Interest Rate Modelsirmuva.rogerlord.com/Slides.pdf · Suppose you are a trader in a bank or a...

October – November 2014

Dr. ir. Roger Lord

(roger (dot) lord (at) gmail (dot) com)

Interest Rate Models

MSc in Stochastics and Financial Mathematics

VU, UvA, Utrecht and Leiden University

October - November 2014 2

Material

The slides, a selection of the book of Filipović,

chapter 2 of my PhD thesis, plus a selection of

scientific articles will be the material;

All material will be made available at:

http://irmuva.rogerlord.com/

http://dso.duisenbergschooloffinance.com/








Material (2)

The material from Filipović will be:

HJM: 6.1 up to and including 6.4

Forward measures: 7.1 – 7.3 (7.1 until remark 7.1)

Market models: 11.1 – 11.7 (without 11.3 - 11.4)

For the last part (affine models) the material is:

Chapter 2 from Lord [2008], to be found here.

Finally, the papers of Brigo and Mercurio [2001],

Andreasen [2005], Mercurio [2009] are recommended

as additional reading material.

http://rogerlord.com/thesis.pdf

http://www.fabiomercurio.it/deterministic_shift.pdf







http://www.risk.net/data/risk/pdf/technical/0905_pg104.pdf




http://www.fabiomercurio.it/LMMpostcrunch5.pdf





Material (3)

Other good books on interest rate models:

http://ecx.images-amazon.com/images/I/81gb062ZJFL._SL1360_.jpg

http://ecx.images-amazon.com/images/I/41Xhdo5mG1L.jpg

http://ecx.images-amazon.com/images/I/41kqGVXXpqL.jpg

http://www.andersen-piterbarg-book.com/



http://ecx.images-amazon.com/images/I/51NPkFPSA+L.jpg

http://ecx.images-amazon.com/images/I/418rDQWfD6L.jpg


Outline

We will deal with the following topics:

1. Why do we need models?

2. Heath-Jarrow-Morton

3. Forward measures

4. Market models

5. Recent developments

6. Affine models

Three sets of homework exercises will be provided, and these will determine 50% of your grade. The other half will be determined by the exam.


Schedule

Date Time Location

Thu 23-10-2014 15:00-18:00 SP G3.05

Thu 30-10-2014 15:00-18:00 SP G3.05

Thu 06-11-2014 09:00-11:00 D1.162

Thu 20-11-2014 09:00-13:00 SP G3.05

Thu 18-12-2014

(Exam)

13:00 -16:00 SP D1.112


Part I – Why do we need models?


The need for models

Suppose you are a trader in a bank or a hedge fund,

and you only trade exchange-traded options. Do you

really need a model?

Suppose you are working in a pension fund, and

you can value your swaps and swaptions portfolio

relatively simply by bootstrapping the interest rate

curve and interpolating between available swaption

volatility quotes. Do you really need a model?


The need for models (2)

We do not always need sophisticated pricing models.

However, they can be of great importance in the

following situations:

Valuing (exotic) contracts for which there is hardly

any market data available;

As a mean of a more sophisticated interpolation

technique – incorporating no-arbitrage restrictions

in volatility interpolation is already quite tricky;

To determine hedges.


Exotic interest rate derivatives

Once we know how to price the plain vanilla options

(caps/floors and swaptions), the next step is of course

to consider more exotic contracts. Reasons for using

swaps and swaptions have been treated. Suppose

further that a company is uncertain about:

whether or not it will require the swap;

from when onwards it will require the swap.

In this situation it may be advisable for the company

to enter into a Bermudan swaption. With this product

the company can decide whether and when it enters

into a swap.


Exotic interest rate derivatives (2)

Bermudan swaptions are characterised by:

Possible exercise dates ex = {t1, …, tM} ;

Typically there are two lockout dates: a date until

which we cannot call, and one specifying after

which date we cannot call anymore.

The most liquid Bermudans have a fixed maturity. If

the swaption is a payer swaption, and we choose to exercise at Ti ex , the payoff is:

ii,i Tat )T(PS



The term fixed maturity refers to the fact that the

maturity date of the swap, i.e. the final payment date,

T, is fixed. Typically the final lockout date is one

period prior to the final payment date, i.e. T-1. With t1

being the first possible exercise date, the deal is often

referred to as a T no-call (nc) t1, or t1 into T

Bermudan swaption.



An example of a Bermudan option in real life:


Part II – Heath-Jarrow-Morton


Recap of short rate models

Short rate models model the short rate (note: this rate

is not directly observable in the market).

Mathematically it is equal to:

where f(t,T) is the instantaneous forward rate:

See also Section 2.2 of Filipović.

)t,t(f)T,t(flim)t(rTt

T

)T,t(Pln)T,t(f


Recap of short rate models (2)

There are a wide variety of short-rate models, for

example the so-called equilibrium/endogenous

models, which try to explain the behaviour of bond

and option prices as a function of the short rate. Some

examples:

Model Specification

Merton (1973) )t(dWdt)t(dr

Vašíček (1977) )t(dWdt)r)t(r(a)t(dr

Dothan (1978) )t(dW)t(rdt)t(r)t(dr

Cox, Ingersoll & Ross (1985) )t(dW)t(rdtr)t(ra)t(dr


Recap of short rate models (3)

Hereafter a number of authors considered no-arbitrage

models in which the term structure is a given that has

to be fitted in order to be able to price other, more

complex derivatives – coinciding with an increased

liquidity in the interest rate market:

Model Specification

Ho-Lee (1986)

Black-Derman-Toy (1990)

Hull-White (1990)

Hull-White Ext-CIR (1990)

Black-Karasinski (1991)

)t(dWdt)t()t(dr

)t(dW)t(r)t(dt)t(r)t()t(dr

)t(dW)t(dt)t()t(r)t(a)t(dr

)t(dW)t(r)t(dt)t()t(r)t(a)t(dr

)t(dW)t(dt)t()t(r)t(a)t(rlnd


An extension of short rate models

There is a general way to come up with a no-arbitrage

extension of an otherwise tractable short rate model.

The idea builds on that of Hull and White, and is

formalised in e.g. Brigo and Mercurio [2001].

The ingredients are:

A short rate model in which we can calculate

Pmodel(t,T) as a function of the state variables;

Market prices Pmarket(0,T) for all T;

In general Pmodel(0,T) Pmarket(0,T)

(otherwise there would be no point to this exercise).











An extension of short rate models (2)

By setting:

we obtain bond prices:

The objective is that the left-hand side equals the

market prices of zero-coupon bonds:

)t(r)t()t(r elmod

)T,t(Pdu)u(exp)T,t(P elmod

T

t

)T,0(Pdu)u(exp)T,0(P elmod

T

0market


An extension of short rate models (3)

Rearranging the equation yields:

)T,0(f)T,0(f

T

)T,0(Pln

T

)T,0(Pln

T

ln)T(

elmodmarket

marketmodel

)T,0(P

)T,0(P

market

model


Shortcomings of short rate models

Suppose we have a one dimensional affine short rate

model:

i.e. the zero bonds are exponentially affine in the short

rate r(t). We can already see that the model is a bit too

simple. Suppose the short rate is initially r and moves

to r+ε. Then the yield curve changes as follows:

)t(r)T,t(B)T,t(Ae)T,t(P

tT

)r()T,t(B)T,t(A

tT

r)T,t(B)T,t(A)T,t(R)T,t(R


Shortcomings of short rate models (2)

The difference is therefore εB(t,T)/(T-t), which means

that only particular changes can be achieved across the

yield curve. For example, consider the Vašíček model:

In this time-homogeneous model we therefore have:

The mean reversion coefficient a is typically just a

couple of percent, so that the yield curve can roughly

only shift parallel.

)a(O)tT(a1)tT/()T,t(B 2

21

)tT(a

a1 e1)T,t(B



An example in the Vašíček model of what can happen

when the short rate moves from 2.5% to 2.7% (quite a

large move):



Furthermore, one can check that in one-factor models

that have an affine term structure zero yields and

instantaneous forward rates are perfectly correlated.

Empirically for zero yields and continuously

compounded annual forward rates:

2

4

6

8

2

4

6

8

10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Co

rre

lati

on

Tenor (yr)

Tenor (yr)

2

4

6

8

2

4

6

8

10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Co

rre

lati

on

Tenor (yr)

Tenor (yr)


Heath-Jarrow-Morton

Before the previously described developments that

allowed short rate models to fit the initial term

structure exactly, the shortcomings of short rate

models were:

Initial term structure could not be fit exactly;

Fact that driving process is one-dimensional is not

very realistic.

For these reasons, Heath, Jarrow and Morton ventured

to investigate whether they could formulate a multi-

dimensional model for instantaneous forward rates.


Heath-Jarrow-Morton (2)

In short rate models we model the short rate. In the

Heath-Jarrow-Morton (HJM) framework, we model

all instantaneous forward rates under Q:

where:

is a one-dimensional adapted stochastic process;

is a d-dimensional adapted stochastic process;

W is a d-dimensional Wiener process;

f(0, T) = fmarket(0, T).

)t(dW)T,t(dt)T,t()T,t(df



The question Heath, Jarrow and Morton asked, is:

What does the absence of arbitrage imply for (t,T)?

Remember we can write zero-coupon bond prices as:

and we know that, in absence of arbitrage, the

following must hold:

T

tdu)u,t(fexp)T,t(P

)t(dWOdt)t(r)T,t(P

)T,t(Pd



We should analyse the dynamics of P(t,T) and see

what this implies about its drift.

First of all notice that the integrated SDE reads:

implying the following for the short rate:

t

0

t

0)s(dW)T,s(ds)T,s()T,0(f)T,t(f

t

0

t

0)s(dW)t,s(ds)t,s()t,0(f)t,t(f)t(r



If we take the logarithm of P(t,T), we obtain:

We assume the conditions on and are such that

Fubini’s theorem (classical and stochastic) apply. See

also paragraph 7.5 of this course for a rigorous proof

of Fubini’s theorem. It follows that:

T

t

T

t

t

0

t

0

T

t

T

t

du)s(dW)u,s(duds)u,s(

du)u,0(fdu)u,t(f)T,t(Pln

t

0

t

s

t

0

T

s

T

t

t

0dsdu)u,s(dsdu)u,s(duds)u,s(

https://staff.fnwi.uva.nl/p.j.c.spreij/onderwijs/master/si.pdf





Collecting results, this yields:

where we defined:

t

0

t

0

T

s

)s(dW)T,s(v

ds)s(rdu)u,s(

)T,0(Pln)T,t(Pln

T

sdu)u,s()T,s(v



Applying Itō’s lemma to – ln P(t,T) yields:

Since the drift should equal r(t), we can answer Heath,

Jarrow and Morton’s question as:

)t(dW)T,t(v

dt)T,t(vdu)u,t()t(r)T,t(P

)T,t(dP 2

21

T

t

2

21

T

t)T,t(vdu)u,t(



When is P(t,T) a true martingale?

Either when the Novikov condition is satisfied:

for all maturities T, or:

If forward rates f(t,T) are nonnegative.

The latter implies 0 P(t, T) 1. Since P(t, T) divided

by the money market account is a local martingale,

and it is now also uniformly bounded (MMA 1), it is

by definition a true martingale.

T

0

2

21 dt)T,t(vexpE


HJM and short rate models

Remember that the short rate process satisfies:

and that:

t

0

t


T

tdu)u,t()T,t(v

2

212

21

T

t)T,t(v

T)T,t()T,t(vdu)u,t(


HJM and short rate models (2)

As an example, consider . Then we have:

We obtain:

)tT()T,t(v

)tT()T,t(vT

)T,t( 22

21

)T,t(

)t(Wt)t,0(f

)s(dWds)t,s()t,0(f)t,t(f)t(r

22

21

t

0

t

0



In SDE form this becomes:

which is exactly the Ho-Lee model with the drift that

is required to fit the initial term structure exactly.

)t(dWdttdt)t,0(f)t(dr 2

t



Let us look at the general case. We start with:

Assuming that (t, T) is differentiable w.r.t. T, as well

as the technical conditions to be allowed to use

Fubini’s theorem (see Proposition 6.1):

t

0

t


t

0

u

0 u

t

0

t

0

t

s u

t

0

t

0

t

0

t

0

du)s(dW)u,s()s(dW)s,s(

)s(dWdu)u,s()s(dW)s,s(

)s(dW)s,s()t,s()s(dW)s,s()s(dW)t,s(



Similarly:

Hence the short rate is an Itō process:

where is defined accordingly.

t

0

u

0 u

t

0

t

0duds)u,s(ds)s,s(ds)t,s(

t

0 u

t

0 udu)u,0(f)0(rdu)u,0(f)0,0(f)t,0(f

t

0

t

0)s(dW)s,s(ds)s()0(r)t(r


HJM – Existence and uniqueness

For modelling purposes one would prefer

The HJM SDE in this case equals:

If the volatility function is uniformly bounded, jointly

continuous and Lipschitz continuous in the last

argument, there exists a unique solution of f(t,T).

)t(dW)T,t(f,T,t

dtdu)u,t(f,u,t)T,t(f,T,t)T,t(dfT

t

)T,t(f,T,t)T,t(


HJM and the curse of dimensionality

Consider a one-factor HJM model:

Whilst this is a one-factor model, its general form

suffers from the curse of dimensionality.

To see the curse of dimensionality in a one-factor

model, let us integrate the SDE to obtain:

)t(dW)T,t(dtdu)u,t()T,t()T,t(dfT

t

t

0)s(dW)T,s(...)T,t(f


HJM and the curse of dimensionality (2)

Focussing on the correlation function:

One can immediately see that for general volatility

functions this leads to an infinite-dimensional process.

t

0

2t

0

2

t

0

0

ds)T,s(ds)S,s(

ds)T,s()S,s()T,t(f),S,t(fCorr



Jamshidian [1991], Cheyette [1992], Babbs [1993],

Ritchen and Subsankramanian [1993] and Carverhill

[1994], see also Andreasen [2005] for more

background, noticed that separable volatility functions

will lead to a Markovian model in a manageable

number of variables. In a one factor setting, this is

achieved by:

Then the stochastic integral immediately becomes:

t

0

t

0)u(dW)u(h)T(g)u(dW)T,u(

)t(h)T(g)T,t(








The correlation function then becomes:

which immediately reduces the dimensionality, and

makes the model amenable to a tree and finite

difference implementation, see also Andreasen [2005].

One can show that we can retrieve the Hull-White

model exactly if we choose the following separable

volatility function in the one factor setup:

1)T,t(f),S,t(fCorr0

)tT(ae)T,t(







HJM – pros and cons

If we return to the initial goals of Heath, Jarrow and

Morton, we can see that they realised them:

The initial term structure can be fitted exactly;

More realistic (multi-dimensional) models of the

term structure can be constructed.

This can also be achieved within short rate models,

though within HJM we only have to worry about

volatilities (and correlations).

At the end of the day, however, neither short rates nor

instantaneous forward rates are market rates – this led

to the development of the so-called market models.


Part III – Forward measures


Forward measures

So far the usual setup has been to price derivatives in

the risk-neutral measure Q, associated with the bank

account as the numeraire asset (we assume B(0) = 1):

The value at time t, (t), of a claim X at time T in this

measure is:

dt)t(B)t(r)t(dB

)T(B

X)t(B)t( t

QE


Forward measures (2)

If in particular we choose X = 1 and t = 0, we obtain:

Since P(0,T) B(T) > 0, we can define an equivalent

probability measure QT Q as follows:

)T(B)T,0(P

)T,T(P)0(B1

)T(B

1)0(B)T,0(P 00

QQ EE

)T(B)T,0(P

1T

dQ

dQ



For t T we have:

Assume we are now in the HJM framework:

)t(B)T,0(P

)T,t(P

)T(B)T,0(P

1t

T

t

T

t

Q

Q

E

dQ

dQE

dQ

dQ

)t(dW)T,t(vdt)t(r)T,t(P

)T,t(dP



Hence we can write:

The Cameron-Martin-Girsanov theorem implies that:

is a Brownian motion under the T-forward measure.

W)T,(v

)u(dW)T,u(vdu)T,u(vexp

t

T

t

T

t

2

21

T

t

dQ

dQ

t

0

T ds)T,s(v)t(W)t(W �



Now consider the ratio of two bonds: P(t,S) / P(t,T). In

the T-forward measure this is a martingale, since it is

the ratio of a tradeable asset, divided by the numeraire.

In formulae, we have:

Q

QE

Q

QEE QQQ

d

d/

)T,t(P

)S,t(P

d

d

)T,t(P

)S,t(P T

u

T

uu

T

)t(B)T,0(P

)T,t(P/

)t(B)T,0(P

)S,t(Puu

QQ EE

)T,u(P

)s,u(P



In the HJM framework, we have:

where:

and WT is a Brownian motion under the T-forward

measure.

T

T,St W)T,0(P

)S,0(P

)T,t(P

)S,t(P

)T,t(v)S,t(v)t(T,S



The T- and S-forward measures are related by:

and WT is a Brownian motion under the T-forward

measure.

T

T,St

T

S

T

S

W

)S,0(P)T,t(P

)T,0(P)S,t(P

)T,t(P

)t(B)T,0(P

)t(B)S,0(P

)S,t(P

ttt

dQ

dQ

dQ

dQ

dQ

dQ


Expectation hypothesis (2)

Changing measure we obtain:

Since we defined: , we obtain:

Hence, f(t, T) is a T-forward martingale.

t

0

Tt

0

t

0

T

s

)s(dW)T,s(ds)T,s(v)T,s(

dsdu)u,s()T,s()T,0(f)T,t(f

�

�

T

tdu)u,t()T,t(v

t

0

T )s(dW)T,s()T,0(f)T,t(f



Moving back to the original claim X:

Using the change of measure, we find:

)T(B

X)t(B)t( t

QE

X)T,t(P

d

d/

)T(B

X)t(B

d

d)t(

T

TT

t

TtTt

Q

QQ

E

Q

QE

Q

QE



This approach might greatly simplify our pricing

formula, since we now do not have to worry about the

covariance between X and the bank account B(T). The

change of measure effectively takes care of this for us.

This result holds in general for any valid numeraire N:

It makes sense to choose that numeraire that will allow

us to calculate the pricing formula as efficiently as

possible.

)T(N

X

)t(N

)t(t

NE



Kanō Jigorō:

In randori we learn to employ

the principle of maximum

efficiency even when we could

easily overpower an opponent.

Damien Hirst:

What I really like is minimum

effort for maximum efficiency.


Expectation hypothesis

Back to the T-forward measure – we can show that the

expectation hypothesis holds:

The long-term rate is determined by the short-term

rate, plus a risk premium

For the proof within the HJM setting, we need the

following two elements:

t

0

T ds)T,s(v)t(W)t(W �

t

0

t

0

T

s)s(dW)T,s(dsdu)u,s()T,s()T,0(f)T,t(f �


Expectation hypothesis (3)

Moreover, since:

we obtain the following version of the expectation

hypothesis:

T

0

T )s(dW)T,s()T,0(f)T,T(f)T(r

)T,t(f

)s(dW)T,s()T,0(f)T(rt

0

T

t

T

QE


Bond option pricing

Let us consider a European option on a bond maturing

at time S. The maturity of the option is T < S, and

strike price K. The price of this option is:

Utilising the T-forward measure, this simplifies to:

)T(B

K)S,T(P)t(B)t( t

QE

K)S,T(P)T,t(P)t(T

t

QE


Bond option pricing (2)

If P(T,S) would be lognormal in the T-forward

measure, this would lead to a Black-Scholes like

formula. Actually, recall Black’s ’76 formula. If:

X is lognormally distributed

and

Then the forward (undiscounted) option price equals:

)d(KN)d(FNKX 21t

E

FXt E )tT()X(lnVar 2

t

tT

)tT(K/Flnd

2

21

2,1



From the discussion on forward measures, we know:

Furthermore, in the HJM framework we have:

If we therefore assume that (t, T) is a vector with

non-stochastic entries (i.e. only time-dependent

volatilities), we have a lognormal bond ratio.

)T,t(P

)S,t(P

)T,T(P

)S,T(PT

t

QE

T

T,St W)T,0(P

)S,0(P

)T,t(P

)S,t(P



The variance of the logarithm of the bond ratio is:

leading to the following (forward) option price:

T

t

2

T,S

T

T,STt du)u(WlnVar

)d(KN)d(N)T,t(P

)S,t(PK)S,T(P 21t

T

QE

T

t

2

T,S

T

t

2

T,S21

)T,t(KP

)S,t(P

2,1

du)u(

du)u(lnd



Consider the Vašíček model:

From exercise 6.6 we know that we have:

leading to the result that we can easily apply Black’s

formula to the pricing of bond options within this

model.

)t(dWdt)r)t(r(a)t(rd

)t(dWedt)T,t()T,t(fd )tT(a



Let us consider the pricing of ATM caplets in the no-

arbitrage Hull-White model, with constant volatilities.

The Black price of a caplet is given by (when IV is

the quoted vol):

when the caplet is ATM, K = Li(0), which yields:

iIV

i2IV2

1i

iIV

i2IV2

1i

T

TK/)0(Lln

T

TK/)0(Lln

i1ii KNN)0(L)T,0(P

1)T(N2)T,0(P)T,0(P

)T(N)T(N)0(L)T,0(P

iIV21

1ii

iIV21

iIV21

i1ii



In the no-arbitrage Hull-White model the price of a

zero-coupon bond put option equals:

where:

where .

)d(NdKN)T,0(P

)T,T(PK)T,0(P

1)T,0(P

)T,0(P

2i

1ii0i

i

1i

iT

QE

)T,a2(B)TT,a(B)T,T(PlnVarT

Tddd

i

2

i1i

2

1ii

T

0i

2

HW

iHW12T

T))T,0(KP/()T,0(Pln

1

i

iHW

i2HW2

1i1i

a/)e1()t,a(B at



To price a caplet with strike K, we end up with:

and for an ATM caplet K = Li(0). Note that:

so that the price of a caplet ends up being:

)T,T(P)T,0(P)K1( 1iiK1

1T

0ii i

iE

1)T,0(P

)T,0(P

)T,0(P

)T,0(P

)T,0(P

))0(L1)(T,0(P

1i

i

i

1i

i

ii1i

1)T(N2)T,0(P

)T(NTN)T,0(P

iHW21

i

iHW21

iHW21

i



Equating the Black price and the Hull-White price:

Now, for small values of x we have:

which leads to the following approximation:

1)T(N2)T,0(P

1)T(N2)T,0(P)T,0(P

iHW21

i

iIV21

1ii

2

x21)x(N

HW

ii

iiIV

)0(L

)0(L1



Remember what we are plotting when we are plotting

the ATM caplet volatility curve:

The Black vol for the ith caplet (on Li, which

sets at Ti and pays at Ti+1), as a function of Ti;

For a fixed tenor, i.e. Ti+1 – Ti = constant (3m, 6m).

The Hull-White vol (HW) is equal to:

)T,a2(B)TT,a(B

)T,T(PlnVar

iT1

i1i

1ii

T

0T1

HW

i

i

i

Fixed



An example of a typical graph of ATM caplet vols is

the following (23 July 1999, USD):



If we look at at-the-money caplet vols we can clearly

see a hump at intermediate maturities. This means that

the implied volatility of intermediate interest rates is

higher. Why?

One possible (plausible) reason is:

Central banks control the short-term rates;

Traders determine the intermediate maturities, who

are myopic, hereby causing volatility;

Mean-reversion for longer-term rates pulls down

the volatilities of longer-term rates.



What we end up with is:

The first term is decreasing when the term structure is

increasing, and vice versa. The second is a purely

decreasing function in Ti (when a > 0). So when the

forward term structure is increasing (which usually is

the case), we can only have a decreasing ATM caplet

volatility structure. So the one-factor Vašíček/Hull-

White model is not rich enough. Note: when a = 0, the

volatilities are flat. So mean reversion is important!

)e1()0(L

)0(L1)T( i

i

aT2

aT21

ii

iiiIV


Part IV – Market models


A little bit of history

When it came to pricing bond options, Black’s 1976

model was considered. There are two obvious

problems that can be signalled immediately:

deterministic discounting: perhaps this is second-

order because the payoff and the discounting cancel

out to some extent?

lognormal asset values: for a bond this means that

the bond price may be larger than the face value, i.e.

negative yields are possible.


A little bit of history (2)

There is however another problem: the pull-to-par

problem. Initially the bond price is known, then the

uncertainty increases gradually. However, at

maturity we know the value of the bond again.



People were quite accustomed to assume that

underlyings were lognormally distributed, and

switched to using rates (LIBORs, swap rates) or yields

as the underlying assets, and assuming these were

lognormal. This got rid of two problems:

The pull-to-par problem;

Bonds were now no longer able to be larger than

their face value;

But rates and yields are not tradables…



Nevertheless, people started using Black’s formula for

rates and yields. Certainly until the ’90s, people were

using Black’s formula as it was convenient, but it was

regarded as being theoretically unsound. Today, caps

and swaptions are still quoted using Black ’76.

First of all, remember that a forward LIBOR rate that

has Ti as its reset date, and Ti+1 as its payment date,

with daycount fraction i, is defined as:

1)T,t(P

)T,t(P1)t(L

1i

i

i

i



The price of a caplet is, using Black’s formula:

Price of a caplet:

where i is the volatility of the ith LIBOR.

Secondly, remember that a swap rate is defined as:

tTddtT

)tT(K/)t(Llnd

)d(KN)d(N)t(L)T,t(P

i

2

i12

ii

i

2

i21

i1

21i1ii

1

i 1ii

,

)T,t(P

)T,t(P)T,t(P)t(S



Price of a swaption:

Here we use the payoff of a payer swaption as a call

option on the swaprate:

We can see this as a security paying (S,(T)-K)+ at

times T+1, …, T. Black’s ’76 price is then:

K)T(S)T,T(P ,

1

i 1ii

tTddtT

)tT(K/)t(Slnd

)d(KN)d(N)t(S)T,t(P

,12

,

2

,21

,

1

1

i 21,1ii



Example of EUR ATM swaption vols at 05-04-2013 1

YR

2 Y

R

3 Y

R

4 Y

R

5 Y

R

6 Y

R

7 Y

R

8 Y

R

9 Y

R

10

YR

12

YR

15

YR

20

YR

25

YR

30

YR

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

1 M

O

6 M

O

1 Y

R

3 Y

R

5 Y

R

7 Y

R

9 Y

R

12

YR

20

YR

30

YR

Swap tenor

Bla

ck v

ola

tilit

y (%

)

Expiry

80.00%-100.00%

60.00%-80.00%

40.00%-60.00%

20.00%-40.00%

0.00%-20.00%



How about a theoretical justification of Black’s

formula? Consider a caplet. Its price at time t, using

the Ti+1-forward measure, can be written as:

One can show that:

so that the only assumption that is effectively being

made, is that the forward LIBOR rate is lognormally

distributed.

K)T(L)T,t(P ii

1i

t1ii E

)]T(L[)t(L ii

1i

ti

E



For the swaption, we again noted that the payoff of a

payer swaption is, at the first reset date T:

The PVBP is a portfolio of zero-coupon bonds, and

hence is a tradable asset with a positive value. We can

use it as the numéraire asset, to find:

where we are working under the (,)-swap measure.

K)T(S)T,T(P ,

1

i 1ii

K)T(S)T,t(P ,

,

t

1

i 1ii E



We know now that in the market:

Caplets are quoted using Black’s price, which in

effect assumes that Li(Ti) is a lognormal martingale

under the Ti+1-forward measure;

Swaptions are quoted using Black’s price, which in

effect assumes that S,(T) is a lognormal

martingale under the (,)-swap measure.

These are all models focussing on one quantity. Can

we however build up a consistent term structure model

that implies the above rates are all lognormal (in their

own measure)?


Market models

The previous modelling approaches were all in terms

of rates that are not directly observable in the market:

Short rates (short rate models);

Instantaneous forward rates (HJM);

The rates which are directly observable, are:

LIBORs (also the underlyings in caps/floors);

Swap rates (also the underlyings in swaptions).

The market models focus on modelling these.


Market models (2)

In 1997, Brace, Gątarek and Musiela, Jamshidian, and

Miltersen, Sandmann and Sondermann came up with

the market models. We will start with the LIBOR

market model (often just LMM or BGM/J model).

Note that a forward LIBOR rate equals:

Since P(t,Ti) / P(t,Ti+1) is a ratio of tradeables, the

quantity must be a martingale under the measure

induced by the denominator, the Ti+1-bond.

1)T,t(P

)T,t(P1)t(L

1i

i

i

i

http://www.cmap.polytechnique.fr/~mazari/NoufelFrikha/Brace_Gatarek_Musiela_1997.pdf








http://www.dm.unipi.it/~pratelli/Finanza/materiale/Jamshidian.pdf

http://www.math.ku.dk/~rolf/teaching/PhDcourse/MSSlibor.pdf









Market models (3)

The forward Li(t) is just an affine combination of this

ratio, which immediately yields, for s ≤ t:

i.e. the forward LIBOR rate is a martingale. This also

shows that the floating leg of a swap can be priced by

discounting the “expected” cash flows.

Consider a swap that has T as its first reset date, and

T as the final payment date. It pays Li(Ti) at Ti+1, for i

= , …, -1.

)s(L)t(L ii

1i

s E


Market models (4)

The value of the swap then becomes:

Unfortunately one cannot always split seemingly

linear payments like that into “expected” cash flows

and discounting. Consider a Libor-in-arrears payment,

where the payment is made at the reset date.

)T,t(P)T,t(P

)T,t(P)T,t(P

)T(L)T,t(P)t(V

1

i 1ii

1

i ii

1i

t1iifloat

E


Market models (5)

Hence, it pays iLi(Ti) at time Ti. We can reinvest this

in a deposit maturing at time Ti+1, and hence its

(undiscounted) value will be equal to:

which will not only depend on forward rates, but also

on the second moment of the Libor rate, which will

depend on its volatility.

)T(L1)T(L iiiiii

1i

t E


Market models (6)

To price caplets/floorlets, Black’s model assumed the

Libor rate had lognormal dynamics. In the Ti+1-

forward measure we therefore should have:

Contrary to the book, we use a one-dimensional

Brownian motion here, and further assume:

A reduction of dimensionality can be achieved by

reducing the rank of the correlation matrix.

)t(dW)t(L)t()t(dL 1i

iiii

dt)t()t(dW)t(dW ij

k

j

k

i


Market models (7)

What about Lj, for j < i? Clearly, Lj(t) cannot be a

martingale under this measure. But it should be under

its “own” measure:

Let us start with j = i-1. We know:

should also be a Ti+1-martingale. Remember:

)t(dW)t(L)t()t(dL 1j

jjjj

)t(L1)t(L1)T,t(P

)T,t(P

)T,t(P

)T,t(Pii1i1i

1i

i

i

1i

y,xddyxdxyxyd


Market models (8)

Hence:

Its drift should be zero. If dWi(t) · dWj(t) = ij(t) dt

(under any measure), we find:

)t(dL)t(dL

)t(dL)t(L1

)t(dL)t(L1)T,t(P

)T,t(Pd

ii1i1i

1i1iii

ii1i1i

1i

1i

)t(L)t(L)t()t()t(

)t(LDrift)t(L1)T,t(P

)T,t(PDrift

i1ii1ii,1ii1i

1i

1i

ii1i

1i

1i1i


Market models (9)

Solving for the drift of Li-1(t), we find:

and we thus end up with:

We can derive the correct dynamics recursively, see

section 11.2 in the book.

)t(L)t(L1

)t(L)t()t()t()t(LDrift 1i

ii

iii1ii,1i1i

1i

)t(dW)t(

dt)t(L1

)t(L)t()t()t(

)t(L

)t(dL

1i

1i1i

ii

iii1ii,1i

1i

1i


Market models (10)

Under the Ti+1-forward measure, we find:

So now we have a consistent no arbitrage model in

which forward LIBORs are modelled. Each forward

LIBOR is a lognormal martingale in its own forward

measure. This means:

The LMM is consistent with Black’s caplet formula.

)t(dW)t(

dt)t(L1

)t(L)t()t()t(

)t(L

)t(dL

1i

jj

i

1jkkk

kkkjjk

j

j


Market models (11)

The derivation till here was due to Jamshidian. Brace,

Gątarek and Musiela, and Miltersen, Sandmann and

Sondermann constructed an HJM setup in which Libor

rates were lognormal. The argument goes roughly as

follows (cf. 11.1 of the book). From Lemma 7.1:

implying the following for Li(t):

)t(dW)T,t(P

)T,t(P)t(

)T,t(P

)T,t(Pd 1i

1ii

T

1i

iT,T

1i

i

)t(dW)t(L1)t(1

)T,t(P

)T,t(Pd)t(dL 1i

1ii

T

iiT,T

i1ii

ii


Market models (12)

Now if we suppose there is an Rd-valued deterministic

function (t,T), such that:

Then we have:

One can prove that such a function exists, see Chapter

11.1 for a sketch.

)T,t()t(L1

)t(L)t( i

ii

iiT,T 1ii

)t(dW)t(L)T,t()t(dL 1iT

iii


Market models (13)

If we take a step back, we have found a way to derive

the dynamics of all Libor rates in one forward

measure. A common choice is the terminal measure,

so that we view all Libors L0, …, LN in the

TN+1-forward measure. Since we calculate terms like:

we numerically blow up each term. Whether this is a

major issue or not depends on the total variance of

each term.

)T,t(P/)t(X 1N


Market models (14)

Numerically it seems to be more advantageous

however to sometimes use a measure that is similar to

the risk neutral measure. We define the spot Libor

measure by defining a discrete version of the money

market account:

What is happening economically is that we are rolling

all our investments into the next maturing zero-coupon

bond.

i

1j j1j

i

*

)T,T(P

1)T(B


Market models (15)

Essentially this boils down to:

Using the T1-forward measure from T0 to T1;

The T2-forward measure from T1 to T2;

etc.

The relevant dynamics for each Libor rate can be

derived by using the appropriate dynamics on each

interval. See also Section 11.4.


Market models (16)

How about pricing swaptions? We know the forward

swap rate is defined as:

and it should be a martingale under the (,)-forward

swap measure.

1

j jj

1

j1j

j

j

1

i 1ii

1jj

1

j 1

i 1ii

1jj

1

i 1ii

,

)t(L)t(w

1)T,t(P

)T,t(P1

)T,t(P

)T,t(P

)T,t(P

)T,t(P)T,t(P

)T,t(P

)T,t(P)T,t(P)t(S


Market models (17)

So we are only interested in the diffusion of the

forward swaprate. Itō’s lemma yields:

The first approach to yield something tractable (by

BGM) first assumed:

This is clearly not the case, but let us start from here.

1

j j

j

,

, )t(dL)t(L

)t(S)dt(O)t(dS

)t(w)t(L

)t(Sj

j

,


Market models (18)

Then:

Defining j(t) = wj(t)Lj(t)/S,(t), if S, would be

lognormal, we would be able to use Black’s formula

with the following total variance:

1

j

,

jj

,

jj

,

,)t(dW)t(

)t(S

)t(L)t(w)dt(O

)t(S

)t(dS

1

k,j

T

0kjjkkj

T

0 )t(S

)t(dS

)t(S

)t(dS2,

Black

dt)t()t()t()t()t(

dtT,

,

,

,


Market models (19)

But the weights j(t) are not deterministic, but

stochastic… what happens to them if we bump the

forward curve up/down by 25 bp?


Market models (20)

These are the percentage differences in i · j for an

upward move of 25 bp:


Market models (21)

The products i· j do not seem too stochastic. So, let

us “freeze” them at their initial value, and obtain:

By calculating the real S,(t)/Li(t) and doing some

additional “freezing” we can get a more accurate

approximation that works in the same way (just a

slightly different, more complicated, ). The latter can

be found in Hull, and is due to Hull and White, and

Rebonato and Jäckel.

1

k,j

T

0kjjkkj

2,

Black dt)t()t()t()0()0(T)(

http://www.quarchome.org/capletswaption.pdf






Market models (22)

Actually, it’s not bad at all…

-35

-30

-25

-20

-15

-10

-5

0

5

10

0 5 10 15 20

Maturity

Dif

fere

nce in

Bla

ck v

ol. (

bp

)

Simple Shape-corrected


Market models (23)

So where has this got us? In the LMM:

We work with N forward LIBOR rates; forward

rates, and hence the corresponding zero-coupon

bonds are input;

Caplets/floorlets are priced via Black’s formula:

Swaptions priced via Black’s formula:

iT

0

2

ii

2i

Black dt)t(T)(

1

k,j

T

0kjjkkj

2,

Black dt)t()t()t()0()0(T)(


Market models (24)

A calibration to ATM caps/floors and swaptions is

therefore much more direct than in other models,

where we typically have a least squares calibration

model in terms of squared differences:

much easier calibration to at-the-money products

N

1i

2

market,ielmod,ii P)(Pwmin θθ


Market models (25)

Some challenges with the LIBOR market model:

Not a model of the full term structure; e.g. if we

model 6m LIBORs, there is no direct way to

recover 3m LIBORs, see e.g. 11.8 for a solution;

Although each LIBOR is lognormal under its own

forward measure, other LIBORs are not. This

complicates a Monte Carlo simulation, see 11.6;

LMM only allows for a perfect fit to volatilities at

one strike; how about the skew/smile?

In many products LIBORs are not (directly) the

underlying rates, but swap rates are.


Market models (26)

To understand the latter problem, consider a

Bermudan swaption. In a Bermudan swaption we can

decide at Ti to enter into a swap with payment dates

Ti+1 through TN. The most important drivers of the

product are therefore:

Si,N for all call dates Ti and their interdependence;

European swaptions with the same strike as the

Bermudan product.

We would certainly like to calibrate exactly to these

core European swaptions.


Market models (27)

For these purposes more generic market models have

been developed. Jamshidian introduced the most

popular of these, the (co-terminal) swap market model

(SMM), which models co-terminal swap rates. Here

each swaprate is assumed to be a lognormal

martingale under its own forward swap measure:

Derivation proceeds much the same as in the LMM.

)t(dW)t()t(S

)t(dSN,i

N,iN,i

N,i

N,i


Part V – Recent developments


Recent developments

What is the value of this structure?


Recent developments (2)

Remember that the value of a floating leg equals:

Clearly this is independent of how the date strip is

structured, but does depend on start date (T) and end

date (T).

)T,t(P)T,t(P

)T,t(P)T,t(P

)T(L)T,t(P)t(V

1

i 1ii

1

i ii

1i

t1iifloat

E



Again - what is the value of this structure?

In our setup, it must be zero.



There was already a basis in some currencies…



Though the basis between deposit rates and overnight

(o/n) rates of the same maturity have exploded.



So how can we explain this? Remember that LIBOR is

the average unsecured funding rate at which a

contributor bank can obtain unsecured funding in the

London interbank market, for a given period, in a

given currency.

What happens if bank A has a cash deposit with bank

B, and bank B defaults? Since it is unsecured funding,

bank A will have a claim on the assets of bank B, but

will have a much larger loss than a secured creditor.

Hence, there is credit risk involved – something we

ignored till now.



Like Mercurio [2009] let us introduce the concept of

default-free zero coupon bonds (P(t,,T)), and

defaultable zero-coupon bonds (D(t, T)). If we define

as the default time of a counterparty, and assume:

That the default process and rates are independent;

We have a constant recovery rate;

Then the price of a defaultable bond follows as:

T)R1(R)T,t(P

1)R1(R)T,t(P)T,t(D

T

t

]T[t

T

Q

EQ








Returning to LIBOR rates – this was defined as the

return on buying the zero-coupon bond maturing at

Ti+1, at Ti, and selling this bond at its maturity. Using

this defaultable zero-coupon bond, the return is:

1

T)R1(R

1

)T,T(P

11

1)T,T(D

11)T(L

1i

T

T1iii

1iii

ii

1i

iQ



To value swaps, we need forward LIBOR rates, which

typically follow from FRAs or are implied from

swaps. Contrary to what many books state (also that of

Filipović), a FRA in practice pays the following

amount at Ti:

In a default-free setting this is equivalent to receiving

the amount of the numerator at Ti+1.

)T(L1

K)T(L

iii

iii



Assuming the FRA itself has no counterparty risk

(more on this later), we obtain as its value:

The FRA rate K which renders the value 0, equals:

1i

T

T0i1ii

1i

T

T1ii0ii

iii

iiiti

T)R1(RK1)T,0(P)T,0(P

T)R1(R)T,T(PK11)T,0(P

)T(L1

K)T(L)T,0(P

1i

i

iT

1i

i

iT

iT

QE

QE

E

Q

Q

Q

1T)R1(R)T,0(P

)T,0(P1K

1i

T

T01i

i

i 1i

i

iT

QEQ



Since 0 R 1, and , we have:

and thus:

The FRA rate is larger than the default-free one.

1T0 1i

T

T1i

i

Q

1T)R1(R 1i

T

T01i

i

iT

QEQ

1)T,0(P

)T,0(P1

1T)R1(R)T,0(P

)T,0(P1K

1i

i

i

1i

T

T01i

i

i 1i

i

iT

QEQ



So how do we proceed? Well, most setups still ignore

the direct credit risk element, and proceed as follows:

Assume the risk-free rate is the overnight rate – we

can estimate our term structure from the overnight

swap (OIS) market;

Strip 3m forward rates from FRAs and 3m swaps,

where the discount rate is the OIS curve;

Strip 6m forward rates from FRAs and 6m swaps,

where the discount rate is the OIS curve;

etc.



Overnight swaps by the way are slightly different than

regular swaps, in that all o/n rates are compounded up

and paid out in one go. In the larger markets this

happens annually. This does not complicate the

valuation greatly.



Clearly there is more to it – if we only pay the

compounded o/n rate on an annual basis, are we not

exposed to a similar credit risk? Yes, though most

trades in the market these days are fully collateralised.

How does collateralisation work?

Let party A and B enter into a par swap;

Initially this has zero market value;

At time t, the contract has a value of V(t); suppose

this is positive to party A;

Party A asks party B to post V(t) of cash or bonds;



How does collateralisation work (continued)?

Party B posts V(t) worth of collateral;

The net exposure is now 0;

This process continues till the end of the contract

(and is typically netted over all outstanding trades).

An interest rate is reimbursed over the posted

collateral – this depends on the credit support annex

(CSA) under which this contract falls, but is often

equal to the o/n rate;



Schematically the following roughly happens:

V(t)

V(t) V(t+)-V(t)

V(t+)

V(t+2)

V(t+2)-V(t+ )

…

Value of the contract

Collateral to be posted

Interest rate accrued over collateral



Should the collateral rate not feature in the pricing

formula? If we have to set some money aside that we

cannot use for other purposes, the cost of this has to be

taken into account. Since the collateral rate before the

credit crunch was close to Libor, and banks were able

to fund themselves at Libor, this was neglected.

How can we incorporate this in a mathematical

model? We refer the interested reader to Piterbarg

[2012], a paper called “Cooking with collateral”, and

only sketch the very rough thought process.

http://www.risk.net/digital_assets/5658/risk_0812_piterbarg.pdf







Suppose we have a collateral account C(t), on which

we earn a (possibly stochastic) rate c(t):

We assume that we can replicate this return in the

market. Now suppose we are at time t, and are pricing

a derivative that pays 1 unit of cash at t + dt. By

purchasing:

we can replicate this cash flow.

dt)t(C)t(c)t(dC

)t(cdtexp



Piterbarg has made this more rigorous, and eventually

ends up at the result that any payoff can be priced as:

where Q is the measure under which the collateral

account is used as the numeraire.

This ties in nicely with Mercurio, in whose work the

OIS curve was used as a proxy for the risk-free rate. It

is not because it is almost risk-free that this is correct,

it is because all quotes are based on fully (cash)

collateralised swaps.

)T(Vdt)t(cexp)t(V i

T

tti

QE



What if there is the option to choose which collateral

one can post? Think of EUR cash, or GBP cash?

It goes too far to describe the full model here, but it

suffices to say that the problem then becomes a lot

more complicated. The current value of such a

structure can be expressed as:

where Pd is a domestic zero-coupon bond, cf and cd the

foreign and domestic collateral rates, and rd,f the drift

of the foreign exchange rate.

)T(Ve)T,0(P)t(V

Tt df,df dt0),t(c)t(r)t(cmaxT,d

td E



This means that a simple EUR 6m swap, collateralised

with EUR and GBP cash (earning respectively the

Eonia and Sonia rates on collateral), depends on:

6m swap rates

Eonia swap rates

Sonia swap rates

EUR/GBP FX forward rates and xccy basis swaps

Potentially EUR and GBP swaptions

Pricing a simple swap has become complicated!



Other recent developments:

What about the option to post EUR or GBP cash

collateral?

CVA / DVA – charges due to expected loss on

portfolio in case of default of counterparty or of

oneself;

Move to central clearing or exchanges – this also

affects the pricing;

There are many, many new challenges in derivative

pricing!


Part VI – Affine models


Affine models

We will now turn to affine models, a class of models

which is very popular thanks to its modelling

flexibility and analytical tractability.

Duffie and Kan [1996] pioneered these models in a

term structure context, and were subsequently treated

in more detail in Duffie, Pan and Singleton [2000] and

Duffie, Filipović and Schachermayer [2003].

Prominent members of this class are Black-Scholes

[1973], the Vašíček [1977] and Cox-Ingersoll-Ross

[1985] interest rate models, as well as the Heston

stochastic-volatility model [1993].


Affine models (2)

The material will be Chapter 2 of Lord [2008], and we

will assume that the processes do not contain a Lévy

component. For the interested reader, Filipović’

Chapter 10 also focuses on affine processes.

Let X be a Markov process in D Rn:

where W is a standard Brownian motion on Rn, and:

: D n

: D nn

)t(d)t(dt)t()t(d WXσXμX

http://www.rogerlord.com/thesis.pdf






The money market or bank account is defined as:

Now, the process X is called affine, if and only if:

, for (m0, m1) n nn

,

for (0, 1) nn nnn

where and 1x is interpreted

as a vector inner product

, for (r0, r1) n

Affine models (3)

dt)t(M)t(r)t(Md X

xmmx 10)(

xxx 10

n

1i ii10

T x)()(

n1111 ΣΣΣ

xrxT

10r)(r


Heston stochastic volatility model

In Heston’s [1993] stochastic volatility model, the

stochastic volatility is modelled by the same square

root process as is used in the Cox-Ingersoll-Ross

interest rate model:

The drift is deterministic, and the variance process

v(t) follows a mean-reverting square root process, with

the rate of mean reversion, the long-term level of

variance, and the volatility of variance.

)t(dW)t(vωdtθ)t(vκ)t(vd

)t(dW)t(S)t(vdt)t(Sμ)t(Sd

v

S


Heston stochastic volatility model (2)

Moreover, both Brownian motions are correlated:

Clearly, (S, v) is not an affine process. We can

however switch to (x, v), where x = ln S:

dtρ)t(dW)t(dW vS

)t(dW)t(v)ρ1(ω

)t(Wd)t(vρωdt)θ)t(v(κ)t(dv

)t(Wd)t(vdt)t(vμ)t(xd

2

2

1

121



Using the notation from before, the drift can be

identified as:

The variance-covariance matrix can be seen to equal:

0

021

10 mm

vωvρω

vρωv

v)ρ1(ωvρω

0v

v)ρ1(ωvρω

0v)()(

2

T

22

Txx



Leading to:

Since the interest rate is taken to be constant, this

finishes the classification of the Heston model as an

affine one.

210

1

00

00

00

00


Affine models (4)

Duffie, Pan and Singleton [2000] studied the extended

characteristic function of affine processes, which is

defined as:

where u Cn. A quick recap on characteristic

functions will follow later, for now it suffices to

remember that the characteristic function determines

the full probability distribution.

X(T)iuexpT)P(t,

X(T)iuexpM(t)X(t)T,t,u,

TT

t

T

M(T)1Q

t

E

E


Affine models (5)

Contrary to the real-valued characteristic function, the

extended version is not always defined. We have:

This equals the –Im(u)th moment, and unfortunately

not every moment is guaranteed to be finite. Hence we

define the strip of regularity as:

In the remainder we will drop the term “extended” and

just use the term characteristic function.

)Im(i]|e|[|]e[||)(| T)iT)i TT

uuX(uX(u EE

)Im(inuuX C


Affine models (6)

The question we are going to pose now is how we can

find the characteristic function of an affine process.

The characteristic function, as it can be viewed as a

special type of payoff, should satisfy the Feynman-

Kac formula:

subject to the obvious boundary condition:

rtr

t

T2

21

T

σσxx

μx T

)T(iexp)t(,T,T, TXuXu


Affine models (7)

Under certain technical regularity conditions, Duffie et

al. [2000] show that the solution is given by:

where A and B are respectively R and Rn-valued

functions. From the ‘ansatz’ we find:

xuBuxuT)T,t,()T,t,(Aexp),T,t,(

T

T

2T

td

d

td

Ad

tBB

xxB

xx

B


Affine models (8)

Using tr(AB) = tr(BA) if AB and BA are well-defined:

Inserting this into the PDE, and dividing by we find:

xBBBBxBBxx

1

T

0

T

10

TT

T

2

)(trtr

xrxBBBB

xmmBxB

T

101

T

0

T

21

10

T

T

r

)(td

d

td

Ad


Affine models (9)

Since the PDE should hold for every x, it must in

particular hold for x = 0, x = (1, 0, …, 0)T, etc. This

leads to the following system of ODEs:

where we denote BT1B for the vector in Cn with ith

element equal to BT1iB. The boundary conditions

carry over: A(u, T, T) = 0 and B(u, T, T) = iu.

11

T

21T

1

00

T

21

0

T

td

d

rtd

Ad

rBBBmB

BBmB


Affine models (10)

For some models the system of ODEs (a system of

Ricatti equations) can be solved explicitly, though in

general they have to be solved numerically. This can

be done quite efficiently.

We return to the technical conditions. If we define:

then the conditions formulated by Duffie et al. are:

)t(M

),T,t,()t(

xu ))t(()t()t()t( T

XσBη

)t(E

T

0

T dt)t()t( ηηE


Affine models (11)

In any model, the undiscounted (forward) value of a

call price on an asset S can be written as:

Very generally, we can write the call price as:

where S indicates a probability in the measure

associated with using the asset as the numeraire, and P

indicates the T-forward measure. Finally, F(t, T) is the

forward price of the underlying asset at time T, as seen

from time t – equal to: .

])K)T(S([T,K),t(SC T

t

E

K)T(SKK)T(S)T,t(FT,K),t(SC PS

)]T(S[)T,t(F T

tE


Affine models (12)

This general form also holds in the Black-Scholes

model, where we can explicitly calculate the two

probabilities as N(d1) and N(d2).

A probability can be calculated by means of Fourier

inversion, an approach dating back to Lévy [1925], see

also Gurland [1948] and Gil-Pelaez [1951]:

where k = ln K, and is the characteristic function of

the logarithm of the underlying asset.

duiu

)u(eReK)T(S

0

iuk

π1

21

P


To obtain the same probability in the stock-price

measure, we can use a measure transform:

In a model without dividends, the scaling is exactly

the inverse of the forward, i.e. 1 / (-i), so that the

full term evaluates to (u-i) / (-i).

Affine models (13)

))T(Sln)iu(iexp()t(S

)T,t(P

))T(Slniuexp()T,t(P/)T,T(P

)t(S/)T(S

))T(Slniuexp(d

d))T(Slniuexp(

t

t

tt

P

P

PS

E

E

P

SEE


The S-probability can therefore be expressed as:

Though this is not the best way to value derivatives

via Fourier inversion (see Lord and Kahl [2007]), it

at least shows that if we can calculate the T-forward

characteristic function in closed-form (which we can

for affine models), we can evaluate European option

prices via numerical integration.

Affine models (14)

du)i(iu

)iu(eReK)T(S

0

iuk

π1

21

S

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