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 Approximations for local vol baskets · November 29, 2014 · Page 1 (30) Mohrenstrasse 39 · 10117 Berlin · Germany · Tel. +49 30 20372 0 · www.wias-berlin.de
Outline Introduction 1 2 Outline of our approach Heat kernel expansions 3 4 Numerical examples Approximations for local vol baskets · November 29, 2014 · Page 2 (30)
Outline Introduction 1 2 Outline of our approach Heat kernel expansions 3 4 Numerical examples Approximations for local vol baskets · November 29, 2014 · Page 3 (30)
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) �� n � + , at least one weight positive ◮ f ( S ) = i = 1 w i S i − K ◮ 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 29, 2014 · Page 4 (30)
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) �� n � + , at least one weight positive ◮ f ( S ) = i = 1 w i S i − K ◮ 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 29, 2014 · Page 4 (30)
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) �� n � + , at least one weight positive ◮ f ( S ) = i = 1 w i S i − K ◮ 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 29, 2014 · Page 4 (30)
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) �� n � + , at least one weight positive ◮ f ( S ) = i = 1 w i S i − K ◮ 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 29, 2014 · Page 4 (30)
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) �� n � + , at least one weight positive ◮ f ( S ) = i = 1 w i S i − K ◮ 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 29, 2014 · Page 4 (30)
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) �� n � + , at least one weight positive ◮ f ( S ) = i = 1 w i S i − K ◮ 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 29, 2014 · Page 4 (30)
Outline Introduction 1 2 Outline of our approach Heat kernel expansions 3 4 Numerical examples Approximations for local vol baskets · November 29, 2014 · Page 5 (30)
Setting ◮ Local volatility model for forward prices dF i ( t ) = σ i ( F i ( t )) dW i ( t ) , i = 1 , . . . , n , � � dW i ( t ) , dW j ( t ) = ρ i j dt �� n � + , at least ◮ Generalized spread option with payoff i = 1 w i F i − K 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 29, 2014 · Page 6 (30)
Setting ◮ Local volatility model for forward prices dF i ( t ) = σ i ( F i ( t )) dW i ( t ) , i = 1 , . . . , n , � � dW i ( t ) , dW j ( t ) = ρ i j dt �� n � + , at least ◮ Generalized spread option with payoff i = 1 w i F i − K 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 29, 2014 · Page 6 (30)
Setting ◮ Local volatility model for forward prices dF i ( t ) = σ i ( F i ( t )) dW i ( t ) , i = 1 , . . . , n , � � dW i ( t ) , dW j ( t ) = ρ i j dt �� n � + , at least ◮ Generalized spread option with payoff i = 1 w i F i − K 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 29, 2014 · Page 6 (30)
Basket Carr-Jarrow formula ◮ Consider the basket (index) � n i = 1 w i F i : n n � � d w i F i ( t ) = w i σ i ( F i ( t )) dW i ( t ) i = 1 i = 1 ◮ Ito’s formula formally implies that ◮ Let H n − 1 be the Hausdorff measure on E ( K ) Approximations for local vol baskets · November 29, 2014 · Page 7 (30)
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 � T � T n 1 � w i F i ( u ) > K dF i ( u ) + 1 � δ � w i F i ( u ) = K σ 2 + w i N , B ( F ( u )) du 2 0 0 i = 1 ◮ Let H n − 1 be the Hausdorff measure on E ( K ) Approximations for local vol baskets · November 29, 2014 · Page 7 (30)
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 + � T n + 1 � � � σ 2 C ( F (0) , K , T ) = w i F i (0) − K E N , B ( F ( u )) δ E ( K ) ( F ( u )) du 2 0 i = 1 ◮ Let H n − 1 be the Hausdorff measure on E ( K ) Approximations for local vol baskets · November 29, 2014 · Page 7 (30)
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 i = 1 � T � + 1 R n σ 2 N , B ( F ) δ E ( K ) ( F ) p ( F 0 , F , u ) d F du 2 0 ◮ Let H n − 1 be the Hausdorff measure on E ( K ) Approximations for local vol baskets · November 29, 2014 · Page 7 (30)
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 } and v ( F ) ≔ � i w i F i that + n � C ( F (0) , K , T ) = w i F i (0) − K i = 1 � T � 1 R n |∇ v ( F ) | σ 2 N , B ( F ) δ 0 ( v ( F ) − K ) p ( F 0 , F , u ) d F du + 2 | w | 0 ◮ Let H n − 1 be the Hausdorff measure on E ( K ) Approximations for local vol baskets · November 29, 2014 · Page 7 (30)
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 } and v ( F ) ≔ � i w i F i that + n � C ( F (0) , K , T ) = w i F i (0) − K i = 1 � T � 1 R n |∇ v ( F ) | σ 2 + N , B ( F ) δ 0 ( v ( F ) − K ) p ( F 0 , F , u ) d F du 2 | w | 0 ◮ Let H n − 1 be the Hausdorff measure on E ( K ) . Recall the co-area formula: � ∞ � � |∇ v ( x ) | g ( x ) dx = g ( x ) H n − 1 ( dx ) ds v − 1 ( { s } ) −∞ Ω Approximations for local vol baskets · November 29, 2014 · Page 7 (30)
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