Different aspects in correlation products pricing Pascal DELANOE, - - PowerPoint PPT Presentation

Different aspects in correlation products pricing

Pascal DELANOE, Structured Equity Derivatives

HSBC

2nd March 2012

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 1 / 46

First part : Local correlation calibration

Outline

1

First part : Local correlation calibration References Dupire general formula Calibration results : local volatility Extension to Stochastic Volatility Calibration results : stochastic volatility Products Study

2

Second part : study of correlation swap product Purpose of the study RCS Risk analysis : Hedge with Basket and stock vanillas Results PnL decomposition RCS Risk analysis : Hedge with Var Swaps Transaction costs Other Basket Main conclusions Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 2 / 46

First part : Local correlation calibration References

Presentation of the main used methodologies introducing local correlation

1 Langnau : Interesting intuitions, copula method 2 Sbai-Jourdan : no explicit local correlation => complex framework

: cf. deduce stocks vols from index and not index from stocks

3 Reghai : convergence using fixed point algorithm, but slow

convergence

4 Guyon/PHL : iterative approach and Dupire Formula (also

available in Avellaneda et al. ) Objectives : Investigate (4) and present alternative ideas Extend to Stochastic Volatility and applications to products

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 3 / 46

First part : Local correlation calibration References

Our framework = Reghai’s

Local Correlation introduced through the use of an overomega approach. What is Overomega ? ρPricing

i,j

= (1 − ω)ρHisto

i,j

+ ω First Model = Simple Local Vol Model with continuous dividends (mix

• f prop and cash dividends).

dSi

t = (rtSi t − Qi t − qi tSi t)dt + σ(t, Si t)Si t(

• 1 − ω(t, IS

t )dW i t +

• ω(t, IS

t )dW ⊥ t )

with Qi

t and qi t deterministic and :

IS

t

=

N

• i=1

wiSi

t

< dW i

t , dW j t >

= ρ0

i,jdt

< dW i

t , dW ⊥ t

> = 0∀i

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 4 / 46

First part : Local correlation calibration Dupire general formula

Local Correlation formula (general case)

Second Model = simple local vol model written on the index(continuous divs) : dIt = (rtIt − Qt − qtIt)dt + Itσ(t, It)dWt with Qt and qt deterministic. Same Basket Call prices in both models (Specific set of wi) : C(K, t) = EQ(exp(− t rsds)(It − K)+) = EQ(exp(− t rsds)(IS

t − K)+)∀t, K

but :

∂C ∂t dt = EQ(exp(− t rsds)((dIS

t − rt (IS t − K)dt)1IS t >K +

1 2 d < IS >t 1IS

t =K )) in basket model

∂C ∂t dt = EQ(exp(− t rsds)((dIt − rt (It − K)dt)1It >K + 1 2 d < I >t 1It =K )) in index model Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 5 / 46

First part : Local correlation calibration Dupire general formula

Local Correlation formula(2)

EQt ((−

• i

wi (Qi

t + qi t Si t ) + rt Kdt)1IS t >K +

1 2 d < IS >t 1IS

t =K ) = EQt

((−(Qt + qt It ) + rt Kdt)1It >K + 1 2 d < I >t 1It =K ) but : EQt (d < I >t 1It =K ) = EQt (d < I >t |It = K) B(0, t) ∂2C ∂K 2 and also : EQt (It 1It >K ) = EQt (IS

t 1IS t >K ) =

1 B(0, t) (C − K ∂C ∂K ) EQt (1It >K ) = EQt (1IS

t >K ) =

1 B(0, t) (− ∂C ∂K ) Condition on the forward (K = 0): Qt =

• i

wi Qi

t

qt = EQt (

i wi qi t Si t )

EQt (It ) Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 6 / 46

First part : Local correlation calibration Dupire general formula

Local Correlation formula(3)

ω(t, K) =

• K 2σ(t, K)2 + 2 ∂C

∂K ∂2C ∂K2

• EQt((qtIt−rtK)1It >K )

EQt(1It >K )

−

EQt((

i wiqi tSi t−rtK)1IS t >K )

EQt(1IS

t >K )

• EQt(

i,j wiwjSi tSj tσ(t, Si t)σ(t, Sj t)(1 − ρ0 i,j)|IS t = K)

− EQt(

i,j wiwjSi tSj tσ(t, Si t)σ(t, Sj t)ρ0 i,j|IS t = K)

EQt(

i,j wiwjSi tSj tσ(t, Si t)σ(t, Sj t)(1 − ρ0 i,j)|IS t = K)

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 7 / 46

First part : Local correlation calibration Dupire general formula

Local Correlation formula : understand each term

Particular cases : no dividends, deterministic interest rates ω(t, K) = K 2σ(t, K)2 − EQ(

i,j wiwjSi tSj tσ(t, Si t)σ(t, Sj t)ρ0 i,j|IS t = K)

EQ(

i,j wiwjSi tSj tσ(t, Si t)σ(t, Sj t)(1 − ρ0 i,j)|IS t = K)

Dupire/Avellaneda/Piterbarg/Guyon-PHL formula. Case where constant vol and null correlation: ω = σ2

I − 1 N σ2 S

σ2

S(1 − 1 N )

Well known formula : see Bossu for example. Idea : (Implied Index Covariance - Minimum Covariance)/(Maximum Covariance - Minimum Covariance)

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 8 / 46

First part : Local correlation calibration Dupire general formula

Local Correlation formula : understand each term(2)

EQt ((qt It − rt K)1It >K ) EQt (1It >K ) − EQt ((

i wi qi t Si t − rt K)1IS t >K )

EQt (1IS

t >K )

Stochastic rate term + Dividend term. Deterministic interest rates : first term vanishes since rt in factor and EQt (1It >K ) = EQt (1IS

t >K ) = 1 B(0,t) (− ∂C ∂K )

Residual term : cf. no arbitrage condition in case of discrete dividends : EQt ((It − K)+) − EQt ((It− − K)+) ≃ EQt ((It − It− )1It >K ) EQt ((IS

t − K)+) − EQt

((IS

t− − K)+)

≃ EQt ((IS

t − IS t− )1IS t >K )

but : It − It− = −(Qt + qt It− ) IS

t − IS t− =

• i

−wi (Qi

t + qt Si t− )

leads to (first order in dividend level) : EQt (qt It 1It >K ) = EQt (

• i

wi qi

t Si t 1IS t >K )

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 9 / 46

First part : Local correlation calibration Dupire general formula

Local Correlation formula : understand each term(3)

If discrete dividends : impossible to achieve for each K if qt constant(except in particular cases : null volatility or qt = qi

t ∀i)

=> two models are generally inconsistent. => Need to use continuous div model One more derivation in K + Same ∂2C

∂K 2 gives :

EQt(qtIt|It = K) = EQt(

• i

wiqi

tSi t|IS t = K)

• cf. Markovian projection : sufficient but not necessary condition

Other possible conditions :

                 qt =

EQ( i wi qi t Si t ) EQ(It )

ω(t, K) =

   K2σ(t,K)2− 2(C−K ∂C ∂K ) ∂2C ∂K2    qt − EQ(( i wi qi t Si t )1IS t >K ) EQ(( i wi Si t )1IS t >K )        −EQ( i,j wi wj Si t Sj t σ(t,Si t )σ(t,Sj t )ρ0 i,j |IS t =K) EQ( i,j wi wj Si t Sj t σ(t,Si t )σ(t,Sj t )(1−ρ0 i,j )|IS t =K)

ω helps recover from the generated error.

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 10 / 46

First part : Local correlation calibration Dupire general formula

Implement the convergence algorithm

Ideas proposed in Guyon/PHL. Formula available for ω(t, K) Fixed point algorithm (cf. Reghai also but mixes implied/local) Hybrid-LSV model : "Kernelize" the drift term introducing Malliavin calculus

ω(n+1)(t, K) = K 2σ(t, K)2 − EQ(

i,j wiwjSi,(n) t

Sj,(n)

t

σ(t, Si,(n)

t

)σ(t, Sj,(n)

t

)ρ0

i,j|IS,(n) t

= K) EQ(

i,j wiwjSi,(n) t

Sj,(n)

t

σ(t, Si,(n)

t

)σ(t, Sj,(n)

t

)(1 − ρ0

i,j)|IS,(n) t

= K)

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 11 / 46

First part : Local correlation calibration Dupire general formula

Implement the convergence algorithm

Alternative ideas: "Kernelize" the drift term by introducing a division by EQ(1IS

t >K) or

EQ(IS

t 1IS

t >K): "natural" kernel for more stability

replace non parametric regression by a parametric regression for the dirac term ex : Case of hybrid model.Use the following formula :

σ(t, K)2 = σDup(t, K)2 + 2

∂C ∂K

K ∂2C

∂K 2

• EQ(exp(−

t

0(rs − r 0 s )ds)(rt − r 0 t )1St>K)

EQ(exp(− t

0(rs − r 0 s )ds)1St>K)

• Evaluate numerator and denominator using same paths

+ Use a fast exponential => no need for Malliavin calculus (multiple diffusions) and numeraire changes. Non parametric regression : complexity in O(Nln(N)) operations if sorted performances.

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 12 / 46

First part : Local correlation calibration Dupire general formula

Discussion : ω ∈ [0; 1] ?

Not theoretically true in practice if ρ0

i,j enough low (=> less than historical a priori)

Explanation ? Possible to infer a positive implied correlation (K,T) for a stock model with ρ0

i,j enough low(even 0) + Gatheral-like

formula.(implied covariance = local covariance integrated along the basket’s most likely path)) Idea (Reghai) : unshift the covariance matrix and reshift it using ω(t, K). Introduction of (continuous) dividends doesn’t change much.

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 13 / 46

First part : Local correlation calibration Dupire general formula

Parametric Regression

Need to estimate : EQ(

i,j wiwjSi tSj tσ(t, Si t)σ(t, Sj t)ρ0 i,j|IS t = K)

EQ(

i,j wiwjSi tSj tσ(t, Si t)σ(t, Sj t)|IS t = K)

What do they look like ?

Figure: Variable To Explain versus Basket

Interest : instead of non parametric regression, natural regression on (1, B, B2, . . . , Bp) can also be used. Proves to be stable and complexity in O(Np) (less than O(Nln(N))).

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 14 / 46

First part : Local correlation calibration Calibration results : local volatility

Results

Application to Eurostoxx smile (market data, 2012/02/20) : only two iterations that need 2000 simulations each. Very fast calibration of the local correlation. To ensure algorithm converge, ρ0

i,j = 0.

Figure: Fitting the Index Smile using Correlation Skew

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 15 / 46

First part : Local correlation calibration Calibration results : local volatility

Zoom : after two iterations

Figure: Fitting the Index Smile using Correlation Skew

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 16 / 46

First part : Local correlation calibration Calibration results : local volatility

Local Correlation Shape

Figure: Local Correlation Smile Figure: ATM Local Correlation Skew

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 17 / 46

First part : Local correlation calibration Extension to Stochastic Volatility

Extension to Stochastic Volatility framework

Chosen volatility model = Bergomi : dSi

t

Si

t

= σ(t, Si

t)

• ξi,t

t (

• 1 − ω(t, IS

t )dW i t +

• ω(t, IS

t )dW ⊥ t )

• r dSi

t

Si

t

=

• ξi,t

t d

W i

t

dξi,T

t

ξi,T

t

= σi

S exp(−κi S(T − t))dW i,S t

+ σi

L exp(−κi L(T − t))dW i,L t

with ξi,T

t

= EQ(V i

T|Ft)

< dW i

t , dW j t >= ρ0 i,jdt

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 18 / 46

First part : Local correlation calibration Extension to Stochastic Volatility

Correlation Structure

dW i,S

t

= ρi

Sd

W i

t +

• 1 − (ρi

S)2(αidZt +

• 1 − (αi)2dW

i,S t

) dW i,L

t

= ρi

Ld

W i

t +

• 1 − (ρi

L)2(βidZt +

• 1 − (βi)2dW

i,L t )

with : αi ∈ [0; 1] βi = 1 αi ρi

SL − ρi Sρi L

• 1 − (ρi

S)2

• 1 − (ρi

L)2

βi ∈ [0; 1]? 3N + 2 brownians required. Parametrization maintains basket volatility skew due to stochastic vol.

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 19 / 46

First part : Local correlation calibration Extension to Stochastic Volatility

Correlation Structure(2)

but 1 + 2ρi

Sρi Lρi SL − (ρi S)2 − (ρi L)2 − (ρi SL)2

≥ so   ρi

SL − ρi Sρi L

• 1 − (ρi

S)2

• 1 − (ρi

L)2

 

2

≤ 1

αiβi ∈ [−1; 1] => if αi = 1 and ρi

SL ≥ ρi Sρi L, we obtain a PSD matrix

remark : αi = 1= situation where basket prices closest with local and stochastic vol.

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 20 / 46

First part : Local correlation calibration Extension to Stochastic Volatility

Calibration formula

Calibration can be done using iteration scheme and parametric (in ln(K) or non-parametric regression). Formula becomes :

ω(n+1)(t, K) = K 2σ(t, K)2 − EQ(

i,j wi wj Si,(n) t

Sj,(n)

t

σ(t, Si,(n)

t

)

• ξi,t

t

σ(t, Sj,(n)

t

)

• ξj,t

t

ρ0

i,j |IS,(n) t

= K) EQ(

i,j wi wj Si,(n) t

Sj,(n)

t

σ(t, Si,(n)

t

)σ(t, Sj,(n)

t

)(1 − ρ0

i,j )|IS,(n) t

= K) = K 2σ(t, K)2 − f (n)(t, K) g(n)(t, K) − f (n)(t, K) Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 21 / 46

First part : Local correlation calibration Calibration results : stochastic volatility

Decorrelation effect

Stochastic Vol Parameters (low vovol to avoid numerical/existence issues) Parameter κS κL σS σL ρS ρL ρSL Value 400% 50% 200% 65% −60% −60% 60% Figure: No Correlation skew : decorrelation Without correlation skew = decorrelation effect

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 22 / 46

First part : Local correlation calibration Calibration results : stochastic volatility

Calibration Results

Application to Eurostoxx smile:

Figure: Fitting Index Smile with SV + Correlation Skew

Comments: not perfect fit but high dimension Stochastic vol induces greater ω (cf. decorrelation effect)

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 23 / 46

First part : Local correlation calibration Products Study

Product Study

Several products have been studied : 1Y Worst Of Calls on Eurostoxx 1Y Best Of Puts on Eurostoxx 1Y Spread Options Total versus Eurostoxx 1Y Correlation Swap on Eurostoxx

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 24 / 46

First part : Local correlation calibration Products Study

Results

Product/Model Without CS With CS With CS and SV / α = 1 With CS and SV / α = 0.5 Forward WO 58.57% 59.08% 56.71% 56.90% WO Call 90 1.31% 0.98% 0.68% 0.85% WO Call 95 0.86% 0.61% 0.45% 0.58% WO Call 100 0.55% 0.37% 0.29% 0.39% WO Call 105 0.35% 0.21% 0.18% 0.24% WO Call 110 0.21% 0.13% 0.12% 0.17% Forward BO 155.81% 156.42% 142.08% 142.79% BO Put 90 0.32% 0.68% 1.27% 1.35% BO Put 95 0.56% 1.04% 1.95% 2.05% BO Put 100 0.92% 1.54% 2.84% 2.96% BO Put 105 1.43% 2.21% 3.94% 4.09% BO Put 110 2.12% 3.03% 5.27% 5.44% Correlation Swap 66.67% 68.31% 72.33% 73.61% Spread Option -10 12.42% 12.40% 12.82% 12.78% Spread Option -5 8.90% 8.85% 9.04% 9.02% Spread Option 0 6.02% 5.90% 5.87% 5.86% Spread Option 5 3.80% 3.64% 3.51% 3.49% Spread Option 10 2.23% 2.07% 1.94% 1.92% Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 25 / 46

First part : Local correlation calibration Products Study

Results : analysis

Correlation Swap : increases when ω increases. Not relevant since basket = not very appropriate as hedge instrument Spread Option : Correlation skew decreases price(expected), no intuition for stochastic vol, but not very appropriate as hedge instrument Worst Of Payoffs : less expensive with Correlation Skew, no intuition with Stochastic Vol, but price uncertainty(considering the number of underlyings) Best Of Payoffs : more expensive with Correlation Skew, no intuition with Stochastic Vol, but price uncertainty(considering the number of underlyings)

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 26 / 46

First part : Local correlation calibration Products Study

Conclusion

Fast and robust way to calibrate local correlation Example of applications : Cancellables on baskets (cf. Worst Of risk + Stoch vol Risk) Beware of dividend term! Can be extended to stochastic correlation Good risk management tool if basket appropriate hedge instrument!

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 27 / 46

First part : Local correlation calibration Products Study

Transition

Are vanilla baskets appropriate hedge instruments? More generally : quality of the projection on vanillas Conclusion : need to backtest hedge strategies using historical data

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 28 / 46

Second part : study of correlation swap product

Outline

1

First part : Local correlation calibration References Dupire general formula Calibration results : local volatility Extension to Stochastic Volatility Calibration results : stochastic volatility Products Study

2

Second part : study of correlation swap product Purpose of the study RCS Risk analysis : Hedge with Basket and stock vanillas Results PnL decomposition RCS Risk analysis : Hedge with Var Swaps Transaction costs Other Basket Main conclusions Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 29 / 46

Second part : study of correlation swap product Purpose of the study

Understand Swap correlation swap

Correlation Swap : simple formula, complex payoff. Observation = basket implied correlation different from RCS implied correlation (basket implied correlation ≃ RCS correlation +5/15%). Different hypotheses : local correlation, vovol, vol-correl correlation, correlation between index VS and average stock VS Best way to understand the product : backtest using historical data.

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 30 / 46

Second part : study of correlation swap product Purpose of the study

Hypothesis

No time zone effect(here, backtest on stocks of Eurostoxx only) Simple evaluation model : Black-Scholes Backtest using historical data

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 31 / 46

Second part : study of correlation swap product Purpose of the study

Basic Definitions

KRCS = 2 N(N − 1)

• i>j

N

k=1(ln( Si

tk+1

Si

tk

) − ln(

Si

tk+1

Si

tk

))(ln(

Sj

tk+1

Sj

tk

) − ln(

Sj

tk+1

Sj

tk

))

• N

k=1(ln( Si

tk+1

Si

tk

) − ln(

Si

tk+1

Si

tk

))2

• N

k=1(ln( Sj

tk+1

Sj

tk

) − ln(

Sj

tk+1

Sj

tk

))2

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 32 / 46

Second part : study of correlation swap product RCS Risk analysis : Hedge with Basket and stock vanillas

Backtest a long RCS short Basket Vanilla position

Question = Being long hedged RCS to take advantage of the difference between basket implied correlation and Implied Realized(RCS) correlation ? Simple Strategy : Long RCS Short ATM Basket Vanilla with same residual maturity to be flat

• veromega

Long ATM underlying Vanillas with same residual maturity to be flat vega position in spot to be flat delta

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 33 / 46

Second part : study of correlation swap product Results

Backtest a long RCS short Basket Vanilla position

For our tests: Only one correlation = basket. Difference between correlations must be added to displayed PnL. basket = eight stocks amongst the biggest Eurostoxx capitalizations : Total/Sanofi/Telefonica/Banco Santander/BNP/Siemens/E ON/Allianz Hedges :each business day in spot/vol/overomega

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 34 / 46

Second part : study of correlation swap product PnL decomposition

PnL evolution

Usually negative. But during crisis : profitable strategy. Here : 1Y Correlation Swap Figure: PnL of the long hedged RCS Strategy Short analysis big jump between 7th and 10th of may 2010 : massive rebound PnL tends to come back to negative values

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 35 / 46

Second part : study of correlation swap product PnL decomposition

PnL decomposition : split PnL between spot, vol and correl effects

We write that :

PnL(S

tk+1 i

, σ

tk+1 i

, ρ

tk+1 i,j

) − PnL(Stk

i , σtk i , ρtk i,j )

= PnL(S

tk+1 i

, σ

tk+1 i

, ρ

tk+1 i,j

) − PnL(S

tk+1 i

, σ

tk+1 i

, ρtk

i,j )

+ PnL(S

tk+1 i

, σ

tk+1 i

, ρtk

i,j ) − PnL(S tk+1 i

, σtk

i , ρtk i,j )

+ PnL(S

tk+1 i

, σtk

i , ρtk i,j ) − PnL(Stk i , σtk i , ρtk i,j )

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 36 / 46

Second part : study of correlation swap product PnL decomposition

PnL decomposition

Figure: PnL Decomposition of long hedged RCS Strategy

Short analysis gives :

correlation PnL very low negative vol PnL, because strategy proves to be volga negative essentially gamma PnL Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 37 / 46

Second part : study of correlation swap product PnL decomposition

Gamma PnL decomposition between diagonal and extra diagonal PnL

2RealizedPnLΓ ≃

• i

Γi,iS2

i r 2 i +

• i=j

Γi,jSiSjrirj leads to : Figure: Decomposition between Diagonal and Extra Diagonal Gamma

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 38 / 46

Second part : study of correlation swap product PnL decomposition

More relevant Gamma PnL decomposition

2RealizedPnLΓ =

• i,j

Γi,j Si Sj ri rj =

• i

(Γi,i S2

i − Γi,i S2 i )r2 i + Γi,i S2 i

• i

r2

i

+

• i=j

(Γi,j Si Sj − Γi,j Si Sj )ri rj + Γi,j Si Sj

• i=j

ri rj 2PnLΓ = A + B + Γi,i S2

i

• i

r2

i + Γi,j Si Sj ((N − 1)

• i

r2

i − N2Ξ)

≃ Γi,i S2

i

• i

r2

i + Γi,j Si Sj ((N − 1)

• i

r2

i − N2Ξ)

with : Ξ = 1 N

• i

(ri − ri )2 :Dispersion. Γβ1 = 1 2

• i,j

Γi,j Si Sj ∗ (1%)2 ΓX = 1 2

• i=j

Γi,j Si Sj ∗ (1%)2 PnLΓ ≃ Γβ1 (1%)2 ( 1 N

• i

r2

i − ImpliedVariance) −

N N − 1 ΓX (1%)2 (Ξ − ΞImplied ) Strategy roughly flat Γβ1 and negative Crossed Gamma position => long dispersion. Example for initial hedge : Γβ1

ΓX

<

1 10

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 39 / 46

Second part : study of correlation swap product PnL decomposition

Dispersion analysis

Figure: History of Day dispersion before/after crisis

Short analysis gives : important dispersion during the crisis dispersion clustering important dispersion when market rebounds : 10/05/2010 - 10/03/2009 - 28/01/2009 - 29/10/2008.

• r idiosyncratic event : 17/12/2008 with BNP results.

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 40 / 46

Second part : study of correlation swap product PnL decomposition

Dispersion implied versus historical

Figure: Implied versus Historical Dispersion

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 41 / 46

Second part : study of correlation swap product RCS Risk analysis : Hedge with Var Swaps

Hedge with Var Swaps

see also Bossu, but too strong approximation for "long" dated RCS(1Y) and Slaoui-Jacquier, but no backtest to compare Gamma and Volga PnL Same methodology but hedge with Var Swaps instead of Vanillas Idea = estimate transaction costs(cf. no need to restrike)

Figure: Var Swap Hedge : PnL Decomposition

Short analysis gives : Evolution similar to Basket Hedge Gamma PnL still the more important vol PnL becomes positive : volga positive strategy Correlation PnL more important Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 42 / 46

Second part : study of correlation swap product Transaction costs

Transaction costs

2 bps transaction cost for deltas 20 bps transaction cost in stock vol 20 bps transaction cost in basket vol leads to -3.50% and -9% (correl points) Initial hedge cost = half of it

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 43 / 46

Second part : study of correlation swap product Other Basket

Other Basket

Basket made of 10 more important Eurostoxx capitalizations(BBVA,Deutsche Bank, etc.) Same kind of results : back to normal after the crisis.

Figure: Other Basket : PnL Decomposition

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 44 / 46

Second part : study of correlation swap product Main conclusions

Reason that explain most of the impact between RCS and Basket Correlation. negative (or positive) volga of the strategy difference between realized and implied dispersion daily restriking => sensitive to short term correlation : term structure effect

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 45 / 46

Second part : study of correlation swap product Main conclusions

References

Bossu : Equity Correlation Swaps : A new approach for modelling and pricing Jacquier-Slaoui : Variance Dispersion and Correlation Swaps Reghai : Using Local Correlation models to improve option hedging Langnau : A dynamic model for correlation risk Sbai-Jourdain : Coupling Index and stocks Guyon/Henry-Labordere : The smile calibration problem solved Piterbarg : Markovian projection for volatility calibration

Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 46 / 46