calibrating local correlations to a basket smile · calibrating local correlations to a basket...

Motivation The two existing models Building all admissible models Numerical examples Price impact of correl Extensions Path-dep vol Conclusion

Calibrating Local Correlations to a Basket Smile

Julien Guyon

Bloomberg L.P.Quantitative Research

Advances in Financial MathematicsChaire Risques FinanciersParis, January 10th, 2014

[email protected]

Julien Guyon Bloomberg L.P.



Outline

Motivation

The two existing admissible local correlation models

How do we build all the admissible local correlation models?

The particle method

Numerical example: joint calibration of EURUSD, GBPUSD and EURGBPsmiles

Impact of correlation on option prices

Extension to stochastic interest rates, stochastic dividend yield, andstochastic volatility

Path-dependent volatility

Open questions and conclusion




Motivation

Multi-asset Dupire local volatility models with constant correlation do notcapture market skew of stock indices

Stocks highly correlated in bearish markets

Local correlation (LC) models incorporate correl variability in option pricesand help traders risk-manage their correl positions during crises

∆P&Lt =1

2

N∑i,j=1

SitSjt ∂

2SiSjP (t, St)

(∆Sit∆S

jt

SitSjt

− ρijσi(t, Sit)σj(t, Sjt )∆t)

LC models also allow to build models consistent with a triangle of marketsmiles of FX rates

LC also used in interest rates modeling to calibrate to spread options prices




Motivation

Only 2 methods so far to calibrate to smile of index It =∑Ni=1 αiS

it :

Local in index volatility of the basket (Langnau):

N∑i,j=1

αiαjρij(t, St)σi(t, Sit)σj(t, S

jt )S

itSjt = f(t, It)

Local in index correlation matrix (Reghai, G. and Henry-Labordere):

ρ(t, St) = f(t, It)

Both methods may lead to correlation candidates that fail to be positivesemi-definite

And anyway why would one undergo either correlation structure?




Motivation

Our goal: build all the LC models calibrated to a basket smile (stockindex, cross FX rate, interest rate spread)

For the first time, one can design a particular calibrated model in order tomatch a view on a correlation skewreproduce some features of historical correlcalibrate to other option prices

We can reconcile static (implied) calibration and dynamic(historical/statistical) calibration




The FX triangle smile calibration problem


LC model for a triangle of FX rates (deterministic rates):

dS1t = (rdt − r1t )S1

t dt+ σ1(t, S1t )S1

t dW1t



t dW2t

d〈W 1,W 2〉t = ρ(t, S1t , S

2t ) dt

Example: S1 = EUR/USD, S2 = GBP/USD and S12 = EUR/GBP

Model calibrated to market smile of cross rate S12 ≡ S1/S2 iff for all t

EQfρ

[σ21(t, S1

t ) + σ22(t, S2

t )− 2ρ(t, S1t , S

2t )σ1(t, S1

t )σ2(t, S2t )

∣∣∣∣S1t

S2t

]= σ2

12

(t,S1t

S2t

)Qf = risk-neutral measure associated to the foreign currency in S2 (GBP):

dQf

dQ=S2T

S20

exp

(∫ T

0

(r2t − rdt

)dt

)






The calibration condition

EQfρ

[σ21(t, S1

t ) + σ22(t, S2

t )− 2ρ(t, S1t , S

2t )σ1(t, S1

t )σ2(t, S2t )

∣∣∣∣S1t

S2t

]= σ2

12

(t,S1t

S2t

)(1)

is equivalent to

EQρ

[S2t

(σ21(t, S1

t ) + σ22(t, S2

t )− 2ρ(t, S1t , S

2t )σ1(t, S1

t )σ2(t, S2t )) ∣∣∣S1

t

S2t

]EQρ

[S2t

∣∣∣S1t

S2t

] = σ212

(t,S1t

S2t

)

Any ρ : [0, T ]× R∗+ × R∗+ → [−1, 1] satisfying (1) is called an “admissiblecorrelation.” Admissible = calibrated to market smile of basket





Existing model #1: Local in cross volatility of the cross (Langnau, 2010,Kovrizhkin, 2012)

Assume that the volatility of the cross is local in cross:

σ21(t, S1) + σ2

2(t, S2)− 2ρ(t, S1, S2)σ1(t, S1)σ2(t, S2) ≡ f(t,S1

S2

)Calibration condition reads

σ21(t, S1

t ) + σ22(t, S2

t )− 2ρ(t, S1t , S

2t )σ1(t, S1

t )σ2(t, S2t ) = σ2

12

(t,S1t

S2t

)=⇒

ρ =σ21 + σ2

2 − σ212

2σ1σ2≡ ρ∗

Does ρ∗ stay in [−1, 1]?





Existing model #2: Local in cross correlation (Reghai, G. andHenry-Labordere, 2011)

Assume that ρ is local in cross: ρ(t, S

1

S2

)(Reghai, G. and

Henry-Labordere, 2011)

Calibration condition reads

EQfρ

[σ21(t, S1

t ) + σ22(t, S2

t )

∣∣∣∣S1t

S2t

]−2ρ

(t,S1t

S2t

)EQfρ

[σ1(t, S1

t )σ2(t, S2t )

∣∣∣∣S1t

S2t

]= σ2

12

(t,S1t

S2t

)=⇒

ρ

(t,S1t

S2t

)=

EQf[σ21(t, S1

t ) + σ22(t, S2

t )∣∣∣S1t

S2t

]− σ2

12

(t,S1t

S2t

)2EQf

[σ1(t, S1

t )σ2(t, S2t )∣∣∣S1t

S2t

]=

EQ[S2t

(σ21(t, S1

t ) + σ22(t, S2

t )) ∣∣∣S1

t

S2t

]− σ2

12

(t,S1t

S2t

)EQ[S2t

∣∣∣S1t

S2t

]2EQ

[S2t σ1(t, S1

t )σ2(t, S2t )∣∣∣S1t

S2t

]Julien Guyon Bloomberg L.P.




Existing model #2: Local in cross correlation (Reghai, G. andHenry-Labordere, 2011)

Calibrated model follows the McKean nonlinear SDE:



t dW1t



t dW2t

d〈W 1,W 2〉t =EQ[S2t

(σ21(t, S1

t ) + σ22(t, S2

t )) ∣∣∣S1

t

S2t

]− σ2

12

(t,S1t

S2t

)EQ[S2t

∣∣∣S1t

S2t

]2EQ

[S2t σ1(t, S1

t )σ2(t, S2t )∣∣∣S1t

S2t

] dt

=⇒ We can build ρ using the particle methodCf. G. and Henry-Labordere, Being Particular About Calibration, Riskmagazine, January 2012; Jourdain and Sbai, Coupling Index and stocks,Quantitative Finance, October 2010.





Particle method for local in cross correl (G. and Henry-Labordere, 2011)

1 Set k = 1 and ρ(t, S1, S2) =σ21(0,S

1)+σ22(0,S

2)−σ212(0,

S1

S2 )

2σ1(0,S1)σ2(0,S2)for t ∈ [t0, t1]

2 Simulate (S1,it , S2,i

t )1≤i≤N from tk−1 to tk using a discretization scheme

3 For all S12 in a grid Gtk of cross rate values, compute non-parametric

kernel estimates Enumtk (S12) and Eden

tk (S12) of EQfρ

[σ21 + σ2

2

∣∣∣S1t

S2t

]and

EQfρ

[σ1σ2

∣∣∣S1t

S2t

]at date tk, define

ρ(tk, S12) =

Enumtk (S12)− σ2

12

(tk, S

12)

2Edentk

(S12)

interpolate ρ(tk, ·), e.g., using cubic splines, extrapolate, and, for all

t ∈ [tk, tk+1], set ρ(t, S1, S2) = ρ(tk,S1

S2 ). If ρ(tk,S1

S2 ) > 1 (resp. < −1),cap it at +1 (resp. floor it at −1) =⇒ imperfect calibration (may beaccurate enough!)

4 Set k := k + 1. Iterate steps 2 and 3 up to the maturity date T




The equity index smile calibration problem


St = (S1t , . . . , S

Nt )

dSit = rtSitdt+ σi(t, S

it)S

itdW

it , d〈W i,W j〉t = ρij(t, St)dt

vρ(t, St) ≡N∑

i,j=1


jt )S

itS

jt

Index It =∑Ni=1 αiS

it made of N weighted stocks

Model calibrated to index smile if and only if

I2t σIDup(t, It)

2 = Eρ [vρ(t, St)| It] (2)

N(N − 1)/2 parameters for 1 scalar equation. Dimension reduction (e.g.,ρ0 = ρhist, ρ1 = 1):

ρ(t, S) = (1− λ(t, S))ρ0 + λ(t, S)ρ1, λ(t, S) ∈ RIf λ ∈ [0, 1], ρ is guaranteed to be a correlation matrixAfter dimension reduction, calibration condition reads

I2t σIDup(t, It)

2 = Eρ[vρ0(t, St) +

(vρ1(t, St)− vρ0(t, St)

)λ(t, St)

∣∣ It]Julien Guyon Bloomberg L.P.




Existing model #1: Local in index volatility of the index (Langnau, 2010)

Assume that vρ is local in index:

vρ(t, S) ≡ f(t, I)

Then the calibration condition reads

I2t σIDup(t, It)

2 = vρ0(t, St) +(vρ1(t, St)− vρ0(t, St)

)λ(t, St)

=⇒

λ(t, S) =I2σIDup(t, I)2 − vρ0(t, S)

vρ1(t, S)− vρ0(t, S)≡ λ∗(t, S)

Does λ∗ stay in [0, 1]?





Existing model #2: Local in index correl for stock indices (G. andHenry-Labordere, 2011)

Recall that, after dimension reduction, calibration condition reads

I2t σIDup(t, It)

2 = Eρ[vρ0(t, St) +


)λ(t, St)

∣∣ It]Assume that λ is local in index: λ(t, I)

=⇒

λ(t, It) =I2t σ

IDup(t, It)

2 − Eρ[vρ0(t, St)

∣∣ It]Eρ[vρ1(t, St)− vρ0(t, St)

∣∣ It]The calibrated model then follows the McKean nonlinear SDE

dSit = rtSitdt+ Sitσi(t, S

it)dW

it , d〈W i,W j〉t = ρij(t, It)dt

λ(t, I) =I2σIDup(t, I)2 −

∑Ni,j=1 αiαjρ

0ijE[σi(t, S

it)σj(t, S

jt )S

itS

jt |It = I]∑N

i,j=1 αiαj(ρ1ij − ρ0ij)E[σi(t, Sit)σj(t, S

jt )S

itS

jt |It = I]

Does λ stay in [0, 1]?




The FX smile triangle problem

Building all admissible local correlation models: FX (G., 2013)

Vol or ρ local in cross/index = a particular modeling choice only guided bycomputational convenience. How do we build all admissible correls?Let ρ be admissible. We can always pick two functions a(t, S1, S2) andb(t, S1, S2) such that b does not vanish and a+ bρ is local in cross:

a(t, S1, S2) + b(t, S1, S2)ρ(t, S1, S2) ≡ f(t,S1

S2

)e.g., b ≡ 1, a(t, S1, S2) = f

(t, S

1

S2

)− ρ(t, S1, S2)

Local in cross correl: a ≡ 0 and b ≡ 1 (Reghai, G. and Henry-Labordere)

Local in cross volatility of the cross: a = σ21 + σ2

2 and b = −2σ1σ2

(Langnau, Kovrizhkin): ρ∗ =σ21+σ

22−σ

212

2σ1σ2

σ212

(t,S1t

S2t

)= EQf

ρ

[σ21 + σ2

2 − 2ρσ1σ2

∣∣∣∣S1t

S2t

]= EQf

ρ

[σ21 + σ2

2+2a

bσ1σ2

∣∣∣∣S1t

S2t

]− 2(a+ bρ)

(t,S1t

S2t

)EQfρ

[σ1σ2

b

∣∣∣∣S1t

S2t

]Julien Guyon Bloomberg L.P.





σ212

(t,S1t

S2t

)= EQf

ρ

[σ21 + σ2

2 + 2a

bσ1σ2

∣∣∣∣S1t

S2t

]− 2 (a+ bρ)

(t,S1t

S2t

)EQfρ

[σ1σ2

b

∣∣∣∣S1t

S2t

]

=⇒ ρ(a,b) =1

b

EQfρ(a,b)

[σ21 + σ2

2 + 2abσ1σ2

∣∣∣S1t

S2t

]− σ2

12

2EQfρ(a,b)

[σ1σ2b

∣∣∣S1t

S2t

] − a

(3)

The affine transform may seem ad hoc at first sight but actually anyadmissible correlation is of the above type

Conversely, if a function ρ(a,b) : [0, T ]× R∗+ × R∗+ → [−1, 1] satisfies (3),then it is an admissible correlation

We call (3) the “local in cross affine transform representation” ofadmissible correlations






Calibrated model follows the McKean nonlinear SDE:



t dW1t



t dW2t

d〈W 1,W 2〉t =

(EQ[S2t

(σ21 + σ2

2 + 2abσ1σ2

) ∣∣∣S1t

S2t

]− σ2

12

(t,S1t

S2t

)EQ[S2t

∣∣∣S1t

S2t

]2EQ

[S2tσ1σ2b

∣∣∣S1t

S2t

]−a(t, S1

t , S2t )

)dt/b(t, S1

t , S2t )

The particle method allows us to estimate the conditional expectations





Building ρ(a,b) using the particle method

1 Set k = 1 and ρ(a,b) =σ21(0,S

1)+σ22(0,S

2)−σ212

(0,S

1

S2

)2σ1(0,S1)σ2(0,S2)

for t ∈ [t0 = 0; t1].

2 Simulate (S1,it , S2,i

t )1≤i≤N from tk−1 to tk using a discretization scheme.

3 For all S12 in a grid Gtk of cross rate values, compute

Enumtk

(S12

)

=

∑Ni=1 S

2,itk

σ21(tk, S1,itk

) + σ22(tk, S2,itk

)+2a(tk,S

1,itk,S

2,itk

)

b(tk,S1,itk,S

2,itk

)σ1(tk, S

1,itk

)σ2(tk, S2,itk

)

δtk,NS1,i

tk

S2,itk

− S12

∑Ni=1

S2,itkδtk,N

S1,itk

S2,itk

− S12





Building ρ(a,b) using the particle method

Edentk

(S12

) =

∑Ni=1 S

2,itk

σ1(tk,S1,itk

)σ2(tk,S2,itk

)

b(tk,S1,itk,S

2,itk

)δtk,N

S1,itk

S2,itk

− S12

∑Ni=1

S2,itkδtk,N

S1,itk

S2,itk

− S12

f(tk, S12

) =Enumtk

(S12) − σ212

(tk, S

12)

2Edentk

(S12)

3 cont’d interpolate and extrapolate f(tk, ·), for instance using cubic splines, and,for all t ∈ [tk, tk+1], set

ρ(a,b)(t, S1, S2) =

1

b(t, S1, S2)

(f

(tk,

S1

S2

)− a(t, S1, S2)

)If ρ(a,b)(t, S

1, S2) > 1 (resp. < −1), cap it at +1 (resp. floor it at −1)=⇒ imperfect calibration (may still be very accurate!)

4 Set k := k + 1. Iterate steps 2 and 3 up to the maturity date T .





Links between local correlations

(a+ bρ(a,b)

)(t,S1t

S2t

)=

EQfρ(a,b)

[(a+ bρ∗)

(t, S1

t , S2t

) σ1(t,S1t )σ2(t,S

2t )

b(t,S1t ,S

2t )

∣∣∣S1t

S2t

]EQfρ(a,b)

[σ1(t,S

1t )σ2(t,S

2t )

b(t,S1t ,S

2t )

∣∣∣S1t

S2t

]ρ(0,1)

(t,S1t

S2t

)=

EQfρ(0,1)

[ρ∗(t, S1

t , S2t

)σ1(t, S1

t )σ2(t, S2t )∣∣∣S1t

S2t

]EQfρ(0,1)

[σ1(t, S1

t )σ2(t, S2t )∣∣∣S1t

S2t

]ρ(0,1) = an average of ρ∗

If ρ(0,1) is admissible then its image is included in the image of ρ∗

τρ∗ ≤ τρ(0,1) with τρ the smallest time at which ρ fails to be a correlationfunction:

τρ = inf{t ∈ [0, T ] | ∃S1, S2 > 0, ρ(t, S1, S2) /∈ [−1, 1]

}Julien Guyon Bloomberg L.P.




Links between local correlations

σ212

(t,S1t

S2t

)= EQf

ρ(a,b)

[σ21(t, S1

t ) + σ22(t, S2

t )− 2ρ(a,b)(t, S1t , S

2t )σ1(t, S1

t )σ2(t, S2t )

∣∣∣∣S1t

S2t

]If no skew on S1 and S2:

ρ∗(t, S1t , S

2t ) = ρ(0,1)

(t,S1t

S2t

)= EQf

ρ(a,b)

[ρ(a,b)(t, S

1t , S

2t )

∣∣∣∣S1t

S2t

]

All admissible correls have same average value over constant cross lines

ρ(0,1) = ρ∗ is among all the admissible correls the one with smallest image

ρ(0,1)

(t,S1t

S2t

)> 1⇐⇒ |σ1(t)− σ2(t)| > σ12

(t,S1t

S2t

)ρ(0,1)

(t,S1t

S2t

)< −1⇐⇒ σ1(t) + σ2(t) < σ12

(t,S1t

S2t

)





Building whole families of admissible local correlation models: equity (G.,2013)

dSit = Sitσi(t, Sit)dW

it , d〈W i,W j〉t = ρij(t, St)dt

vρ(t, St) ≡N∑

i,j=1


jt )S

itS

jt

After dimension reduction, calibration condition reads

I2t σIDup(t, It)

2 = Eρ[vρ0(t, St) +


)λ(t, St)

∣∣ It]We can always pick two functions a and b such that b does not vanish anda(t, St) + b(t, St)λ(t, St) ≡ f (t, It) is local in index. Then

I2t σIDup(t, It)

2 = (a+ bλ) (t, I)Eρ[

1

b

(vρ1 − vρ0

)∣∣∣∣ It]+Eρ[vρ0 −

a

b

(vρ1 − vρ0

)∣∣∣ It]

=⇒ λ(a,b) =1

b

(I2t σ

IDup(t, It)

2 − Eρ(a,b)[vρ0−ab

(vρ1 − vρ0

)∣∣ It]Eρ(a,b)

[1b

(vρ1 − vρ0

)∣∣ It] − a

)Julien Guyon Bloomberg L.P.




Building families of admissible local correlation models: equity (G., 2013)

λ(a,b) =1

b

(I2t σ

IDup(t, It)

2 − Eρ(a,b)[vρ0 − a

b

(vρ1 − vρ0

)∣∣ It]Eρ(a,b)

[1b

(vρ1 − vρ0

)∣∣ It] − a

)(4)

If a function λ(a,b) satisfies (4) and is s.t. ρ(a,b) ≡ (1−λ(a,b))ρ0 +λ(a,b)ρ

1

is positive semi-definite, then ρ(a,b) is an admissible correlation

We call this procedure the local in index a+ bλ method.

Local in index λ method: a ≡ 0 and b ≡ 1 (Reghai, G. andHenry-Labordere)

Local in index volatility method: a = vρ0 and b = vρ1 − vρ0 (Langnau,Kovrizhkin)

Particle method =⇒ λ(a,b)




Numerical examples: FX

S1 = EURUSD, S2 = GBPUSD, S12 = S1/S2 = EURGBP (March2012)

T = 1

N = 10, 000 particles

∆t = 180

K(x) = (1− x2)21{|x|≤1}

Bandwidth h = κσ12S120

√max(t, tmin)N−

15 , σ12 = 10%, tmin = 0.25 and

κ ≈ 3.

The constant correlation that fits ATM implied volatility of cross rate atmaturity = 72%




a = 0, b = 1 (local in cross correlation)




a = σ21 + σ22, b = −2σ1σ2 (local in cross volatility of the cross)




a = 0, b = σ1σ2 (local in cross covariance)




a = 0, b =√S1S2




a = 0, b = min(S1, S2)




a = 0, b = max(S1, S2)




a = 0, b = (S1S2)1/4




a = 0, b =√

min(S1, S2)




a = 0, b =√

max(S1, S2)




a = 0, b = 1√S1S2




a = 0, b = 1.5 + 1{S1

S10+S2

S20>2

}




a = 0, b = 1.5 + 12

(1 + tanh

(10(S1

S10+ S2

S20− 2)))




a = 0, b = 0.08 +(S1

S10− 1)2

+(S2

S20− 1)2




a = 0, b = 0.02 +(S1

S10− 1)2

+(S2

S20− 1)2




a = 3√S1S2, b =

√S1S2




Historical local correlation EUR/USD & GBP/USD, 2007-13 vs 2011-13




The formulas

Impact of correlation on option prices

Price impact formula (similar to El Karoui, Dupire for vol):

Eρt [g(S1T , S

2T )]− P0(0, S1

0 , S20)

= Eρt[∫ T

0

(ρt − ρ0(t, S1t , S

2t ))σ1(t, S1

t )σ2(t, S2t )S1

t S2t ∂

2S1S2P0(t, S1

t , S2t )dt

]Implied correlation (cf Dupire for implied vol):

ρ(T, g) =Eρt

[∫ T0ρtσ1(t, S1

t )σ2(t, S2t )S1

t S2t ∂

2S1S2Pρ(T,g)(t, S

1t , S

2t )dt

]Eρt

[∫ T0σ1(t, S1

t )σ2(t, S2t )S1

t S2t ∂

2S1S2Pρ(T,g)(t, S

1t , S

2t )dt

]Implied correl is the fixed point of

ρ 7→∫ T

0

∫ ∞0

∫ ∞0

ρloc(t, S1, S2)qρ(t, S

1, S2)dS1dS2dt

qρ(t, S1, S2) =

σ1(t, S1)σ2(t, S2)S1S2∂2S1S2Pρ(t, S

1, S2)p(t, S1, S2)∫ T0

∫∞0

∫∞0σ1(t∗, S1

∗)σ2(t∗, S2∗)S1∗S2∗∂2S1S2Pρ(t∗, S

1∗ , S2∗)p(t∗, S1

∗ , S2∗)dS1

∗dS2∗dt∗

p→ p =⇒ ρ(T, g)→ ρ(T, g) (cf G. and Henry-Labordere, 2011)Julien Guyon Bloomberg L.P.



The formulas

Graphs of (S1, S2) 7→ qρ(t, S1, S2) for different values of t

Figure : Black-Scholes model: σ1 = 20%, σ2 = 20%, ρ = 0, S10 = 100, S2

0 = 100.

Payoff g(S1T , S

2T ) = (S1

T −KS2T )+, K = 1.3, T = 1.




The formulas

Following Gatheral, we get another expression for implied correl byconsidering a time-dependent ρ(t)

Models with ρt and ρ(t) give same price to the option iff∫ T

0

Eρt[(ρt − ρ(t))σ1(t, S1

t )σ2(t, S2t )S1

t S2t ∂

2S1S2Pρ(t)(t, S

1t , S

2t )]dt = 0

The integrand vanishes for each time slice t =⇒

ρ(t;T, g) =Eρt

[ρtσ1(t, S1

t )σ2(t, S2t )S1

t S2t ∂

2S1S2Pρ(t;T,g)(t, S

1t , S

2t )]

Eρt[σ1(t, S1

t )σ2(t, S2t )S1

t S2t ∂


1t , S

2t )]

When S1 and S2 have no skew,

ρ(T, g) =

∫ T0ρ(t;T, g)σ1(t)σ2(t)dt∫ T

0σ1(t)σ2(t)dt

=

∫ T0

Eρt [ρtS1t S

2t ∂

2S1S2Pρ(t;T,g)(t,S

1t ,S

2t )]

Eρt[S1t S

2t ∂

2S1S2Pρ(t;T,g)(t,S

1t ,S

2t )] σ1(t)σ2(t)dt∫ T

0σ1(t)σ2(t)dt

When σ1 and σ2 are constant, this reads (cf Gatheral for implied vol)

ρ(T, g) =1

T

∫ T

0

Eρt[ρtS

1t S

2t ∂


1t , S

2t )]

Eρt[S1t S

2t ∂


1t , S

2t )] dt




Numerical tests

Impact of admissible ρ(a,b) on option prices

Options considered:

Min of calls : g(S1T , S

2T ) = min

((S1T

K1− 1

)+

,

(S2T

K2− 1

)+

), Ki = Si0

Put on worst : g(S1T , S

2T ) =

(K −min

(S1T

S10

,S2T

S20

))+

, K = 0.95

Put on basket : g(S1T , S

2T ) =

(K −

(S1T

S10

+S2T

S20

))+

, K = 1.8

Their cross gamma at maturity is proportional to

Min of calls : δ

(S2

K2− S1

K1

)1{ S1

K1≥1}

Put on worst : −δ(S2

S20

− S1

S10

)1{

S1TS10

≤K}

Put on basket : δ

(S1

S10

+S2

S20

−K)




Numerical tests

Impact of admissible ρ(a,b) on option prices




Numerical tests

a b Min of calls Put on worst Put on basket

Standard deviation ≈ 0.020 ≈ 0.027 ≈ 0.027

Constant correlation 72% 2.59 3.47 1.88

0 1 2.65 3.49 1.91

σ21 + σ2

2 −2σ1σ2 2.53 3.37 1.990 σ1σ2 2.91 3.70 1.78

0√S1S2 2.56 3.41 1.95

0 max(S1, S2) 2.56 3.40 1.95

0 min(S1, S2) 2.56 3.41 1.95

0 (S1S2)1/4 2.61 3.45 1.93

0√

max(S1, S2) 2.61 3.45 1.93

0√

min(S1, S2) 2.61 3.45 1.93

0 1√S1S2

2.74 3.56 1.87

0 1.5 + 1{S1

S10

+S2

S20

>2

} 2.37 3.25 2.06

0 2 + 12th(

10(S1

S10

+ S2

S20− 2))

2.42 3.28 2.04




Extension to stochastic interest rates, stochastic dividend yield, andstochastic volatility (G., 2013)

Our method is robust and easily handles stoch rates, stoch div yield, andstoch vol. For example in FX:


t dt+ σ1(t, S1t )a1tS

1t dW

1t


t dt+ σ2(t, S2t )a2tS

2t dW

2t

d〈W 1,W 2〉t = ρ(t, S1t , S

2t , a

1t , a

2t , D

d0t, D

10t, D

20t) dt ≡ ρ(t,Xt) dt

Easy case: (W 1,W 2) indepdt of (W 3,W 4, . . .). First calibrate σ1 andσ2 using the particle method (cf. calibration condition in the article).Then calibrate ρ by picking a and b such that b does not vanish anda(t,X) + b(t,X)ρ(t,X) is local in cross.

If (W 1,W 2) not indepdt of (W 3,W 4, . . .), be careful! Fix all the correlsthat are needed to first calibrate σ1 and σ2. Then both ρ0 and ρ1 musthave those fixed correl values, so that calibration of ρ does not destroycalibration of σ1 and σ2.




Path-dependent volatility (G., 2013)

Our method is robust and also works for path-dep correlation models:

a(t, S1, S2, X) + b(t, S1, S2, X)ρ(t, S1, S2, X) ≡ f(t,S1

S2

)X can be any path-dep variable: running averages, moving averages,running maximums/mimimums, moving maximums/minimums, realizedcorrelations/variances over the last month (cf GARCH), etc.

What works for correl works for vol! Path-dep vol model:

dStSt

= (rt − qt) dt+ σ(t, St, Xt) dWt

Determ. rates and div yield: model calibrated to the smile iff for all t

E[σ(t, St, Xt)2|St] = σ2

Dup(t, St)

All calibrated models can be built by picking a and b such that

a(t, S,X) + b(t, S,X)σ2(t, S,X) ≡ f(S)

and using the particle method




Path-dependent volatility (G., 2013)

Calibrated model follows the McKean nonlinear SDE

dStSt

= (rt−qt) dt+

√√√√ 1

b(t, St, Xt)

(σ2Dup(t, St) + E

[ab

∣∣St]E[

1b

∣∣St] − a(t, St, Xt)

)dWt

Complete model: prices are uniquely defined. Asset and implied voldynamics richer than in the local vol model. Possibly better fit to exoticoption prices (work in progress)Extension to stochastic vol, stochastic rates, stochastic div yield is easy:

dStSt

= (rt − qt) dt+ σ(t, St, Xt)αt dWt

Model calibrated to the smile iff for all t,K

E[D0tσ2(t, St, Xt)α

2t |St = K]

E[D0t|St = K]= σ2

Dup(t,K)

−E[D0t

(rt − qt − (r0t − q0t )

)1St>K

]12K∂2

KC(t,K)+

E[D0t

(qt − q0t

)(St −K)+

]12K2∂2

KC(t,K)

where σDup is the Dupire local volatility computed using r0t and q0tJulien Guyon Bloomberg L.P.



Conclusion

Only two methods proposed in the past to calibrate a LC to smile of abasket (stock index, cross FX rate, interest rate spread...). Both may failto generate a true correl + they impose particular shape of correl matrix

We suggest a general method that produces a whole family ofadmissible local correlations. It spans all the admissible localcorrelations such that ρ ∈ (ρ0, ρ1). The two existing methods are justparticular points.

No added complexity: the particle method does the job!

The huge number of degrees of freedom (represented by two functions aand b) allows one to pick one’s favorite correl with desirable propertiesamong this family of admissible correls. It reconciles static calibration(calibration from snapshot of prices of options on basket) anddynamic calibration (calibration from historical state-dependency ofcorrelation)

Numerical tests show the wide variety of admissible correlations andgive insight on lower bounds/upper bounds on option prices given smileof basket and smiles of its constituents




Open questions

Under which condition are the 3 surfaces of implied volatility of a triangleof FX rates jointly arbitrage-free? How to detect a arbitrage?

Under which necessary and sufficient conditions on σ1, σ2 and σ12 doesthere exist an admissible local correlation?

What are the lower bound and upper bound on the price of g(S1T , S

2T ) (or

more complex payoffs) given the 3 surfaces of implied volatility of atriangle of FX rates?




Acknowledgements. I would like to thank Bruno Dupire, Sylvain Corlay,Alexey Polishchuk, Fabio Mercurio, Oleg Kovrizhkin and PierreHenry-Labordere for fruitful discussions. I am grateful to Oleg Kovrizhkin andSylvain Corlay for providing me with the data.




References

Ahdida, A. and Alfonsi, A., A mean-reverting SDE on correlation matrices,Stochastic Processes and their Applications, 123(4):1472-1520, 2013.

Avellaneda M., Boyer-Olson D., Busca J. and Friz P., ReconstructingVolatility, Risk Magazine, October, 2002.

Cont R. and Deguest R., Equity correlations implied by index options:estimation and model uncertainty analysis, SSRN, 2010.

Delanoe P., Local Correlation with Local Vol and Stochastic Vol: TowardsCorrelation Dynamics?, Presentation at Global Derivatives, April 2013.

Dupire B., A new approach for understanding the impact of volatility onoption prices, Presentation at Risk conference, October 30, 1998.

Durrleman V. and El Karoui N., Coupling Smiles, Quantitative Finance,vol. 8 (6), 573-590, 2008.

El Karoui N., Jeanblanc M. and Shreve S.E., Robustness of the Black andScholes formula, Math. Finance, 8(2):93-126, 1998.




References

Gatheral J., The volatility surface, a practitioner’s guide, Wiley, 2006.

Guyon J., Calibrating Local Correlations to Basket Smiles, Risk Magazine,February 2014.

Guyon J. and Henry-Labordere P., From spot volatilities to impliedvolatilities, Risk Magazine, June 2011.

Guyon J. and Henry-Labordere P., Being Particular About Calibration,Risk Magazine, January 2012. Longer version published in Post-CrisisQuant Finance, Risk Books, 2013.

Jourdain B. and Sbai M., Coupling Index and stocks, QuantitativeFinance, October, 2010.

Kovrizhkin O., Local Volatility + Local Correlation Multicurrency Model,Presentation at Global Derivatives, April 2012.

Langnau A., A dynamic model for correlation, Risk magazine, April, 2010.

Reghai A., Breaking correlation breaks, Risk magazine, October, 2010.Julien Guyon Bloomberg L.P.



K16480

copy to come

Mathematics

Nonlinear Option Pricing

Nonlinear Option Pricing

Nonlinear Option Pricing Julien Guyon and Pierre Henry-Labordère

Guyon and Henry-Labordère

K16480_Cover.indd 1 4/11/13 8:36 AM




The field of mathematics is very wide and it is not easy to predictwhat happens next, but I can tell you it is alive and well. Two generaltrends are obvious and will surely persist. In its pure aspect, thesubject has changed, much for the better I think, by moving to moreconcrete problems. In both its pure and applied aspects, an equallybeneficial shift to nonlinear problems can be seen. Mostmathematical questions suggested by Nature are genuinely nonlinear,meaning very roughly that the result is not proportional to the cause,but varies with it as the square or the cube, or in some morecomplicated way. The study of such questions is still, after two orthree hundred years, in its infancy. Only a few of the simplestexamples are understood in any really satisfactory way. I believe thisdirection will be a principal theme in the future.

— Henry P. McKean, Some Mathematical Coincidences (May 2003)



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