Local Correlation with Local Vol and Stochastic Vol ... · PDF fileLocal Correlation with...

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

Local Correlation : where are we ?

Outline

1 Local Correlation : where are we ?

2 PnL equation

3 Observe correlationEvidence of Correlation Skew

4 Model correlation ?Introduce DecorrelationNew Methods in FinanceLocal Formulae : Derivate Market Information

5 Why basket local correlation ?

6 Calibration results : Local Volatility

7 Extension to Stochastic VolatilityNeed to introduce specific parametrizationDecorrelation with Multi-Underlying Stochastic VolatilityUsual values of correl between vols

8 Focus on correlation products

9 Main conclusions

10 References



Recent (or less recent) developments in localcorrelation

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, toomany 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 historicalparameter 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 forspecific points.Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ?10th January 2014 3 / 55


"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, alsoused in shrinkage methods)

We introduce "Overomega" (not a standard notation) :ρ

Pricingi,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. crossedgamma sign), with PricingMatrix = (1− ω)InitMatrix + ωTargetMatrix

But Issues when Crossed gammas change sign locally (=⇒ uncertaincorrelation pricing)


PnL equation

Outline


2 PnL equation







9 Main conclusions

10 References


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

∂xirt xi +

∑i,j

1

2

∂2P

∂xi ∂xjρi,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

∂x2i

(∆S2i − σ

2i (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


Observe correlation

Outline


2 PnL equation







9 Main conclusions

10 References


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 betweenbrownians (generally, "Gaussian copula")

Example of issue : correlation can be more than 1 due to time zones (Bergomi) orother (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 VolRebonato :"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 rightprice"


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 ATMFimplied Vols

Based on only 50 underlyings of SP500 (liquidity issues)


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

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 SkewStructured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ?10th January 2014 11 / 55


Evidence based on Implied Data(1)

Figure: Basket Smile versus Index Smile : SMI case

Consequence : market expects more correlation on the downside, andless on the upside.Structured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ?10th January 2014 12 / 55


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)



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, allstocks impacted

Upside : generally decreases, but (sometimes) systemic "rescue".When good news concerning the economy (rates decrease, centralbank actions,...), all stocks impacted (and correlation increases).


Model correlation ?

Outline


2 PnL equation







9 Main conclusions

10 References


Model correlation ?

The purpose of modelling correlation

Different situations :Our focus : liquid basket optionsHowever, 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


Model correlation ? Introduce Decorrelation

Decorrelation Step

Ideas :Decorrelate initial correlation matrix

use parametric local overomega to recorrelate the matrix

Decorrelation :

ρDi,j = (1− ω1)ρ

Hi,j + ω1 with ω1 < 0

⇐⇒ ρHi,j = (1− ω0)ρ

Di,j + ω0 with ω0 =

ω1

ω1 − 1

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


Model correlation ? New Methods in Finance

Foreword

Standard Models are simpler to handle with local vol, local correladjustments :

Fixed Point algorithm (Reghai) +

Local Formulae (Dupire) +

Numerical Evaluation of conditional expectations (not specificallylinked to finance) =

Local fixed-point methods (particular case for explicit schemeswith one iteration = Particle method)

Fixed Point problem : contracting function(?) on a space of stochasticprocesses. Existence still needs to be solved theoretically.


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

∂tdt

= EQd(St − K )+

= EQ(dSt 1St >K +1

2d < S >t δSt =K )

= EQ((rt − qt )St 1St >K − Qt 1St >K +1

2σ(t, K )2K 2

δSt =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

2σ(t, K )2K 2 ∂2C

∂K 2

And finally : σ(t, K )2 =∂C∂t − (rt − qt )(C − K ∂C

∂K )− Qt∂C∂K

12 K 2 ∂2C

∂K 2



Our framework = Reghai’s

Local Correlation introduced through the use of an overomegaapproach.What is Overomega ? ρ

Pricingi,j = (1− ω)ρHisto

i,j + ωFirst Model = Simple Local Vol Model with continuous dividends (mixof prop and cash dividends).

dSit = (rtSi

t −Q it − q i

t Sit )dt + σ(t , Si

t )Sit (√

1− ω(t , ISt )dW i

t +√

ω(t , ISt )dW⊥

t )

with Q it and qi

t deterministic and :

ISt =

N∑i=1

wiSit

< dW it , dW j

t > = ρ0i,jdt

< dW it , dW⊥

t > = 0∀i



Local Correlation formula (general case)Second Model = simple local vol model written on the index(continuous divs) :

dIt = (rt It −Qt − qt It)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

0rsds)(It − K )+)

= EQ(exp(−∫ t

0rsds)(IS

t − K )+)∀t , K

but :

∂C

∂tdt = EQ(exp(−

∫ t

0rsds)((dISt − rt (I

St − K )dt)1

ISt >K+

1

2d < IS >t δ

ISt =K)) in basket model

∂C

∂tdt = EQ(exp(−

∫ t

0rsds)((dIt − rt (It − K )dt)1It >K +

1

2d < I >t δIt =K )) in index model



Local Correlation formula(2)

EQt((−

∑i

wi (Qit + qi

t Sit ) + rt Kdt)1

ISt >K+

1

2d < IS >t δ

ISt =K) = EQt

((−(Qt + qt It ) + rt Kdt)1It >K +1

2d < 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

(ISt 1ISt >K

) =1

B(0, t)(C − K

∂C

∂K)

EQt(1It >K ) = EQt

(1ISt >K

) =1

B(0, t)(−

∂C

∂K)

Condition on the forward (K = 0):

Qt =∑

i wi Qit

qt =EQt

(∑

i wi qit Si

t )

EQt(It )

(1)



Local Correlation formula(3)

ω(t , K ) =

(K 2σ(t , K )2 +

2 ∂C∂K

∂2C∂K 2

(EQt

((qt It−rt K )1It >K )

EQt (1It >K )−

EQt((

∑i wi q

it S

it−rt K )1

ISt >K)

EQt (1ISt >K

)

))EQt (

∑i,j wiwjSi

tSjtσ(t , Si

t )σ(t , Sjt )(1− ρ0

i,j)|ISt = K )

−EQt

(∑

i,j wiwjSitS

jtσ(t , Si

t )σ(t , Sjt )ρ

0i,j |IS

t = K )

EQt (∑

i,j wiwjSitS

jtσ(t , Si

t )σ(t , Sjt )(1− ρ0

i,j)|ISt = K )



Local Correlation formula : simplest formula

Particular cases : no dividends, deterministic interest rates

ω(t , K ) =K 2σ(t , K )2 − EQ(

∑i,j wiwjSi

tSjtσ(t , Si

t )σ(t , Sjt )ρ

0i,j |IS

t = K )

EQ(∑

i,j wiwjSitS

jtσ(t , Si

t )σ(t , Sjt )(1− ρ0

i,j)|ISt = K )

Dupire/Avellaneda/Piterbarg/Guyon-PHL formula.

Case where constant vol and null correlation:

ω =σ2

I −1N σ2

S

σ2S(1− 1

N )

Well known formula : see Bossu for example.

Idea = Depends on covariance : Implied−MinimumMaximum−Minimum



Local Correlation formula : focus on dividends

EQt((qt It − rt K )1It >K )

EQt(1It >K )

−EQt

((∑

i wi qit S

it − rt K )1

ISt >K)

EQt(1

ISt >K)

Stochastic rate term + Dividend term.

Deterministic interest rates : first term vanishes since rt in factor and EQt(1It >K ) = EQt

(1ISt >K

) = 1B(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((ISt − K )+)− EQt

((ISt− − K )+) ' EQt

((ISt − ISt− )1

ISt >K)

but :

It − It− = −(Qt + qt It− )

ISt − ISt− =

∑i

−wi (Qit + qt S

it− )

leads to (first order in dividend level) :

EQt(qt It 1It >K ) = EQt

(∑

i

wi qit S

it 1ISt >K

)



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(qt It |It = K ) = EQt

(∑

i

wi qit S

it |I

St = K )

cf. Markovian projection : sufficient but not necessary conditionOther possible conditions :

qt =EQ(

∑i wi q

it Si

t )

EQ(It )

ω(t, K ) =

K 2σ(t,K )2−2(C−K ∂C

∂K )

∂2C∂K 2

qt−EQ((

∑i wi q

it Si

t )1ISt >K)

EQ((∑

i wi Sit )1ISt >K

)

−EQ(

∑i,j wi wj S

it Sj

t σ(t,Sit )σ(t,Sj

t )ρ0i,j |I

St =K )

EQ(∑

i,j wi wj Sit Sj

t σ(t,Sit )σ(t,Sj

t )(1−ρ0i,j )|I

St =K )

Comments :

ω helps recover from the generated error

in practice, prop divs smile correction can be neglected


Why basket local correlation ?

Outline


2 PnL equation







9 Main conclusions

10 References



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

Sit

= (rt − qit )dt + σi (t , Si

t )(

√1− ω(t , W̃Ot )dW i

t +

√ω(t , W̃Ot )dW⊥

t )

with : W̃Ot = mini (Si

t

Sit0

)

and : < dW it , dW i

t > = ρ0i,j (t)dt



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

WO model :∂CWO

∂t= −rt C

WO + (rt − qt )(CWO − K

∂CWO

∂K) +

1

2K 2

σ2(t, K )

∂2CWO

∂K 2

WO local correl model :∂CWO

∂t= −rt C

WO + (rt − qWOt (K ))(CWO − K

∂CWO

∂K) +

1

2K 2EQ(σW̃O (t, K )2|W̃O = K )

∂2CWO

∂K 2

−1

2

∑i>j

EQ(d < Si

t , Sit > +d < Sj

t , Sjt > −2d < Si

t , Sjt >

dtδ

Sit =Sj

t1

W̃Ot >K1

W̃Ot =Sit)

with : qWOt (K ) =

EQ(qW̃OW̃O1W̃O>K

)

EQ(W̃O1W̃O>K

)

Condition on WO Forward : qt = qWOt (0)

If qWOt (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 − ρ0

i,j (t)), one more K derivation gives :

ω(t, K ) =

∂∂K

(K 2 ∂2CWO

∂K 2

(EQ(σW̃O (t, K )2|W̃O = K )− σ2(t, K )

))K 2 ∑

i>j EQ(2(1 − ρ0i,j (t))σi (t, K )σj (t, K )δ

Sit =Sj

t1

W̃Ot >K1

W̃Ot =Sit)

−K 2 ∑

i>j EQ((2ρ0i,j (t)σi (t, K )σj (t, K )− σ2

i (t, K )− σ2j (t, K ))δ

Sit =Sj

t1

W̃Ot >K1

W̃Ot =Sit)

K 2 ∑i>j EQ(2(1 − ρ0

i,j (t))σi (t, K )σj (t, K )δSi

t =Sjt1

W̃Ot >K1

W̃Ot =Sit)



Worst Of Correlation (3)Two important quantities :

Switching Local Time : δSi

t =Sjt

Local Dispersion :d<Si

t ,Sit >+d<Sj

t ,Sjt >−2d<Si

t ,Sjt >

dt =d<Si

t−Sjt ,S

it−Sj

t >

dt = (σiS i)2 + (σjS j)2 − 2ρi,jσiσjS iS j

Local Dispersion = short correl, long volatility, positive quantitycf. −2ρi,jσ

iσjS iS j + (σiS i)2 + (σjS j)2 = (σiS i − σjS j)2 + 2(1− ρi,j)σiσjS iS j

Note : local dispersion in Spread Option Local equation :

∂CSpread

∂t= −rt C

Spread + (rt − qSpreadt )(CSpread − K

∂CSpread

∂K)

+1

2EQ(

d < S1t , S1

t > +d < S2t , S2

t > −2d < S1t , S2

t >

dtδSpread=K )

Remember also Margrabe formula : σ =√

(σi)2 + (σj)2 − 2ρi,jσiσj

=⇒ WO Call short disp product, spread option long disp product.



Local Correlation Models LimitsWO local Correlation :

no real observable smile for WO vanillas

Dynamic issue : only valid at inception (local vol -and forward- of WO modelshould 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 weightsshould 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


Discussion : ω ∈ [0; 1] ?

Cf. Guyon/Henry-Labordere remark.

Not theoretically (no static arbitrage)

True in practice if ρ0i,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 ifρ0

i,j = 0 for ex.Gatheral-like formula :ρI

i,j (T , m)σIi (T , m)σI

j (T , m) ' 1T

∫ T0 ρL

i,j (t ,mtT )σL

i (t , mtT )σL

j (t , mtT )dt

Introduction of drift (continuous dividends) still OK.



Parametric RegressionNeed to estimate :

EQ(∑

i,j wiwjS it S

jtσ(t , S i

t )σ(t , S jt )ρ

0i,j |IS

t = K )

EQ(∑

i,j wiwjS it S

jtσ(t , S i

t )σ(t , S jt )|I

St = 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)


Calibration results : Local Volatility

Outline


2 PnL equation







9 Main conclusions

10 References



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 SkewStructured Equity Research (HSBC) Local Correlation with Local Vol and Stochastic Vol :Towards Correlation dynamics ?10th January 2014 35 / 55


Local Correlation Shape

Figure: Local CorrelationSmile Figure: ATM Local Correlation Skew


Extension to Stochastic Volatility

Outline


2 PnL equation







9 Main conclusions

10 References


Extension to Stochastic Volatility Need to introduce specific parametrization

Extension to Stochastic Volatility frameworkChosen volatility model = (continuous) Bergomi model :

dSit

Sit

= σ(t, Sit )

√ξ

i,tt (

√1 − ω(t, ISt )dW i

t +√

ω(t, ISt )dW⊥t )

ordSi

t

Sit

=

√ξ

i,tt dW̃ i

t

with : dW̃ it =

√1 − ω(t, ISt )dW i

t +√

ω(t, ISt )dW⊥t

dξi,Tt

ξi,Tt

= σiS exp(−κ

iS(T − t))dW i,S

t + σiL exp(−κ

iL(T − t))dW i,L

t

with : ξi,Tt = EQ(V i

T |Ft )

< dW it , dW j

t > = ρ0i,j dt

dW i,St = ρ

iSdW̃ i

t +√

1 − (ρiS)2(αdZt +

√1 − (α)2dW i,S

t )

dW i,Lt = ρ

iLdW̃ i

t +√

1 − (ρiL)2(βi dZt +

√1 − (βi )

2dW i,Lt )

< dW i,St , dW i,L

t > = ρiSLdt

Comments :

3N + 2 brownians required (NW ,NW L,NW S , Z , W⊥)

Parametrization maintains mono underlying volatility skew due to stochastic vol.


Extension to Stochastic Volatility Need to introduce specific parametrization

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

βi =1

α

ρiSL − ρi

SρiL√

1− (ρiS)2√

1− (ρiL)

2

but :

1 + 2ρiSρi

LρiSL − (ρi

S)2 − (ρiL)

2 − (ρiSL)

2 ≥ 0 (cf. PSD for each underlying)

⇐⇒

ρiSL − ρi

SρiL√

1− (ρiS)2√

1− (ρiL)

2

2

≤ 1

⇐⇒ αβi ∈ [−1; 1]

Comments :α = 1 : OKρi

SL = ρiSρi

L : OK for any α

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


Extension to Stochastic Volatility Decorrelation with Multi-Underlying Stochastic Volatility

Decorrelation Effect

Observation = "Decorrelation Effect with Stochastic Volatility"

If ∀i , V Stoi (K ) = V Loc

i (K ) then V Stobskt(K ) ≤ V Loc

bskt(K ) where V= Call or Put

Why ? Heuristic :

EQ(∑i,j

wi wj

√Vi

√Vj S

it Sj

t ρi,j |∑

iwi S

it =

∑i

wi Fit ) ' EQ(

∑i,j

1

N2

√Vi

√Vj ρi,j |∀k, Sk

t = Fkt )

and :EQ(∑i,j

wi wj σi (t, Sit )σj (t, Sj

t )Sit Sj

t ρi,j |∑

iwi S

it =

∑i

wi Fit ) '

∑i,j

1

N2σi (t, F i

t )σj (t, F jt )ρi,j

Cauchy-Schwarz :EQ(√

Vi√

Vj |Sit = F i

t , Sjt = F j

t ) .

√EQ(Vi |S

it = F i

t )EQ(Vj |Sjt = F j

t )

or :EQ(√

Vi√

Vj |Sit = F i

t , Sjt = F j

t ) . σi (t, F it )σj (t, F j

t )

so that :EQ(∑i,j

wi wj

√Vi

√Vj S

it Sj

t ρi,j |∑

iwi S

it =

∑i

wi Fit ) . EQ(

∑i,j

wi wj σi (t, Sit )σj (t, Sj

t )Sit Sj

t ρi,j |∑

iwi S

it =

∑i

wi Fit )


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


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 ,ViρSi ,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

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 )

√ξ

i,tt σ(t, Sj,(n)

t )√

ξj,tt 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 − ρ0i,j )|I

S,(n)t = K )

=K 2σ(t, K )2 − f (n)(t, K )

g(n)(t, K )− f (n)(t, K )


Focus on correlation products

Outline


2 PnL equation







9 Main conclusions

10 References



Analysis of correlation product prices in differentmodelsWorld 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 = 0.5Forward 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 = 0.5Spread 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 ρSL400.0% 12.5% 350% 100% -50% -50% 50%



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 60Model 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


Main conclusions

Outline


2 PnL equation







9 Main conclusions

10 References


Main conclusions

Main conclusions

Main issues with correlation in equity :

Constraints : must remain between -1 and 1, must be part of PSDmatrices

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.


References

Outline


2 PnL equation







9 Main conclusions

10 References


References

References

I Reghai : Using Local Correlation models to improve optionhedging

I Reghai : Breaking correlation breaksI Avellaneda/Boyer-Olson/Busca/Friz : Reconstructing VolatilityI Dupire : Pricing with a smileI Langnau : Introduction into Local Correlation ModellingI Sbai-Jourdain : Coupling Index and stocksI Guyon/Henry-Labordere : The smile calibration problem solvedI Christoph Burgard : New Developments in Vol and Var ProductsI Lee et all. : Index Volatility surface via moment-matching

techniquesI Piterbarg : Markovian projection for volatility calibration


References

Appendix 1 : Pathwise equalityTrue 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 Sit σi (t, Si

t )dW it

One can write :

EQ0 (C(T , IT )) = C(0, I0) +

∫ T

0EQ

0 (dC)

= C(0, I0) +1

2

∫ T

0EQ

0 (∂2C

∂x2(d < It , It > −d < It , It >))

Pathwise equality :d < It , It >= d < It , It >.Or :∑

i,j wi wj Sit S

jt σi (t, Si

t )σj (t, Sjt )ρi,j = σ2(t, It ).

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

EQ0 (

∑i,j

wi wj Sit S

jt σi (t, Si

t )σj (t, Sjt )ρ

Loci,j (S1

t , . . . , Snt )|S1

t , . . . , Snt ) = EQ

0 (∑i,j

wi wj

√V i

t

√V j

t ρStoi,j |S

1t , . . . , Sn

t )


References

Appendix 2 : WO formula demonstration(1)

Ito-Tanaka to WO Call (abusive notations) :

f (x1, . . . , xn) = (∑

l

xl

∏k 6=l

1xl<xk − K )+

∂f∂xi

=∏k 6=i

1xi<xk 1xi≥K (cf. other terms cancel out)

∂2f∂x2

i

=∏k 6=i

1xi<xk 1xi=K − (∑j 6=i

(∏

k 6=i,k 6=j

1xi<xk )1xi=xj )1xi≥K

∂2f∂xi∂xj

=∏

k 6=i,k 6=j

1xi<xk 1xi=xj 1xi≥K


References

Appendix 2: WO formula demonstration(2)

Deterministic interest rates :

dEQ(

exp(−∫ t

0rsds)f (S1

t , . . . , S1t )

)=

∂C

∂tdt

= −rt Cdt +∑

i

EQ(

exp(−∫ t

0rsds)

∂f

∂xi(rt − qi

t )Sit

)dt

+1

2

∑i

EQ

(exp(−

∫ t

0rsds)

∂2f

∂x2i

d < Sit >

)

+∑j>i

EQ

(exp(−

∫ t

0rsds)

∂2f

∂xi∂xjd < Si

t , Sjt >

)

Rearranging terms give final result.


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

2K 2 ∂2CBO

∂K 2EQ(σB̃O (t, K )2|B̃O = K )

+1

2

∑i>j

EQ

d < Sit − Sj

t , Sit − Sj

t >

dtδ

Sit =Sj

t

1B̃Ot >K

1B̃Ot =Si

t

and Rainbow (calculation is awful) :

Rbw(S1, . . . , Sn) =

∑j

wj S(j) with :S(1) ≥ . . . ≥ S(n)

∂CRbw

∂t= −rt CRbw + rt (C

Rbw − K∂CRbw

∂K) − EQ(

∑i

∑j

wj qit S(j)1

S(j)=Si 1̃Rbw>K

)

+1

2

∂2CRbw

∂K 2EQ

∑i,j

∑k,l

wk wl 1S(k)=Si 1S(l)=Sj

d < Sit , Sj

t >

dt

∣∣∣∣∣R̃bw = K

−1

2

∑i>j

EQ

d < Sit − Sj

t , Sit − Sj

t >

dtδ

Sit =Sj

t

1̃Rbwt >K

∑l

(wl+1 − wl )1S(l)=Si

Last term disappears and other terms equal to basket equation for equally weighted baskets.


References

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