Asymptotics beats Monte Carlo: The case of correlated local vol baskets

Transcription

1 Approximations for local vol baskets November 28, 2013 Page 1 (32) Mohrenstrasse Berlin Germany Tel Weierstrass Institute for Applied Analysis and Stochastics Asymptotics beats Monte Carlo: The case of correlated local vol baskets Christian Bayer and Peter Laurence WIAS Berlin and Università di Roma

2 Outline 1 Introduction 2 Outline of our approach 3 Heat kernel expansions 4 Numerical examples Approximations for local vol baskets November 28, 2013 Page 2 (32)

3 Outline 1 Introduction 2 Outline of our approach 3 Heat kernel expansions 4 Numerical examples Approximations for local vol baskets November 28, 2013 Page 3 (32)

4 Methods of European option pricing u(t, S t ) = e r(t t) E [ ] f (S T ) S t Example (Example treated in this work) f (S) = ( ni=1 w i S i K ) +, at least one weight positive n large (e.g., n = 500 for SPX) PDE methods (Quasi) Monte Carlo method Fourier transform based methods Approximation formulas Approximations for local vol baskets November 28, 2013 Page 4 (32)

5 Methods of European option pricing u(t, S t ) = e r(t t) E [ ] f (S T ) S t Example (Example treated in this work) f (S) = ( ni=1 w i S i K ) +, at least one weight positive n large (e.g., n = 500 for SPX) PDE methods Pros: fast, general Cons: curse of dimensionality, path-dependence may or may not be easy to include (Quasi) Monte Carlo method Fourier transform based methods Approximation formulas Approximations for local vol baskets November 28, 2013 Page 4 (32)

6 Methods of European option pricing u(t, S t ) = e r(t t) E [ ] f (S T ) S t Example (Example treated in this work) f (S) = ( ni=1 w i S i K ) +, at least one weight positive n large (e.g., n = 500 for SPX) PDE methods (Quasi) Monte Carlo method Pros: very general, easy to adapt, no curse of dimensionality Cons: slow, quasi MC may be difficult in high dimensions Fourier transform based methods Approximation formulas Approximations for local vol baskets November 28, 2013 Page 4 (32)

7 Methods of European option pricing u(t, S t ) = e r(t t) E [ ] f (S T ) S t Example (Example treated in this work) f (S) = ( ni=1 w i S i K ) +, at least one weight positive n large (e.g., n = 500 for SPX) PDE methods (Quasi) Monte Carlo method Fourier transform based methods Pros: very fast to evaluate ( explicit formula ) Cons: only available for affine models, difficult to generalize, curse of dimensionality Approximation formulas Approximations for local vol baskets November 28, 2013 Page 4 (32)

8 Methods of European option pricing u(t, S t ) = e r(t t) E [ ] f (S T ) S t Example (Example treated in this work) f (S) = ( ni=1 w i S i K ) +, at least one weight positive n large (e.g., n = 500 for SPX) PDE methods (Quasi) Monte Carlo method Fourier transform based methods Approximation formulas Pros: very fast evaluation Cons: derived on case by case basis, therefore very restrictive Approximations for local vol baskets November 28, 2013 Page 4 (32)

9 Methods of European option pricing u(t, S t ) = e r(t t) E [ f (S T ) S t ] Example (Example treated in this work) f (S) = ( ni=1 w i S i K ) +, at least one weight positive n large (e.g., n = 500 for SPX) PDE methods (Quasi) Monte Carlo method Fourier transform based methods Approximation formulas Work horse methods: PDE methods and (in particular) (Q)MC Particular models allowing approximation formulas (e.g., SABR formula) or FFT (Heston model) very popular Approximations for local vol baskets November 28, 2013 Page 4 (32)

10 Approximation formulas based on expansions in option parameters Expansions in large/small strike or large/small maturity Example (Large strike expansion, Lee formula) K Call(S 0, T, K) = P (S T K) For K 1, this is a rare event (large deviation) Lee formula: m log(s 0 /K), β related to moment explosion T lim m ± m σ2 I (S 0, T, K) = β ± Extensions e.g., by P. Friz et al. Approximations for local vol baskets November 28, 2013 Page 5 (32)

11 Approximation formulas based on expansions in option parameters Expansions in large/small strike or large/small maturity Example (Large strike expansion, Lee formula) K Call(S 0, T, K) = P (S T K) For K 1, this is a rare event (large deviation) Lee formula: m log(s 0 /K), β related to moment explosion T lim m ± m σ2 I (S 0, T, K) = β ± Extensions e.g., by P. Friz et al. Approximations for local vol baskets November 28, 2013 Page 5 (32)

12 Approximation formulas based on expansions in option parameters Expansions in large/small strike or large/small maturity Example (Large strike expansion, Lee formula) K Call(S 0, T, K) = P (S T K) For K 1, this is a rare event (large deviation) Lee formula: m log(s 0 /K), β related to moment explosion T lim m ± m σ2 I (S 0, T, K) = β ± Extensions e.g., by P. Friz et al. Approximations for local vol baskets November 28, 2013 Page 5 (32)

13 Small noise expansion (stochastic approach) Consider the (one-dimensional) model ds t = σ(t, S t )dw t, S 0 R Expansion: S ɛ t = S 0 + ɛs 1,t ɛ2 S 2,t + o(ɛ 2 ), with S 1,t = S 2,t = 2 t 0 t Wiener chaos decomposition 0 σ(s, S 0 )dw s, x σ(s, S 0 )S 1,s dw s E [ f (S T ) ] f (S 0 ) + ɛ f (S 0 )E [ S 1,T ] ɛ2 ( f (S 0 )E [ S 2,T ] + f (S 0 )E [ S 2 1,T ]) + For non-smooth payoffs, extensions possible by Malliavin weights. Approximations for local vol baskets November 28, 2013 Page 6 (32)

14 Small noise expansion (stochastic approach) Consider the (one-dimensional) model ds ɛ t = ɛσ(t, S ɛ t )dw t, S ɛ 0 = S 0 R Expansion: S ɛ t = S 0 + ɛs 1,t ɛ2 S 2,t + o(ɛ 2 ), with S 1,t = S 2,t = 2 t 0 t Wiener chaos decomposition 0 σ(s, S 0 )dw s, x σ(s, S 0 )S 1,s dw s E [ f (S T ) ] f (S 0 ) + ɛ f (S 0 )E [ S 1,T ] ɛ2 ( f (S 0 )E [ S 2,T ] + f (S 0 )E [ S 2 1,T ]) + For non-smooth payoffs, extensions possible by Malliavin weights. Approximations for local vol baskets November 28, 2013 Page 6 (32)

15 Small noise expansion (stochastic approach) Consider the (one-dimensional) model ds ɛ t = ɛσ(t, S ɛ t )dw t, S ɛ 0 = S 0 R Expansion: S ɛ t = S 0 + ɛs 1,t ɛ2 S 2,t + o(ɛ 2 ), with S 1,t = S 2,t = 2 t 0 t Wiener chaos decomposition 0 σ(s, S 0 )dw s, x σ(s, S 0 )S 1,s dw s E [ f (S T ) ] f (S 0 ) + ɛ f (S 0 )E [ S 1,T ] ɛ2 ( f (S 0 )E [ S 2,T ] + f (S 0 )E [ S 2 1,T ]) + For non-smooth payoffs, extensions possible by Malliavin weights. Approximations for local vol baskets November 28, 2013 Page 6 (32)

16 Small noise expansion (stochastic approach) Consider the (one-dimensional) model ds ɛ t = ɛσ(t, S ɛ t )dw t, S ɛ 0 = S 0 R Expansion: S ɛ t = S 0 + ɛs 1,t ɛ2 S 2,t + o(ɛ 2 ), with S 1,t = S 2,t = 2 t 0 t Wiener chaos decomposition 0 σ(s, S 0 )dw s, x σ(s, S 0 )S 1,s dw s E [ f (S T ) ] f (S 0 ) + ɛ f (S 0 )E [ S 1,T ] ɛ2 ( f (S 0 )E [ S 2,T ] + f (S 0 )E [ S 2 1,T ]) + For non-smooth payoffs, extensions possible by Malliavin weights. Approximations for local vol baskets November 28, 2013 Page 6 (32)

17 Small noise expansion (PDE approach) Price u ɛ (t, S 0 ) solves L ɛ u = 0 with L ɛ = t ɛ2 σ 2 2 x L 0 + ɛ 2 L 2 Ansatz u ɛ = u 0 + ɛu ɛ2 u 2 + gives (regular perturbation) L 0 u 0 + ɛl 0 u 1 + ɛ 2 ( 1 2 L 0u 2 + L 2 u 0 ) + o(ɛ 2 ) = 0 Formally, we get u 0 (T, S 0 ) = f (S 0 ), L 0 u 0 = 0, L 0 u 1 = 0, u 0 (t, S 0 ) = f (S 0 ), u 1 = 0, u 2 (t, S 0 ) = L 0u 2 + L 2 u 0 = 0,... T t L 2 u 0 (s, S 0 )ds For non-smooth payoffs, a singular perturbation in the fast variable y (x K)/ɛ can be used Approximations for local vol baskets November 28, 2013 Page 7 (32)

18 Small noise expansion (PDE approach) Price u ɛ (t, S 0 ) solves L ɛ u = 0 with L ɛ = t ɛ2 σ 2 2 x L 0 + ɛ 2 L 2 Ansatz u ɛ = u 0 + ɛu ɛ2 u 2 + gives (regular perturbation) L 0 u 0 + ɛl 0 u 1 + ɛ 2 ( 1 2 L 0u 2 + L 2 u 0 ) + o(ɛ 2 ) = 0 Formally, we get u 0 (T, S 0 ) = f (S 0 ), L 0 u 0 = 0, L 0 u 1 = 0, u 0 (t, S 0 ) = f (S 0 ), u 1 = 0, u 2 (t, S 0 ) = L 0u 2 + L 2 u 0 = 0,... T t L 2 u 0 (s, S 0 )ds For non-smooth payoffs, a singular perturbation in the fast variable y (x K)/ɛ can be used Approximations for local vol baskets November 28, 2013 Page 7 (32)

19 Small noise expansion (PDE approach) Price u ɛ (t, S 0 ) solves L ɛ u = 0 with L ɛ = t ɛ2 σ 2 2 x L 0 + ɛ 2 L 2 Ansatz u ɛ = u 0 + ɛu ɛ2 u 2 + gives (regular perturbation) L 0 u 0 + ɛl 0 u 1 + ɛ 2 ( 1 2 L 0u 2 + L 2 u 0 ) + o(ɛ 2 ) = 0 Formally, we get u 0 (T, S 0 ) = f (S 0 ), L 0 u 0 = 0, L 0 u 1 = 0, u 0 (t, S 0 ) = f (S 0 ), u 1 = 0, u 2 (t, S 0 ) = L 0u 2 + L 2 u 0 = 0,... T t L 2 u 0 (s, S 0 )ds For non-smooth payoffs, a singular perturbation in the fast variable y (x K)/ɛ can be used Approximations for local vol baskets November 28, 2013 Page 7 (32)

20 Small noise expansion (PDE approach) Price u ɛ (t, S 0 ) solves L ɛ u = 0 with L ɛ = t ɛ2 σ 2 2 x L 0 + ɛ 2 L 2 Ansatz u ɛ = u 0 + ɛu ɛ2 u 2 + gives (regular perturbation) L 0 u 0 + ɛl 0 u 1 + ɛ 2 ( 1 2 L 0u 2 + L 2 u 0 ) + o(ɛ 2 ) = 0 Formally, we get u 0 (T, S 0 ) = f (S 0 ), L 0 u 0 = 0, L 0 u 1 = 0, u 0 (t, S 0 ) = f (S 0 ), u 1 = 0, u 2 (t, S 0 ) = L 0u 2 + L 2 u 0 = 0,... T t L 2 u 0 (s, S 0 )ds For non-smooth payoffs, a singular perturbation in the fast variable y (x K)/ɛ can be used Approximations for local vol baskets November 28, 2013 Page 7 (32)

21 Outline 1 Introduction 2 Outline of our approach 3 Heat kernel expansions 4 Numerical examples Approximations for local vol baskets November 28, 2013 Page 8 (32)

22 Setting Local volatility model for forward prices df i (t) = σ i (F i (t))dw i (t), i = 1,..., n, dwi (t), dw j (t) = ρ i j dt ( Generalized spread option with payoff ni=1 w i F i K ) +, at least one w i positive Goal: fast and accurate approximation formulas, even for high n n = 100 or n = 500 not uncommon (index options) Example Black-Scholes model: σ i (F i ) = σ i F i CEV model: σ i (F i ) = σ i F β i i Approximations for local vol baskets November 28, 2013 Page 9 (32)

23 Setting Local volatility model for forward prices df i (t) = σ i (F i (t))dw i (t), i = 1,..., n, dwi (t), dw j (t) = ρ i j dt ( Generalized spread option with payoff ni=1 w i F i K ) +, at least one w i positive Goal: fast and accurate approximation formulas, even for high n n = 100 or n = 500 not uncommon (index options) Example Black-Scholes model: σ i (F i ) = σ i F i CEV model: σ i (F i ) = σ i F β i i Approximations for local vol baskets November 28, 2013 Page 9 (32)

24 Setting Local volatility model for forward prices df i (t) = σ i (F i (t))dw i (t), i = 1,..., n, dwi (t), dw j (t) = ρ i j dt ( Generalized spread option with payoff ni=1 w i F i K ) +, at least one w i positive Goal: fast and accurate approximation formulas, even for high n n = 100 or n = 500 not uncommon (index options) Example Black-Scholes model: σ i (F i ) = σ i F i CEV model: σ i (F i ) = σ i F β i i Approximations for local vol baskets November 28, 2013 Page 9 (32)

25 Basket Carr-Jarrow formula Consider the basket (index) n i=1 w i F i : d n w i F i (t) = i=1 n w i σ i (F i (t))dw i (t) i=1 Ito s formula formally implies that Let p(f 0, F, t) P (F(t) df F(0) = F 0 ) and H n 1 be the Hausdorff measure on E(K), then we have the Carr-Jarrow formula n + C(F 0, K, T) = w i F i (0) K T 0 1 w i=1 E(K) i, j=1 n w i w j σ i (F i )σ j (F j )ρ i j p(f 0, F, u)h n 1 (df)du. Approximations for local vol baskets November 28, 2013 Page 10 (32)

26 Basket Carr-Jarrow formula Consider the basket (index) n i=1 w i F i : Ito s formula formally implies that n + n w i F i (t) K = w i F i (0) K + i=1 i=1 n T w i 1 w i Fi(u)>KdF i (u) i=1 0 T δ w i F i (u)=kσ 2 N,B (F(u))du Let p(f 0, F, t) P (F(t) df F(0) = F 0 ) and H n 1 be the Hausdorff measure on E(K), then we have the Carr-Jarrow formula n + C(F 0, K, T) = w i F i (0) K T 1 w i=1 n w i w j σ i (F i )σ j (F j )ρ i j p(f 0, F, u)h n 1 (df)du. 0 E(K) i, j=1 Approximations for local vol baskets November 28, 2013 Page 10 (32)

27 Basket Carr-Jarrow formula Consider the basket (index) n i=1 w i F i : Ito s formula formally implies (with E(K) = {F w i F i = K}) that n + C(F(0), K, T) = w i F i (0) K + 1 T E [ σ 2 N,B 2 (F(u))δ E(K)(F(u)) ] du i=1 Let p(f 0, F, t) P (F(t) df F(0) = F 0 ) and H n 1 be the Hausdorff measure on E(K), then we have the Carr-Jarrow formula n + C(F 0, K, T) = w i F i (0) K T 0 1 w i=1 E(K) i, j=1 0 n w i w j σ i (F i )σ j (F j )ρ i j p(f 0, F, u)h n 1 (df)du. Approximations for local vol baskets November 28, 2013 Page 10 (32)

28 Basket Carr-Jarrow formula Consider the basket (index) n i=1 w i F i : Ito s formula formally implies (with E(K) = {F w i F i = K}) that n + C(F(0), K, T) = w i F i (0) K + 1 T E [ σ 2 N,B 2 (F(u))δ E(K)(F(u)) ] du i=1 Let p(f 0, F, t) P (F(t) df F(0) = F 0 ) and H n 1 be the Hausdorff measure on E(K), then we have the Carr-Jarrow formula n + C(F 0, K, T) = w i F i (0) K T 0 1 w i=1 E(K) i, j=1 0 n w i w j σ i (F i )σ j (F j )ρ i j p(f 0, F, u)h n 1 (df)du. Approximations for local vol baskets November 28, 2013 Page 10 (32)

29 Approximations Heat kernel expansion (to be discussed in detail later): σ 2 N,B (F)p(F 1 0, F, t) exp ( d(f 0, F) 2 ) C(F (2πt) n/2 0, F) 2t By change of variables F n = 1 w n ( K n 1 i=1 w if i ) on EK : H n 1 (df) = w w n df 1 df n 1 Laplace approximation: with F = argmin F EK d(f 0, F) and G K = {(F 1,..., F n 1 ) n 1 i=1 w if i < K} d(f 0,F) 2 2t C(F 0,F) df 1 df n 1 e d(f 0,F ) 2 2t C(F 0,F ) G K e = t n 1 2 e d(f 0,F ) 2 2t C(F 0,F ) We rely on the principle of not feeling the boundary. R n 1 e n 1 (2π) 2 det Q z T Qz 2t dz Approximations for local vol baskets November 28, 2013 Page 11 (32)

30 Approximations Heat kernel expansion (to be discussed in detail later): σ 2 N,B (F)p(F 1 0, F, t) exp ( d(f 0, F) 2 ) C(F (2πt) n/2 0, F) 2t By change of variables F n = 1 w n ( K n 1 i=1 w if i ) on EK : H n 1 (df) = w w n df 1 df n 1 Laplace approximation: with F = argmin F EK d(f 0, F) and G K = {(F 1,..., F n 1 ) n 1 i=1 w if i < K} d(f 0,F) 2 2t C(F 0,F) df 1 df n 1 e d(f 0,F ) 2 2t C(F 0,F ) G K e = t n 1 2 e d(f 0,F ) 2 2t C(F 0,F ) We rely on the principle of not feeling the boundary. R n 1 e n 1 (2π) 2 det Q z T Qz 2t dz Approximations for local vol baskets November 28, 2013 Page 11 (32)

31 Approximations Heat kernel expansion (to be discussed in detail later): σ 2 N,B (F)p(F 1 0, F, t) exp ( d(f 0, F) 2 ) C(F (2πt) n/2 0, F) 2t By change of variables F n = 1 w n ( K n 1 i=1 w if i ) on EK : H n 1 (df) = w w n df 1 df n 1 Laplace approximation: with F = argmin F EK d(f 0, F) and G K = {(F 1,..., F n 1 ) n 1 i=1 w if i < K} d(f 0,F) 2 2t C(F 0,F) df 1 df n 1 e d(f 0,F ) 2 2t C(F 0,F ) G K e = t n 1 2 e d(f 0,F ) 2 2t C(F 0,F ) We rely on the principle of not feeling the boundary. R n 1 e n 1 (2π) 2 det Q z T Qz 2t dz Approximations for local vol baskets November 28, 2013 Page 11 (32)

32 Approximations Heat kernel expansion (to be discussed in detail later): σ 2 N,B (F)p(F 1 0, F, t) exp ( d(f 0, F) 2 ) C(F (2πt) n/2 0, F) 2t By change of variables F n = 1 w n ( K n 1 i=1 w if i ) on EK : H n 1 (df) = w w n df 1 df n 1 Laplace approximation: with F = argmin F EK d(f 0, F) and G K = {(F 1,..., F n 1 ) n 1 i=1 w if i < K} d(f 0,F) 2 2t C(F 0,F) df 1 df n 1 e d(f 0,F ) 2 2t C(F 0,F ) G K e = t n 1 2 e d(f 0,F ) 2 2t C(F 0,F ) We rely on the principle of not feeling the boundary. R n 1 e n 1 (2π) 2 det Q z T Qz 2t dz Approximations for local vol baskets November 28, 2013 Page 11 (32)

33 Matching to implied volatilities Theorem n C B (F 0, K, T) = w i F i (0) K i= π w n d(f 0, F ) 2 det Q e C(F 0,F d(f ) 0,F ) 2T T 3/2 +o(t 3/2 ), as T 0. Bachelier implied vol (with F 0 = n i=1 w i F 0,i ): F 0 K σ B σ B,0 + Tσ B,1 with σ B,0 = d(f 0, F ) F0, σ B,1 = Black-Scholes implied voila: log ( F 0 /K ) σ BS σ BS,0 + Tσ BS,1 with σ BS,0 = d(f 0, F, σ BS,1 = ) + Approximations for local vol baskets November 28, 2013 Page 12 (32)

34 Matching to implied volatilities Theorem n C B (F 0, K, T) = w i F i (0) K i= π w n d(f 0, F ) 2 det Q e C(F 0,F d(f ) 0,F ) 2T T 3/2 +o(t 3/2 ), as T 0. Bachelier implied vol (with F 0 = n i=1 w i F 0,i ): F 0 K σ B σ B,0 + Tσ B,1 with σ B,0 = d(f 0, F ) F0, σ B,1 = Black-Scholes implied voila: log ( F 0 /K ) σ BS σ BS,0 + Tσ BS,1 with σ BS,0 = d(f 0, F, σ BS,1 = ) + Approximations for local vol baskets November 28, 2013 Page 12 (32)

35 Matching to implied volatilities Theorem n C B (F 0, K, T) = w i F i (0) K i= π w n d(f 0, F ) 2 det Q e C(F 0,F d(f ) 0,F ) 2T T 3/2 +o(t 3/2 ), as T 0. Bachelier implied vol (with F 0 = n i=1 w i F 0,i ): F 0 K σ B σ B,0 + Tσ B,1 with σ B,0 = d(f 0, F ) F0, σ B,1 = Black-Scholes implied voila: log ( F 0 /K ) σ BS σ BS,0 + Tσ BS,1 with σ BS,0 = d(f 0, F, σ BS,1 = ) + Approximations for local vol baskets November 28, 2013 Page 12 (32)

36 Greeks Goal: sensitivity w. r. t. model parameter κ of the option price C B (F 0, K, T) C BS (F 0, K, σ BS, T) Sensitivity: κ C }{{} BS (F 0, K, σ BS, T) + ν BS (F }{{} 0, K, σ BS, T) κ σ BS BS greek BS vega Recall that σ BS,0, σ BS,1 explicit up to F By the minimizing property: Fi d 2 (F 0, F K (G)) G=G = 0 Differentiating with respect to κ gives κ Fi d 2 (F 0, F K (G)) n 1 G + Fl Fi d 2 (F 0, F K (G)) G κ Fl = 0 l=1 Up to the above system of linear equations for κ F, there are explicit expression for the sensitivities of the approximate option prices. Approximations for local vol baskets November 28, 2013 Page 13 (32)

37 Greeks Goal: sensitivity w. r. t. model parameter κ of the option price C B (F 0, K, T) C BS (F 0, K, σ BS, T) Sensitivity: κ C BS (F 0, K, σ BS, T) + ν BS (F 0, K, σ BS, T) κ σ BS Recall that σ BS,0, σ BS,1 explicit up to F By the minimizing property: Fi d 2 (F 0, F K (G)) G=G = 0 Differentiating with respect to κ gives κ Fi d 2 (F 0, F K (G)) n 1 G + Fl Fi d 2 (F 0, F K (G)) G κ Fl = 0 l=1 Up to the above system of linear equations for κ F, there are explicit expression for the sensitivities of the approximate option prices. Approximations for local vol baskets November 28, 2013 Page 13 (32)

38 Outline 1 Introduction 2 Outline of our approach 3 Heat kernel expansions 4 Numerical examples Approximations for local vol baskets November 28, 2013 Page 14 (32)

39 Heat kernels and geometry dx t = b(x t )dt + σ(x t )dw t, L = 1 2 ai, j 2 x i x j + bi x i, a = σt σ Heat kernel: fundamental solution p(x, y, t) of t u = Lu Transition density of X t "Can you hear the shape of the drum?"(kac 66) Take L = on a domain D and relate: Geometrical properties of the domain D Partition function Z = k N e γ kt Heat kernel E.g. γ k C(n)(k/ vol D) 2/n (Weyl, 46) E.g. (for n = 2): Z = area 4πt circ. 4πt + O(1) (McKean & Singer, 67) Approximations for local vol baskets November 28, 2013 Page 15 (32)

40 Heat kernels and geometry dx t = b(x t )dt + σ(x t )dw t, L = 1 2 ai, j 2 x i x j + bi x i, a = σt σ Heat kernel: fundamental solution p(x, y, t) of t u = Lu Transition density of X t "Can you hear the shape of the drum?"(kac 66) Take L = on a domain D and relate: Geometrical properties of the domain D Partition function Z = k N e γ kt Heat kernel E.g. γ k C(n)(k/ vol D) 2/n (Weyl, 46) E.g. (for n = 2): Z = area 4πt circ. 4πt + O(1) (McKean & Singer, 67) Approximations for local vol baskets November 28, 2013 Page 15 (32)

41 Heat kernels and geometry dx t = b(x t )dt + σ(x t )dw t, L = 1 2 ai, j 2 x i x j + bi x i, a = σt σ Heat kernel: fundamental solution p(x, y, t) of t u = Lu Transition density of X t "Can you hear the shape of the drum?"(kac 66) Take L = on a domain D and relate: Geometrical properties of the domain D Partition function Z = k N e γ kt Heat kernel E.g. γ k C(n)(k/ vol D) 2/n (Weyl, 46) E.g. (for n = 2): Z = area 4πt circ. 4πt + O(1) (McKean & Singer, 67) Approximations for local vol baskets November 28, 2013 Page 15 (32)

42 The Riemannian metric associated to a diffusion dx t = b(x t )dt + σ(x t )dw t, L = 1 2 ai j 2 x i x j + bi x i, a = σt σ On R n (or a submanifold), introduce g i j a i j, Riemannian metric tensor (g i j (x)) n i, j=1 ( (g i j (x)) n i, j=1 Geodesic distance: d(x, y) 1 inf z(0)=x, z(1)=y 0 ) 1 g i j (z(t))ż i (t)ż j (t)dt inf attained by a smooth curve, the geodesic Laplace-Beltrami operator: g = ( det(g i j ) ) 2 1 ( det(gi x i j ) ) 1 2 g i j x j L = 1 2 ai j 2 x i x j + bi x i = 1 2 g + h i x i Approximations for local vol baskets November 28, 2013 Page 16 (32)

43 The Riemannian metric associated to a diffusion dx t = b(x t )dt + σ(x t )dw t, L = 1 2 ai j 2 x i x j + bi x i, a = σt σ On R n (or a submanifold), introduce g i j a i j, Riemannian metric tensor (g i j (x)) n i, j=1 ( (g i j (x)) n i, j=1 Geodesic distance: d(x, y) 1 inf z(0)=x, z(1)=y 0 ) 1 g i j (z(t))ż i (t)ż j (t)dt inf attained by a smooth curve, the geodesic Laplace-Beltrami operator: g = ( det(g i j ) ) 1 ( 2 det(gi x i j ) ) 1 2 g i j x j L = 1 2 ai j 2 x i x j + bi x i = 1 2 g + h i x i Approximations for local vol baskets November 28, 2013 Page 16 (32)

44 The Riemannian metric associated to a diffusion dx t = b(x t )dt + σ(x t )dw t, L = 1 2 ai j 2 x i x j + bi x i, a = σt σ On R n (or a submanifold), introduce g i j a i j, Riemannian metric tensor (g i j (x)) n i, j=1 ( (g i j (x)) n i, j=1 Geodesic distance: d(x, y) 1 inf z(0)=x, z(1)=y 0 ) 1 g i j (z(t))ż i (t)ż j (t)dt inf attained by a smooth curve, the geodesic Laplace-Beltrami operator: g = ( det(g i j ) ) 1 ( 2 det(gi x i j ) ) 1 2 g i j x j L = 1 2 ai j 2 x i x j + bi x i = 1 2 g + h i x i Approximations for local vol baskets November 28, 2013 Page 16 (32)

45 The Riemannian metric associated to a diffusion dx t = b(x t )dt + σ(x t )dw t, L = 1 2 ai j 2 x i x j + bi x i, a = σt σ On R n (or a submanifold), introduce g i j a i j, Riemannian metric tensor (g i j (x)) n i, j=1 ( (g i j (x)) n i, j=1 Geodesic distance: d(x, y) 1 inf z(0)=x, z(1)=y 0 ) 1 g i j (z(t))ż i (t)ż j (t)dt inf attained by a smooth curve, the geodesic Laplace-Beltrami operator: g = ( det(g i j ) ) 1 ( 2 det(gi x i j ) ) 1 2 g i j x j L = 1 2 ai j 2 x i x j + bi x i = 1 2 g + h i x i Approximations for local vol baskets November 28, 2013 Page 16 (32)

46 Heat kernel expansion p N (x 0, x, T) = det(g(x) i j )U N (x 0, x, T) e d2 (x 0, x) 2T (2πT) n 2 U N (x 0, x, T) = N k=0 u k (x 0, x)t k, the heat kernel coefficients u 0 (x 0, x) = (x 0, x)e z h(z(t)), ż(t) gdt is the Van Vleck-DeWitt determinant: 1 (x 0, x) = det ( 1 2 d 2 det(g(x0 ) i j ) det(g(x) i j ) 2 x 0 x). e z h(z(t)), ż(t) gdt is the exponential of the work done by the vector field h along the geodesic [ z joining x 0 to x with h i = b i 1 det(gi x j j )g i j] 2 det(g i j ) Approximations for local vol baskets November 28, 2013 Page 17 (32)

47 Heat kernel expansion p N (x 0, x, T) = det(g(x) i j )U N (x 0, x, T) e d2 (x 0, x) 2T (2πT) n 2 U N (x 0, x, T) = N k=0 u k (x 0, x)t k, the heat kernel coefficients u 0 (x 0, x) = (x 0, x)e z h(z(t)), ż(t) gdt is the Van Vleck-DeWitt determinant: 1 (x 0, x) = det ( 1 2 d 2 det(g(x0 ) i j ) det(g(x) i j ) 2 x 0 x). e z h(z(t)), ż(t) gdt is the exponential of the work done by the vector field h along the geodesic [ z joining x 0 to x with h i = b i 1 det(gi x j j )g i j] 2 det(g i j ) Approximations for local vol baskets November 28, 2013 Page 17 (32)

48 Heat kernel expansion 2 Assumption The cut-locus of any point is empty, i.e., any two points are connected by a unique minimizing geodesic. Theorem (Varadhan 67) b = 0, σ uniformly Hölder continuous, system uniformly elliptic, then lim T 0 T log p(x, y, T) = 1 2 d(x, y)2. Theorem (Yosida 53) On a compact Riemannian manifold, assume smooth vector fields and an ellipticity property. Then p(x, y, T) p N (x, y, T) = O(T N ) as T 0. Theorem (Azencott 84) For a locally elliptic system in an open set U R n, x, y U s. t. d(x, y) < d(x, U) + d(y, U), we have p(x, y, T) p N (x, y, T) = O(T N ) as T 0. Approximations for local vol baskets November 28, 2013 Page 18 (32)

49 Heat kernel expansion 2 Assumption The cut-locus of any point is empty. Theorem (Varadhan 67) b = 0, σ uniformly Hölder continuous, system uniformly elliptic, then lim T 0 T log p(x, y, T) = 1 2 d(x, y)2. Theorem (Yosida 53) On a compact Riemannian manifold, assume smooth vector fields and an ellipticity property. Then p(x, y, T) p N (x, y, T) = O(T N ) as T 0. Theorem (Azencott 84) For a locally elliptic system in an open set U R n, x, y U s. t. d(x, y) < d(x, U) + d(y, U), we have p(x, y, T) p N (x, y, T) = O(T N ) as T 0. Approximations for local vol baskets November 28, 2013 Page 18 (32)

50 Heat kernel expansion 2 Assumption The cut-locus of any point is empty. Theorem (Varadhan 67) b = 0, σ uniformly Hölder continuous, system uniformly elliptic, then lim T 0 T log p(x, y, T) = 1 2 d(x, y)2. Theorem (Yosida 53) On a compact Riemannian manifold, assume smooth vector fields and an ellipticity property. Then p(x, y, T) p N (x, y, T) = O(T N ) as T 0. Theorem (Azencott 84) For a locally elliptic system in an open set U R n, x, y U s. t. d(x, y) < d(x, U) + d(y, U), we have p(x, y, T) p N (x, y, T) = O(T N ) as T 0. Approximations for local vol baskets November 28, 2013 Page 18 (32)

51 Heat kernel expansion 2 Assumption The cut-locus of any point is empty. Theorem (Varadhan 67) b = 0, σ uniformly Hölder continuous, system uniformly elliptic, then lim T 0 T log p(x, y, T) = 1 2 d(x, y)2. Theorem (Yosida 53) On a compact Riemannian manifold, assume smooth vector fields and an ellipticity property. Then p(x, y, T) p N (x, y, T) = O(T N ) as T 0. Theorem (Azencott 84) For a locally elliptic system in an open set U R n, x, y U s. t. d(x, y) < d(x, U) + d(y, U), we have p(x, y, T) p N (x, y, T) = O(T N ) as T 0. Approximations for local vol baskets November 28, 2013 Page 18 (32)

52 The local vol case Domain R n +, df i(t) = σ i (F i (t))dw i (t), L = 1 2 ρ i jσ i (x i )σ j (x j ) 2 x i x j i = 1,..., n Let A R n n be such that AρA T = I n. Change variables F y x according to y i = Fi 0 du σ i (u), i = 1,..., n, x = Ay, L x 2 i 1 2 A ikσ k (F k) x i Isomorphic (up to boundary) to Euclidean geometry: Geodesics known in closed form d(f 0, F) = x 0 x CEV case: σ i (F i ) = σ i F β i i, zeroth and first order heat kernel coefficients given explicitly Approximations for local vol baskets November 28, 2013 Page 19 (32)

53 The local vol case Domain R n +, df i(t) = σ i (F i (t))dw i (t), L = 1 2 ρ i jσ i (x i )σ j (x j ) 2 x i x j i = 1,..., n Let A R n n be such that AρA T = I n. Change variables F y x according to y i = Fi 0 du σ i (u), i = 1,..., n, x = Ay, L x 2 i 1 2 A ikσ k (F k) x i Isomorphic (up to boundary) to Euclidean geometry: Geodesics known in closed form d(f 0, F) = x 0 x CEV case: σ i (F i ) = σ i F β i i, zeroth and first order heat kernel coefficients given explicitly Approximations for local vol baskets November 28, 2013 Page 19 (32)

54 The local vol case Domain R n +, df i(t) = σ i (F i (t))dw i (t), L = 1 2 ρ i jσ i (x i )σ j (x j ) 2 x i x j i = 1,..., n Let A R n n be such that AρA T = I n. Change variables F y x according to y i = Fi 0 du σ i (u), i = 1,..., n, x = Ay, L x 2 i 1 2 A ikσ k (F k) x i Isomorphic (up to boundary) to Euclidean geometry: Geodesics known in closed form d(f 0, F) = x 0 x CEV case: σ i (F i ) = σ i F β i i, zeroth and first order heat kernel coefficients given explicitly Approximations for local vol baskets November 28, 2013 Page 19 (32)

55 The local vol case Domain R n +, df i(t) = σ i (F i (t))dw i (t), L = 1 2 ρ i jσ i (x i )σ j (x j ) 2 x i x j i = 1,..., n Let A R n n be such that AρA T = I n. Change variables F y x according to y i = Fi 0 du σ i (u), i = 1,..., n, x = Ay, L x 2 i 1 2 A ikσ k (F k) x i Isomorphic (up to boundary) to Euclidean geometry: Geodesics known in closed form d(f 0, F) = x 0 x CEV case: σ i (F i ) = σ i F β i i, zeroth and first order heat kernel coefficients given explicitly Approximations for local vol baskets November 28, 2013 Page 19 (32)

56 Outline 1 Introduction 2 Outline of our approach 3 Heat kernel expansions 4 Numerical examples Approximations for local vol baskets November 28, 2013 Page 20 (32)

57 Implementation Optimization problem for F is non-linear with a linear constraint With q i F i du F 0,i σ i (u), it is a quadratic optimization problem with non-linear constraint Fast convergence of Newton iteration Given F, C(F 0, F ) is a line integral along the geodesic; this integral can be calculated in closed form in the CEV model. Formulas can be evaluated in less than 2 seconds for n = 100 Our work relies on the principle of not feeling the boundary. Approximations for local vol baskets November 28, 2013 Page 21 (32)

58 Implementation Optimization problem for F is non-linear with a linear constraint With q i F i du F 0,i σ i (u), it is a quadratic optimization problem with non-linear constraint Fast convergence of Newton iteration Given F, C(F 0, F ) is a line integral along the geodesic; this integral can be calculated in closed form in the CEV model. Formulas can be evaluated in less than 2 seconds for n = 100 Our work relies on the principle of not feeling the boundary. Approximations for local vol baskets November 28, 2013 Page 21 (32)

59 Implementation Optimization problem for F is non-linear with a linear constraint With q i F i du F 0,i σ i (u), it is a quadratic optimization problem with non-linear constraint Fast convergence of Newton iteration Given F, C(F 0, F ) is a line integral along the geodesic; this integral can be calculated in closed form in the CEV model. Formulas can be evaluated in less than 2 seconds for n = 100 Our work relies on the principle of not feeling the boundary. Approximations for local vol baskets November 28, 2013 Page 21 (32)

60 Implementation Optimization problem for F is non-linear with a linear constraint With q i F i du F 0,i σ i (u), it is a quadratic optimization problem with non-linear constraint Fast convergence of Newton iteration Given F, C(F 0, F ) is a line integral along the geodesic; this integral can be calculated in closed form in the CEV model. Formulas can be evaluated in less than 2 seconds for n = 100 Our work relies on the principle of not feeling the boundary. Approximations for local vol baskets November 28, 2013 Page 21 (32)

61 The initial guess in the Newton iteration Change of variable: q i = F1 β i i Λ 1 = (σ i σ j ρ i j ) n i, j=1 F 1 β i 0,i 1 β i, F i = ( F 1 β i 0,i + (1 β i )q i ) 1/(1 βi ) Optimization problem: min q T Λq : n i=1 w i F i (q i ) = K n ( Linearized constraint: w i F0,i + F β i 0,i q i) = K i=1 Minimizer q 0 = K F 0 F T 0 Λ 1 F 0 Λ 1 F 0 with Lagrange multiplier q 0 λ = 2 K F 0 F T 0 Λ 1 F 0, where F 0,i = w i F 0,i not good enough (unless coupled with 1/2-slope rule ) Use as initial guess in Newton iteration Approximations for local vol baskets November 28, 2013 Page 22 (32)

62 The initial guess in the Newton iteration Change of variable: q i = F1 β i i Λ 1 = (σ i σ j ρ i j ) n i, j=1 F 1 β i 0,i 1 β i, F i = ( F 1 β i 0,i + (1 β i )q i ) 1/(1 βi ) Optimization problem: min q T Λq : n i=1 w i F i (q i ) = K n ( Linearized constraint: w i F0,i + F β i 0,i q i) = K i=1 Minimizer q 0 = K F 0 F T 0 Λ 1 F 0 Λ 1 F 0 with Lagrange multiplier q 0 λ = 2 K F 0 F T 0 Λ 1 F 0, where F 0,i = w i F 0,i not good enough (unless coupled with 1/2-slope rule ) Use as initial guess in Newton iteration Approximations for local vol baskets November 28, 2013 Page 22 (32)

63 The initial guess in the Newton iteration Change of variable: q i = F1 β i i Λ 1 = (σ i σ j ρ i j ) n i, j=1 F 1 β i 0,i 1 β i, F i = ( F 1 β i 0,i + (1 β i )q i ) 1/(1 βi ) Optimization problem: min q T Λq : n i=1 w i F i (q i ) = K n ( Linearized constraint: w i F0,i + F β i 0,i q i) = K i=1 Minimizer q 0 = K F 0 F T 0 Λ 1 F 0 Λ 1 F 0 with Lagrange multiplier q 0 λ = 2 K F 0 F T 0 Λ 1 F 0, where F 0,i = w i F 0,i not good enough (unless coupled with 1/2-slope rule ) Use as initial guess in Newton iteration Approximations for local vol baskets November 28, 2013 Page 22 (32)

64 Numerical examples CEV model framework For CEV, the formulas are fully explicit apart from the minimizing configuration F We observe very fast convergence of the iteration, but the initial guess is crucial. Reference values obtained using: Ninomiya Victoir discretization Quasi Monte Carlo based on Sobol numbers, Monte Carlo for very high dimensions (n 100) Variance (dimension) reduction using Mean value Monte Carlo based on one-dimensional Black-Scholes prices Approximations for local vol baskets November 28, 2013 Page 23 (32)

65 Numerical examples CEV model framework For CEV, the formulas are fully explicit apart from the minimizing configuration F We observe very fast convergence of the iteration, but the initial guess is crucial. Reference values obtained using: Ninomiya Victoir discretization Quasi Monte Carlo based on Sobol numbers, Monte Carlo for very high dimensions (n 100) Variance (dimension) reduction using Mean value Monte Carlo based on one-dimensional Black-Scholes prices Approximations for local vol baskets November 28, 2013 Page 23 (32)

66 CEV index implied vol three-dimensional visualization Implied vol Basket St. 1 (F 0 = 10, β = 0.3, σ = 0.9) St. 2 (F 0 = 11, β = 0.2, σ = 0.7) St. 3 (F 0 = 17, β = 0.2, σ = 0.9) log(k F 0 ) Approximations for local vol baskets November 28, 2013 Page 24 (32)

67 CEV index implied vol three-dimensional visualization Optimal component strike St. 1 (F 0 = 10, β = 0.3, σ = 0.9) St. 2 (F 0 = 11, β = 0.2, σ = 0.7) St. 3 (F 0 = 17, β = 0.2, σ = 0.9) Basket strike Approximations for local vol baskets November 28, 2013 Page 24 (32)

68 Spread option in dimension 10 Recall: df i (t) = σ i F i (t) β i dw i (t) β = (0.7, 0.2, 0.8, 0.3, 0.5, 0.5, 0.6, 0.6, 0.3, 0.3) σ = (0.8, 0.6, 0.9, 0.6, 0.8, 0.4, 0.9, 0.9, 0.3, 0.8) F 0 = (10, 13, 11, 18, 9, 10, 17, 16, 13, 17) w = ( 1, 1, 1, 1, 1, 1, 1, 1, 1, 1) Approximations for local vol baskets November 28, 2013 Page 25 (32)

69 Spread option in dimension 10 T K = 32.9 K = 33.8 K = 34.1 K = 34.4 K = Table : Quasi Monte Carlo prices. T K = 32.9 K = 33.8 K = 34.1 K = 34.4 K = Table : Zero order asymptotic prices. Approximations for local vol baskets November 28, 2013 Page 25 (32)

70 Spread option in dimension 10 T K = 32.9 K = 33.8 K = 34.1 K = 34.4 K = Table : Quasi Monte Carlo prices. T K = 32.9 K = 33.8 K = 34.1 K = 34.4 K = Table : First order asymptotic prices. Approximations for local vol baskets November 28, 2013 Page 25 (32)

71 Normalized errors Approximation error supposed to depend on dimension-free time to maturity σ 2 T Use σ σ N,B (F 0 )/ ( ni=1 w i F 0,i ) as proxy in local vol framework Normalized error: Rel. error σ 2 T T Dim. 5 Dim. 10 Dim. 15 Dim σ Table : Normalized relative error of the zero-order asymptotic prices. Approximations for local vol baskets November 28, 2013 Page 26 (32)

72 Normalized errors Approximation error supposed to depend on dimension-free time to maturity σ 2 T Use σ σ N,B (F 0 )/ ( ni=1 w i F 0,i ) as proxy in local vol framework Normalized error: Rel. error σ 2 T T Dim. 5 Dim. 10 Dim. 15 Dim σ Table : Normalized error of the first order asymptotic prices. Approximations for local vol baskets November 28, 2013 Page 26 (32)

73 First order prices Price T = 0.5 T = 1 T = 2 T = 5 T = Strike Approximations for local vol baskets November 28, 2013 Page 27 (32)

74 Relative errors Relative error of price 1e 06 1e 04 1e Strike Zeroth order First order Approximations for local vol baskets November 28, 2013 Page 28 (32)

75 Delta F 0 = 9, ξ = 0.7, β = 0.7, w = ρ = Objective: Compute the sensitivity (delta) w.r.t.f 0,3. Note that the option payoff is P(F) = (F 1 + F 2 F 3 K) + Approximations for local vol baskets November 28, 2013 Page 29 (32)

76 Delta T = 0.5 T = MC Delta 0 order Delta 1 order Delta MC Delta 0 order Delta 1 order Delta Strike Strike Approximations for local vol baskets November 28, 2013 Page 30 (32)

77 Relative error of delta T = 0.5 T = 5 1e 06 1e 04 1e 02 0 order Delta 1 order Delta 1e 06 1e 04 1e 02 0 order Delta 1 order Delta Strike Strike Approximations for local vol baskets November 28, 2013 Page 31 (32)

78 References M. Avellaneda, D. Boyer-Olson, J. Busca, P. Friz: Application of large deviation methods to the pricing of index options in finance, C. R. Math. Acad. Sci. Paris, 336(3), R. Azencott: Densité des diffusions en temps petit: développements asymptotiques I, Seminar on probability XVIII, L. N. M. 1059, C. Bayer, P. Laurence: Asymptotics beats Monte Carlo: The case of correlated local vol baskets, to appear in Comm. Pure Appl. Math. J. Gatheral, E. P. Hsu, P. Laurence, C. Ouyang, T.-H. Wang: Asymptotics of implied volatility in local volatility models, Math. Fin., P. Henry-Labordère: Analysis, geometry, and modeling in finance, CRC Press, R. S. Varadhan: Diffusion processes in a small time interval, Comm. Pure Appl. Math. 20, K. Yosida: On the fundamental solution of the parabolic equation in a Riemannian space, Osaka Math. J. 5, Approximations for local vol baskets November 28, 2013 Page 32 (32)