Valuing Retail Structured Products

Jos van Bommel Luxembourg School of Finance

What are Retail Structured Products?

Of-the-shelve financial securities sold to private investors, mostly by commercial banks, traded either over the counter, or through a stock exchange. They can be classified as i) guaranteed products, ii) leverage products, and iii) others. Guaranteed products are long term (3-10 years), and offer investors a guaranteed payoff plus an upside linked to the performance of an index. They are mostly sold over the counter and unlisted. Leverage products offer accelerated participation in stocks or indices (a.o.). They are short term investments, listed on a stock market.

Retail Structured Products, raison d´être

Market Completion They offer plays that are not available, or difficult to implement, for private investors. The demand for Guaranteed Products derives from loss aversion, either as a behavioral bias, or as a genuine financial constraint. The demand for Leveraged Products derives from the desire for speculation. Both motives, have, IMHO, merit. The key question is Is the product fairly priced? A follow up question would be whether they are fairly marketed…

Outline of the talk

1) Example of a Guaranteed Product.

Valuation by replication Valuation by simulation 2) Example of a Leverage Product (a barrier option) Valuation in a Black Scholes world (Merton, 1977).

Valuation by simulation 3) Empirical findings on the pricing of Leverage

Products

Guaranteed Products

Offer a guaranteed payoff, with an upside tied to one or more stocks or stock market indices.

Example of a simple GP: BNP Fonds a Capital Protégé “X2”

BNP X2 Avec X2, profitez d’une hausse à 4 ans de l’indice DJ Euro Stoxx 501 dans la limite de 60%, et récupérez à l’échéance, jusqu’à 150% de votre capital investi2. L’indice DJ EuroStoxx 501, diversifié géographiquement et sectoriellement, est constitué des 50 valeurs phares de la zone euro. Le 7 avril 2014, on calcule la performance de l’indice DJ Euro Stoxx 50 par rapport à son cours d’origine3 du 7 avril 2010. La Performance de l’Indice est alors égale à la performance ainsi calculée dans la limite de 60%, et sans pouvoir être négative (« Performance Finale de l’Indice ») A l’échéance, le 17 avril 2014, la Valeur Liquidative Finale du Fonds est égale à 90% du capital investi majoré de la Performance Finale de l’indice: elle sera donc comprise entre 90% et 150% (90% + 60%) de la Valeur Liquidative de Référence4, soit un rendement actuariel maximum de 10,55%5.

BNP X2 To make a long story short.. The payoff per €1000 investment is, after four years:

500

1000

1500

-20% 0% 20% 40% 60%

4 year return on Stoxx50

X2 Payoff

Or, payoff = €900 + €max(0,min(600,rstoxx×1000))

BNP X2 Clearly, we can replicate the payoff of the BNP X2 with

500

1000

1500

-20% 0% 20% 40% 60%

4 year return on Stoxx50

X2 Payoff

A 4-yr deposit of €PV(900)

A Call Option on the Stoxx with X = S

Less a Call with X = 1.6 S

We know that the value of a Call is C(S,X,∆t,σ,rf)

Value of the BNP X2

σ, anual volatility

- Option values are computed with the Black Scholes formula - At issue, the long term implied volatility of the Stoxx was about 23% - The 3 year swap rate was 1.96%, the 5 year rate was 2.46% - Assumption: BNPs credit risk is zero…

With ∆T and rf (≈2%) given, the replication value of the BNP X2 depends on σ only:

900 €

910 €

920 €

930 €

940 €

950 €

960 €

970 €

980 €

990 €

0% 10% 20% 30% 40% 50% 60% 70% 80%

Value of the BNP X2

€965,00

€970,00

€975,00

€980,00

€985,00

€990,00

18% 20% 22% 24% 26% 28% 30% 32% 34%σ, anual volatility

With ∆T and rf (≈2%) given, the replication value of the BNP X2 depends on σ only:

Value of the BNP X2

The BNP X2 is overpriced by about 2%. (compared to the replication value)

Which, IMHO, is not unreasonable for a 4 year security.

Constructing the replicating portfolio would probably cost cost much more than 2% for most retail investors:

To replicate a €1000 investment in BNP X2s, you would need to trade options for approximately €275 .. ( buy €215 worth of at-the-money Calls, and sell €60 worth of X = 1.6S Calls. )

Complex Guaranteed Products

Locally Capped products are popular in the U.S.

These pay a garantee, plus the sum of series of (e.g. monthly) capped returns of an underlying.

Ex.: A five year contract, guarantee = 100%, with an upside computed as the sum of monthly returns, capped at 5%.

the maximum payoff is 1+60×5% = 400% the original investment.

These products also come in compounded forms. If the above were a compounded product, the maximum payoff would be (1+5%)60 = 18,679 times the investment !!!

Locally Capped Products

Because their payoffs are path dependent, these products are difficult to value.

Using a Monte Carlo method of Rossetto and van Bommel (2009), Bernard and Boyle (2010) show that LCPs are overpriced by 14% or more, and have expected returns < rf

Marketing is very agressive, through unrealistic scenarios: Prospectuses contain approx. 7 scenarios, of which 5 or 6 are extremely unrealistic.

Selling commisions to (brokers) are very high (average 6%!!).

Evidence (and theory) suggests that overpricing increases in complexity (Stoimenov and Wilkens (2007).

Excerpt from the Prospectus of the NAS.2 (monthly compounded LCP), issued by a Citigroup

Excerpt from the prospectus of the NAS If the Nasdaq 100 would appreciate by 3% every month…

Your return would be > 600%

Source: Bernard et al. (2009)

Excerpt from the Prospectus of the NAS.2 (monthly compounded LCP), issued by a Citigroup

If the Nasdaq 100 would appreciate by more than 5.5% (the cap) every month…

Your return would be 3325%!!

Source: Bernard et al. (2009)

Bernard and Boyle show that the probability of this is lower than 10-19 …

Locally Capped Products Five additional examples are given, three of which are (also) unreasably optimistic.

Representative scenarios such as +14%, -9%, +11%, -12%, etc. Are left out because these return 0, due to the capped increases and the uncapped decreases..

Complex Garanteed Products in Europe

For every month that Apple, Vodafone and Microsoft go up, you accumulate 0.5%

Three years, return is minimum 0% maximum 18% (AER 5.66%)

Locally Capped Products in Europe

9 year “FCP” (Fonds Commun de Placement), essentially an ETF. Value at maturity = maximum value during its life. (“Cliquet”) E.g. if after 3 months, the FCP trades at €109, you get at least €109 back at maturity. The “click” effect does not apply to a visible index or basket, but to the fund itself. Of which AXA is manager and market maker... They invest in environment stocks, employing a complex stop-loss strategy.

Axa´s Stop-Loss strategy ???

Method 1 “safe” If V0 is fund size at day 0, invest the PV(V0) in bonds, the remainder in (environment) stocks. After three months, invest PV(V1) in bonds, the rest in stocks, etc.

Method 2 “Risky” Invest all monies in stocks. Switch to all bonds as soon as value falls below PV(Vt)..

- From the prospectus it is not clear whether they choose 1) or 2)…

+ Funds are “ringfenced”: no transfer to AXA other than management costs (max. 2.2% according to prospectus). Discussion question: what if everybody follows stop-loss strategies…

Leverage Products First there were warrants (covered warrants, option-scheinen), essentially bank-issued European style options on stocks, indices, baskets, etc.

Very popular since the mid 90s.

Mostly because they offer, due to issuer market making, lower bid-ask spreads than exchange initiated (American-style) option contracts.

Traded mostly by speculating “hobbyists”.

Around the turn of the millenium, two new types of products: 1) barrier options (down and out calls, up and out puts) 2) endless leverage certificates appeared

Barrier Options The first generation had a B > X (for calls). They could be interpreted as levered positions:

If stock goes to €25.50 (+2%)

Turbo goes to €5.50 (+10%, if you paid €5)

If stock goes to €24.50 (-2%) Turbo goes to €4.50 (-10%)

The first such instruments were called Turbos. They had a knockout level slightly above X, say B = €21. If the stock drops to this level, the issuer would redeem the instrument, and return €1.

Since 2004 (or so) most banks issue barrier options with B = X. (and without residual value..)

Stock, Trading at €25

Turbo

Loan, Face value X = €20

Barrier Options (B=X)

X=B S

Payoff, Value

Max(0,S-X)

S-PV(X)

If the underlying is traded in ‘continuous time’ Merton (1973)

Barrier Options (B=X)

S

Payoff, Value

Max(0,S-X)

Max(0,S-PV(X))

With jumps, days and nights.

However, in reality, trade is not continuous…

Barrier Options (B=X)

X=B S

Payoff, Value

Max(0,S-X)

Max(0,S-PV(X))

At 9:05

The Gap Risk depends on the time of day

Barrier Options (B=X) The Gap Risk depends on the time of day

X=B S

Payoff, Value

Max(0,S-X)

Max(0,S-PV(X))

At 17:25

Barrier Options (B=X) How do we find the true value?

Impossible to find a closed form solution…

So, we use a Monte Carlo Simulation,

Monte Carlo Valuation (Boyle, 1977)

Simulate price paths of S, under the risk neutral probability measure,

compute the payoff,

and discount at the riskfree rate.

Do this 1,000,000 times,

The average is the value of the Derivative.

Advantage of MC: We can accomodate i) any payoff function, (e.g. look back options, shout options), and ii) any prices process (e.g. Geometric BM, Jump-Diffusion, ARCH-GARCH, periodic jumps).

For path dependent options MC is very expensive..

So we use “tricks”.

I have developed two such tricks

The Price Process of the Underlying Early models assume Geometric Brownian Motion (Black Scholes is based on this)

td dWdtS

dSσ+ν=

ν = intraday drift

σd = intraday volatility

W = Wiener Process

The Price Process of the Underlying Early models assume Geometric Brownian Motion

Later models assume Jump Processes

ttjtd dPdWdtS

dSφσ+σ+ν=~

ν = intraday drift

σd = intraday volatility

W = Wiener Process

σj = Jump standard deviation

φt = Normal Random Variable; N(0,1); i.i.d. over time

P = Poisson Process, intensity λ

The Price Process of the Underlying Early models assume Geometric Brownian Motion

Later Models assume Jump Processes

No model (that I know of), consider periodic jumps..

dNdNdPdWdt

SdS

ntnttjtd ν+φσ+φσ+σ+ν=~~

σn = Overnight Standard Deviation

N = Counting Process (days)

νn = Overnight drift

ν = intraday drift

σd = intraday volatility

W = Wiener Process

σj = Jump standard deviation

φt = Normal Random Variable; N(0,1); i.i.d. over time

P = Poisson Process, intensity λ

Compounding Interest The exsting literature always assumes continuously compounded interest..

However, in the real world, interest is compounded discretely!!!

It accrues on overnight periods only!!

This fact makes “closed form” formulas more complicated, but makes Monte Carlo more powerful

First Trick Hence, in real life we have the following Value function:

V = V(S, X, r, σd, λ, σj, σn, t, T)

Where T denotes the number of days until maturity, and t ∈ [0,1) denotes the time (in days) until the end of the trading day.

Trick: If S is far away from X, take large (simulation) steps, if S is close to X, take small steps:

Choose δt so that the expected overshooting is fixed (e.g. the maximum overshooting if the step is 0,01 second).

Second Trick: Extreme Simulation We sample extreme values (either the minimum or the maximum) in a continuous interval.

For each continuous interval, we denote the return to the minimum value µ ∈ (−∞,0], and the return from the minimum to the ending price ω ∈ [0,∞).

Although they are non-overlapping return from a Martingale, µ and ω are dependent: If µ is very small (very negative), the miminum is more likely to reached late in the interval, so that ω is likely to be small.

It turns out that µ and ω follow a joint probability distribution: (proposition I)

( )2

2

2)()(),( σ

µ−ω−µ+ω

µ−ω⋅σ=ωµ eKf

Where K(σ) is a proportionality constant.

Min-End Distribution

f(µ, ω) for the case where σ = 20% (i.e. yearly intervals).

The asymmetry along the diagonal is due to the fact that the process is a Geometric BM. For a “Arithmetic” BM, there would be no asymmetry.

For small (i.e. daily) σ´s the asymmetry is hardly noticable.

µ-0.2

0.4

0.6

ω

0.2

( )2

2

2)()(),( σ

µ−ω−µ+ω

µ−ω⋅σ=ωµ eKf

0.0

0.1

0.2

0.3

100 101 102 103 104

KB

O -

(S-X

)

S

t

1 second

1 minute

15 minutes

1 hour

2 hours

4 hours

1 day

ASM 1 day

Merton

Value diagram as a function of t.

(time left until overnight jump)

X = 100, r = 2.5%, σd =1%, σn=0.7% (historical averages), λ = 0, (no intraday jumps), T = 20 (days), t ∈ (1 – 0.0001)

An typical Index KBO

Typical Index vs. Typical Stock

0.0

0.1

0.2

0.3

100 101 102 103 104

KB

O -

(S-X

)

S

t

1 second

1 minute

15 minutes

1 hour

2 hours

4 hours

1 day

ASM 1 day

Merton

0

0.1

0.2

0.3

0.4

0.5

100 105 110 115

1 minute

15 minutes

1 hour

3 hours

1 day

Merton

σd =1%, σn=0.7%, λ = 0, (annual σ ≈ 20%)

σd =1.5%, σn=1%, λ = 0.1, σj =2% (annual σ ≈ 30%)

Valuation - Summary We suggest new tricks to value Path Dependent Options

with Monte Carlo.

First trick is to adapt the step as a function of the distance to critical values (the barrier(s)).

Second trick is to simulate minimum and ending returns of the continuous intervals.

Due to overnight jumps KBOs are more valuable than implied by the Merton (1977) formula.

And are worth more in the afternoon than in the morning, especially if they trade close to the barrier.

Are Leverage Products fairly priced?

Yes!! Their values are very close to the intrinsic value.

What´s more important: the bid-ask spreads are very low, lowest of all high-risk derivates:

30 stocks

grossleverage adjusted gross

leverage adjusted

average 6.4 97.7 24.3 655.3 46.4

min 1.3 2.6 1.1 7.9 2.125%-percentile 3.2 36.6 12.4 109.3 18.5median 4.7 62.4 17.8 235.3 29.575%-percentile 6.8 115.1 26.2 606.1 50.8max 37.9 1,639.3 292.5 14,782.6 4,607.6stdev 6.0 112.5 24.2 1,244.1 106.4

Rel

ativ

e Sp

read

(b

asis

poin

ts)

18,003 covered warrants

5,129 ELCs

“Playing” with leveraged products is not clever (neither is going to the casino). But if you want some excitement, knockout barrier options have a lower “cut” than regular options

Summary Retail Structured Products

It´s a huge and growing business!

In theory, they can provide market completion.

The key question is whether they are fairly priced.

We can figure this out with replication, calculation and with simulation.

Findings: Some are fairly fairly priced, others are not. Simple products (the BNP X2) are better.

For leverage products, look at Bid/Ask spreads.

References Black, Fisher, and Scholes, Myron (1973) “The pricing of options and corporate liabilities”, Journal of Political Economy,(May-June).

Bernard, Carole, and Boyle, Phelim (2008) “Structured products with caps and floors”, Unpublished working paper, University of Waterloo.

Entrop Oliver, Scholz, Hendrik, and Wilkens, Marco (2008) “The price setting behavior of banks: an analysis of open-end leverage certificates on the german market” Unpublished working paper, University of Ingolstadt.

Merton, Robert C. (1973) “Theory of Rational Option Pricing”, Bell Journal of Economics and Management Science, 4, 1, 141-183.

Rossetto, Silvia, and van Bommel, Jos (2009) “Endless Leverage Certificates”, Journal of Banking and Finance, 33, 1543-1553.

Stoimenov, Pavel A., and, Wilkens, Sacha (2005) “Are structured products fairly priced? An analysis of the German market for equity-linked instruments”, Journal of Banking and Finance, 29, 2971-2993.