[PPT] - Local Correlation with Local Vol and Stochastic Vol : Towards PowerPoint Presentation

SLIDE 1

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 :Towards Correlation dynamics ? 10th January 2014 1 / 55

SLIDE 2

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 :Towards Correlation dynamics ? 10th January 2014 2 / 55

SLIDE 3

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 :Towards Correlation dynamics ? 10th January 2014 3 / 55

SLIDE 4

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) : ρPricing

i,j

= (1 − ω)ρHisto

i,j

+ ω 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 :Towards Correlation dynamics ? 10th January 2014 4 / 55

SLIDE 5

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 :Towards Correlation dynamics ? 10th January 2014 5 / 55

SLIDE 6

PnL equation

Why does correlation matters : PnL Equation

Consider a product with value P that we buy. Pricing equation

rt P = ∂P ∂t +

i

∂P ∂xi rt xi +

i,j

1 2 ∂2P ∂xi ∂xj ρi,j σi σj xi xj

PnL equation (integrated = "tracking error"):

∆P − rP∆t −

i

∂P ∂xi (∆Si − rSi ∆t) = ∂P ∂t ∆t +

i

∂P ∂xi ∆Si +

i,j

1 2 ∂2P ∂xi ∂xj ∆Si ∆Sj − rP∆t −

i

∂P ∂xi (∆Si − rSi ∆t = 1 2

i

∂2P ∂x2

i

(∆S2

i − σ2 i (Si )2∆t) +

i>j

∂2P ∂xi ∂xj (∆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 :Towards Correlation dynamics ? 10th January 2014 6 / 55

SLIDE 7

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 :Towards Correlation dynamics ? 10th January 2014 7 / 55

SLIDE 8

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

ther (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"

Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 8 / 55

SLIDE 9

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)

Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 9 / 55

SLIDE 10

Observe correlation

Implied Correlation Data(2)

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

Figure: Evidence of Correlation Skew

First observation = stochastic parameter!

Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 10 / 55

SLIDE 11

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 ? 10th January 2014 11 / 55

SLIDE 12

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 ? 10th January 2014 12 / 55

SLIDE 13

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 ? 10th January 2014 13 / 55

SLIDE 14

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).

Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 14 / 55

SLIDE 15

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 ? 10th January 2014 15 / 55

SLIDE 16

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

Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 16 / 55

SLIDE 17

Model correlation ? Introduce Decorrelation

Decorrelation Step

Ideas :

Decorrelate initial correlation matrix use parametric local overomega to recorrelate the matrix

Decorrelation :

ρD

i,j = (1 − ω1)ρH i,j + ω1 with ω1 < 0

⇐ ⇒ ρH

i,j = (1 − ω0)ρD i,j + ω0 with ω0 =

ω1 ω1 − 1

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

Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 17 / 55

SLIDE 18

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.

Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 18 / 55

SLIDE 19

Model correlation ? Local Formulae : Derivate Market Information

Remember : how to prove Dupire’s formula ?

Idea : Derive Market Information/Observables

dSt = (rt St − Qt − qt St )dt + σ(t, St )St dWt so that (undiscounted calls) dC = dEQ(St − K)+ = ∂C ∂t dt = EQd(St − K)+ = EQ(dSt 1St >K + 1 2 d < S >t δSt =K ) = EQ((rt − qt )St 1St >K − Qt 1St >K + 1 2 σ(t, K)2K 2δSt =K )dt But : ∂C ∂K = −EQ(1St >K ) Or : C − K ∂C ∂K = EQ(St 1St >K ) And : ∂2C ∂K 2 = EQ(δSt =K ) So that : ∂C ∂t = (rt − qt )(C − K ∂C ∂K ) + Qt ∂C ∂K + 1 2 σ(t, K)2K 2 ∂2C ∂K 2 And finally : σ(t, K)2 =

∂C ∂t − (rt − qt )(C − K ∂C ∂K ) − Qt ∂C ∂K 1 2 K 2 ∂2C ∂K2

Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 19 / 55

SLIDE 20

Model correlation ? Local Formulae : Derivate Market Information

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

SLIDE 21

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 = (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 δIS

t =K )) in basket model

∂C ∂t dt = EQ(exp(− t rsds)((dIt − rt (It − K)dt)1It >K + 1 2 d < I >t δIt =K )) in index model Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 21 / 55

SLIDE 22

Model correlation ? Local Formulae : Derivate Market Information

Local Correlation formula(2)

EQt ((−

i

wi (Qi

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

1 2 d < IS >t δIS

t =K ) = EQt

((−(Qt + qt It ) + rt Kdt)1It >K + 1 2 d < I >t δIt =K ) but : EQt (d < I >t δIt =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 )

(1) Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 22 / 55

SLIDE 23

Model correlation ? Local Formulae : Derivate Market Information

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

SLIDE 24

Model correlation ? Local Formulae : Derivate Market Information

Local Correlation formula : simplest formula

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 = Depends on covariance :

Implied−Minimum Maximum−Minimum

Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 24 / 55

SLIDE 25

Model correlation ? Local Formulae : Derivate Market Information

Local Correlation formula : focus on dividends

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

SLIDE 26

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 = qi

t ∀i)

= ⇒ two models are generally inconsistent. = ⇒ Need to use continuous div model One more derivation in K + same density ( ∂2C

∂K 2 ) give :

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)

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 ? 10th January 2014 26 / 55

SLIDE 27

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 ? 10th January 2014 27 / 55

SLIDE 28

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 = (rt − qt)dt + σ(t, WOt)dWt Standard Model : dSi

t

Si

t

= (rt − qi

t)dt + σi(t, Si t)(

1 − ω(t,

WOt)dW i

t +

ω(t,

WOt)dW ⊥

t )

with : WOt = mini( Si

t

Si

t0

) and : < dW i

t , dW i t >

= ρ0

i,j(t)dt Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 28 / 55

SLIDE 29

Why basket local correlation ?

Worst Of Correlation (2)

Derive WO Calls in both models (calculation a little heavy): WO model : ∂CWO ∂t = −rt CWO + (rt − qt )(CWO − K ∂CWO ∂K ) + 1 2 K 2σ2(t, K) ∂2CWO ∂K 2 WO local correl model : ∂CWO ∂t = −rt CWO + (rt − qWO

t

(K))(CWO − K ∂CWO ∂K ) + 1 2 K 2EQ(σ

WO(t, K)2|

WO = K) ∂2CWO ∂K 2 − 1 2

i>j

EQ( d < Si

t , Si t > +d < Sj t , Sj t > −2d < Si t , Sj t >

dt δ

Si t =Sj t

1

WOt >K 1 WOt =Si t

) with : qWO

t

(K) = EQ(q

WO

WO1

WO>K )

EQ( WO1

WO>K )

Condition on WO Forward : qt = qWO

t

(0) If qWO

t

(K) = qt (not true in general, else add corrective term to overomega like in basket formula) and ρi,j (t, K) = ρ0

i,j (t) + ω(t, K)(1 − ρ0 i,j (t)), one more K derivation gives :

ω(t, K) =

∂ ∂K

K 2 ∂2CWO

∂K2

EQ(σ
WO(t, K)2|

WO = K) − σ2(t, K)

K 2

i>j EQ(2(1 − ρ0 i,j (t))σi (t, K)σj (t, K)δ Si t =Sj t

1

WOt >K 1 WOt =Si t

) − K 2

i>j EQ((2ρ0 i,j (t)σi (t, K)σj (t, K) − σ2 i (t, K) − σ2 j (t, K))δ Si t =Sj t

1

WOt >K 1 WOt =Si t

) K 2

i>j EQ(2(1 − ρ0 i,j (t))σi (t, K)σj (t, K)δ Si t =Sj t

1

WOt >K 1 WOt =Si t

) Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 29 / 55

SLIDE 30

Why basket local correlation ?

Worst Of Correlation (3)

Two important quantities : Switching Local Time : δSi

t =Sj t

Local Dispersion :

d<Si

t ,Si t >+d<Sj t ,Sj t >−2d<Si t ,Sj t >

dt

=

d<Si

t −Sj t ,Si t −Sj t >

dt

= (σiSi)2 + (σjSj)2 − 2ρi,jσiσjSiSj Local Dispersion = short correl, long volatility, positive quantity

cf. −2ρi,jσiσjSiSj + (σiSi)2 + (σjSj)2 = (σiSi − σjSj)2 + 2(1 − ρi,j)σiσjSiSj

Note : local dispersion in Spread Option Local equation :

∂CSpread ∂t = −rtCSpread + (rt − qSpread

t

)(CSpread − K ∂CSpread ∂K ) + 1 2 EQ( d < S1

t , S1 t > +d < S2 t , S2 t > −2d < S1 t , S2 t >

dt δSpread=K )

Remember also Margrabe formula : σ =

(σi)2 + (σj)2 − 2ρi,jσiσj

= ⇒ 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 ? 10th January 2014 30 / 55

SLIDE 31

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 ? 10th January 2014 31 / 55

SLIDE 32

Why basket local correlation ?

Discussion : ω ∈ [0; 1] ?

Cf. Guyon/Henry-Labordere remark.

Not theoretically (no static arbitrage) True in practice if ρ0

i,j enough low (Ex: ρ0 i,j = 0∀i, j)

Explanation ? Possible to infer a positive implied correlation ωI(K, T) for a standard model if ρ0

i,j = 0 for ex.

Gatheral-like formula : ρI

i,j(T, m)σI i (T, m)σI j (T, m) ≃ 1 T

T

0 ρL i,j(t, mt T )σL i (t, mt T )σL j (t, mt T )dt

Introduction of drift (continuous dividends) still OK.

Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 32 / 55

SLIDE 33

Why basket local correlation ?

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)

Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 33 / 55

SLIDE 34

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 ? 10th January 2014 34 / 55

SLIDE 35

Calibration results : Local Volatility

Results

Application to Eurostoxx smile. Only two iterations that need 2000 simulations each : quick calibration. Here, ρ0

i,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 ? 10th January 2014 35 / 55

SLIDE 36

Calibration results : Local Volatility

Local Correlation Shape

Figure: Local Correlation Smile Figure: ATM Local Correlation Skew

Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 36 / 55

SLIDE 37

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 ? 10th January 2014 37 / 55

SLIDE 38

Extension to Stochastic Volatility Need to introduce specific parametrization

Extension to Stochastic Volatility framework

Chosen volatility model = (continuous) Bergomi model : 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

with : d W i

t

=

1 − ω(t, IS

t )dW i t +

ω(t, IS

t )dW ⊥ 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,j dt

dW i,S

t

= ρi

Sd

W i

t +

1 − (ρi

S)2(αdZt +

1 − (α)2dW i,S

t

) dW i,L

t

= ρi

Ld

W i

t +

1 − (ρi

L)2(βi dZt +

1 − (βi )2dW i,L

t

) < dW i,S

t

, dW i,L

t

> = ρi

SLdt

Comments : 3N + 2 brownians required (NW,NW L,NW S, Z, W ⊥) Parametrization maintains mono underlying volatility skew due to stochastic vol. Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 38 / 55

SLIDE 39

Extension to Stochastic Volatility Need to introduce specific parametrization

Correlation Structure

Complicated PSD conditions ? No, because direct use of brownians = ⇒ no cholesky = ⇒ computation time gain But : α and βi ∈ [−1; 1] Notice that : βi = 1 α ρi

SL − ρi Sρi L

1 − (ρi

S)2

1 − (ρi

L)2

but : 1 + 2ρi

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

≥ 0 (cf. PSD for each underlying) ⇐ ⇒    ρi

SL − ρi Sρi L

1 − (ρi

S)2

1 − (ρi

L)2

  

2

≤ 1 ⇐ ⇒ αβi ∈ [−1; 1] Comments : α = 1 : OK ρi

SL = ρi Sρi L : OK for any α

Remember α = 1 : basket prices closest with LV and SV

Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 39 / 55

SLIDE 40

Extension to Stochastic Volatility Decorrelation with Multi-Underlying Stochastic Volatility

Decorrelation Effect

Observation = "Decorrelation Effect with Stochastic Volatility" If ∀i, V Sto

i

(K) = V Loc

i

(K) then V Sto

bskt(K) ≤ V Loc bskt(K) where V= Call or Put

Why ? Heuristic :

EQ(

i,j

wi wj

Vi
Vj Si

t Sj t ρi,j |

i

wi Si t =

i

wi Fi t ) ≃ EQ(

i,j

1 N2

Vi
Vj ρi,j |∀k, Sk

t = Fk t ) and :EQ(

i,j

wi wj σi (t, Si t )σj (t, Sj t )Si t Sj t ρi,j |

i

wi Si t =

i

wi Fi t ) ≃

i,j

1 N2 σi (t, Fi t )σj (t, Fj t )ρi,j Cauchy-Schwarz :EQ(

Vi
Vj |Si

t = Fi t , Sj t = Fj t )

EQ(Vi |Si

t = Fi t )EQ(Vj |Sj t = Fj t )

r :EQ(
Vi
Vj |Si

t = Fi t , Sj t = Fj t )

σi (t, Fi

t )σj (t, Fj t ) so that :EQ(

i,j

wi wj

Vi
Vj Si

t Sj t ρi,j |

i

wi Si t =

i

wi Fi t )

EQ(
i,j

wi wj σi (t, Si t )σj (t, Sj t )Si t Sj t ρi,j |

i

wi Si t =

i

wi Fi t )

Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 40 / 55

SLIDE 41

Extension to Stochastic Volatility Decorrelation with Multi-Underlying Stochastic Volatility

Decorrelation Effect

The decorrelation effect depends a great deal on the value of α (correlation between vols) and a little on the size of the basket.

Figure: Decorrelation Effect depending on basket size

Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 41 / 55

SLIDE 42

Extension to Stochastic Volatility Usual values of correl between vols

Historical value of implied volatilities

Figure: Historical correlation between vols for main indices

= ⇒ High level of correlation between vols. Link between Correl and α ? One factor case : ρVi ,Vj = ρSi ,Vi ρSi ,Sj ρSj ,Vj + α2 1 − ρ2

Si ,Vi

1 − ρ2

Sj ,Vj

Standard values : α ≃ 1 or α > 1 = ⇒ need for level correction.

Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 42 / 55

SLIDE 43

Extension to Stochastic Volatility Usual values of correl between vols

Calibration formula

Calibration (no div case) using fixed point algorithm with parametric (polynomial in basket moneyness) or non-parametric regression. Formula :

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) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 43 / 55

SLIDE 44

Focus on correlation products

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 ? 10th January 2014 44 / 55

SLIDE 45

Focus on correlation products

Analysis of correlation product prices in different models

World Basket as of 05/04/2013. Maturity = 1Y. Strikes in Forward Basket Moneyness. Product/Model Without CS With CS With CS and SV / α = 1 With CS and SV / α = 0.5 Forward WO 90.15% 90.25% 90.48% 90.52% WO Call 90 8.73% 8.45% 8.53% 8.54% WO Call 95 6.27% 5.97% 6.00% 6.01% WO Call 100 4.28% 3.98% 3.98% 3.98% WO Call 105 2.76% 2.49% 2.47% 2.47% WO Call 110 1.66% 1.14% 1.42% 1.42% Forward BO 109.49% 109.27% 109.34% 109.32% BO Put 90 1.88% 2.34% 2.28% 2.32% BO Put 95 2.81% 3.32% 3.24% 3.28% BO Put 100 4.11% 4.64% 4.55% 4.58% BO Put 105 5.84% 6.38% 6.27% 6.30% BO Put 110 8.09% 8.62% 8.48% 8.51% Spread Option Eurostoxx versus SP500. Strikes in Forward Spread Moneyness. Product/Model Without CS With CS With CS and SV / α = 1 With CS and SV / α = 0.5 Spread Option -10 12.46% 12.31% 12.45% 12.92% Spread Option -5 9.00% 8.85% 8.87% 9.47% Spread Option 0 6.15% 6.03% 6.20% 6.72% Spread Option 5 3.96% 3.87% 4.01% 4.52% Spread Option 10 2.40% 2.34% 2.53% 2.90% Call on Spread : Long Vovol, Short Correl between vols. Stochastic Vol parameters (for three underlyings): κS κL σS σL ρS ρL ρSL 400.0% 12.5% 350% 100%

50%
50%

50% Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 45 / 55

SLIDE 46

Focus on correlation products

Zoom on Cancellable

Product generally considered as a simple product but : interest rate risk dividend risk volatility risk (with vanna and volga changing signs!) correlation risk pin risk (need for smoothing) Case of 3Y autocall product : 3Y product three underlyings (World Basket) can be cancelled every year at 100 short Put 100 Discrete DI 60

Model LV LSV (α = 1) LV + CS LSV + CS, α = 1 LSV + CS, α = 0.5 Basket Cancellable 93.88% 94.14% 93.61% 94.04% 94.02% WO Cancellable 86.15% 85.94% 87.15% 87.44% 87.55% Two main conclusions: Correlation Skew and Stochastic Volatility don’t add (cross effect cannot be neglected) Price doesn’t depend on correlation between vols

Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 46 / 55

SLIDE 47

Main conclusions

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 ? 10th January 2014 47 / 55

SLIDE 48

Main conclusions

Main issues with correlation in equity : Constraints : must remain between -1 and 1, must be part of PSD matrices Numerous elements compared to volatility Illiquid parameter Difficult to integrate new dimensions (overlap between baskets) Next parameter to gain in complexity, but long evolution. Currently = essentially studied for improved Macro Risk Management.

Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 48 / 55

SLIDE 49

References

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 ? 10th January 2014 49 / 55

SLIDE 50

References

◮ Reghai : Using Local Correlation models to improve option

hedging

◮ Reghai : Breaking correlation breaks ◮ Avellaneda/Boyer-Olson/Busca/Friz : Reconstructing Volatility ◮ Dupire : Pricing with a smile ◮ Langnau : Introduction into Local Correlation Modelling ◮ Sbai-Jourdain : Coupling Index and stocks ◮ Guyon/Henry-Labordere : The smile calibration problem solved ◮ Christoph Burgard : New Developments in Vol and Var Products ◮ Lee et all. : Index Volatility surface via moment-matching

techniques

◮ Piterbarg : Markovian projection for volatility calibration

Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 50 / 55

SLIDE 51

References

Appendix 1 : Pathwise equality

True Model = believed to be Index model with Local Vol : EQ

t ((IT − K)+) = C(t, It , K, T)

dIt = It σ(t, It )dWt dIt =

i

wi Si

t σi (t, Si t )dW i t

One can write : EQ

0 (C(T, IT ))

= C(0, I0) + T EQ

0 (dC)

= C(0, I0) + 1 2 T EQ

0 (

∂2C ∂x2 (d < It , It > −d < It , It >)) Pathwise equality : d < It , It >= d < It , It >. Or :

i,j wi wj Si

t Sj t σi (t, Si t )σj (t, Sj t )ρi,j = σ2(t, It ).

Sufficient condition, but not necessary. Other sufficient condition (but still not necessary for models like in Lucic 2009) : EQ

0 (

i,j

wi wj Si

t Sj t σi (t, Si t )σj (t, Sj t )ρLoc i,j (S1 t , . . . , Sn t )|S1 t , . . . , Sn t ) = EQ 0 (

i,j

wi wj

V i

t

V j

t ρSto i,j |S1 t , . . . , Sn t )

Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 51 / 55

SLIDE 52

References

Appendix 2 : WO formula demonstration(1)

Ito-Tanaka to WO Call (abusive notations) : f(x1, . . . , xn) = (

l

xl

k=l

1xl<xk − K)+ ∂f ∂xi =

k=i

1xi<xk1xi≥K (cf. other terms cancel out) ∂2f ∂x2

i

=

k=i

1xi<xk1xi=K − (

j=i

(

k=i,k=j

1xi<xk)1xi=xj)1xi≥K ∂2f ∂xi∂xj =

k=i,k=j

1xi<xk1xi=xj1xi≥K

Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 52 / 55

SLIDE 53

References

Appendix 2: WO formula demonstration(2)

Deterministic interest rates : dEQ

exp(−

t rsds)f(S1

t , . . . , S1 t )

=

∂C ∂t dt = −rtCdt +

i

EQ

exp(−

t rsds) ∂f ∂xi (rt − qi

t)Si t

dt

+ 1 2

i

EQ

exp(−

t rsds) ∂2f ∂x2

i

d < Si

t >

+
j>i

EQ

exp(−

t rsds) ∂2f ∂xi∂xj d < Si

t, Sj t >

Rearranging terms give final result.

Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 53 / 55

SLIDE 54

References

Appendix 2: WO formula demonstration(3)

Same formula for BO options : ∂CBO ∂t = −rt CBO + (rt − qBO t (K))(CBO − K ∂CBO ∂K ) + 1 2 K2 ∂2CBO ∂K2 EQ(σ

BO(t, K)2|

BO = K) + 1 2

i>j

EQ   d < Si t − Sj t , Si t − Sj t > dt δ Si t =Sj t 1 BOt >K 1 BOt =Si t   and Rainbow (calculation is awful) : Rbw(S1, . . . , Sn) =

j

wj S(j) with :S(1) ≥ . . . ≥ S(n) ∂CRbw ∂t = −rt CRbw + rt (CRbw − K ∂CRbw ∂K ) − EQ(

i
j

wj qi t S(j)1S(j)=Si 1

Rbw>K )

+ 1 2 ∂2CRbw ∂K2 EQ   

i,j
k,l

wk wl 1S(k)=Si 1S(l)=Sj d < Si t , Sj t > dt

Rbw = K

   − 1 2

i>j

EQ   d < Si t − Sj t , Si t − Sj t > dt δ Si t =Sj t 1

Rbwt >K
l

(wl+1 − wl )1S(l)=Si   Last term disappears and other terms equal to basket equation for equally weighted baskets.

Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 54 / 55

SLIDE 55

References

Disclaimer

This document is issued in the France by HSBC France. HSBC France is authorised by the CECEI and regulated by the Autorité des Marchés Financiers ("AMF") and the Authorité de Contrôle Prudentielle. HSBC France is a member of the HSBC Group of companies ("HSBC Group"). Any member of the HSBC Group, together with their directors, officers and employees may have traded for their own account as principal, underwritten an issue within the last 36 months, or have a long or short position in any related instrument mentioned in this material. HSBC France or any of its affiliate ("HSBC") may be solicited in the course of its placement efforts for a new issue by investment clients for whom it already provides other services. HSBC may allocate securities to its own proprietary book or to an associate of HSBC Group. This represents a potential conflict of

interest. HSBC has internal arrangements designed to ensure that it will give unbiased and full advice to a corporate finance client about valuation and

pricing of the Issue and internal systems, controls and procedures to identify and to manage potential conflicts of interest. This document is for information and convenient reference, and is not intended as an offer or solicitation of the purchase or sale of any security or other

investment. Except in the case of fraudulent misrepresentation, HSBC does not make any representation or warranty (express or implied) of any nature or

accept any responsibility or liability of any kind for accuracy or sufficiency of any information, statement, assumption or projection in this document, or for any loss or damage (whether direct, indirect, consequential or other) arising out of reliance upon this document. Information in this document has not been independently verified by HSBC. Any reference in this document to particular proposed terms of issue is intended as a summary and not a complete

description. Terms or characteristics are subject to change. You are solely responsible for making your own independent appraisal of and investigations into

the products, investments and transactions referred to in this document and you should not rely on any information in this document as constituting investment advice. Neither HSBC nor any of its affiliates are responsible for providing you with legal, tax or other specialist advice and you should make your own arrangements in respect of this accordingly. Any projection, forecast, estimate or other "forward-looking" statement in this document only illustrates hypothetical performance under specified assumptions of events or conditions, which may include (but are not limited to) prepayment expectations, interest rates, collateral and volatility. Such projections, forecasts, estimates or other "forward-looking" statements are not reliable indicators of future performance. As with any mathematical model that calculates results from inputs, results may vary significantly according to the values input. You should understand the assumptions and evaluate whether they are appropriate for their purposes. Some relevant events or conditions may not have been considered in such assumptions. Actual events or conditions may differ materially from assumptions. Past performance is not a reliable indicator of future performance. This document is intended for persons who are professional clients or eligible counterparties (as defined in the rules of the AMF) only and is not intended for distribution to, or use by, retail clients. This document also is not intended for distribution to, or use by, any person or entity in any jurisdiction or country where such distribution or use would be contrary to law or regulation. In particular, this document and the information contained herein do not constitute an

ffer of securities for sale in the United States and are not for publication or distribution to persons in the United States (within the meaning of Regulation S

under the Unites States Securities Act of 1933, as amended). Information in this document is confidential. Distribution of this document, or information in this document, to any person other than an original recipient (or to such recipient’s advisors) is prohibited. Reproduction of this document, in whole or in part, or disclosure of any of its contents, without prior consent of HSBC or an associate, is prohibited. This document should be read in its entirety. This document remains the property of HSBC and on request this document, and all other materials provided by the HSBC Group relating to proposals contained herein, must be returned and any copies destroyed. The issue of this document shall not be regarded as creating any form of contractual relationship. This document is a "financial promotion" within the scope of the rules of the AMF .

Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ? 10th January 2014 55 / 55