion of default intensity can be extendedto the case in which we have several accounting reports, as follows. At time tn < t < tn+1

the Ht-conditional density of Zt is given by

f(t, x, ω) =

∫ ∞zb

f(t− tn, x, zn)gtn(zn|Y (n)(ω), τb(ω) > tn)dzn,

where as before f(t, ·, z0) is the Ht-conditional density before the first accounting report,given by equation (3.9). The density f(t, ·, ω) again has a derivative with respect to xgiven by

fx(t, x, ω) =

∫ ∞zb

fx(t− tn, x, zn)gtn(zn|Y (n)(ω), τb(ω) > tn)dzn.

39

Then also the default intensity is extended to the case in which we have several account-ing reports, i.e. the default intensity λ with respect to Ht is given by

λt(ω) =1

2σ2fx(t, zb, ω)1{τb>t}(ω), for t > 0. (3.14)

The proof of this result immediately follows from the proof of theorem 3.2, because weonly have a finite number of accounting reports.

3.5 Valuation of CoCos with a principal write-down

In this section, the model with several noisy accounting reports of the previous sectionwill be applied to the valuation of contingent convertible bonds with a (partial) principalwrite-down at converison. It is assumed that, additionally to the straight bonds, the firmissues contingent convertible debt with a total par value outstanding of P and maturitytime T .

3.5.1 CoCos with a regulatory trigger

Banks have the obligation to report it to their supervisor at the moment they are ap-proaching a trigger. Then the regulator will call for conversion, this is called a point ofnon-viability. Of course, this type of conversion can also happen in between accountingreport dates. This type of conversion thus occurs when the value of assets falls for thefirst time below some threshold vc, i.e. the conversion time is given by

τc = inf{t ≥ 0 : Vt ≤ vc}.

Also, the firm pays coupons continuously at rate c until either maturity or conversion.To start with, we assume that the CoCo bond suffers a principal write-down uponconversion. Later on, in Section 3.6, the case of a conversion into a fixed number ofshares will be considered. For now, a fraction 1 − R of the principal value is writtendown at conversion, while a fraction R is recovered to the bond holder, for R ∈ [0, 1).Furthermore it is assumed that the risk free rate is constant, denoted by r.Of course, the bond investors can observe whether the CoCos have converted. Thismeans that Ht is now defined as

Ht = σ({Yt1 , . . . Ytn ,1{τc≤s} : s ≤ t}).

Denote by pc(t, s) the Ht-conditional probability that the CoCos did not convert untiltime s > t, that is

pc(t, s) = P(τc > s|Ht).

As before, for zc = log vc, we can also write

τc = inf{t ≥ 0 : Zt ≤ zc}.

40

In analogy to equation (3.13), we have for tn < τc and s > tn

pc(tn, s) = P(τc > s|τc > tn, Y(n)) =

∫ ∞zc

(1− π(s− tn, zn − zc))gtn(zn|Y (n), τc > tn)dzn.

For a more general time t such that tn ≤ t < tn+1 and τc > t, this conditional probabilityreads

pc(t, s) =

∫ ∞zc

(1− π(s− t, x− zc))f(t, x, ω)dx, (3.15)

for

f(t, x, ω) =

∫ ∞zc

f(t− tn, x, zn)gtn(zn|Y (n)(ω), τc(ω) > tn)dzn, (3.16)

where f(t, x, z0) is, in analogy to equation (3.9), given by

f(t, x, z0) =1

σ√t

exp(−m(z0−x)

σ2 − m2t2σ2

)(φ(z0−xσ√t

)− φ

(−z0−x+2zc

σ√t

))Φ(z0−zc+mt

σ√t

)− e−2m(z0−zc)/σ2Φ

(zc−z0+mt

σ√t

) .

Now the value at time t < τc of the CoCos, given imperfect accounting information Ht,is given by

C(t) = E(Pe−r(T−t)1{τc>T}|Ht

)+ E

(∫ T

tcPe−r(u−t)1{τc>u}du|Ht

)+ E

(RPe−r(τc−t)1{τc≤T}|Ht

)= Pe−r(T−t)P (τc > T |Ht) + cP

∫ T

te−r(u−t)P (τc > u|Ht) du

+RP

∫ T

te−r(u−t)P(τc ∈ du|Ht)

= Pe−r(T−t)pc(t, T ) + cP

∫ T

te−r(u−t)pc(t, u)du−RP

∫ T

te−r(u−t)pc(t,du).

(3.17)

Here the first term represents the payment of the principal, in case conversion does nothappen before maturity, while the second term accounts for the payment of couponsuntil either conversion or maturity. The last term values the recovery of the principal atconversion. The integral in this last term can be written as∫ T

te−r(u−t)pc(t,du) =

∫ T

te−r(u−t)

∂

∂upc(t, u)du

=

∫ T

te−r(u−t)

∫ ∞zc

∂

∂u(1− π(u− t, x− zc))f(t, x, ω)dxdu

41

=

∫ ∞zc

f(t, x, ω)

∫ T

te−r(u−t)

∂

∂u(1− π(u− t, x− zc))dudx

=

∫ ∞zc

f(t, x, ω)I(x)dx, (3.18)

where

I(x) =

∫ T

te−r(u−t)

∂

∂u(1− π(u− t, x− zc))du. (3.19)

Furthermore, the integral in the second term of equation (3.17) can be written as∫ T

te−r(u−t)pc(t, u)du =

∫ T

te−r(u−t)

∫ ∞zc

(1− π(u− t, x− zc))f(t, x, ω)dxdu

=

∫ ∞zc

f(t, x, ω)

∫ T

te−r(u−t)(1− π(u− t, x− zc))dudx

=

∫ ∞zc

f(t, x, ω)I(x)dx, (3.20)

where

I(x) =

∫ T

te−r(u−t)(1− π(u− t, x− zc))du

=

[−1

re−r(u−t)(1− π(u− t, x− zc))

]Tu=t

+1

rI(x)

= −1

re−r(T−t)(1− π(T − t, x− zc)) +

1

r+

1

rI(x). (3.21)

Putting equations (3.17), (3.19) and (3.20) together allows us to write the CoCo priceC(t) as a single integral, weighted by the density f(t, x, ω), as follows

C(t) =

∫ ∞zc

(Pe−r(T−t)(1− π(T − t, x− zc)) + cP I(x)−RPI(x)

)f(t, x, ω)dx (3.22)

=

∫ ∞zc

(r − cr

Pe−r(T−t)(1− π(T − t, x− zc)) +c

rP +

(cP

r−RP

)I(x)

)f(t, x, ω)dx.

(3.23)

It now remains to find an analytical expression for I(x). First consider

∂

∂u(1− π(u− t, x− zc))

=∂

∂u

(Φ

(x− zc +m(u− t)

σ√u− t

)− e−2m(x−zc)/σ2

Φ

(−(x− zc) +m(u− t)

σ√u− t

))= φ

(x− zc +m(u− t)

σ√u− t

)(m

2σ√u− t

− x− zc2σ(u− t)3/2

)

42

− e−2m(x−zc)/σ2φ

(−(x− zc) +m(u− t)

σ√u− t

)(m

2σ√u− t

+x− zc

2σ(u− t)3/2

)=

zc − xσ(u− t)3/2

φ

(x− zc +m(u− t)

σ√u− t

),

which implies

I(x) =

∫ T

te−r(u−t)

zc − xσ(u− t)3/2

1√2π

exp

(−(x− zc +m(u− t))2

2σ2(u− t)

)du

=zc − x√

2πσ2exp

(−m(x− zc)

σ2

)∫ T−t

0e−ru

1

u3/2exp

(−(x− zc)2

2σ2u− m2u

2σ2

)du

=zc − x√

2πσ2exp

(−m(x− zc)

σ2

)∫ T−t

0

1

u3/2exp

(−(x− zc)2

2σ2

1

u−(m2

2σ2+ r

)u

)du

= 2zc − x√

2πσ2exp

(−m(x− zc)

σ2

)∫ ∞(T−t)−1/2

exp

(−Av2 −B 1

v2

)dv, (3.24)

where the last line follows by substitution of v = u−1/2 and by setting

A =(x− zc)2

2σ2, B =

m2

2σ2+ r.

Now, by noting that (Av2 + B/v2) = (√Av −

√B/v)2 + 2

√AB, as well as (Av2 +

B/v2) = (√Av +

√B/v)2 − 2

√AB, the remaining integral can be evaluated, by doing

the substitutions u =√Av −

√B/v and u =

√Av +

√B/v, as follows∫ ∞

(T−t)−1/2

exp

(−Av2 −B 1

v2

)dv

=1

2√A

∫ ∞(T−t)−1/2

exp(−(√Av −

√B/v)2 − 2

√AB)

(√A+√B

1

v2)dv

+1

2√A

∫ ∞(T−t)−1/2

exp(−(√Av +

√B/v)2 + 2

√AB)

(√A−√B

1

v2)dv

=1

2√Ae−2√AB

∫ ∞√A/(T−t)−

√B(T−t)

e−u2du

+1

2√Ae2√AB

∫ ∞√A/(T−t)+

√B(T−t)

e−u2du

=

√π

4√A

(e−2√AB erfc

(√A/(T − t)−

√B(T − t)

)+ e2

√AB erfc

(√A/(T − t) +

√B(T − t)

)), (3.25)

where erfc(x) is the complementary error function, which is defined by

erfc(x) :=2√π

∫ ∞x

e−u2du

43

and satisfies1

2erfc(x/

√2) = 1− Φ(x).

Combining equations (3.24) and (3.25) and substituting back the expressions for A andB, finally leads to the following expression for I(x):

I(x) = 2zc − x√

2πσ2exp

(−m(x− zc)

σ2

)∫ ∞(T−t)−1/2

exp

(−Av2 −B 1

v2

)dv

= exp

(−m(x− zc)

σ2

)(−e−2

√AB 1

2erfc

(√A/(T − t)−

√B(T − t)

)− e2

√AB 1

2erfc

(√A/(T − t) +

√B(T − t)

))= exp

(−m(x− zc) + (x− zc)

√m2 + 2rσ2

σ2

)(Φ

(x− zc −

√m2 + 2rσ2(T − t)σ√T − t

)− 1

)

+ exp

(−m(x− zc)− (x− zc)

√m2 + 2rσ2

σ2

)(Φ

(x− zc +

√m2 + 2rσ2(T − t)σ√T − t

)− 1

).

(3.26)

To summarize, in the above we have proven the following theorem.

Theorem 3.3 (Price of a CoCo with a regulatory trigger and a principal write-down).The secondary market price of the CoCo at time t < τc is given by

C(t) =

∫ ∞zc

h(x)f(t, x)dx, (3.27)

where f(t, x) is given by equation (3.16) for tn ≤ t < tn+1 and h(x) is defined as

h(x) :=r − cr

Pe−r(T−t)(1− π(T − t, x− zc)) +c

rP +

(cP

r−RP

)I(x), (3.28)

in which I(x) is given by equation (3.26).

Now we can move on with the computation of this integral. Let us first considerthe computation of C(tn), i.e. the price of the CoCo at the time of the nth accountingreport. In this case, the conditional density f(tn, ·) is equal to the marginal densitygtn(·|τc > tn, Y

(n)), as can be seen from equation (3.16). Recall from equation (3.12)that the marginal density gtn(zn|τc > tn, Y

(n)) is given by

gtn(zn|y(n), τc > tn) =

∫An−1

gtn(z(n)|y(n), τc > tn)dz(n−1),

where An is defined as

An = {z = (z1, . . . , zn) ∈ Rn : z1 ≥ zc, . . . , zn ≥ zc}. (3.29)

44

So it follows that C(tn) satisfies

C(tn) =

∫ ∞zc

h(zn)f(tn, zn)dzn

=

∫An

h(zn)gtn(z(n)|y(n), τc > tn)dz(n). (3.30)

This integral cannot be evaluated analytically, so this will be done using Monte Carlosimulation. If we can simulate a sample ((z(n))1, . . . , (z(n))G) from gtn(z(n)|y(n), τc > tn),it is possible to approximate C(tn) as

C(tn) ≈ 1

G

G∑g=1

h(zgn), (3.31)

where zgn denotes the nth coordinate of (z(n))g, for g = 1, . . . , G.The sample ((z(n))1, . . . , (z(n))G) is obtained by execution of the following MCMC-algorithm.

Algorithm 3.1 (Random walk Metropolis-Hastings).

1. In each iteration g, g = 1, . . . , n0 +G, given the current value (z(n))g, the proposal(zn)′ is drawn according to

(z(n))′ = (z(n))g +X, for X ∼ Nn(0,Σ),

where the n × n-covariance matrix Σ is chosen to reach some desired acceptancerate.

2. Set

(z(n))(g+1) =

{(z(n))′ with prob. α((z(n))g, (z(n))′)

z(n) with prob. 1− α((z(n))g, (z(n))′),

where the acceptance-probability α(z(n), (z(n))′) is given by

α(z(n), (z(n))′) = min

{1,gtn((z(n))′|y(n), τc > tn)

gtn(z(n)|y(n), τc > tn)

}= min

{1,bn((z(n))′|y(n))

bn(z(n)|y(n))

}.

3. Discard the draws from the first n0 iterations (because the Markov chain needs aburn-in period to converge to the target distribution) and save the sample(z(n))n0+1, . . . , (z(n))n0+G.

In order to compute C(t) for tn ≤ t < tn+1 a similar procedure can be followed. Firstnote that we can write

C(t) =

∫ ∞zc

h(x)f(t, x)dx

45

=

∫ ∞zc

h(x)

∫ ∞zc

f(t− tn, x, zn)gtn(zn|Y (n), τc > tn)dzndx

=

∫ ∞zc

∫An

h(x)f(t− tn, x, zn)gtn(z(n)|Y (n), τc > tn)dz(n)dx

=

∫An+1

h(zn+1)f(t− tn, zn+1, zn)gtn(z(n)|Y (n), τc > tn)dz(n+1). (3.32)

So now we need a sample ((z(n+1))1, . . . , (zn+1)G) from the (n + 1)-dimensional dis-tribution on An+1 with density f(t − tn, zn+1, zn)gtn(z(n)|Y (n), τc > tn), in order toapproximate C(t) as

C(t) ≈ 1

G

G∑g=1

h(zgn+1). (3.33)

The algorithm used to obtain the sample, now modifies into

Algorithm 3.2 (Random walk Metropolis-Hastings).

1. In each iteration g, g = 1, . . . n0 +G, given the current value (z(n+1))g, the proposal(zn+1)′ is drawn according to

(z(n+1))′ = (z(n+1))g +X, for X ∼ Nn+1(0,Σ),

where the (n + 1) × (n + 1)-covariance matrix Σ is chosen to reach some desiredacceptance rate.

2. Set

(z(n+1))(g+1) =

{(z(n+1))′ with prob. α((z(n+1))g, (z(n+1))′)

z(n+1) with prob. 1− α((z(n+1))g, (z(n+1))′),

where the acceptance-probability α(z(n+1), (z(n+1))′) is given by

α(z(n+1), (z(n+1))′) = min

{1,f(t− tn, z′n+1, z

′n)gtn((z(n))′|y(n), τc > tn)

f(t− tn, zn+1, zn)gtn(z(n)|y(n), τc > tn)

}

= min

{1,f(t− tn, z′n+1, z

′n)bn((z(n))′|y(n))

f(t− tn, zn+1, zn)bn(z(n)|y(n))

}.

3. Discard the draws from the first n0 iterations and save the sample(z(n+1))n0+1, . . . , (z(n+1))n0+G.

In both algorithms the acceptance probability involves the termbn((z(n))′|y(n))

bn(z(n)|y(n)). It

follows from equations (3.2) and (3.10) that this fraction is explicitly given by

bn((z(n))′|y(n))

bn(z(n)|y(n))=

∏ni=1 ψ(z′i−1 − zc, z′i − zc, σ

√ti − ti−1)pZ(z′i|z′i−1)pU (yi − z′i|yi−1 − z′i−1)∏n

i=1 ψ(zi−1 − zc, zi − zc, σ√ti − ti−1)pZ(zi|zi−1)pU (yi − zi|yi−1 − zi−1)

,

46

under the convention that t0 = 0 and pU (·|u0) = pU (·) is a Gaussian density with meanu and variance a2 (as in section 3.2). Note that pZ(zi|zi−1) is a Gaussian density withmean zi−1 +m(ti−ti−1) and variance σ2(ti−ti−1), that pU (ui|ui−1) is a Gaussian densitywith mean κui−1 + µε and variance σ2

ε and that ψ is given in Lemma 3.1.

Early cancelling of coupons

In the valuation of the firm’s convertible debt in equation (3.17), it is assumed thatcoupons are paid until conversion. However, as pointed out in the Introduction, CoCosare affected by the Maximum Distributable Amount (MDA), which requires regulators tostop earnings distributions when the firm’s total capital falls below some trigger, higherthan the conversion trigger. This we could model by introducing a trigger zcc > zc. IfZ is below zcc the firm will not pay coupons, while if Z is above zcc the firm still payscoupons. To value the CoCo in this case, only the second term in equation (3.17) needsto be adjusted. In this case, coupons are only paid at time u if Zu > zcc, so the term

E(∫ T

tcPe−r(u−t)1{τc>u}du|Ht

),

needs to be replaced with

E(∫ T

tcPe−r(u−t)1{τc>u,Zu>zcc}du|Ht

). (3.34)

For τc > t and tn ≤ t < tn+1 this term equals

cP

∫ T

te−r(u−t)P

(τc > u,Zu > zcc|Y (n), τc > t

)du.

So now we need to compute P(τc > u,Zu > zcc|Y (n), τc > t

). In order to compute this

conditional probability, we first need the following result.

Lemma 3.4. The joint probability π(t, x, y) that Z, starting from x > 0, does not hit 0before time t and that Zt > y is given by

π(t, x, y) := P( inf0≤s≤t

Zs > 0, Zt > y) = Φ

(x− y +mt

σ√t

)− e−2mx/σ2

Φ

(−x− y +mt

σ√t

).

(3.35)

Proof. To prove this lemma, we will rely on the following result by Harrison [14, Chapter1.8]. For X a Brownian motion with drift µ, variance σ2 and X0 = 0, define Mt =sup0≤s≤tXs. Then the joint distribution of Mt and Xt is given by

P(Xt ≤ x,Mt ≤ y) = Φ

(x− µtσ√t

)− e2µy/σ2

Φ

(x− 2y − µt

σ√t

). (3.36)

47

Now define Xt = x− Zt, then Xt is a Brownian motion with drift −m, variance σ2 andX0 = 0. And Mt = x− inf0≤s≤t Zs. Hence

π(t, x, y) := P( inf0≤s≤t

Zs > 0, Zt > y) = P (Mt ≤ x,Xt ≤ x− y) ,

which proves the result, by applying equation (3.36).

Now, similarly to equation (3.15), we can write

P(τc > u,Zu > zcc|Y (n), τc > t

)=

∫ ∞zc

π(u− t, x− zc, zcc − zc)f(t, x)dx. (3.37)

As in equation (3.32), this can be written as∫An+1

π(u− t, zn+1 − zc, zcc − zc)f(t− tn, zn+1, zn)gtn(z(n)|Y (n), τc > tn)dz(n+1).

So, for a sample ((z(n+1))1, . . . , (zn+1)G) from the (n+ 1)-dimensional distribution withdensity f(t − tn, zn+1, zn)gtn(z(n)|Y (n), τc > tn), we can approximate the desired condi-tional probability by

P(τc > u,Zu > zcc|Y (n), τc > t

)≈ 1

G

G∑g=1

π(u− t, zgn+1 − zc, zcc − zc).

The necessary sample is again obtained using Algorithm 3.2.Recall that we wanted to compute

cP

∫ T

te−r(u−t)P

(τc > u,Zu > zcc|Y (n), τc > t

)du,

which now can be done by performing numerical integration over u.The other two terms in equation (3.17) do not change, so the CoCo price at time

t < τc is given by the sum of the new term in (3.34) and the unchanged part

Pe−r(T−t)pc(t, T )−RP∫ T

te−r(u−t)pc(t,du).

By an adaption of equation (3.27) it is seen that this unchanged part can be written as∫ ∞zc

h(x)f(t, x)dx,

whereh(x) = Pe−r(T−t)(1− π(T − t, x− zc))−RPI(x),

in which I(x) is given by equation (3.26). So this part can be computed performing thesame computations as in the old case, by replacing h(x) with h(x).

48

3.5.2 CoCos with only a book value trigger

There exists also CoCos which conversion trigger solely depends on accounting reports,for example the CoCos issued by Barclays. This means that conversion happens whenthe reported value of the capital ratio falls below some threshold and hence conversioncan only happen at one of the accounting report dates t1, t2, . . . . This corresponds to asetting in which the conversion time is defined as

τc = inf{ti ≥ 0 : Yti ≤ yc},

for some threshold yc ≥ 0. In this case the available information at time t would reduceto

Ht = σ(Yt1 , . . . , Ytn),

for the largest n such that tn ≤ t. This means that we are interested in the probabilitythat, given n accounting reports and τc > tn, the (n+ i)th accounting report will causea trigger event, for i = 1, 2, . . . . So we are primarily interested in the conditional densitypY (yn+i, . . . , yn+1|y(n)) of Yn+i, . . . , Yn+1, given Y (n) = y(n), where yi > yc, for 1 ≤ i ≤ n.

Denote by ∆t the time between to succesive accounting reports and recall from section3.4 the notation Yi = Yti , Zi = Zti , Ui = Uti and that Yi = Zi + Ui, where

Ui = κUi−1 + εi,

for some fixed κ ∈ R and independent and identically distributed ε1, ε2, . . . , which havea normal distribution, say with mean µε and variance σ2

ε , and are independent of Z.This allows us to write

Yn+1 = Zn+1 + Un+1

= Zn+1 + κUn + εn+1

= Zn+1 − κZn + εn+1 + κYn,

which leads to the following expression(Yn+2

Yn+1

)=

(Zn+2 − κ2Zn + εn+2 + κεn+1 + κ2Yn

Zn+1 − κZn + εn+1 + κYn

)

=

(1 1 1 κ0 1 0 1

)Zn+2 − Zn+1

Zn+1 − Znεn+2

εn+1

+

(κ2

κ

)Yn +

(1− κ2

1− κ

)Zn.

Now note that Zn+2−Zn+1, Zn+1−Zn, εn+2 and εn+1 are all Gaussian and independentof Y (n), Z(n) and each other. Hence, conditional on Y (n) = y(n), Z(n) = z(n), the vector(Zn+2 − Zn+1, Zn+1 − Zn, εn+2, εn+1) follows a multivariate normal distribution. So theabove implies that the conditional density pY (yn+2, yn+1|y(n), z(n)) of yn+2, yn+1 givenY (n) = y(n) and Z(n) = z(n), is the density of a bivariate normal distribution with certain

49

mean vector and covariance matrix, which is presented below for the general case.The above can be extended, for i = 1, 2, . . . , as follows

Yn+i...

Yn+1

= M

Zn+i − Zn+i−1...

Zn+1 − Znεn+i

...εn+1

+

κi

...κ

Yn +

1− κi...

1− κ

Zn,

where M denotes the (i× 2i)-matrix defined by

M =

i components︷ ︸︸ ︷1 1 · · · 1 10 1 · · · 1 1...

. . .. . .

......

0 · · · 0 1 10 · · · 0 0 1

i components︷ ︸︸ ︷1 κ κ2 · · · κi−1

0 1 κ · · · κi−2

.... . .

. . .. . .

...0 · · · 0 1 κ0 · · · 0 0 1

and where the vector (Zn+i−Zn+i−1, . . . , Zn+1−Zn, εn+i, . . . , εn+1) follows a multivari-ate normal distribution with 2i-dimensional mean vector µ′i and (2i × 2i)-dimensionalcovariance matrix Σ′i, defined by

µ′i =

m∆t...

m∆tµε...µε

, Σ′i = Diag(σ2∆t, . . . σ2∆t, σ2

ε , . . . , σ2ε ).

Hence it follows that the conditional density pY (yn+i, . . . , yn+1|y(n), z(n)) of yn+i, . . . , yn+1

given Y (n) = y(n) and Z(n) = z(n), is the density of a multivariate normal distributionwith mean vector

µi = Mµ′i +

κi

...κ

yn +

1− κi...

1− κ

zn

and covariance matrixΣi = MΣ′iM

>.

Recall that the goal was to find an expression for pY (yn+i, . . . , yn+1|y(n)), which cannow be done as follows

pY (yn+i, . . . , yn+1|y(n)) =

∫RnpY (yn+i, . . . , yn+1|y(n), z(n))pZ(z(n)|y(n))dz(n),

50

where the conditional density pZ(z(n)|y(n)) of Z(n), given Y (n) = y(n), can be computedin the same way as bn(z(n)|y(n)) in section 3.4, which leads to

pZ(z(n)|y(n)) =pZ(zn|zn−1)pU (yn − zn|yn−1 − zn−1)pZ(z(n−1)|y(n−1))

pY (yn|y(n−1))

=

∏ni=1 pZ(zi|zi−1)pU (yi − zi|yi−1 − zi−1)

pY (yn|y(n−1)), (3.38)

under the convention that t0 = 0 and pU (·|u0) = pU (·) is a Gaussian density with mean uand variance a2 (as in section 3.2). Also, note that pZ(zi|zi−1) is a Gaussian density withmean zi−1 + m(ti − ti−1) and variance σ2(ti − ti−1) and that pU (ui|ui−1) is a Gaussiandensity with mean κui−1 + µε and variance σ2

ε .This leads to an expression for the survival probability until time tn+i, given survivaluntil time tn ≤ t < tn+1, that is

P(τc > tn+i|Y (n) = y(n)

)=

∫(yc,∞)i

pY (yn+i, . . . , yn+1|y(n))dyn+1, . . . ,dyn+i

=

∫Rn

P(ξ ∈ (yc,∞)i)pZ(z(n)|y(n))dz(n), (3.39)

where ξ denotes a multivariate normal distributed random variable with mean vector µiand covariance matrix Σi. To indicate the dependence of ξ on zn, we denote it as ξ(zn).For a sample ((z(n))1, . . . , (z(n))G) from pZ(z(n)|y(n)), this survival probability can becomputed as

P(τc > tn+i|Y (n) = y(n)

)≈ 1

G

G∑g=1

P(ξ(zgn) ∈ (yc,∞)i). (3.40)

The necessary sample is again obtained using a MCMC-algorithm, as follows.

Algorithm 3.3 (Random walk Metropolis-Hastings).

1. In each iteration g, g = 1, . . . , n0 +G, given the current value (z(n))g, the proposal(zn)′ is drawn according to

(z(n))′ = (z(n))g +X, for X ∼ Nn(0,Σ),

where the n × n-covariance matrix Σ is chosen to reach some desired acceptancerate.

2. Set

(z(n))(g+1) =

{(z(n))′ with prob. α((z(n))g, (z(n))′)

z(n) with prob. 1− α((z(n))g, (z(n))′),

where the acceptance-probability α(z(n), (z(n))′) is given by

α(z(n), (z(n))′) = min

{1,pZ((z(n))′|y(n))

pZ(z(n)|y(n))

}

51

= min

{1,

∏ni=1 pZ(z′i|z′i−1)pU (yi − z′i|yi−1 − z′i−1)∏ni=1 pZ(zi|zi−1)pU (yi − zi|yi−1 − zi−1)

}.

3. Discard the draws from the first n0 iterations (because the Markov chain needs aburn-in period to converge to the target distribution) and save the sample(z(n))n0+1, . . . , (z(n))n0+G.

Finally, it is now possible to value the contingent convertible bond with, as before,principal P , continuous coupon rate c, maturity T and a principal write-down withrecovery rate R. As in equation (3.17), this CoCo has secondary market price

C ′(t) = E(Pe−r(T−t)1{τc>T}|Ht

)+ E

(∫ T

tcPe−r(u−t)1{τc>u}du|Ht

)+ E

(RPe−r(τc−t)1{τc≤T}|Ht

). (3.41)

For tn ≤ t < tn+1, T = tn+m for some m ∈ N and Y (n) = y(n), where yi > yc, 1 ≤ i ≤ n,this can be written as

C ′(t) = Pe−r(T−t)P(τc > T |Y (n) = y(n)) +

∫ T

tcPe−r(u−t)P(τc > u|Y (n) = y(n))du

+RP

m∑i=1

e−r(tn+i−t)P(τc = tn+i|Y (n) = y(n))

= Pe−r(T−t)P(τc > tn+m|Y (n) = y(n))

+ cP

(m−1∑i=1

∫ tn+i+1

tn+i

e−r(u−t)duP(τc > tn+i|Y (n) = y(n)) +

∫ tn+i

te−r(u−t)du

)

+RPm∑i=1

e−r(tn+i−t)(P(τc > tn+i−1|Y (n) = y(n))− P(τc > tn+i|Y (n) = y(n))

)= Pe−r(T−t)P(τc > tn+m|Y (n) = y(n))

+

m−1∑i=1

cP

r(e−r(tn+i−t) − e−r(tn+i+1−t))P(τc > tn+i|Y (n) = y(n))

+cP

r(1− e−r(tn+1−t))

+RP

m∑i=1

e−r(tn+i−t)(P(τc > tn+i−1|Y (n) = y(n))− P(τc > tn+i|Y (n) = y(n))

)= (1−R)Pe−r(T−t)P(τc > tn+m|Y (n) = y(n))

+

m−1∑i=1

(cP

r−RP

)(e−r(tn+i−t) − e−r(tn+i+1−t))P(τc > tn+i|Y (n) = y(n))

+cP

r(1− e−r(tn+1−t)) +RPe−r(tn+1−t), (3.42)

52

where the only things left to compute are of the form P(τc > tn+i|Y (n) = y(n)) for1 ≤ i ≤ m, which can be computed using equation (3.40).

Early cancelling of coupons

As in the previous subsection, we can also consider the case in which coupons are alreadycancelled at a moment before the conversion date. It is now assumed that coupons overthe time interval [ti, ti+1) are only paid if Yi > ycc, for some trigger level ycc > yc. Tovalue the CoCo in this case, the second term in equation (3.41) needs to be changed to

E

(m−1∑i=1

∫ tn+i+1

tn+i

cPe−r(u−t)1{τc>u,Yn+i>ycc}du+ 1{Yn>ycc}

∫ tn+1

tcPe−r(u−t)du

∣∣∣Ht) ,where tn ≤ t < tn+1, T = tn+m for some m ∈ N.For Y (n) = y(n), where yi > yc, 1 ≤ i ≤ n, this can be written as

m−1∑i=1

∫ tn+i+1

tn+i

e−r(u−t)P(τc > u, Yn+i > ycc|Y (n) = y(n))du+ 1{Yn>ycc}

∫ tn+1

tcPe−r(u−t)du

=

m−1∑i=1

cP

r(e−r(tn+i−t) − e−r(tn+i+1−t))P(τc > tn+i, Yn+i > ycc|Y (n) = y(n))

+ 1{Yn>ycc}cP

r(1− e−r(tn+1−t)), (3.43)

where, similar to equation (3.39),

P(τc > tn+i, Yn+i > ycc|Y (n) = y(n)) = P(Yn+1 > yc, . . . , Yn+i−1 > yc, Yn+i > ycc|Y (n) = y(n))

=

∫Rn

P(ξ ∈ (yc,∞)i−1 × (ycc,∞))pZ(z(n)|y(n))dz(n).

This integral can as before be computed by

P(τc > tn+i, Yn+i > ycc|Y (n) = y(n)) ≈ 1

G

G∑g=1

P(ξ(zgn) ∈ (yc,∞)i−1 × (ycc,∞)),

for a sample ((z(n))1, . . . , (z(n))G) from pZ(z(n)|y(n)).The CoCo price of equation (3.42) then modifies for the current setting into

C ′(t) = Pe−r(T−t)P(τc > tn+m|Y (n) = y(n)) + 1{Yn>ycc}cP

r(1− e−r(tn+1−t))

+

m−1∑i=1

cP

r(e−r(tn+i−t) − e−r(tn+i+1−t))P(τc > tn+i, Yn+i > ycc|Y (n) = y(n))

+RPm∑i=1

e−r(tn+j−t)(P(τc > tn+i−1|Y (n) = y(n))− P(τc > tn+i|Y (n) = y(n))

).

(3.44)

53

3.6 Valuation of CoCos with a conversion into shares

In this section we consider the valuation of contingent convertible bonds which convertinto equity at the conversion date. As before the firm issues two types of debt; straightdebt and contingent convertible debt. The total par value of straight debt outstanding,is denoted by P1, over which coupons are paid continuously at rate c1. Furthermore, thestraight bonds have a perpetual maturity and, as before, default occurs at

τb = inf{t ≥ 0 : Zt ≤ zb}.

At the moment of default a fraction (1 − α), for α ∈ (0, 1), of the firm’s asset value islost to bankruptcy costs, so a fraction α of the asset value is recovered and distributedamong the senior debt holders.

The total par value of CoCos outstanding is denoted by P2, over which coupons arepaid continuously at rate c2. Furthermore, the maturity of the contingent convertiblebonds is denoted by T . As in Subsection 3.5.1 the conversion date is denoted by

τc = inf{t ≥ 0 : Zt ≤ zc},

where zc > zb, to ensure that conversion happens before default. Note that we considerhere a regulatory trigger, which means that conversion can happen in between accountingdates, as in subsection 3.5.1. As in Chapter 2, following [6], the CoCo holders receive ∆shares for every dollar of principal at the moment of conversion. This means that, if wenormalize the number of shares before conversion to 1, the CoCo holders own a fraction

∆P2∆P2+1 of the firm’s equity after conversion.As before, the information in the market at time t is described by

Ht = σ({Yt1 , . . . Ytn ,1{τc≤s},1{τb≤s} : s ≤ t}), for tn ≤ t < tn+1.

In analogy to equation (3.17), the market price of the CoCos is given by

C(t) = E(P2e−r(T−t)1{τc>T}|Ht

)+ E

(∫ T

tc2P2e

−r(u−t)1{τc>u}du|Ht)

+ E(

∆P2

∆P2 + 1EPC(τc)e

−r(τc−t)1{τc≤T}|Ht). (3.45)

Of course only the third term has changed compared to equation (3.17), because thisterm describes what happens at the moment of conversion (note that the second termneeds to be replaced by the corresponding term in equation (3.34), if we want to includeearly cancelling of coupons). The third term now describes that the CoCo holdersobtain a fraction ∆P2

∆P2+1 of the firms post-conversion equity, denoted by EPC(τc). Thispost conversion equity satisfies

EPC(τc) = Vτc −D(τc)− E(e−r(τb−τc)(1− α)Vτb |Hτc

).

54

That is, the firm’s value of assets minus the value of straight debt, denoted by D(τc),and bankruptcy costs, described by the last term. Note that the value of straight debtat conversion is given by

D(τc) = E(∫ ∞

τc

c1P1e−r(u−τc)1{τb>u}du|Hτc

)+ E

(αVτbe

−r(τb−τc)|Hτc),

where the first term accounts for the continuous payment of coupons and the secondterm describes the payment at default. It follows that the post-conversion equity valueis given by

EPC(τc) = Vτc − E(∫ ∞

τc

c1P1e−r(u−τc)1{τb>u}du|Hτc

)− E

(e−r(τb−τc)Vτb |Hτc

)= ezc − E

(∫ ∞t

c1P1e−r(u−τc)1{τc≤u,τb>u}du|Hτc

)− ezbE

(e−r(τb−τc)|Hτc

).

So for τc > t, the third term in equation (3.45) can be written as

E(

∆P2

∆P2 + 1EPC(τc)e

−r(τc−t)1{τc≤T}|Ht)

=∆P2

∆P2 + 1ezcE

(e−r(τc−t)1{τc≤T}|Ht

)− ∆P2

∆P2 + 1E(∫ ∞

tc1P1e

−r(u−t)1{τc≤T∧u,τb>u}du|Ht)

− ∆P2

∆P2 + 1ezbE

(e−r(τb−t)1{τc≤T}|Ht

)=

∆P2

∆P2 + 1ezc∫ T

te−r(u−t)P(τc ∈ du|τc > t, Y (n))

− ∆P2c1P1

∆P2 + 1

∫ ∞t

e−r(u−t)P(τc ≤ T ∧ u, τb > u|τc > t, Y (n))du

− ∆P2

∆P2 + 1ezb∫ ∞t

e−r(u−t)P(τc ≤ T, τb ∈ du|τc > t, Y (n)). (3.46)

Note that the first integral in the last line of this equation is already computed insubsection 3.5.1 and given by equation (3.18) as

ezc∫ T

te−r(u−t)P(τc ∈ du|τc > t, Y (n)) = −ezc

∫ ∞zc

f(t, x)I(x)dx, (3.47)

where I(x) is given by equation (3.26).To compute the other integrals in equation (3.46), it is sufficient to find expressions for

P(τc ≤ T, τb > u|τc > t, Y (n) = y(n)) and P(τc ≤ u, τb > u|τc > t, Y (n) = y(n)).

In order to find expressions for this joint probabilities, we first need the following lemma.

55

Lemma 3.5. The joint probability γ(x, y, z, t1, t2) that Z, starting from x, does not hitz before time t1 but does hit y before time t2, is for x > y > z given by

γ(x, y, z, t1, t2) = P( inf0≤s≤t1

Zs > z, inf0≤s≤t2

Zs ≤ y)

=

{π(t2, x− y)− π(t1, x− z) for t1 ≤ t2,

1− π(t1, x− z)−∫∞y (1− π(t1 − t2, z − z))f(x, y, z, t2)dz for t1 > t2,

where

f(x, y, z, t2) =1

σ√t2

exp

(−m(x− z)

σ2− m2t2

2σ2

)(φ

(x− zσ√t2

)− φ

(−x− z + 2y

σ√t2

))(3.48)

Proof.

• For t1 ≤ t2, we can write

P( inf0≤s≤t1

Zs > z, inf0≤s≤t2

Zs ≤ y) = P( inf0≤s≤t2

Zs ≤ y)− P( inf0≤s≤t1

Zs ≤ z, inf0≤s≤t2

Zs ≤ y)

= P( inf0≤s≤t2

Zs ≤ y)− P( inf0≤s≤t1

Zs ≤ z)

= π(t2, x− y)− π(t1, x− z).

• For t1 > t2, note that

P( inf0≤s≤t1

Zs > z, inf0≤s≤t2

Zs ≤ y) = P( inf0≤s≤t1

Zs > z)− P( inf0≤s≤t1

Zs > z, inf0≤s≤t2

Zs > y)

= 1− π(t1, x− z)− P( inf0≤s≤t1

Zs > z, inf0≤s≤t2

Zs > y),

where

P( inf0≤s≤t1

Zs > z, inf0≤s≤t2

Zs > y)

=

∫ ∞y

P(

inft2≤s≤t1

Zs > z, inf0≤s≤t2

Zs > y|Zt2 = z

)P(Zt2 ∈ dz)

=

∫ ∞y

P(

inft2≤s≤t1

Zs − Zt2 > z − z)P( inf

0≤s≤t2Zs > y,Zt2 ∈ dz)

=

∫ ∞y

P(

inf0≤s≤t1−t2

Zs > z − z + x

)P( inf

0≤s≤t2Zs > y,Zt2 ∈ dz)

=

∫ ∞y

(1− π(t1 − t2, z − z))P( inf0≤s≤t2

Zs > y,Zt2 ∈ dz),

where is used that Z has independent and stationary increments.Now the result follows by noting that by a modification of equation (3.8) to thecurrent setting, it holds that

56

P( inf0≤s≤t2

Zs > y,Zt2 ∈ dz) = f(x, y, z, t2)dz.

Now the desired probabilities are, in analogy to equation (3.15), given by

P(τc ≤ T, τb > u|τc > t, Y (n) = y(n)) =

∫ ∞zc

γ(x, zc, zb, u− t, T − t)f(t, x)dx

and

P(τc ≤ u, τb > u|τc > t, Y (n) = y(n)) =

∫ ∞zc

γ(x, zc, zb, u− t, u− t)f(t, x)dx.

Recall that the objective was to compute the last two integrals in equation (3.46). Letus first consider the second one, that is

−∫ ∞t

e−r(u−t)P(τc ≤ T, τb ∈ du|τc > t, Y (n)) =

∫ ∞t

e−r(u−t)∂

∂uP(τc ≤ T, τb > u|τc > t, Y (n))du

= (I) + (II),

where

(I) =

∫ T

te−r(u−t)

∂

∂uP(τc ≤ T, τb > u|τc > t, Y (n))du

=

∫ ∞zc

f(t, x)

∫ T

te−r(u−t)

∂

∂uγ(x, zc, zb, u− t, T − t)dudx

=

∫ ∞zc

f(t, x)

∫ T

te−r(u−t)

∂

∂u(−π(u− t, x− zb))dudx

=

∫ ∞zc

f(t, x)Ib(x)dx,

in which

Ib(x) =

∫ T

te−r(u−t)

∂

∂u(−π(u− t, x− zb))du

= exp

(−m(x− zb) + (x− zb)

√m2 + 2rσ2

σ2

)(Φ

(x− zb −

√m2 + 2rσ2(T − t)σ√T − t

)− 1

)

+ exp

(−m(x− zb)− (x− zb)

√m2 + 2rσ2

σ2

)(Φ

(x− zb +

√m2 + 2rσ2(T − t)σ√T − t

)− 1

),

(3.49)

which follows from equation (3.26), by replacing zc by zb. Furthermore, we have

(II) =

∫ ∞T

e−r(u−t)∂

∂uP(τc ≤ T, τb > u|τc > t, Y (n))du

57

=

∫ ∞zc

f(t, x)

∫ ∞T

e−r(u−t)∂

∂uγ(x, zc, zb, u− t, T − t)dudx

=

∫ ∞zc

f(t, x)

∫ ∞T

e−r(u−t)∂

∂u(−π(u− t, x− zb))dudx

−∫ ∞zc

∫ ∞zc

f(t, x)f(x, zc, z, T − t)∫ ∞T

e−r(u−t)∂

∂u(1− π(u− T, z − zb))dudzdx

=

∫ ∞zc

f(t, x)(Jb(x)− Ib(x))dx

−∫ ∞zc

∫ ∞zc

f(t, x)f(x, zc, z, T − t)e−r(T−t)Jb(z, T )dzdx,

where

Jb(x) =

∫ ∞t

e−r(u−t)∂

∂u(1− π(u− t, x− zb))du

= − exp

(−m(x− zb) + (x− zb)

√m2 + 2rσ2

σ2

), (3.50)

where the last line follows by taking T →∞ in equation (3.49). This leaves us with anexpression for the last integral in equation (3.46).

Similarly, the other integral satisfies∫ ∞t

e−r(u−t)P(τc ≤ T ∧ u, τb > u|τc > t, Y (n))du = (III) + (IV ),

where

(III) =

∫ T

te−r(u−t)P(τc ≤ u, τb > u|τc > t, Y (n))du

=

∫ ∞zc

f(t, x)

∫ T

te−r(u−t)γ(x, zc, zb, u− t, u− t)dudx

=

∫ ∞zc

f(t, x)

∫ T

te−r(u−t)(π(u− t, x− zc)− π(u− t, x− zb))dudx

=

∫ ∞zc

f(t, x)(Ib(x)− I(x))dx

in which I(x) is defined in equation (3.21) and Ib(x) is equivalently defined as

Ib(x) =

∫ T

te−r(u−t)(1− π(u− t, x− zb))du

=

[−1

re−r(u−t)(1− π(u− t, x− zb))

]Tu=t

+1

rIb(x)

= −1

re−r(T−t)(1− π(T − t, x− zb)) +

1

r+

1

rIb(x). (3.51)

58

Furthermore, we have

(IV ) =

∫ ∞T

e−r(u−t)P(τc ≤ T, τb > u|τc > t, Y (n))du

=

∫ ∞zc

∫ ∞T

e−r(u−t)γ(x, zc, zb, u− t, T − t)dudx

=

∫ ∞zc

f(t, x)

∫ ∞T

e−r(u−t)(1− π(u− t, x− zb))dudx

−∫ ∞zc

∫ ∞zc

f(t, x)f(x, zc, z, T − t)∫ ∞T

e−r(u−t)(1− π(u− T, z − zb))dudzdx

=

∫ ∞zc

f(t, x)(Jb(x)− Ib(x))dx

−∫ ∞zc

∫ ∞zc

f(t, x)f(x, zc, z, T − t)e−r(T−t)Jb(z)dzdx,

in which

Jb(x) =

∫ ∞t

e−r(u−t)(1− π(u− t, x− zb))du

=

[−1

r(1− π(u− t, x− zb)

]∞u=t

+1

rJb(x)

=1

r+

1

rJb(x). (3.52)

Putting all the above together leads to an expression for the last two integrals in equation(3.46), given by

− c1P1

∫ ∞t

e−r(u−t)P(τc ≤ T ∧ u, τb > u|τc > t, Y (n))du

− ezb∫ ∞t

e−r(u−t)P(τc ≤ T, τb ∈ du|τc > t, Y (n))

= ezb((I) + (II))− c1P1((III) + (IV ))

=

∫ ∞zc

f(t, x)(ezbJb(x) + c1P1I(x)− c1P1Jb(x)

)dx

+

∫ ∞zc

∫ ∞zc

f(t, x)f(x, zc, z, T − t)e−r(T−t)(c1P1Jb(z)− ezbJb(z))dzdx. (3.53)

Finally, by combining equations (3.46), (3.47) and (3.53), it follows that the thirdterm in equation (3.45), i.e.

E(

∆P2

∆P2 + 1EPC(τc)e

−r(τc−t)1{τc≤T}|Ht),

is given by ∫ ∞zc

f(t, x)h1(x)dx+

∫ ∞zc

∫ ∞zc

f(t, x)f(x, zc, z, T − t)h2(z)dzdx, (3.54)

59

where

h1(x) =∆P2

∆P2 + 1

(ezbJb(x) + c1P1I(x)− c1P1Jb(x)− ezcI(x)

),

h2(z) =∆P2

∆P2 + 1e−r(T−t)(c1P1Jb(z)− ezbJb(z)),

in which I(x), I(x), Jb(x) and Jb(x) are respectively given by equations (3.21), (3.26),(3.50) and (3.52). Also, note that the first two terms of equation (3.45) are equal to theintegral in equation (3.27), by taking R = 0, c = c2 and P = P2 in equation (3.28)(wedenote the function corresponding to this choice by h0 in the following theorem).To summarize, in the above we have proven the following theorem.

Theorem 3.4 (Price of a CoCo with a regulatory trigger and a conversion into shares).The secondary market price at time t < τc of the CoCo with a regulatory trigger and aconversion into shares is given by

C(t) =

∫ ∞zc

(h0(x) + h1(x))f(t, x)dx+

∫ ∞zc

∫ ∞zc

f(t, x)f(x, zc, z, T − t)h2(z)dzdx,

(3.55)

where f(x, y, z, t2) is given by equation (3.48) and

h0(x) =r − c2

rP2e−r(T−t)(1− π(T − t, x− zc)) +

c2P2

r+c2P2

rI(x),

h1(x) =∆P2

∆P2 + 1

(ezbJb(x) + c1P1I(x)− c1P1Jb(x)− ezcI(x)

),

h2(z) =∆P2

∆P2 + 1e−r(T−t)(c1P1Jb(z)− ezbJb(z)),

in which I(x), I(x), Jb(x) and Jb(x) are respectively given by equations (3.21), (3.26),(3.50) and (3.52).

Now note that the first integral in equation (3.55) can be approximated in the sameway as in equations (3.32) and (3.33), by replacing h(x) with h0(x)+h1(x). Furthermore,the second integral in equation (3.55) can, in analogy to equation (3.32), be written as∫ ∞zc

∫An+1

h2(z)f(zn+1, zc, z, T − t)f(t− tn, zn+1, zn)gtn(z(n)|Y (n), τc > tn)dz(n+1)dz

=

∫An+2

h2(zn+2)f(zn+1, zc, zn+2, T − t)f(t− tn, zn+1, zn)gtn(z(n)|Y (n), τc > tn)dz(n+2),

(3.56)

for An as defined in equation (3.29).Now note that, by definition of f , it holds that∫ ∞zc

f(zn+1, zc, zn+2, T − t)dzn+2 =

∫ ∞zc

P(

inf0≤s≤T−t

Zs > zc, ZT−t ∈ dzn+2

∣∣∣Z0 = zn+1

)

60

= P(

inf0≤s≤T−t

Zs > zc|Z0 = zn+1

)= 1− π(T − t, zn+1 − zc).

Hence, f(zn+1, zc, zn+2, T − t)f(t − tn, zn+1, zn)gtn(z(n)|Y (n), τc > tn) is not a densityfunction on An+2, so it is not possible to proceed in the same way as before. However,by the above we know that

f(zn+1, zc, zn+2, T − t)f(t− tn, zn+1, zn)gtn(z(n)|Y (n), τc > tn)

1− π(T − t, zn+1 − zc)

is a density function on An+2.So if we have a sample ((z(n+2))1, . . . , (zn+2)G) from the (n+ 2)-dimensional distribu-

tion with this density, we can approximate the integral in equation (3.56) by

1

G

G∑g=1

h2(zgn+2)(1− π(T − t, zgn+1 − zc)). (3.57)

This sample is, in analogy to Algorithm 3.2, obtained by the following MCMC-algorithm.

Algorithm 3.4 (Random walk Metropolis-Hastings).

1. In each iteration g, g = 1, . . . n0 +G, given the current value (z(n+2))g, the proposal(zn+2)′ is drawn according to

(z(n+2))′ = (z(n+2))g +X, for X ∼ Nn+2(0,Σ),

where the (n + 2) × (n + 2)-covariance matrix Σ is chosen to reach some desiredacceptance rate.

2. Set

(z(n+2))(g+1) =

{(z(n+2))′ with prob. α((z(n+2))g, (z(n+2))′)

z(n+2) with prob. 1− α((z(n+2))g, (z(n+2))′),

where the acceptance-probability α(z(n+2), (z(n+2))′) is given by

min

{1,f(z′n+1, zc, z

′n+2, T − t)f(t− tn, z′n+1, z

′n)bn((z(n))′|y(n))(1− π(T − t, zn+1 − zc))

f(zn+1, zc, zn+2, T − t)f(t− tn, zn+1, zn)bn(z(n)|y(n))(1− π(T − t, z′n+1 − zc))

}.

3. Discard the draws from the first n0 iterations and save the sample(z(n+2))n0+1, . . . , (z(n+2))n0+G.

61

4 Applications

4.1 Impact of model parameters

In this section the impact of several model parameters on the CoCo price is investigated.Unless stated otherwise the parameters are as in Table 4.1 and a CoCo has a regulatorytrigger as in Section 3.6. For the choice of the base case parameters, some restraintsshould be taken into account. For example, the conversion trigger should be higher thanthe default trigger. Also, a CoCo should pay a higher coupon than straight debt, tocompensate for the higher risk. We have no empirical evidence for a reasonable levelof accounting noise, so the volatility of accounting noise is set equal to the base caseparameter chosen by Duffie and Lando [10].Furthermore, in Section 3.6 it is assumed that the CoCo holders own a fraction ∆P2

∆P2+1of the firm after conversion, in this section we will denote this fraction by ρ and call itthe dilution ratio, as in [6]. A dilution ratio of ρ = 0 means that the CoCo suffers aprincipal write-down (PWD) at conversion, while ρ = 1 corresponds to the extreme casethat the original shareholders are completely wiped out at conversion.To compute prices for PWD CoCos, we make use of Theorem 3.3. The involving integralis approximated as in equation (3.33), for which the necessary sample is obtained by usingAlgorithm 3.2. To compute prices for CoCos with a conversion into shares, we makeuse of Theorem 3.4, where the first term in the pricing formula follows again by usingAlgorithm 3.2 and the second term is approximated as in equation (3.57), for whichthe necessary sample is obtained by execution of Algorithm 3.4. Then the figures areproduced by repeatedly following this procedures for different values of the parameters.

62

Parameter Value

Initial asset value, V0 100Number of acc. reports until time t, n 2Conversion trigger, vc 80Default trigger, vb 65Recovery rate at default, α 0.5Total principal straight debt, P1 50Coupon straight debt, c1 0.04Total principal CoCos, P2 5Coupon CoCos, c2 0.07Maturity CoCos, T t+5Drift asset process, m 0.01Volatility asset process, σ 0.1Mean accounting noise, µε 0Volatility accounting noise, σε 0.1Risk free interest rate, r 0.03

Table 4.1: Base case parameters.

4.1.1 Volatility

In Figure 4.1 several CoCo prices are plotted against the volatility of assets, σ. The solidline corresponds to a PWD CoCo. Clearly, the price of a PWD CoCo decreases whenassets become more volatile. This is of course as one would expect, as an increasing σincreases the probability of the principal write-down happening, causing the CoCo priceto decrease. The dashed line, corresponding to ρ = 0.5, shows already that this negativeeffect from volatility on the CoCo price is weaker when terms of conversion are morefavorable to the CoCo investor. In the extreme case that shareholders are completelywiped out at conversion, corresponding to the dashed-dotted line, this negative effectis even partially reversed. In this case, the price first increases with volatility, as theprice increases when the favorable conversion becomes more likely. However, for highervolatility levels the increasing probability of default pushes the price down again.

4.1.2 Accounting noise

Recall that the size of the noise of the noisy accounting reports is determined by thevolatility σε. Now let us consider the relationship between the amount of accountingnoise and the price of a CoCo. In the upper panel of Figure 4.2 the price of a PWDCoCo is plotted against σε. The figure shows that the CoCo price decreases when theamount of accounting noise increases. This is in line with the results of Duffie and Lando[10], as they find that the default probability decreases when the reports become morenoisy. In our CoCo setting, this would mean that the probability of conversion increaseswhen σε increases, causing the CoCo price to decrease. In the upper panel of Figure 4.2 aCoCo with a regulatory trigger is considered, but it is also interesting to look at a CoCo

63

which conversion trigger only depends on the noisy accounting reports, as in subsection3.5.2. This type of CoCo is priced by the formula given in equation (3.42), which iscomputed using the approximation in equation (3.40), for which the necessary samplesare obtained by using Algorithm 3.3. In the lower panel of Figure 4.2, the dashed lineshows that for a trigger which only depends on accounting reports, the CoCo price ismuch more sensitive to accounting noise than in case of a regulatory trigger.

Volatility of Assets0 0.05 0.1 0.15 0.2 0.25 0.3

CoC

o pr

ice

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

; = 0 (PWD); = 0.5; = 1

Figure 4.1: CoCo price versus σ, for different values of ρ.

64

Volatility accounting noise0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

CoC

o pr

ice

4.96

4.98

5

5.02

5.04

5.06

5.08

5.1

5.12

5.14

5.16

Volatility accounting noise0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

CoC

o pr

ice

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5

5.2 Regulatory triggerAccounting trigger

Figure 4.2: PWD CoCo price versus σε, the volatility of accounting noise. The upperpanel considers only a regulatory trigger, the lower panel considers both aregulatory trigger and a book value trigger.

65

4.1.3 The conversion trigger

In Figure 4.3, the CoCo price is plotted against the conversion trigger. The solid linecorresponds to a PWD CoCo. As one would expect, the price of a PWD CoCo is lowerfor a higher conversion trigger, as a higher conversion trigger increases the probabilityof a principal write-down. However, the other lines show that if conversion terms arefavorable enough to the coco investor, the price will increase with the conversion trigger.In the extreme case that the dilution ratio ρ equals one, represented by the dashed line,the CoCo price increases strongly with the conversion trigger, while for other cases thispositive effect is weaker (ρ = 0.5, dashed-dotted line) or terms of conversion are notfavorable enough for CoCo holders to create this effect (ρ = 0.1, dotted line).

Conversion trigger75 80 85 90 95 100

CoC

o pr

ice

4

6

8

10

12

14

16

18

20

22

; = 1; = 0.5; = 0.1; = 0 (PWD)

Figure 4.3: CoCo price versus vc, for different values of ρ.

4.1.4 The number of shares received at conversion

In Figure 4.4, the price of a CoCo is plotted against ∆, the number of shares received atconversion per unit of principal, for different values of straight debt in the firm’s capitalstructure. ∆ = 0 corresponds to a principal write-down CoCo, while ∆ =∞ correspondsto the case in which all of the original shareholders are wiped out at conversion and theCoCo investors are the only shareholders left. As becomes clear from Figure 4.4, theCoCo price increases with ∆. This is of course as expected, as a higher ∆ means a higherpayout at conversion. Furthermore, the figure shows that a CoCo with a conversion intoshares has a higher price when there is a lower amount of straight debt issued. Hence

66

the CoCo is more valuable when the firm has a lower leverage. This can also be easilyexplained, as the CoCo investors receive a fraction of the firm’s equity value at conversionand of course the equity value is higher in case there are less liabilities.

"

0 100 200 300 400 500 600 700 800 900 1000

CoC

o pr

ice

4

4.5

5

5.5

6

6.5

7

7.5

8

8.5

Debt: 40, CoCo: 5Debt: 45, CoCo: 5Debt: 50, CoCo: 5

Figure 4.4: CoCo price versus ∆, for different values of P1.

4.2 Impact CoCos on capital structure

In this section we examine the impact of the issuance of CoCos on the capital structureof the bank and on incentives for shareholders. The computation of the prices and theproduction of the figures is performed following the same procedures as in Section 4.1.

4.2.1 Replacing debt with CoCos

First consider the case in which straight debt is replaced with CoCos. In Figure 4.5 thechange in equity value occuring when 5 units of straight debt are replaced with 5 unitsof CoCos, is plotted against the conversion trigger. It is seen from the solid line andthe dotted line that shareholders only benefit from replacing debt with CoCos, whenthe terms of conversion are favorable enough to the shareholders and the trigger is highenough. The dashed line and the dashed-dotted line show that shareholders have noincentive to issue highly dilutive CoCos.

67

Conversion trigger75 80 85 90 95 100

Cha

nge

in e

quity

val

ue

-20

-15

-10

-5

0

5

; = 0 (PWD); = 0.1; = 0.5; = 1

Figure 4.5: Change in equity value when 5 units of debt are replaced with 5 units ofCoCos (in market value).

4.2.2 Replacing equity with CoCos

Similarly, we could consider the case in which equity is replaced with CoCos and theproceeds of the CoCo issue are used to buy back equity. This case gives similar resultsas when we replace debt with CoCos. Again, the shareholders only have incentive toissue CoCos, when the conversion terms are not too favorable to the CoCo investors, aswe see by Figure 4.6. By comparing the Figures 4.5 and 4.6, it is seen that shareholdershave a higher incentive to replace debt with PWD CoCos than to replace equity withPWD CoCos. However, if they have to issue more dilutive CoCos, they should chooseto replace equity with CoCos, instead of replacing debt with CoCos.

4.2.3 Risk taking incentives

We can examine the impact of the introduction of Contingent Convertible bonds on therisk taking incentives of the shareholders. This can be done by looking at the relationbetween equity value and σ, the volatility of the asset process. If equity value increaseswith volatility, this would mean that the equity holders have an incentive to take onmore risk. In the upper panel of Figure 4.7, the value of equity is plotted against σ fora total straight debt of P1 = 50. The solid line corresponds to a firm with no CoCos,which shows that in this situation equity value decreases when volatility increases. Thatis, in this case the shareholders have no risk taking incentive. However, if 5 units of

68

Conversion trigger75 80 85 90 95 100

Cha

nge

in e

quity

val

ue

-16

-14

-12

-10

-8

-6

-4

-2

0

2

4

; = 0 (PWD); = 0.1; = 0.5; = 1

Figure 4.6: Change in equity value when 5 units of equity are replaced with 5 units ofCoCos (in market value).

equity are replaced with 5 units of principal write-down CoCos, then the firm’s valueof equity increases with volatility, as shown by the dashed line. This suggests that theissuance of principal write-down CoCos could give shareholders an incentive to take onmore risk. The dashed-dotted line corresponds to an issuance of CoCos with a conversioninto shares for a dilution ratio ρ = 1, which means that the original shareholders arecompletely wiped out. The dotted line corresponds to a situation including CoCos witha conversion into shares for a dilution ratio ρ = 0.5. These two lines show that this risktaking incentive does not appear in case of an issuance of CoCos with a conversion intoshares. This difference can of course be explained by the fact that the write-down of aCoCo is beneficial to the shareholders, as equity will raise in this case, while a conversioninto shares may be disadvantageous to the original shareholders.In the lower panel of Figure 4.7 the same situation is considered, now for a firm witha higher leverage; where the total amount of straight debt is 60. Again, the solid linecorresponds to the case where no CoCos are issued. In contrast to the case with lowerleverage, the shareholders already have an incentive to take on more risk without CoCosin the capital structure. It is seen by the dashed line that PWD CoCos certainly are notable to reverse this incentive, while the CoCos with a conversion into shares are only ableto neutralize this incentive for higher volatility levels, as can be seen from the dashedline and the dashed-dotted line. This is not in line with Chen et al [6], who find thatCoCos are able to reverse this risk taking incentive. This difference could be explainedby the fact that in their model, the maturity of debt is rolling over, while we use a fixed

69

perpetual maturity. Also, they make a distinction between jump risk and diffusion risk,which is not possible in our model. To summarize, we see that at the lower leverage levelthere is no risk taking incentive for the shareholders in a firm without CoCos, but theissuance of PWD CoCos could create such an incentive. The incentive does not appearwhen conversion terms of CoCos are beneficial enough to the CoCo investors. In case ofa higher leverage, the risk taking incentive is already there in a firm without CoCos andit can only be partially neutralized by issuing CoCos with a conversion into shares anda sufficiently high dilution ratio.

4.2.4 Investment incentives

In this subsection it is investigated if shareholders have an incentive to invest in thefirm. In a structural model without CoCos, the shareholders do not have an incentiveto invest exactly at the moment the firm most needs an investment: when the firm isnear bankruptcy. In this case, the firm would need an investment to avoid bankruptcy,however almost all of the value of the investment will be captured by the debt holders,as the value of debt increases when the probability of a bankruptcy is reduced. Thisphenomenon is called debt overhang. This problem of debt overhang could possibly besolved by the issuance of CoCos as, in the presence of CoCos, the shareholders may havean incentive to make an investment, to avoid conversion.The investment incentives of shareholders can be investigated by looking at what happenswhen assets are increased by one unit, financed through one unit of equity (in marketvalue). If equity raises by more that one unit, the shareholders would make a profitwhen they invest, giving them an incentive to do so. However, when equity increasesby less than one unit, investment is not beneficial to shareholders. Consider the case inwhich a new accounting report has just be released, with an asset value of Ytn = 100,we can then examine what happens when this asset value increases by one unit. Theprofit of this investment of one unit is plotted against the conversion trigger in Figure4.8. The solid black line shows the concept of debt overhang: in the case of no CoCosthe shareholders do not make a profit when they invest, they actually suffer a smallloss. Furthermore, the dashed-dotted lines show that when the terms of conversion arefavorable to shareholders, the shareholders do not have an incentive to invest in the firm.The black dashed-dotted line corresponds to the existence of a PWD CoCo in the capitalstructure of the firm and shows that the PWD CoCo even makes the investment incentivefor shareholders more negative, especially close to the conversion trigger. This showsthat CoCos with a principal write-down or less dilutive CoCos are not capable of solvingthe problem of debt overhang. However, highly dilutive CoCos do give the shareholdersan incentive to invest to avoid conversion. This is shown by the dashed lines in Figure4.8, which correspond to highly dilutive CoCos and show that the shareholders makea profit when they invest. Especially close to the conversion trigger, the shareholdershave a big incentive to invest in a last attempt to avoid conversion. To summarize,when terms of conversion are beneficial enough to CoCo investors, CoCos are capable ofcreating an investment incentive for the shareholders. However, PWD CoCos and lessdilutive CoCos are certainly not capable of creating such an incentive.

70

Volatility of Assets0 0.05 0.1 0.15 0.2 0.25 0.3

Equ

ity

26

27

28

29

30

31

32

33

34

35

Debt: 50, no CoCosDebt: 50, CoCo: 5, ; = 0 (PWD)Debt: 50, CoCo: 5, ; = 0.5Debt: 50, CoCo: 5, ; = 1

Volatility of Assets0 0.05 0.1 0.15 0.2 0.25 0.3

Equ

ity

14

16

18

20

22

24

26

28

Debt: 60, no CoCosDebt: 60, CoCo: 5, ; = 0 (PWD)Debt: 60, CoCo: 5, ; = 0.5Debt: 60, CoCo: 5, ; = 1

Figure 4.7: Equity value versus volatility, where equity is replaced by different types ofCoCos. The upper panel corresponds to a lower level of leverage than thelower panel.

71

Conversion trigger75 80 85 90 95 100

Pro

fit o

f inv

estm

ent

-1

0

1

2

3

4

5

No CoCos; = 0 (PWD); = 0.1; = 1; = 0.5

Figure 4.8: Profit (a negative profit means a loss) made by the shareholders, followingfrom the investment of one unit, plotted against the conversion trigger.

4.3 The role of the MDA trigger

In this section we will investigate the role of the MDA trigger in the valuation of theCoCos. This means that in the valuation of a CoCo, we apply Theorem 3.4 and Algo-rithm 3.2, but with a different coupon term, as in equation (3.34). This computationis performed as described under the header Early cancelling of coupons at the end ofSubsection 3.5.1, by using Algorithm 3.4. To demonstrate the relevance of the inclusionof this trigger in the valuation of CoCos, we will look at the big price drop that the Co-Cos of Deutsche Bank suffered at the beginning of 2016. On January 28 Deutsche Bankreported a net loss of 2.1 Billion EUR over the last quarter of 2015. The relevant reportfurthermore reported for its Risk-Weighted Assets a value of 397 Billion EUR, downfrom 408 Billion EUR in the previous accounting report. Also, the Common Equity Tier1 (CET1) ratio (defined as the fraction of the common equity and the RWA) fell from11.5% to 11.1%, primarily reflecting the net loss over the quarter 1 . The particularreport caused a big downward move in the price of the CoCos of Deutsche Bank. At thistime, Deutsche Bank had four different CoCos issued (two in USD, one in EUR, one inGBP, all PWD CoCos).

To simplify the case, we will only consider the EUR CoCo. This CoCo’s write-down is

1Information comes from the Financial Data Supplement 4Q2015, which can be found athttps://www.db.com/ir/en/download/FDS_4Q2015_11_03_2016.pdf

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triggered when the CET1-ratio hits the level of 5.125% and it pays a coupon of 6%. Asis clear from the above, the CET1-ratio did not even come close to the low trigger level.Still, the CoCo price tumbled 19.5% percent within a week after the announcement ofthe report. Probably, this happened out of fear for reaching the MDA trigger and thesubsequent cancelling of coupon payments. The model proposed in Chapter 3 seemsparticularly relevant to analyze this case, as we can include the announcement of a badaccounting report in the valuation, as well as the early cancelling of coupons when theMDA trigger is hit. The precise value of the MDA trigger is not publicly known, so itis not possible to use the real value of the MDA trigger. However, it is still interestingto examine how much of a price drop the model can explain by taking the MDA triggerclose to the reported values.Unless stated otherwise, we use the same parameters as in Table 4.1. Before the badaccounting report arrives, we are in the case that there is one accounting report, witha value Yt1 = 408 bn, after the new accounting report arrives, we have two accountingreports with values Yt1 = 408 bn and Yt2 = 397 bn. The triggers are chosen such thatthey correspond with CET1 ratios at the moment of the accounting report. That is, wechoose vc such that it corresponds to a CET1 ratio of 5.125%. As we know the CET1ratio is 11.1% where RWA is 397, this means that the total amount of debt (only CoCosand straight debt in the model) is 397 × 0.889 = 352.93. So a CET1 ratio of 5.125%would then correspond to a RWA value of 352.93/(1− 0.05125) = 372, which is thus thevalue of the conversion trigger vc. The value of the MDA trigger vcc can be chosen in thesame way, a MDA trigger at a CET1 ratio of 10% would correspond to a RWA value of352.93/(1−0.1) = 392. The coupon of the CoCo is of course chosen as c2 = 0.06. As therelevant CoCo has a perpetual maturity, we choose the first call date, 10/10/18, as thematurity. Because we assumed that the second accounting report arrives at 01/28/16,t = 0 corresponds to 07/28/15. Hence T = 3 + 2/12 + 13/365. In Figure 4.9 the pricechange after the announcement of a bad accounting report is illustrated for differentchoices of the MDA trigger. The solid line corresponds to the case that the MDA-triggeris not included in the model, in this case only a drop of 14.3% in the CoCo price occurs,when looking at the price just before the release of the accounting report and afterwards.However, if we add the MDA trigger to the model, a more negative price change can befound. The dashed line corresponds to the case that we take the MDA trigger at 11%,i.e. just beneath the reported CET1 value. This gives a price drop of 18.7%. If we takethe MDA trigger to be 10%, the price drops by 17.3%, which is illustrated by the dottedline. However, we have to take the MDA trigger above the reported CET1 ratio of 11.1%to create a price drop of 19.5%, this is showed by the dashed-dotted line. That is, aprice drop of 19.5% corresponds in the model to the situation that the MDA trigger isalready breached, which was not the case. However, it is clear that a significant part ofthe price change is driven by the MDA trigger, not by the conversion trigger. The aboveillustrates the added value of taking the MDA trigger into account in the valuation ofa CoCo, especially when the MDA trigger is coming close, but the conversion trigger isstil far away.

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Weeks after release of accounting report-3 -2 -1 0 1 2 3

CoC

o pr

ice

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

No MDA trigger10%11%12%

Figure 4.9: CoCo price around the release of the bad accounting report, for differentchoices of the MDA trigger. The downfall in the middle of the figure reflectsthe influence of the accounting report. The solid line suffers a downfall of14.3%, the dotted line drops by 17.3%, the dashed line by 18.7% and thedashed-dotted line drops by 19.5%.

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Popular summary

In this thesis different pricing models are studied for the pricing of Contingent Convert-ible bonds (CoCos). These are special type of bonds, which convert into equity or arewritten down, when the capital of the issuing bank becomes too low. In this way, out-standing debt is reduced and capital is raised, to strengthen the capital position of thebank. In practice, this conversion is typically triggered by the capital ratio of the issuingbank falling below some threshold or by a regulator calling for conversion. However, inall of the existing pricing models the conversion of CoCos is triggered by a market value,like a stock price, falling below some threshold. In the first two chapters of this thesis theexisting literature is examined and one particular model is studied in detail. In the thirdchapter of this thesis a new CoCo pricing model is proposed which takes into accountthe difference between market values and book values. In this model, the market cannotobserve the true asset value process, but it only has access to noisy accounting reports,which are only published at discrete moments in time, typically every three months inpractice. In this way, the price of CoCos can only be based on the information from theaccounting reports, not on the true asset process. The model does not lead to closedform solutions for CoCo prices, but Markov Chain Monte Carlo methods are used tocompute prices.

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