American Options Pricing American options using martingale bases An optimization point of view How to effectively solve the optimization problem Numerical experiments Pricing American options using martingale bases Jérôme Lelong Grenoble Alpes University ETH Zürich – May 19, 2016 J. Lelong (Grenoble Alpes University) ETH Zürich – May 19, 2016 1 / 32
American Options Pricing American options using martingale bases An optimization point of view How to effectively solve the optimization problem Numerical experiments Outline American Options 1 An optimization point of view 2 How to effectively solve the optimization problem 3 Numerical experiments 4 J. Lelong (Grenoble Alpes University) ETH Zürich – May 19, 2016 2 / 32
American Options Pricing American options using martingale bases An optimization point of view American Options How to effectively solve the optimization problem Numerical experiments Framework (1) ◮ Consider a multi–dimensional financial market driven by a d − dimensional Brownian motion B . ◮ The process ( S t ) t ≤ T is the underlying asset with values in R d ′ , d ′ ≤ d . � � � t 0 r s ds φ ( S t ) Z t = e − ◮ The discounted payoff process writes t ≤ T . � � sup t Z 2 < ∞ . Assume E t ◮ Consider an American option. Its discounted price at time t of the Bermudan option is given by U t = esssup τ ∈T t E [ Z τ |F t ] where T t is the set of all F− stopping times with values in [ t , T ] . J. Lelong (Grenoble Alpes University) ETH Zürich – May 19, 2016 3 / 32
American Options Pricing American options using martingale bases An optimization point of view American Options How to effectively solve the optimization problem Numerical experiments Framework (2) ◮ The Snell envelope process ( U t ) 0 ≤ t ≤ T admits a Doob–Meyer decomposition U t = U 0 + M ⋆ t − A ⋆ t where M ⋆ is a martingale and A ⋆ a predictable increasing process both vanishing at zero and square integrable. J. Lelong (Grenoble Alpes University) ETH Zürich – May 19, 2016 4 / 32
American Options Pricing American options using martingale bases An optimization point of view American Options How to effectively solve the optimization problem Numerical experiments Dual price (1) We know from [Rogers, 2002] that � � � � ( Z t − M ⋆ U 0 = inf E sup ( Z t − M t ) = E sup t ) M ∈ H 1 0 ≤ t ≤ T 0 ≤ t ≤ T 0 ◮ This problem admits more than a single solution. ◮ Some of the martingales M attaining the infimum are surely optimal ( Z t − M t ) U 0 = sup a . s . 0 ≤ t ≤ T ◮ Let τ be an optimal stopping time, then � � � � ( Z t − M ⋆ ( Z t − M t ) U 0 = inf E sup = E sup t ) . M ∈ H 1 τ ≤ t ≤ T τ ≤ t ≤ T 0 J. Lelong (Grenoble Alpes University) ETH Zürich – May 19, 2016 5 / 32
American Options Pricing American options using martingale bases An optimization point of view American Options How to effectively solve the optimization problem Numerical experiments Dual price (2) ◮ From [Schoenmakers et al., 2013], any martingale satisfying � � Var sup ( Z t − M t ) = 0 0 ≤ t ≤ T is surely optimal. ◮ Let τ be an optimal stopping time, then for any surely optimal martingale M , ( M t ∧ τ ) t = ( M ⋆ t ∧ τ ) t . See [Jamshidian, 2007]. J. Lelong (Grenoble Alpes University) ETH Zürich – May 19, 2016 6 / 32
American Options Pricing American options using martingale bases An optimization point of view American Options How to effectively solve the optimization problem Numerical experiments Dual price (3) With our square integrability assumption, we can rewrite the minimization problem as � � U 0 = inf E sup ( Z t − E [ X |F t ]) . X ∈ L 2 (Ω , F T , P ) 0 ≤ t ≤ T s.t. E [ X ] = 0 How to approximate L 2 (Ω , F T , P ) by a finite dimensional vector space in which conditional expectations are tractable in a closed form? J. Lelong (Grenoble Alpes University) ETH Zürich – May 19, 2016 7 / 32
American Options Pricing American options using martingale bases An optimization point of view An optimization point of view How to effectively solve the optimization problem Numerical experiments Wiener chaos expansion ( d = 1) Let H i be the i − th Hermite polynomial defined by H i ( x ) = ( − 1 ) i e x 2 / 2 d i dx i ( e − x 2 / 2 ) , for i ≥ 1 . H 0 ( x ) = 1 ; ◮ H ′ i = H i − 1 with the convention H − 1 = 0. ◮ If X , Y ∼ N ( 0 , 1 ) and form a Gaussian vector, E [ H i ( X ) H j ( Y )] = i ! ( E [ XY ]) i 1 { i = j } . For i ≥ 0, the L 2 closure of the space � �� T � � : f ∈ L 2 ([ 0 , T ]) H i = H i f t dB t 0 corresponds to the Wiener chaos of order i . J. Lelong (Grenoble Alpes University) ETH Zürich – May 19, 2016 8 / 32
American Options Pricing American options using martingale bases An optimization point of view An optimization point of view How to effectively solve the optimization problem Numerical experiments Truncated Wiener chaos expansion ( d = 1) Take a regular grid 0 = t 0 < t 1 < · · · < t n with step h and consider � T f i ( t ) = 1 { ] t i − 1 , t i ] } ( t ) f i ( t ) dB t = B t i − B t i − 1 √ √ = G i ∼ N ( 0 , 1 ) . ; h h 0 For F ∈ L 2 (Ω , F T ) , we introduce the truncated chaos expansion of order p � � C p , n ( F ) = λ α H α i ( G i ) α ∈ A p , n i ≥ 1 where A p , n = { α ∈ N n : � α � 1 ≤ p } with � α � 1 = � i ≥ 0 α i . In the following we write, � λ α � C p , n ( F ) = H α ( G 1 , . . . , G n ) α ∈ A p , n H α ( x ) = � with � i ≥ 1 H α i ( x i ) . J. Lelong (Grenoble Alpes University) ETH Zürich – May 19, 2016 9 / 32
American Options Pricing American options using martingale bases An optimization point of view An optimization point of view How to effectively solve the optimization problem Numerical experiments Key properties of the truncated Wiener chaos expansion ( d = 1) ◮ Since the Hermite polynomials are orthogonal � � F � H α ( G 1 , · · · , G n ) E λ α = �� � . i ≥ 1 α i ! ◮ For k ≤ n , � λ α � E [ C p , n ( F ) |F t k ] = H α ( G 1 , . . . , G n ) α ∈ A k p , n p , n = { α ∈ N n : � α � 1 ≤ p , α ℓ = 0 ∀ ℓ > k } . with A k “Computing E [ ·|F t k ] ” ⇔ “Dropping all non F t k − measurable terms” J. Lelong (Grenoble Alpes University) ETH Zürich – May 19, 2016 10 / 32
American Options Pricing American options using martingale bases An optimization point of view An optimization point of view How to effectively solve the optimization problem Numerical experiments Extension to the multi–dimensional case (1) Take i ( t ) = 1 { ] t i − 1 , t i ] } ( t ) h j √ e j , i = 1 , . . . , n , j = 1 , . . . , d h where ( e 1 , . . . , e d ) denotes the canonical basis of R d . The truncated Wiener chaos of order p ≥ 0 is given by � d H α j ( G j � 1 , . . . , G j n ) : α ∈ ( N n ) d , � α � 1 ≤ p j = 1 where � α � 1 = � n � d j = 1 α j i . i = 1 J. Lelong (Grenoble Alpes University) ETH Zürich – May 19, 2016 11 / 32
American Options Pricing American options using martingale bases An optimization point of view An optimization point of view How to effectively solve the optimization problem Numerical experiments Extension to the multi–dimensional case (2) With the concise notation � d � H α j ( G j � H ⊗ d 1 , . . . , G j ∀ α ∈ ( N n ) d . α ( G 1 , . . . , G n ) = n ) j = 1 We introduce the truncated chaos expansion of order p of F ∈ L 2 (Ω , F T ) � λ α � H ⊗ d C p , n ( F ) = α ( G 1 , . . . , G n ) α ∈ A ⊗ d p , n � � α ∈ ( N n ) d : � α � 1 ≤ p where A ⊗ d p , n = . With an obvious abuse of notation, we write, for λ ∈ R A ⊗ d p , n , � λ α � H ⊗ d C p , n ( λ ) = α ( G 1 , . . . , G n ) . α ∈ A ⊗ d p , n J. Lelong (Grenoble Alpes University) ETH Zürich – May 19, 2016 12 / 32
American Options Pricing American options using martingale bases An optimization point of view An optimization point of view How to effectively solve the optimization problem Numerical experiments Return to the American option price We approximate the original problem � � ( Z t − E [ X |F t ]) inf E sup X ∈ L 2 (Ω , F T , P ) 0 ≤ t ≤ T s.t. E [ X ] = 0 by inf V p , n ( λ ) (1) λ ∈ R A ⊗ d p , n s.t. λ 0 = 0 with � � V p , n ( λ ) = E 0 ≤ k ≤ n ( Z t k − E [ C p , n ( λ ) |F t k ]) max J. Lelong (Grenoble Alpes University) ETH Zürich – May 19, 2016 13 / 32
American Options Pricing American options using martingale bases An optimization point of view An optimization point of view How to effectively solve the optimization problem Numerical experiments Properties of the minimization problem (1) Proposition 1 The minimization problem (1) has at least one solution. ◮ The function V p , n is clearly convex (maximum of affine funtions). ◮ Not strongly convex but, V p , n ( λ ) ≥ E [( C p , n ( λ )) − ] ≥ 1 2 E [ | C p , n ( λ ) | ] , E [ | C p , n ( λ ) | ] = | λ | E [ | C p , n ( λ/ | λ | ) | ] ≥ | λ | E [ | C p , n ( µ ) | ] . inf µ ∈ R A ⊗ d p , n , | µ | = 1 J. Lelong (Grenoble Alpes University) ETH Zürich – May 19, 2016 14 / 32
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