Local Correlation with Local Vol and Stochastic Vol : Towards Correlation dynamics ? Pascal DELANOE, Structured Equity Derivatives HSBC

10th January 2014

Structured Equity Research (HSBC)

Local Correlation with Local Vol and Stochastic Vol :Towards10th Correlation ?1 / 55 Januarydynamics 2014

Local Correlation : where are we ?

Outline 1

Local Correlation : where are we ?

2

PnL equation

3

Observe correlation Evidence of Correlation Skew

4

Model correlation ? Introduce Decorrelation New Methods in Finance Local Formulae : Derivate Market Information

5

Why basket local correlation ?

6

Calibration results : Local Volatility

7

Extension to Stochastic Volatility Need to introduce specific parametrization Decorrelation with Multi-Underlying Stochastic Volatility Usual values of correl between vols

8

Focus on correlation products

9

Main conclusions

10 References Structured Equity Research (HSBC)

Local Correlation with Local Vol and Stochastic Vol :Towards10th Correlation ?2 / 55 Januarydynamics 2014

Local Correlation : where are we ?

Recent (or less recent) developments in local correlation Avellaneda : local formula + approximation Reghai : based on fixed point algorithm, but slow convergence (cf. based on implied vols) Langnau : pathwise equality of covariance to calibrate local correl, too many constraints ? (cf. sufficient but not necessary condition) Sbai-Jourdain : top-down approach (insert index in stock diffusion) instead of usual bottom up, but issues since introduces historical parameter β Piterbarg : markovian projection, calibration based on approximations (not specific to correlation) Guyon- Henry-Labordere : "Particle Methods"(not specific to correlation) Our approach = similar to Particle Methods, but method slightly differs for specific points. Structured Equity Research (HSBC)

Local Correlation with Local Vol and Stochastic Vol :Towards10th Correlation ?3 / 55 Januarydynamics 2014

Local Correlation : where are we ?

"Overomega" Definition Banks usually short correlation (cf. sell basket calls/puts, sell WO Calls,...)=> need to overprice Correlation. Constraint : needs to remain PSD! Solution : use the convexity for space of correlation matrix (standard, also used in shrinkage methods) We introduce "Overomega" (not a standard notation) : Pricing ρi,j = (1 − ω)ρHisto +ω i,j Generally ω ' 15%. Used to give conservative prices. Remember : not always true (sell spread options,...)! Conservative pricing : Need to choose adapted target matrix (cf. crossed gamma sign), with PricingMatrix = (1 − ω)InitMatrix + ωTargetMatrix But Issues when Crossed gammas change sign locally (=⇒ uncertain correlation pricing) Structured Equity Research (HSBC)

Local Correlation with Local Vol and Stochastic Vol :Towards10th Correlation ?4 / 55 Januarydynamics 2014

PnL equation

Outline 1

Local Correlation : where are we ?

2

PnL equation

3

Observe correlation Evidence of Correlation Skew

4

Model correlation ? Introduce Decorrelation New Methods in Finance Local Formulae : Derivate Market Information

5

Why basket local correlation ?

6

Calibration results : Local Volatility

7

Extension to Stochastic Volatility Need to introduce specific parametrization Decorrelation with Multi-Underlying Stochastic Volatility Usual values of correl between vols

8

Focus on correlation products

9

Main conclusions

10 References Structured Equity Research (HSBC)

Local Correlation with Local Vol and Stochastic Vol :Towards10th Correlation ?5 / 55 Januarydynamics 2014

PnL equation

Why does correlation matters : PnL Equation Consider a product with value P that we buy. Pricing equation rt P =

∂P ∂t

+

X ∂P ∂xi

i

rt xi +

X 1 ∂2P i,j

2 ∂xi ∂xj

ρi,j σi σj xi xj

PnL equation (integrated = "tracking error"): ∆P − rP∆t −

X ∂P i

∂xi

(∆Si − rSi ∆t)

=

=

∂P ∂t

∆t +

X ∂P i

1 X ∂2P 2

i

∂xi2

∂xi 2

∆Si +

X 1 ∂2P i,j

2

2

2 ∂xi ∂xj

(∆Si − σi (Si ) ∆t) +

∆Si ∆Sj − rP∆t −

i

X ∂2P i>j

X ∂P

∂xi ∂xj

∂xi

(∆Si − rSi ∆t

(∆Si ∆Sj − ρi,j σi σj Si Sj ∆t)

Analysis : Link between Cegas (Correlation Greeks) and Crossed Gammas. Short Crossed Gamma and correlated movement, loses money Need to use a model with a theta coherent with these crossed gammas Structured Equity Research (HSBC)

Local Correlation with Local Vol and Stochastic Vol :Towards10th Correlation ?6 / 55 Januarydynamics 2014

Observe correlation

Outline 1

Local Correlation : where are we ?

2

PnL equation

3

Observe correlation Evidence of Correlation Skew

4

Model correlation ? Introduce Decorrelation New Methods in Finance Local Formulae : Derivate Market Information

5

Why basket local correlation ?

6

Calibration results : Local Volatility

7

Extension to Stochastic Volatility Need to introduce specific parametrization Decorrelation with Multi-Underlying Stochastic Volatility Usual values of correl between vols

8

Focus on correlation products

9

Main conclusions

10 References Structured Equity Research (HSBC)

Local Correlation with Local Vol and Stochastic Vol :Towards10th Correlation ?7 / 55 Januarydynamics 2014

Observe correlation

What is correlation ? Correlation not a "clean" quantity, more adequate quantity = covariance. Correlation = for given vol and given covariance, way to introduce link between brownians (generally, "Gaussian copula") Example of issue : correlation can be more than 1 due to time zones (Bergomi) or other (model) reasons. No way (that I know of) to deal with this issue in Monte-Carlo. (and seems to present numerical issues in PDE and Fourier) Natural question : what is Implied Correlation ? Implied Vol Rebonato :"wrong number to put in the wrong formula to get the right price" Implied Correlation "wrong number to put in the wrong pricer given a wrong volatility model to get the right price"

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Local Correlation with Local Vol and Stochastic Vol :Towards10th Correlation ?8 / 55 Januarydynamics 2014

Observe correlation

Implied Correlation Data

Example : ICJ/JCJ/KCJ rotating indexes. Currently : JCJ (Jan. 2014) or KCJ Index (Jan. 2015). Different issues Reference Vol Model = Black-Scholes Based on approximate formula (most likely path) Implied Volatility = for stocks, ATM Spot Implied Vol and not ATMF implied Vols Based on only 50 underlyings of SP500 (liquidity issues)

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Local Correlation with Local Vol and Stochastic Vol :Towards10th Correlation ?9 / 55 Januarydynamics 2014

Observe correlation

Implied Correlation Data(2) Interpolated 1Y Implied Correlation from ICJ/JCJ/KCJ (since 2007). Evolution.

Figure: Evidence of Correlation Skew Structured Equity Research (HSBC)

Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 10 ? / 55 10th January 2014

Observe correlation

Evidence of Correlation Skew

Evidence of correlation Skew based on Historical Data Interpolated 1Y Implied Correlation from ICJ/JCJ/KCJ (since 2007)

Figure: Evidence of Correlation Skew Structured Equity Research (HSBC)

Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 11 ? / 55 10th January 2014

Observe correlation

Evidence of Correlation Skew

Evidence based on Implied Data(1)

Figure: Basket Smile versus Index Smile : SMI case

Consequence : market expects more correlation on the downside, and less on the upside. Structured Equity Research (HSBC)

Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 12 ? / 55 10th January 2014

Observe correlation

Evidence of Correlation Skew

Evidence based on Implied Data(2)

Figure: Index Implied Correlation =⇒ Correlation increases when basket decreases. Note : Here, Overomega skew and not Correlation Skew (ratio 1 − ρ between both) Structured Equity Research (HSBC)

Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 13 ? / 55 10th January 2014

Observe correlation

Evidence of Correlation Skew

Rationale behind correlation skew

Correlation Skew = market evidence. Main reasons : Law of demand and supply : more buyers on the downside (protection) Systemic risk : big downward moves, risk linked to economy, all stocks impacted Upside : generally decreases, but (sometimes) systemic "rescue". When good news concerning the economy (rates decrease, central bank actions,...), all stocks impacted (and correlation increases).

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Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 14 ? / 55 10th January 2014

Model correlation ?

Outline 1

Local Correlation : where are we ?

2

PnL equation

3

Observe correlation Evidence of Correlation Skew

4

Model correlation ? Introduce Decorrelation New Methods in Finance Local Formulae : Derivate Market Information

5

Why basket local correlation ?

6

Calibration results : Local Volatility

7

Extension to Stochastic Volatility Need to introduce specific parametrization Decorrelation with Multi-Underlying Stochastic Volatility Usual values of correl between vols

8

Focus on correlation products

9

Main conclusions

10 References Structured Equity Research (HSBC)

Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 15 ? / 55 10th January 2014

Model correlation ?

The purpose of modelling correlation

Different situations : Our focus : liquid basket options However, no real hedge strategy since basket composition changes : =⇒ essentially Macro Management Tool. Steps : 1

Decorrelate initial correlation matrix

2

Write Local Formula linking two different models

3

Use fixed-point algorithm (or particle method) to calibrate

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Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 16 ? / 55 10th January 2014

Model correlation ?

Introduce Decorrelation

Decorrelation Step Ideas : Decorrelate initial correlation matrix use parametric local overomega to recorrelate the matrix

Decorrelation : H ρD i,j = (1 − ω1 )ρi,j + ω1 with ω1 < 0 ω1 D ⇐⇒ ρH i,j = (1 − ω0 )ρi,j + ω0 with ω0 = ω1 − 1

In practice, maximize |ω1 | so that matrix remains PSD and with positive correlation.

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Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 17 ? / 55 10th January 2014

Model correlation ?

New Methods in Finance

Foreword

Standard Models are simpler to handle with local vol, local correl adjustments : Fixed Point algorithm (Reghai) + Local Formulae (Dupire) + Numerical Evaluation of conditional expectations (not specifically linked to finance) = Local fixed-point methods (particular case for explicit schemes with one iteration = Particle method) Fixed Point problem : contracting function(?) on a space of stochastic processes. Existence still needs to be solved theoretically.

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Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 18 ? / 55 10th January 2014

Model correlation ?

Local Formulae : Derivate Market Information

Remember : how to prove Dupire’s formula ? Idea : Derive Market Information/Observables dSt

=

so that (undiscounted calls) dC

=

∂C

But : Or : C − K And :

∂K ∂C ∂K

∂2C ∂K 2

So that :

∂C ∂t

And finally : σ(t, K )

Structured Equity Research (HSBC)

2

(rt St − Qt − qt St )dt + σ(t, St )St dWt ∂C Q + dE (St − K ) = dt ∂t Q

+

=

E d(St − K )

=

E (dSt 1S >K + t

=

E ((rt − qt )St 1S >K − Qt 1S >K + t t

=

−E (1S >K ) t

=

E (St 1S >K ) t

=

E (δS =K ) t

=

(rt − qt )(C − K

=

Q

1 2

d < S >t δS =K ) t

Q

1 2

2

2

2

∂2C

σ(t, K ) K δS =K )dt t

Q

Q

Q

∂C ∂t

∂C

) + Qt

∂C

+

1

2

σ(t, K ) K

∂K ∂K 2 ∂C ) − Q ∂C − (rt − qt )(C − K ∂K t ∂K

∂K 2

1 K 2 ∂2 C 2 ∂K 2

Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 19 ? / 55 10th January 2014

Model correlation ?

Local Formulae : Derivate Market Information

Our framework = Reghai’s Local Correlation introduced through the use of an overomega approach. Pricing What is Overomega ? ρi,j = (1 − ω)ρHisto +ω i,j First Model = Simple Local Vol Model with continuous dividends (mix of prop and cash dividends). q q dSti = (rt Sti − Qti − qti Sti )dt + σ(t, Sti )Sti ( 1 − ω(t, ItS )dWti + ω(t, ItS )dWt⊥ )

with Qti and qti deterministic and : N X

ItS

=

wi Sti

< dWti , dWtj >

=

ρ0i,j dt

< dWti , dWt⊥ >

=

0∀i

i=1

Structured Equity Research (HSBC)

Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 20 ? / 55 10th January 2014

Model correlation ?

Local Formulae : Derivate Market Information

Local Correlation formula (general case) Second Model = simple local vol model written on the index(continuous divs) : dIt = (rt It − Qt − qt It )dt + It σ(t, It )dWt with Qt and qt deterministic. Same Basket Call prices in both models (Specific set of wi ) : Z t C(K , t) = EQ (exp(− rs ds)(It − K )+ ) 0

=

EQ (exp(−

t

Z

rs ds)(ItS − K )+ )∀t, K

0

but : ∂C ∂t ∂C ∂t

dt dt

= =

Q

Z t

E (exp(− Q

0

S

S

rs ds)((dIt − rt (It − K )dt)1 S + I >K t

Z t

E (exp(−

Structured Equity Research (HSBC)

0

rs ds)((dIt − rt (It − K )dt)1I >K + t

1 2

1 2

d t δ S )) in basket model I =K t

d < I >t δI =K )) in index model t

Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 21 ? / 55 10th January 2014

Model correlation ?

Local Formulae : Derivate Market Information

Local Correlation formula(2) Qt

E

((−

X

i

i

i

wi (Qt + qt St ) + rt Kdt)1 S

I >K t

i

+

1 2

d t δ S

I =K t

)=E

((−(Qt + qt It ) + rt Kdt)1I >K + t

1 2

d < I >t δI =K ) t

but : Qt

E

t

EQ (d < I >t |It = K ) ∂ 2 C B(0, t) ∂K 2

(d < I >t δI =K ) = t

and also : Qt

E

(It 1I >K ) t

Qt

E

(1I >K ) t

=

Qt

E

t

=

Qt

E

1

S

(It 1 S )= I >K (1 S

I >K t

)=

B(0, t) 1

B(0, t)

(C − K

(−

∂C ∂K

∂C ∂K

)

)

Condition on the forward (K = 0):

Structured Equity Research (HSBC)

Qt

=

qt

=

i i wi Qt t P EQ ( i wi qti Sti ) t EQ (It )

P

(1)

Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 22 ? / 55 10th January 2014

Model correlation ?

Local Formulae : Derivate Market Information

Local Correlation formula(3)

2

2

K σ(t, K ) + ω(t, K )

=

EQt (

t

t

∂C 2 ∂K

EQ ((qt It −rt K )1It >K )

∂2 C ∂K 2

EQt (1It >K )

−

EQ ((

j i i j i,j wi wj St St σ(t, St )σ(t, St )(1

P

P

i

wi qti Sti −rt K )1I S >K )

!!

t

EQt (1I S >K ) t

ρ0i,j )|ItS

− P j j EQ ( i,j wi wj Sti St σ(t, Sti )σ(t, St )ρ0i,j |ItS = K ) P EQt ( i,j wi wj Sti Stj σ(t, Sti )σ(t, Stj )(1 − ρ0i,j )|ItS = K )

= K)

t

−

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Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 23 ? / 55 10th January 2014

Model correlation ?

Local Formulae : Derivate Market Information

Local Correlation formula : simplest formula Particular cases : no dividends, deterministic interest rates P K 2 σ(t, K )2 − EQ ( i,j wi wj Sti Stj σ(t, Sti )σ(t, Stj )ρ0i,j |ItS = K ) ω(t, K ) = P EQ ( i,j wi wj Sti Stj σ(t, Sti )σ(t, Stj )(1 − ρ0i,j )|ItS = K ) Dupire/Avellaneda/Piterbarg/Guyon-PHL formula. Case where constant vol and null correlation: ω=

σI2 − σS2 (1

1 2 N σS − N1 )

Well known formula : see Bossu for example. Idea = Depends on covariance : Structured Equity Research (HSBC)

Implied−Minimum Maximum−Minimum

Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 24 ? / 55 10th January 2014

Model correlation ?

Local Formulae : Derivate Market Information

Local Correlation formula : focus on dividends t

EQ ((

t

EQ ((qt It − rt K )1I >K ) t t EQ (1

P

−

i

) wi qti Sti − rt K )1 S I >K t

t

EQ (1 S

It >K )

I >K t

Stochastic rate term + Dividend term.

)

t

t

1 (− ∂C ) Deterministic interest rates : first term vanishes since rt in factor and EQ (1I >K ) = EQ (1 S ) = B(0,t) ∂K t I >K t Residual term linked to dividends : cf. no arbitrage condition in case of discrete dividends : Qt

E

Qt

E

+

Qt

+

Qt

((It − K ) ) − E

S ((It

− K) ) − E

+

((It − − K ) ) S ((I − t

+

− K) )

Qt

'

E

Qt

'

E

((It − It − )1I >K ) t S

S

((It − I − )1 S ) t I >K t

but :

S

S

It − It − = −(Qt + qt It − ) X i i −wi (Qt + qt St − )

It − I − = t

i

leads to (first order in dividend level) : Qt

E

Qt

(qt It 1I >K ) = E t

(

X i

Structured Equity Research (HSBC)

i

i

wi qt St 1 S

I >K t

)

Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 25 ? / 55 10th January 2014

Model correlation ?

Local Formulae : Derivate Market Information

Local Correlation formula : focus on dividends(2) If discrete dividends : impossible to achieve for each K if qt constant(except in particular cases : null volatility or qt = qti ∀i) =⇒ two models are generally inconsistent. =⇒ Need to use continuous div model 2

∂ C One more derivation in K + same density ( ∂K 2 ) give : t

t

EQ (qt It |It = K ) = EQ (

X

wi qti Sti |ItS = K )

i

cf. Markovian projection : sufficient but not necessary condition Other possible conditions :    qt      

=

       ω(t, K )

=

P EQ ( i wi qti Sti ) Q  E (It )

  P ) EQ (( i wi qti Sti )1 S I >K  2(C−K ∂C )   2 P j 0 S i j i t ∂K q − K σ(t,K )2 − −EQ ( P t i,j wi wj St St σ(t,St )σ(t,St )ρi,j |It =K )   ∂2 C EQ (( i wi S i )1 S )  t I >K 2 ∂K t P j j EQ ( i,j wi wj S i S σ(t,S i )σ(t,S )(1−ρ0 )|I S =K ) t t t t i,j t

Comments : ω helps recover from the generated error in practice, prop divs smile correction can be neglected Structured Equity Research (HSBC)

Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 26 ? / 55 10th January 2014

Why basket local correlation ?

Outline 1

Local Correlation : where are we ?

2

PnL equation

3

Observe correlation Evidence of Correlation Skew

4

Model correlation ? Introduce Decorrelation New Methods in Finance Local Formulae : Derivate Market Information

5

Why basket local correlation ?

6

Calibration results : Local Volatility

7

Extension to Stochastic Volatility Need to introduce specific parametrization Decorrelation with Multi-Underlying Stochastic Volatility Usual values of correl between vols

8

Focus on correlation products

9

Main conclusions

10 References Structured Equity Research (HSBC)

Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 27 ? / 55 10th January 2014

Why basket local correlation ?

Other possible local correlations! Reghai : Consider WO, BO, Rainbow local correlation to handle chewing-gum effect Example : Worst Of Local Correlation (for WO products) Two models : Worst Of Model and standard model with WO local correlation WO Model :

dWOt WOt

Standard Model :

dSti Sti

gt with : WO

=

(rt − qt )dt + σ(t, WOt )dWt

=

(rt − qti )dt + σ i (t, Sti )(

=

mini (

Sti Sti

q q g t )dW i + ω(t, WO g t )dW ⊥ ) 1 − ω(t, WO t t

)

0

and : < dWti , dWti >

=

Structured Equity Research (HSBC)

ρ0i,j (t)dt

Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 28 ? / 55 10th January 2014

Why basket local correlation ?

Worst Of Correlation (2) Derive WO Calls in both models (calculation a little heavy):

WO model : WO local correl model :

∂C WO ∂t ∂C WO ∂t

=

−rt C

−

j j j 1 X Q d < Sti , Sti > +d < St , St > −2d < Sti , St > 1g E ( δ i j 1g i) S =S WO t >K WO t =St 2 i>j dt t t

+ (rt −

(K )

=

∂K

WO WO qt (K ))(C

g EQ (q WO WO1 g

WO>K

g EQ (WO1 g

−K

WO>K

−K

)+

2

∂C WO ∂K

2

2

∂ 2 C WO

WO

g

WO

with : qt

1

WO

+ (rt − qt )(C

WO

∂C WO

−rt C

=

K σ (t, K ) )+

1 2

2 Q

∂K 2

K E (σ

g WO

2 g (t, K ) |WO = K)

∂ 2 C WO ∂K 2

)

)

Condition on WO Forward : qt = qtWO (0) If qtWO (K ) = qt (not true in general, else add corrective term to overomega like in basket formula) and ρi,j (t, K ) = ρ0i,j (t) + ω(t, K )(1 − ρ0i,j (t)), one more K derivation gives :    2 WO g ∂ g = K ) − σ 2 (t, K ) K2 ∂ C 2 EQ (σ WO (t, K )2 |WO ∂K ∂K ω(t, K ) = P K 2 i>j EQ (2(1 − ρ0i,j (t))σi (t, K )σj (t, K )δ i j 1 g 1g i) S =S WO t >K WO t =St t

K2

P

i>j

−

t

1g EQ ((2ρ0i,j (t)σi (t, K )σj (t, K ) − σi2 (t, K ) − σj2 (t, K ))δ i j 1 g i) S =S WO t >K WO t =St t t P 0 2 Q K 1g j 1g i) i>j E (2(1 − ρi,j (t))σi (t, K )σj (t, K )δ i S =S WO t >K WO t =St t t

Structured Equity Research (HSBC)

Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 29 ? / 55 10th January 2014

Why basket local correlation ?

Worst Of Correlation (3) Two important quantities : Switching Local Time : δS i =S j t

t

Local Dispersion : j

j

j

d+d−2d dt

j

=

j

d dt

= (σ i S i )2 + (σ j S j )2 − 2ρi,j σ i σ j S i S j

Local Dispersion = short correl, long volatility, positive quantity cf. −2ρi,j σ i σ j S i S j + (σ i S i )2 + (σ j S j )2 = (σ i S i − σ j S j )2 + 2(1 − ρi,j )σ i σ j S i S j Note : local dispersion in Spread Option Local equation : ∂C Spread ∂t

∂C Spread ) ∂K d < St1 , St1 > +d < St2 , St2 > −2d < St1 , St2 > Spread

=

−rt C Spread + (rt − qt

+

1 Q E ( 2

)(C Spread − K

dt

δSpread=K )

p Remember also Margrabe formula : σ = (σ i )2 + (σ j )2 − 2ρi,j σ i σ j =⇒ WO Call short disp product, spread option long disp product. Structured Equity Research (HSBC)

Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 30 ? / 55 10th January 2014

Why basket local correlation ?

Local Correlation Models Limits WO local Correlation : no real observable smile for WO vanillas Dynamic issue : only valid at inception (local vol -and forward- of WO model should change dynamically but how ?) more complex and less stable numerically not much financial sense : how to infer a historical WO local correlation skew ? but "chewing gum" effect Basket local Correlation : not many observables but more precise idea of hypothetic smile Dynamic issue : only valid at inception (local vol of basket with changed weights should change dynamically but how ?) simple and stable numerically financial sense (cf. historical observations) => we will study Basket Local Correlation. Structured Equity Research (HSBC)

Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 31 ? / 55 10th January 2014

Why basket local correlation ?

Discussion : ω ∈ [0; 1] ?

Cf. Guyon/Henry-Labordere remark. Not theoretically (no static arbitrage) True in practice if ρ0i,j enough low (Ex: ρ0i,j = 0∀i, j) Explanation ? Possible to infer a positive implied correlation ω I (K , T ) for a standard model if ρ0i,j = 0 for ex. Gatheral-like formula : R ρIi,j (T , m)σiI (T , m)σjI (T , m) ' T1 0T ρLi,j (t, mt )σiL (t, mt )σjL (t, mt )dt T T T Introduction of drift (continuous dividends) still OK.

Structured Equity Research (HSBC)

Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 32 ? / 55 10th January 2014

Why basket local correlation ?

Parametric Regression Need to estimate : P EQ ( i,j wi wj Sti Stj σ(t, Sti )σ(t, Stj )ρ0i,j |ItS = K ) P EQ ( i,j wi wj Sti Stj σ(t, Sti )σ(t, Stj )|ItS = K ) What do they look like ?

Figure: Variable To Explain versus Basket Interest : instead of non parametric regression, natural regression on (1, B, B 2 , . . . , B p ) can also be used. Proves to be stable and complexity in O(Np) Structured Equity Research (HSBC)

Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 33 ? / 55 10th January 2014

Calibration results : Local Volatility

Outline 1

Local Correlation : where are we ?

2

PnL equation

3

Observe correlation Evidence of Correlation Skew

4

Model correlation ? Introduce Decorrelation New Methods in Finance Local Formulae : Derivate Market Information

5

Why basket local correlation ?

6

Calibration results : Local Volatility

7

Extension to Stochastic Volatility Need to introduce specific parametrization Decorrelation with Multi-Underlying Stochastic Volatility Usual values of correl between vols

8

Focus on correlation products

9

Main conclusions

10 References Structured Equity Research (HSBC)

Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 34 ? / 55 10th January 2014

Calibration results : Local Volatility

Results Application to Eurostoxx smile. Only two iterations that need 2000 simulations each : quick calibration. Here, ρ0i,j = 0.

Figure: Fitting the Index Smile using Correlation Skew Structured Equity Research (HSBC)

Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 35 ? / 55 10th January 2014

Calibration results : Local Volatility

Local Correlation Shape

Figure: Local Correlation Smile

Structured Equity Research (HSBC)

Figure: ATM Local Correlation Skew

Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 36 ? / 55 10th January 2014

Extension to Stochastic Volatility

Outline 1

Local Correlation : where are we ?

2

PnL equation

3

Observe correlation Evidence of Correlation Skew

4

Model correlation ? Introduce Decorrelation New Methods in Finance Local Formulae : Derivate Market Information

5

Why basket local correlation ?

6

Calibration results : Local Volatility

7

Extension to Stochastic Volatility Need to introduce specific parametrization Decorrelation with Multi-Underlying Stochastic Volatility Usual values of correl between vols

8

Focus on correlation products

9

Main conclusions

10 References Structured Equity Research (HSBC)

Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics 37 ? / 55 10th January 2014

Extension to Stochastic Volatility

Need to introduce specific parametrization

Extension to Stochastic Volatility framework Chosen volatility model = (continuous) Bergomi model : dSti Sti or

dSti Sti i

e with : d W t

i

q q q i,t i ⊥ ξt ( 1 − ω(t, ItS )dWt + ω(t, ItS )dWt )

=

σ(t, St )

=

q i,t e i ξt d W t

=

q q i ⊥ 1 − ω(t, ItS )dWt + ω(t, ItS )dWt

=

σS exp(−κS (T − t))dWt

=

E (VT |Ft )

i,T

dξt

i,T

i

ξt

i,T

with : ξt


=

i,S dWt

=

i ei + ρS d W t

i,L

=

q q i,S 1 − (ρiS )2 (αdZt + 1 − (α)2 dW t ) q q i,L i ei 1 − (ρiL )2 (βi dZt + 1 − (βi )2 dW t ) ρL d W t +

>

=

ρSL dt

i,L i,S dWt , dWt

i,L

+ σL exp(−κL (T − t))dWt

0 ρi,j dt

dWt