25 Eb09 d NoureddineElHadjBraiek Structured Products

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Management of Structured Products

Noureddine EL HADJ BRAIEK

Credit Agricole SA

Eurobanking Dusseldorf May

Noureddine EL HADJ BRAIEK Management of Structured Products
http://find/http://goback/


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Structured Products : Overview

A structured Product is a complex deal that pays exoticcoupons Ci in future dates Ti. The deal could involve anoption that allows to cancel the deal before its maturity.

The coupons are mathematical functions of one or manymarket indices : Stock Index, Swap Rates, FX rates... Usually

the underlying indices are transparently observed by the twocounter-parties

Ci = (Ik(Tj),j i, k= 1..n)

The dealers : Investment banks (Exotic trading desks)The investors : Corporate, Insurance, Pension funds, HedgeFunds..



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Structured Products : Pricing

The price that the investor pays includes the hedging costand the commercial margin.

Pricing and Hedging are intimately related, they should bedetermined simultaneously : the dealer may sell at aconservative price and loose because of a miss-hedgingstrategy.

In order to get the price and the hedging strategy, we usestochastic models:

dS(t) = dt+ dW(t)

The main question : how should we choose our models?



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Single Asset Structured Products : delta hedging

Let us consider a structured product indexed on a single asset: Ci = (I(Tj),j

i), with a fair price (t)

The first order risk : Delta risk

(t) = I(t)(t)

Once the delta risk is hedged, it remains a second order risk :

Gamma risk

(t) S(t) = t+ 2

(S(t))2 + o(t)

In order to get a price, we need to estimate the quadratic risk(gamma risk) = we use gaussian or Log-normal diffusions,and the volatility parameter determines the Gamma cost

Gaussian Model :dSt = dt+ dwt

Log Normal Model :dStSt

= dt+ dwt



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Single Asset Structured Products : Gamma hedging

The first class of exotic products are Calls and Puts : called

Vanilla Options Or Gamma Options or Volatility OptionsThe dealers created new complex products, and used Vanillaoptions to hedge their Gamma Risks

But, as the quantity of the Gamma hedge is not stable and

depends for some exotic products on the level of theunderlying or the the implied volatility, we need stochasticvariance models (Heston example):

dSt

St= dt+VtdWt

dVt = k( Vt)dt+ VtdZt

where W and Z are two correlated brownians



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Multi Asset Structured Products : Delta hedging

Let us consider a structured product indexed on N assets :Ci = (I1(Tj)..IN(Tj),j i), with a fair price (t)The first order risk : Delta risk

k(t) = Ik(t)(t)

Once the delta risk is hedged, it remains a second order risk :Cross Gamma risk

(t)kIk(t) = t+jk2 Ij(t)Ij(t) + o(t)



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Multi Asset Structured Products : Cross Gamma Hedging

The single gamma risk could be hedged by Gamma Options

The cross gamma risk depends on the volatilities and thecorrelation

Historical analysis shows that Correlation is as volatile asvolatilities

A suitable modeling framework will consider a stochastic

variance-covariance matrix (Example Wishart):

dSt =VtdWt

dVt = (

KVt

VtK

T)dt+ QdZtVt +VtdZTt Q

T

where S is a vector of N assets prices, W is a N-brownianvector and Z is a N-brownian matrix.

Unfortunately, this modeling framework is not widely used inexotic desks



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Multi Asset Structured Products (Equity Derivatives):Market Practice

Dealers are globally sellers of Basket structured Products

They Hedge their first order risks : Deltas, but it remains

second order risks : Gamma and Cross GammaThat leads to an exposition to the realizations of the Varianceand Covariance between the assets returns

To hedge these toxic risks, dealers use Variance or Volatility

Swaps and Correlation Swaps



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What is wrong with the actual hedge?

Dealers are exposed to the product of the correlation andvolatility

So the needed quantities of both correlation and volatilityswaps depend on the market conditions

= Exposition to the volatilities of both Volatility andCorrelationBut Dealers didnt price these toxic risks : modeling weakness

Correlation and Volatility Swaps hedged a significant part foradvanced desks, but there still a big problem : How should wemodel the assets diffusion in order to take into account thetoxic risks? Risk magazine July 2008 : Sunk by correlation


Hi i l A l i V l ili d C l i


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Historical Analysis : Volatility and Correlation

We analyze the average of 1M volatilities and the average of1M correlations between 10 Big cap firms from the CAC40

IndexThe Volatility of Volatility=24%The volatility of Correlation=63%The correlation between volatility and correlation =50%

Figure:Noureddine EL HADJ BRAIEK Management of Structured Products


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C l i


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Conclusion

The hypothesis of constant correlation leads to hedgingmismatch

But how could we model stochastic correlation matrix?

= it is a hard task to model stochastic matrices under the

correlation constraintsAs a conclusion, What should we learn from the crisis?

We should shift from volatility and correlation description oftoxic risks, to Variance-Covariance one

We should use stochastic Variance Covariance Models, insteadof deterministic correlation models (even with stochasticcorrelation)


E l f St h ti V i C i M d l


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Example of Stochastic Variance Covariance Model :Dynamic

The variance covariance matrix evolves stochastically under thedynamic

dVt = (V Vt)dt+VtdWtQ+ Q

TdWTt

Vt

V0 = v0

with :

Q is n invertible matrix

> 0v0,V are symmetric definite matrix

W is a Brownian motion matrix


Th j i t D i


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The joint Dynamic

Under the forward measure we introduce correlations betweenthe assets and the variance-covariance matrix noises :

dSt =VtdZt

dVt = (VVt)dt+VtdWtQ+ Q

TdWTt VtThe only way to have an affine model is to introducecorrelation such as :

dZt =

1

2dBt + dW

Tt

where is a vector such that < 1 and Bt is a vectorialBrownian motion independent ofWt


The Basket Process


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The Basket Process

This dynamic implies that each basket process Pt =n

i=1 wiSit

follows a Gaussian model with CIR process for volatility under theforward measure :

dPt = vtdztdvt = ( vt)dt+

vtdwt

d < z,w >t= dt

Use the FFT technique for European basket options pricing


Example : Spread options


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Example : Spread options

The dynamic of the spread process Yt = S2t S1t is :

dYt = vtdztdvt = ( vt)dt+

vtdwt

where

= V11

+ V22 2V12

= 2

(q11 q12)2 + (q22 q21)2

= 1(q11 q12) + 2(q21 q22)(q11 q12)2 + (q22 q21)2

v0 = V11

0 + V22

0 2V120


Spread options : calibration Process


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Spread options : calibration Process


Spread options : Underlyings/Covariance Correlation Effect


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Spread options : Underlyings/Covariance Correlation Effect


Spread options : Volatility of Volatility Effect


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Spread options : Volatility of Volatility Effect


Spread options : Covariance Effect


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Spread options : Covariance Effect


Spread options : Correlation between volatility processes


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Spread options : Correlation between volatility processeseffect


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25 Eb09 d NoureddineElHadjBraiek Structured Products