Particle methods with applications in finance Peng HU ICERM, - PowerPoint PPT Presentation

Particle methods with applications in finance Peng HU ICERM, Providence September 5, 2012 P. HU (ICERM) Brown University 1 / 49

Outline Introduction 1 Particle methods for pricing 2 Broadie-Glasserman methods 3 Genealogical/Ancestral tree based method 4 Snell envelope with small probability criteria 5 P. HU (ICERM) Brown University 2 / 49

Summary Introduction 1 Particle methods for pricing 2 3 Broadie-Glasserman methods 4 Genealogical/Ancestral tree based method 5 Snell envelope with small probability criteria P. HU (ICERM) Brown University 3 / 49

Feynman-Kac particle models and financial mathematics Concentration analysis of interacting process 1 Empirical process analysis Path space measures backward particle measures Genealogical tree measures Applications in mathematical finance 2 Sensitivity computation Partial observation problem Parameter inference Option pricing Robustness of Snell envelope Broadie-Glasserman method analysis Genealogical tree based method Snell envelope with Small probability P. HU (ICERM) Brown University 4 / 49

Summary Introduction 1 Particle methods for pricing 2 Some notation Path space models Snell envelope Robustness lemma Examples Small probability criteria Exponential concentration inequalities 3 Broadie-Glasserman methods Genealogical/Ancestral tree based method 4 Snell envelope with small probability criteria 5 P. HU (ICERM) Brown University 5 / 49

Some notation E state space, P ( E ) proba. on E & B ( E ) bounded functions � ( µ, f ) ∈ P ( E ) × B ( E ) − → µ ( f ) = µ ( dx ) f ( x ) M ( x , dy ) integral operator over E � M ( f )( x ) = M ( x , dy ) f ( y ) � [ µ M ]( dy ) = µ ( dx ) M ( x , dy ) (= ⇒ [ µ M ]( f ) = µ [ M ( f )] ) Markov chain X n with transitions M n ( x n − 1 , dx n ) from E n − 1 to E n � E P η 0 { f n ( X n ) | X 0 , . . . , X k } = M k , n ( f n )( X k ) := M k , n ( X k , dx n ) f n ( x n ) E n with M k , n ( x k , dx n ) = ( M k +1 M k +2 . . . M n )( x k , dx n ) = P ( X n ∈ dx n | X k = x k ) P. HU (ICERM) Brown University 6 / 49

Path space models Path space notations Given a elementary X � k Markov chain with transitions M � k ( x � k − 1 , dx � k ) from E � k − 1 into E � k . The historical process X k = ( X � 0 , . . . , X � k ) ∈ E k = ( E � 0 × · · · × E � k ) can be seen as a Markov chain with transitions M k ( x k − 1 , dx k ) P. HU (ICERM) Brown University 7 / 49

Snell envelope Description For 0 ≤ k ≤ n , some process Z k (gain) with F k available information on k , T k set of stopping times taking value in(k,k+1.. . n) Purpose: find sup τ ∈T k E ( Z τ |F k ) Y k the Snell envelope of Z k : Y n = Z n Y k = Z k ∨ E ( Y k +1 |F k ) Main property of the Snell envelope: τ ∗ Y k = sup E ( Z τ |F k ) = E ( Z τ ∗ k |F k ) k = min { k ≤ j ≤ n : Y j = Z j } ∈ T k τ ∈T k P. HU (ICERM) Brown University 8 / 49

Snell envelope Assumption Some Markov chain ( X k ) 0 ≤ k ≤ n , with η 0 ∈ P ( E 0 ), M n ( x n − 1 , dx n ) from E n − 1 to E n on filtered space (Ω , F , P η 0 ), F k associated natural filtration. For f k ∈ B ( E k ), assume Z k = f k ( X k ) (payoff) Then Y k = u k ( X k ) Snell envelope recursion: u k = f k ∨ M k +1 ( u k +1 ) with u n = f n A NSC for the existence of the Snell envelope M k , l f l ( x ) < ∞ for any 1 ≤ k ≤ l ≤ n , and any state x ∈ E k . To check this claim, we simply notice that � f k ≤ u k ≤ f k + M k +1 u k +1 = ⇒ f k ≤ u k ≤ M k , l f l k ≤ l ≤ n P. HU (ICERM) Brown University 9 / 49

Preliminary Numerical solution Replacing ( f k , M k ) 0 ≤ k ≤ n by some approximation model ( � f k , � M k ) 0 ≤ k ≤ n on some possibly reduced measurable subsets � E k ⊂ E k . u k = � f k ∨ � u n = � M k +1 ( � u k +1 ) with terminal condition � f n for 0 ≤ k ≤ n � A robustness/continuity lemma For any 0 ≤ k < n , on the state space � E k , we have that n − 1 n � � M k , l | f l − � � M k , l | ( M l +1 − � � | u k − � u k | ≤ f l | + M l +1 ) u l +1 | l = k l = k Proof: By inequality | ( a ∨ b ) − ( a � ∨ b � ) | ≤ | a ∨ a � | + | b ∨ b � | and induction. P. HU (ICERM) Brown University 10 / 49

Application examples of the lemma Deterministic methods Cut-off type methods Euler approximation methods Interpolation type methods Quantization tree methods Monte Carlo methods ( stoch. N-grid approximation) � Broadie-Glasserman methods [ N 2 ] � BG type adapted mean-field particle method [ N 2 ] � Importance sampling method for path dependent case [ N 2 ] Genealogical tree based method [ N ] P. HU (ICERM) Brown University 11 / 49

Path dependent case Problematic Given gain functions ( f k ) 0 ≤ k ≤ n and obstacle functions ( G k ) 0 ≤ k ≤ n Snell envelope of f k ( X k ) � k − 1 p =0 G p ( X p ) = F k ( X 0 , . . . , X k ) ? Impossible to compute if G k too small on typical trajectories. New recursion Original Snell envelope : u k ( X 0 , . . . , X k ) = F k ( X 0 , . . . , X k ) ∨ E ( u k +1 ( X 0 , . . . , X k +1 ) |F k ) with u n ( X 0 , . . . , X n ) = F n ( X 0 , . . . , X n ) We provide a new recursion v k = f k ∨ ( G k M k +1 ( v k +1 )) with v n = f n u k ( x 0 , . . . , x k ) = v k ( x k ) � k − 1 p =0 G p ( x p ) P. HU (ICERM) Brown University 12 / 49

Exponential concentration inequalities Important constants a (2 p ) 2 p = (2 p ) p 2 − p a (2 p + 1) 2 p +1 = (2 p +1) p +1 p +1 / 2 2 − ( p +1 / 2) √ ∀ p ≥ 0 and Proposition If we have a Khinchine’s type L p -mean error bounds in the following form: ∀ integer p ≥ 1 and constant c √ N sup || u k ( x ) − � u k ( x ) || L p ≤ a ( p ) c x ∈ E k then we have the following exponential concentration inequality � � � − N � 2 / c 2 � c sup | u k ( x k ) − � u k ( x k ) | > √ + � ≤ exp P N x ∈ E k P. HU (ICERM) Brown University 13 / 49

Summary Introduction 1 Particle methods for pricing 2 Broadie-Glasserman methods 3 Original Broadie-Glasserman BG adapted mean-field particle method 4 Genealogical/Ancestral tree based method 5 Snell envelope with small probability criteria P. HU (ICERM) Brown University 14 / 49

� � � � � � � � � � � � � � � � � � � � � � � � � Broadie-Glasserman methods M. Broadie and P. Glasserman. A Stochastic Mesh Method for Pricing High- Dimensional American Options Journal of Computational Finance (04) Original Broadie-Glasserman methods (hyp : M � k � η k ) � N η k = 1 k ) 1 ≤ i ≤ N ∼ i.i.d. N -grid η k on � E � k = E � k where ξ k := ( ξ i η k � � i =1 δ ξ i k N M � k +1 ( x � k , dx � k +1 ) � � M � k +1 ( x � k , dx � η k +1 ( dx � k +1 ) R k +1 ( x � k , x � k +1 ) = � k +1 ) � �� � k +1 ) dM � k +1 ( x � k , . ) η k +1 ( dx � ( x � = k +1 ) � d η k +1 ( N = 3 n = 3) • • • • � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � N 2 computations / time units � • � • • • � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � • • • • P. HU (ICERM) Brown University 15 / 49

Broadie-Glasserman methods By Khintchine’s inequality we notice: � � � � � � √ � � � � l , . ) f ) p ] 1 M � l +1 − � M � ( f )( x � L p ≤ 2 a ( p ) η l +1 [( R l +1 ( x � l ) N p � � � � l +1 We provide the following non asymptotic convergence estimate Theorem For any integer p ≥ 1 , we denote by p � the smallest even integer greater than p. Then for any time horizon 0 ≤ k ≤ n, and any x � k ∈ E � k , we have √ N || u � k ( x � u � k ( x � k ) − � k ) || L p �� l , . ) u l +1 ) p � �� 1 � � p � M � k , l ( x � k , dx � ( R l +1 ( x � ≤ 2 a ( p ) l ) η l +1 k ≤ l < n P. HU (ICERM) Brown University 16 / 49

BG adapted mean-field particle method New ( N 2 ) algorithm with the choice η k = Law ( X � k ) = η k − 1 M � k Description (hyp. : M � k � λ k ) � N η k = 1 k with i.i.d. copies ξ i k of X � η k � � i =1 δ ξ i N k H k +1 ( x � k , x � k +1 ) M � k +1 ( x � k , dx � k +1 ) � � M � k +1 ( x � k , dx � η k +1 ( dx � k +1 ) = k +1 ) � η k ( H k +1 ( . , x � k +1 )) � with n − 1 , . ) n ) = dM � n ( x � H n ( x � n − 1 , x � ( x � ( H ) 0 n ) d λ n P. HU (ICERM) Brown University 17 / 49