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 Mathematics Chaire Risques Financiers Paris, January 10th, 2014 [email protected]

Julien Guyon Calibrating Local Correlations to a Basket Smile

Bloomberg L.P.

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

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 EURGBP smiles Impact of correlation on option prices Extension to stochastic interest rates, stochastic dividend yield, and stochastic volatility Path-dependent volatility Open questions and conclusion

Julien Guyon Calibrating Local Correlations to a Basket Smile

Bloomberg L.P.

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

Motivation

Multi-asset Dupire local volatility models with constant correlation do not capture market skew of stock indices Stocks highly correlated in bearish markets Local correlation (LC) models incorporate correl variability in option prices and help traders risk-manage their correl positions during crises ∆P&Lt =

  N ∆Sti ∆Stj 1 X i j 2 i j St St ∂S i S j P (t, St ) − ρ ij σi (t, St )σj (t, St )∆t j 2 i,j=1 Sti St

LC models also allow to build models consistent with a triangle of market smiles of FX rates LC also used in interest rates modeling to calibrate to spread options prices

Julien Guyon Calibrating Local Correlations to a Basket Smile

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

Motivation

Only 2 methods so far to calibrate to smile of index It =

PN

i=1

αi Sti :

Local in index volatility of the basket (Langnau): N X

αi αj ρij (t, St )σi (t, Sti )σj (t, Stj )Sti Stj = f (t, It )

i,j=1

Local in index correlation matrix (Reghai, G. and Henry-Labord` ere): ρ(t, St ) = f (t, It )

Both methods may lead to correlation candidates that fail to be positive semi-definite And anyway why would one undergo either correlation structure?

Julien Guyon Calibrating Local Correlations to a Basket Smile

Bloomberg L.P.

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

Motivation

Our goal: build all the LC models calibrated to a basket smile (stock index, cross FX rate, interest rate spread) For the first time, one can design a particular calibrated model in order to match a view on a correlation skew reproduce some features of historical correl calibrate to other option prices

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

Julien Guyon Calibrating Local Correlations to a Basket Smile

Bloomberg L.P.

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

The FX triangle smile calibration problem

The FX triangle smile calibration problem LC model for a triangle of FX rates (deterministic rates): dSt1

=

(rtd − rt1 )St1 dt + σ1 (t, St1 )St1 dWt1

dSt2 dhW 1 , W 2 it

=

(rtd − rt2 )St2 dt + σ2 (t, St2 )St2 dWt2

=

ρ(t, St1 , St2 ) dt

Example: S 1 = EUR/USD, S 2 = GBP/USD and S 12 = EUR/GBP

f

EQ ρ

Model calibrated to market smile of cross rate S 12 ≡ S 1 /S 2 iff for all t 1    S S1 2 σ12 (t, St1 ) + σ22 (t, St2 ) − 2ρ(t, St1 , St2 )σ1 (t, St1 )σ2 (t, St2 ) t2 = σ12 t, t2 St St Qf = risk-neutral measure associated to the foreign currency in S 2 (GBP): Z T    dQf S2 = T2 exp rt2 − rtd dt dQ S0 0

Julien Guyon Calibrating Local Correlations to a Basket Smile

Bloomberg L.P.

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

The FX triangle smile calibration problem

The FX triangle smile calibration problem

The calibration condition EQ ρ

f



1   S S1 2 σ12 (t, St1 ) + σ22 (t, St2 ) − 2ρ(t, St1 , St2 )σ1 (t, St1 )σ2 (t, St2 ) t2 = σ12 t, t2 St St

(1)

is equivalent to
St1 i 2 2 1 2 2 1 2 1 2   EQ ρ St σ1 (t, St ) + σ2 (t, St ) − 2ρ(t, St , St )σ1 (t, St )σ2 (t, St ) S 2 St1 2 t h 1i = σ t, 12 St2 2 St EQ ρ St S 2 h

t

R∗+

R∗+

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

Julien Guyon Calibrating Local Correlations to a Basket Smile

Bloomberg L.P.

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

The FX triangle smile calibration problem

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: σ12 (t, S 1 ) + σ22 (t, S 2 ) − 2ρ(t, S 1 , S 2 )σ1 (t, S 1 )σ2 (t, S 2 ) ≡ f

 t,

S1 S2



Calibration condition reads   S1 2 σ12 (t, St1 ) + σ22 (t, St2 ) − 2ρ(t, St1 , St2 )σ1 (t, St1 )σ2 (t, St2 ) = σ12 t, t2 St =⇒ ρ=

2 σ12 + σ22 − σ12 ≡ ρ∗ 2σ1 σ2

Does ρ∗ stay in [−1, 1]?

Julien Guyon Calibrating Local Correlations to a Basket Smile

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

The FX triangle smile calibration problem

Existing model #2: Local in cross correlation (Reghai, G. and Henry-Labord`ere, 2011)  Assume that ρ is local in cross: ρ t,

S1 S2



(Reghai, G. and

Henry-Labord`ere, 2011)

f

EQ ρ

Calibration condition reads 1 1       S f S1 St1 1 2 St 2 σ12 (t, St1 ) + σ22 (t, St2 ) t2 −2ρ t, t2 EQ σ (t, S )σ (t, S ) = σ t, 1 2 ρ t t 2 12 St St St St2 =⇒ 1i   S S1 2 σ12 (t, St1 ) + σ22 (t, St2 ) St2 − σ12 t, St2 t t 1i h S 2EQf σ1 (t, St1 )σ2 (t, St2 ) St2 t h   h 1i
S 1 i S S1 2 EQ St2 σ12 (t, St1 ) + σ22 (t, St2 ) St2 − σ12 t, St2 EQ St2 St2 t t t h i S1 2EQ St2 σ1 (t, St1 )σ2 (t, St2 ) St2 f

S1 ρ t, t2 St 

EQ

 =

=

h

t

Julien Guyon Calibrating Local Correlations to a Basket Smile

Bloomberg L.P.

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

The FX triangle smile calibration problem

Existing model #2: Local in cross correlation (Reghai, G. and Henry-Labord`ere, 2011)

Calibrated model follows the McKean nonlinear SDE: dSt1

=

(rtd − rt1 )St1 dt + σ1 (t, St1 )St1 dWt1

dSt2

=

dhW 1 , W 2 it

=

(rtd − rt2 )St2 dt + σ2 (t, St2 )St2 dWt2 h   h 1i
S 1 i S S1 2 EQ St2 σ12 (t, St1 ) + σ22 (t, St2 ) St2 − σ12 t, St2 EQ St2 St2 t t t 1i h dt S 2EQ St2 σ1 (t, St1 )σ2 (t, St2 ) St2 t

=⇒ We can build ρ using the particle method Cf. G. and Henry-Labord`ere, Being Particular About Calibration, Risk magazine, January 2012; Jourdain and Sbai, Coupling Index and stocks, Quantitative Finance, October 2010.

Julien Guyon Calibrating Local Correlations to a Basket Smile

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

The FX triangle smile calibration problem

Particle method for local in cross correl (G. and Henry-Labord`ere, 2011) 1

1 2 3

Set k = 1 and ρ(t, S 1 , S 2 ) =

2 2 2 σ1 (0,S 1 )+σ2 (0,S 2 )−σ12 (0, S 2 )

2σ1

Simulate (St1,i , St2,i )1≤i≤N from tk−1 For all S 12 in a grid Gtk of cross rate Etnum (S 12 ) k

(0,S 1 )σ

2

(0,S 2 )

S

to tk using a discretization scheme values, compute 1i h non-parametric S Qf of Eρ σ12 + σ22 St2 and

Etden (S 12 ) k

kernel estimates and 1i h St Qf Eρ σ1 σ2 S 2 at date tk , define

for t ∈ [t0 , t1 ]

t

t

2 Etnum (S 12 ) − σ12 tk , S 12 ρ(tk , S ) = k den 12 2Etk (S )




12

interpolate ρ(tk , ·), e.g., using cubic splines, extrapolate, and, for all S1 S1 t ∈ [tk , tk+1 ], set ρ(t, S 1 , S 2 ) = ρ(tk , S 2 ). If ρ(tk , S 2 ) > 1 (resp. < −1), cap it at +1 (resp. floor it at −1) =⇒ imperfect calibration (may be accurate enough!) 4

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

Julien Guyon Calibrating Local Correlations to a Basket Smile

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

The equity index smile calibration problem

The equity index smile calibration problem St

=

(St1 , . . . , StN )

dSti

=

rt Sti dt + σi (t, Sti )Sti dWti ,

vρ (t, St )

≡

N X

dhW i , W j it = ρij (t, St )dt

αi αj ρij (t, St )σi (t, Sti )σj (t, Stj )Sti Stj

i,j=1

PN

Index It = i=1 αi Sti made of N weighted stocks Model calibrated to index smile if and only if I It2 σDup (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) ∈ R

If λ ∈ [0, 1], ρ is guaranteed to be a correlation matrix After dimension reduction, calibration condition reads  
I It2 σDup (t, It )2 = Eρ vρ0 (t, St ) + vρ1 (t, St ) − vρ0 (t, St ) λ(t, St ) It Julien Guyon Calibrating Local Correlations to a Basket Smile

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

The equity index smile calibration problem

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
I It2 σDup (t, It )2 = vρ0 (t, St ) + vρ1 (t, St ) − vρ0 (t, St ) λ(t, St ) =⇒ λ(t, S) =

I I 2 σDup (t, I)2 − vρ0 (t, S) ≡ λ∗ (t, S) vρ1 (t, S) − vρ0 (t, S)

Does λ∗ stay in [0, 1]?

Julien Guyon Calibrating Local Correlations to a Basket Smile

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

The equity index smile calibration problem

Existing model #2: Local in index correl for stock indices (G. and Henry-Labord`ere, 2011) Recall that, after dimension reduction, calibration condition reads 
 I It2 σDup (t, It )2 = Eρ vρ0 (t, St ) + vρ1 (t, St ) − vρ0 (t, St ) λ(t, St ) It Assume that λ is local in index: λ(t, I) =⇒ λ(t, It ) =

  I It2 σDup (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 dSti

=

λ(t, I)

=

rt Sti dt + Sti σi (t, Sti )dWti , dhW i , W j it = ρij (t, It )dt PN 2 I 2 0 I σDup (t, I) − i,j=1 αi αj ρij E[σi (t, Sti )σj (t, Stj )Sti Stj |It = I] PN j 1 0 i i j i,j=1 αi αj (ρij − ρij )E[σi (t, St )σj (t, St )St St |It = I]

Does λ stay in [0, 1]? Julien Guyon Calibrating Local Correlations to a Basket Smile

Bloomberg L.P.

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

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 by computational convenience. How do we build all admissible correls? Let ρ be admissible. We can always pick two functions a(t, S 1 , S 2 ) and b(t, S 1 , S 2 ) such that b does not vanish and a + bρ is local in cross:   S1 a(t, S 1 , S 2 ) + b(t, S 1 , S 2 )ρ(t, S 1 , S 2 ) ≡ f t, 2 S   1 S − ρ(t, S 1 , S 2 ) e.g., b ≡ 1, a(t, S 1 , S 2 ) = f t, S 2 Local in cross correl: a ≡ 0 and b ≡ 1 (Reghai, G. and Henry-Labord`ere) Local in cross volatility of the cross: a = σ12 + σ22 and b = −2σ1 σ2 σ 2 +σ 2 −σ 2 (Langnau, Kovrizhkin): ρ∗ = 1 2σ12σ2 12 2 σ12

1   S St1 Qf 2 2 t, 2 = Eρ σ1 + σ2 − 2ρσ1 σ2 t2 St St 1     f a St1 σ1 σ2 St 2 2 Qf + σ +2 = EQ σ σ σ − 2(a + bρ) t, E 1 2 ρ 1 2 ρ S2 b St2 b t 

Julien Guyon Calibrating Local Correlations to a Basket Smile

1 St S2 t Bloomberg L.P.

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

The FX smile triangle problem

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

1       S f f S1 S1 a σ1 σ2 2 σ12 t, t2 = EQ σ12 + σ22 + 2 σ1 σ2 t2 − 2 (a + bρ) t, t2 EQ ρ ρ St b St St b

=⇒ ρ(a,b)

1i  f h  St 2 2 Q 2 a 1  Eρ(a,b) σ1 + σ2 + 2 b σ1 σ2 St2 − σ12 1i h = − a f σ1 σ2 St b 2EQ ρ (a,b)

b

1 St S2 t

(3)

St2

The affine transform may seem ad hoc at first sight but actually any admissible 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” of admissible correlations

Julien Guyon Calibrating Local Correlations to a Basket Smile

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

The FX smile triangle problem

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

Calibrated model follows the McKean nonlinear SDE: dSt1

=

(rtd − rt1 )St1 dt + σ1 (t, St1 )St1 dWt1

dSt2

=

(rtd − rt2 )St2 dt + σ2 (t, St2 )St2 dWt2 h 
S 1 i 2 EQ St2 σ12 + σ22 + 2 ab σ1 σ2 St2 − σ12 t, t 1i h S 2EQ St2 σ1bσ2 St2 t !

dhW 1 , W 2 it

=

St1 St2



h 1i S EQ St2 St2 t

−a(t, St1 , St2 ) dt/b(t, St1 , St2 ) The particle method allows us to estimate the conditional expectations

Julien Guyon Calibrating Local Correlations to a Basket Smile

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

The FX smile triangle problem

Building ρ(a,b) using the particle method

1 Set k = 1 and ρ(a,b) = 2 3

  1 2 2 2 0, S 2 σ1 (0,S 1 )+σ2 (0,S 2 )−σ12 S

2σ1 (0,S 1 )σ2 (0,S 2 )

Simulate (St1,i , St2,i )1≤i≤N from tk−1 For all S 12 in a grid Gtk of cross rate

for t ∈ [t0 = 0; t1 ].

to tk using a discretization scheme. values, compute

num 12 Et (S ) k 

   1,i 2,i 1,i a(tk ,St ,St ) S PN  tk 1,i 2,i 1,i 2,i  2,i  2 2 12  k k  i=1 Stk σ1 (tk , Stk ) + σ2 (tk , Stk )+2 1,i 2,i σ1 (tk , Stk )σ2 (tk , Stk ) δtk ,N  2,i − S b(tk ,St ,St ) St k k k  = 1,i S PN  tk  2,i S δ − S 12  i=1 tk tk ,N  2,i St k

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

The FX smile triangle problem

Building ρ(a,b) using the particle method

den 12 Et (S ) k

f (tk , S

12

)

=

=

  1,i 1,i 2,i S σ (t ,S )σ2 (tk ,St ) PN  tk 2,i 1 k tk 12  k δ  tk ,N  2,i − S i=1 Stk 1,i 2,i b(tk ,St ,St ) St k k   k 1,i S PN  tk  2,i S δ − S 12  i=1 tk tk ,N  2,i St k   num (S 12 ) − σ 2 12 Et 12 tk , S k den (S 12 ) 2Et k

3 cont’d interpolate and extrapolate f (tk , ·), for instance using cubic splines, and, for all t ∈ [tk , tk+1 ], set     S1 1 1 2 ρ(a,b) (t, S 1 , S 2 ) = f t , − a(t, S , S ) k b(t, S 1 , S 2 ) S2 If ρ(a,b) (t, S 1 , S 2 ) > 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 . Julien Guyon Calibrating Local Correlations to a Basket Smile

Bloomberg L.P.

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

The FX smile triangle problem

Links between local correlations

 
S1 a + bρ(a,b) t, t2 St

=

  S1 ρ(0,1) t, t2 St

=

1i h 1 2
f ∗ 1 2 σ1 (t,St )σ2 (t,St ) St EQ S2 1 2 ρ(a,b) (a + bρ ) t, St , St b(t,St ,St ) t i h 1 )σ (t,S 2 ) S 1 f σ (t,S Q 1 2 t t t Eρ(a,b) S2 b(t,St1 ,St2 ) t 1i h
f S ∗ 1 2 1 Q Eρ(0,1) ρ t, St , St σ1 (t, St )σ2 (t, St2 ) St2 t 1i h f 1 2 St EQ σ (t, S )σ (t, S ) 1 2 ρ(0,1) t t S2 t

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 correlation function:  τρ = inf t ∈ [0, T ] | ∃S 1 , S 2 > 0, ρ(t, S 1 , S 2 ) ∈ / [−1, 1]

Julien Guyon Calibrating Local Correlations to a Basket Smile

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

The FX smile triangle problem

Links between local correlations

2 σ12



S1 t, t2 St

 =

f EQ ρ(a,b)



σ12 (t, St1 )

+

σ22 (t, St2 )

S1 2ρ(a,b) (t, St1 , St2 )σ1 (t, St1 )σ2 (t, St2 ) t2 St

−



If no skew on S 1 and S 2 : ρ

∗

(t, St1 , St2 )

1    St1 Qf 1 2 St = ρ(0,1) t, 2 = Eρ(a,b) ρ(a,b) (t, St , St ) 2 St St

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     S1 S1 ρ(0,1) t, St2 > 1 ⇐⇒ |σ1 (t) − σ2 (t)| > σ12 t, St2 t t     St1 St1 ρ(0,1) t, S 2 < −1 ⇐⇒ σ1 (t) + σ2 (t) < σ12 t, S 2 t

Julien Guyon Calibrating Local Correlations to a Basket Smile

t

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

The equity index smile calibration problem

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

=

vρ (t, St )

≡

Sti σi (t, Sti )dWti , N X

dhW i , W j it = ρij (t, St )dt

αi αj ρij (t, St )σi (t, Sti )σj (t, Stj )Sti Stj

i,j=1

After dimension reduction, calibration condition reads  
I It2 σDup (t, It )2 = Eρ vρ0 (t, St ) + vρ1 (t, St ) − vρ0 (t, St ) λ(t, St ) It We can always pick two functions a and b such that b does not vanish and a(t, St ) + b(t, St )λ(t, St ) ≡ f (t, It ) is local in index. Then  h

i a 1 2 I 2 It σDup (t, It ) = (a + bλ) (t, I)Eρ vρ1 − vρ0 It +Eρ vρ0 − vρ1 − vρ0 It b b !
  2 I 2 a 1 It σDup (t, It ) − Eρ(a,b) vρ0 − b vρ1 − vρ0 It 
 =⇒ λ(a,b) = −a 1 b Eρ vρ1 − vρ0 It (a,b)

Julien Guyon Calibrating Local Correlations to a Basket Smile

b

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The equity index smile calibration problem

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

λ(a,b)

1 = b

!
  I It2 σDup (t, It )2 − Eρ(a,b) vρ0 − ab vρ1 − vρ0 It
  −a Eρ(a,b) 1b vρ1 − vρ0 It

(4)

If a function λ(a,b) satisfies (4) and is s.t. ρ(a,b) ≡ (1 − λ(a,b) )ρ0 + λ(a,b) ρ1 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. and Henry-Labord`ere) Local in index volatility method: a = vρ0 and b = vρ1 − vρ0 (Langnau, Kovrizhkin) Particle method =⇒ λ(a,b)

Julien Guyon Calibrating Local Correlations to a Basket Smile

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Numerical examples: FX

S 1 = EURUSD, S 2 = GBPUSD, S 12 = S 1 /S 2 = EURGBP (March 2012) T =1 N = 10, 000 particles ∆t =

1 80

K(x) = (1 − x2 )2 1{|x|≤1} p 1 Bandwidth h = κ¯ σ 12 S012 max(t, tmin )N − 5 , σ ¯ 12 = 10%, tmin = 0.25 and κ ≈ 3. The constant correlation that fits ATM implied volatility of cross rate at maturity = 72%

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

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

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

a = σ12 + σ22 , b = −2σ1 σ2 (local in cross volatility of the cross)

Julien Guyon Calibrating Local Correlations to a Basket Smile

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

a = 0, b = σ1 σ2 (local in cross covariance)

Julien Guyon Calibrating Local Correlations to a Basket Smile

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

a = 0, b =

√

S1S2

Julien Guyon Calibrating Local Correlations to a Basket Smile

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

a = 0, b = min(S 1 , S 2 )

Julien Guyon Calibrating Local Correlations to a Basket Smile

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a = 0, b = max(S 1 , S 2 )

Julien Guyon Calibrating Local Correlations to a Basket Smile

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a = 0, b = (S 1 S 2 )1/4

Julien Guyon Calibrating Local Correlations to a Basket Smile

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

a = 0, b =

p min(S 1 , S 2 )

Julien Guyon Calibrating Local Correlations to a Basket Smile

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

a = 0, b =

p max(S 1 , S 2 )

Julien Guyon Calibrating Local Correlations to a Basket Smile

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a = 0, b =

√ 1 S1S2

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a = 0, b = 1.5 + 1 S1



S2 1 + S 2 >2 S0 0

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a = 0, b = 1.5 +

1 2



  1 1 + tanh 10 SS 1 +

Julien Guyon Calibrating Local Correlations to a Basket Smile

0

S2 S02

−2



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a = 0, b = 0.08 +



S1 S01

2  2 2 − 1 + SS 2 − 1

Julien Guyon Calibrating Local Correlations to a Basket Smile

0

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

a = 0, b = 0.02 +



S1 S01

2  2 2 − 1 + SS 2 − 1

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0

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

√ √ a = 3 S1S2, b = S1S2

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

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

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

The formulas

Impact of correlation on option prices Price impact formula (similar to El Karoui, Dupire for vol): Eρt [g(ST1 , ST2 )] − P0 (0, S01 , S02 ) Z T  = Eρ t (ρt − ρ0 (t, St1 , St2 ))σ1 (t, St1 )σ2 (t, St2 )St1 St2 ∂S2 1 S 2 P0 (t, St1 , St2 )dt 0

Implied correlation (cf Dupire for implied vol): hR i T Eρt 0 ρt σ1 (t, St1 )σ2 (t, St2 )St1 St2 ∂S2 1 S 2 Pρ(T,g) (t, St1 , St2 )dt hR i ρ(T, g) = T Eρt 0 σ1 (t, St1 )σ2 (t, St2 )St1 St2 ∂S2 1 S 2 Pρ(T,g) (t, St1 , St2 )dt Implied correl is the fixed point of Z T Z ∞Z ∞ ρloc (t, S 1 , S 2 )qρ (t, S 1 , S 2 )dS 1 dS 2 dt ρ 7→ 0 1

0

0

σ1 (t, S 1 )σ2 (t, S 2 )S 1 S 2 ∂S2 1 S 2 Pρ (t, S 1 , S 2 )p(t, S 1 , S 2 )

2

qρ (t, S , S ) = R T R ∞ R ∞ 0

0

0

σ1 (t∗ , S∗1 )σ2 (t∗ , S∗2 )S∗1 S∗2 ∂S2 1 S 2 Pρ (t∗ , S∗1 , S∗2 )p(t∗ , S∗1 , S∗2 )dS∗1 dS∗2 d

p → pˆ =⇒ ρ(T, g) → ρˆ(T, g) (cf G. and Henry-Labord`ere, 2011) Julien Guyon Calibrating Local Correlations to a Basket Smile

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

The formulas

Graphs of (S 1 , S 2 ) 7→ qρ (t, S 1 , S 2 ) for different values of t

Figure : Black-Scholes model: σ1 = 20%, σ2 = 20%, ρ = 0, S01 = 100, S02 = 100. Payoff g(ST1 , ST2 ) = (ST1 − KST2 )+ , K = 1.3, T = 1. Julien Guyon Calibrating Local Correlations to a Basket Smile

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The formulas

Following Gatheral, we get another expression for implied correl by considering a time-dependent ρ(t) Models with ρt and ρ(t) give same price to the option iff Z T   Eρt (ρt − ρ(t))σ1 (t, St1 )σ2 (t, St2 )St1 St2 ∂S2 1 S 2 Pρ(t) (t, St1 , St2 ) dt = 0 0

The integrand vanishes for each time slice t =⇒   Eρt ρt σ1 (t, St1 )σ2 (t, St2 )St1 St2 ∂S2 1 S 2 Pρ(t;T,g) (t, St1 , St2 )   ρ(t; T, g) = Eρt σ1 (t, St1 )σ2 (t, St2 )St1 St2 ∂S2 1 S 2 Pρ(t;T,g) (t, St1 , St2 ) When S 1 and S 2 have no skew, RT RT 0 ρ(t; T, g)σ (t)σ (t)dt 1 2 ρ(T, g) = 0 R T = σ1 (t)σ2 (t)dt 0

Eρt [ρt St1 St2 ∂ 2 1 2 Pρ(t;T ,g) (t,St1 ,St2 )] S S h i σ1 (t)σ2 (t)dt Eρt St1 St2 ∂ 2 1 2 Pρ(t;T ,g) (t,St1 ,St2 ) S S

RT 0

σ1 (t)σ2 (t)dt

When σ1 and σ2 are constant, this reads (cf Gatheral for implied vol)   Z 1 2 2 1 2 1 T Eρt ρt St St ∂S 1 S 2 Pρ(t;T,g) (t, St , St )   ρ(T, g) = dt T 0 Eρt St1 St2 ∂S2 1 S 2 Pρ(t;T,g) (t, St1 , St2 ) Julien Guyon Calibrating Local Correlations to a Basket Smile

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

Numerical tests

Impact of admissible ρ(a,b) on option prices Options considered:   2  ! ST1 ST min −1 −1 , K i = S0i , K1 K2 + +   1  ST ST2 K − min , , K = 0.95 S01 S02 +   1  S2 ST K− + T2 , K = 1.8 1 S0 S0 + 

Min of calls :

g(ST1 , ST2 )

=

Put on worst :

g(ST1 , ST2 )

=

Put on basket :

g(ST1 , ST2 )

=

Their cross gamma at maturity is proportional to  2  S S1 Min of calls : δ − 1n S1 ≥1o K2 K1 K1  2  S S1 ) − 1 1( S1 Put on worst : −δ T ≤K S02 S0 1 S0

 Put on basket Julien Guyon Calibrating Local Correlations to a Basket Smile

:

δ

1

2

S S + 2 −K S01 S0

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Numerical tests

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

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Numerical tests

a

b Standard deviation Constant correlation 72% 0 1 σ12 + σ22 −2σ1 σ2 0 σ1 σ2 √ 0 S1S2 0 max(S 1 , S 2 ) 0 min(S 1 , S 2 ) 1

2 1/4

0 0 0 0

p(S S ) 1 2 pmax(S , S ) 1 2 min(S , S )

0

1.5 +  S01 S0 2  2 + 12 th 10 S +S −2 S1 S2

0

√ 1 S1 S2 1 S1 S2  1 + 2 >2

0

Julien Guyon Calibrating Local Correlations to a Basket Smile

Min of calls ≈ 0.020 2.59 2.65 2.53 2.91

Put on worst ≈ 0.027 3.47 3.49 3.37 3.70

Put on basket ≈ 0.027 1.88 1.91 1.99 1.78

2.56 2.56 2.56

3.41 3.40 3.41

1.95 1.95 1.95

2.61 2.61 2.61 2.74

3.45 3.45 3.45 3.56

1.93 1.93 1.93 1.87

2.37

3.25

2.06

2.42

3.28

2.04

0

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

Extension to stochastic interest rates, stochastic dividend yield, and stochastic volatility (G., 2013) Our method is robust and easily handles stoch rates, stoch div yield, and stoch vol. For example in FX:

1

dSt1

=

(rtd − rt1 )St1 dt + σ1 (t, St1 )a1t St1 dWt1

dSt2

=

(rtd − rt2 )St2 dt + σ2 (t, St2 )a2t St2 dWt2

2

=

d 1 2 ρ(t, St1 , St2 , a1t , a2t , D0t , D0t , D0t ) dt ≡ ρ(t, Xt ) dt

dhW , W it

Easy case: (W 1 , W 2 ) indepdt of (W 3 , W 4 , . . .). First calibrate σ1 and σ2 using the particle method (cf. calibration condition in the article). Then calibrate ρ by picking a and b such that b does not vanish and a(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 correls that are needed to first calibrate σ1 and σ2 . Then both ρ0 and ρ1 must have those fixed correl values, so that calibration of ρ does not destroy calibration of σ1 and σ2 . Julien Guyon Calibrating Local Correlations to a Basket Smile

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

Path-dependent volatility (G., 2013) Our method is robust and also works for path-dep correlation models:   S1 a(t, S 1 , S 2 , X) + b(t, S 1 , S 2 , X)ρ(t, S 1 , S 2 , X) ≡ f t, 2 S X can be any path-dep variable: running averages, moving averages, running maximums/mimimums, moving maximums/minimums, realized correlations/variances over the last month (cf GARCH), etc. What works for correl works for vol! Path-dep vol model: dSt = (rt − qt ) dt + σ(t, St , Xt ) dWt St Determ. rates and div yield: model calibrated to the smile iff for all t 2 E[σ(t, St , Xt )2 |St ] = σDup (t, St )

All calibrated models can be built by picking a and b such that a(t, S, X) + b(t, S, X)σ 2 (t, S, X) ≡ f (S) and using the particle method Julien Guyon Calibrating Local Correlations to a Basket Smile

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

Path-dependent volatility (G., 2013) Calibrated model follows the McKean nonlinear SDE v !   u 2 u σDup (t, St ) + E ab St 1 dSt t   = (rt −qt ) dt+ − a(t, St , Xt ) dWt St b(t, St , Xt ) E 1b St Complete model: prices are uniquely defined. Asset and implied vol dynamics richer than in the local vol model. Possibly better fit to exotic option prices (work in progress) Extension to stochastic vol, stochastic rates, stochastic div yield is easy: dSt = (rt − qt ) dt + σ(t, St , Xt )αt dWt St Model calibrated to the smile iff for all t, K E[D0t σ 2 (t, St , Xt )αt2 |St = K] 2 = σDup (t, K) E[D0t |St = K] 
 
 E D0t qt − qt0 (St − K)+ E D0t rt − qt − (rt0 − qt0 ) 1St >K − + 1 1 2 2 K∂K C(t, K) K 2 ∂K C(t, K) 2 2 where σDup is the Dupire local volatility computed using rt0 and qt0 Julien Guyon Calibrating Local Correlations to a Basket Smile

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

Conclusion Only two methods proposed in the past to calibrate a LC to smile of a basket (stock index, cross FX rate, interest rate spread...). Both may fail to generate a true correl + they impose particular shape of correl matrix We suggest a general method that produces a whole family of admissible local correlations. It spans all the admissible local correlations such that ρ ∈ (ρ0 , ρ1 ). The two existing methods are just particular points. No added complexity: the particle method does the job! The huge number of degrees of freedom (represented by two functions a and b) allows one to pick one’s favorite correl with desirable properties among this family of admissible correls. It reconciles static calibration (calibration from snapshot of prices of options on basket) and dynamic calibration (calibration from historical state-dependency of correlation) Numerical tests show the wide variety of admissible correlations and give insight on lower bounds/upper bounds on option prices given smile of basket and smiles of its constituents Julien Guyon Calibrating Local Correlations to a Basket Smile

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

Open questions

Under which condition are the 3 surfaces of implied volatility of a triangle of FX rates jointly arbitrage-free? How to detect a arbitrage? Under which necessary and sufficient conditions on σ1 , σ2 and σ12 does there exist an admissible local correlation? What are the lower bound and upper bound on the price of g(ST1 , ST2 ) (or more complex payoffs) given the 3 surfaces of implied volatility of a triangle of FX rates?

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

Acknowledgements. I would like to thank Bruno Dupire, Sylvain Corlay, Alexey Polishchuk, Fabio Mercurio, Oleg Kovrizhkin and Pierre Henry-Labord`ere for fruitful discussions. I am grateful to Oleg Kovrizhkin and Sylvain Corlay for providing me with the data.

Julien Guyon Calibrating Local Correlations to a Basket Smile

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

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., Reconstructing Volatility, 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: Towards Correlation Dynamics?, Presentation at Global Derivatives, April 2013. Dupire B., A new approach for understanding the impact of volatility on option 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 and Scholes formula, Math. Finance, 8(2):93-126, 1998. Julien Guyon Calibrating Local Correlations to a Basket Smile

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

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-Labord`ere P., From spot volatilities to implied volatilities, Risk Magazine, June 2011. Guyon J. and Henry-Labord`ere P., Being Particular About Calibration, Risk Magazine, January 2012. Longer version published in Post-Crisis Quant Finance, Risk Books, 2013. Jourdain B. and Sbai M., Coupling Index and stocks, Quantitative Finance, 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 Calibrating Local Correlations to a Basket Smile

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

Mathematics

Julien Guyon and Pierre Henry-Labordère

Nonlinear Option Pricing Nonlinear Option Pricing

copy to come

Nonlinear Option Pricing

Guyon and Henry-Labordère

K16480

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4/11/13 8:36 AM

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

The field of mathematics is very wide and it is not easy to predict what happens next, but I can tell you it is alive and well. Two general trends are obvious and will surely persist. In its pure aspect, the subject has changed, much for the better I think, by moving to more concrete problems. In both its pure and applied aspects, an equally beneficial shift to nonlinear problems can be seen. Most mathematical 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 more complicated way. The study of such questions is still, after two or three hundred years, in its infancy. Only a few of the simplest examples are understood in any really satisfactory way. I believe this direction will be a principal theme in the future. — Henry P. McKean, Some Mathematical Coincidences (May 2003)

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