Rolling Adjoints Fast Greeks along Monte Carlo scenarios for - PowerPoint PPT Presentation

Rolling Adjoints Fast Greeks along Monte Carlo scenarios for early-exercise options Shashi Jain, ´ Alvaro Leitao and Cornelis W. Oosterlee QuantMinds International - Lisbon May 16, 2018 S. Jain & A. Leitao & Kees Oosterlee Rolling Adjoints May 16, 2018 1 / 31

Motivation Efficient calculation of option sensitivities is a problem of practical importance. For many pricing problems, Monte Carlo is the only feasible choice, as typically for early-exercise options. Usual finite differences approach ( bump-and-revalue ) provides poor estimations at high computational cost. Sensitivities along the paths, i.e. at intermediate times, is even more involved. “Generalization” of the Smoking adjoints technique by Giles and Glasserman to a generic interval. Sensitivities required for MVA calculations. Hedging in energy markets: multiple exercise contracts. S. Jain & A. Leitao & Kees Oosterlee Rolling Adjoints May 16, 2018 2 / 31

Outline Problem formulation 1 Stochastic Grid Bundling Method (SGBM) 2 Sensitivities along the paths with SGBM 3 Numerical results 4 Conclusions 5 S. Jain & A. Leitao & Kees Oosterlee Rolling Adjoints May 16, 2018 3 / 31

Problem formulation d − dimensional Bermudan option pricing problem. X t = ( X 1 t , . . . , X d t ) ∈ R d , depending on parameters θ = { θ 1 , . . . , θ N θ } . Let h t := h ( X t ) the intrinsic value of the option at time t . The holder receives max( h t , 0), if the option is exercised. The problem is to compute � h ( X τ ) � V t 0 ( X t 0 ) = max , E B t 0 B τ τ where B t is the risk-free saving account process and τ is a stopping time. Optimization problem: determine the early-exercise policy. S. Jain & A. Leitao & Kees Oosterlee Rolling Adjoints May 16, 2018 4 / 31

Problem formulation It can be solved by the dynamic programming principle. The option value at the terminal time T is V T ( X T ) = max( h ( X T ) , 0) . We solve the problem recursively, moving backwards in time. The continuation value Q t m − 1 is given by � V t m ( X t m ) � � � � Q t m − 1 ( X t m − 1 ) = B t m − 1 E � X t m − 1 . B t m The Bermudan option value at time t m − 1 and state X t m − 1 reads V t m − 1 ( X t m − 1 ) = max( h ( X t m − 1 ) , Q t m − 1 ( X t m − 1 )) . We are interested in V t 0 . S. Jain & A. Leitao & Kees Oosterlee Rolling Adjoints May 16, 2018 5 / 31

Stochastic Grid Bundling Method (SGBM) SGBM is based on N independent paths, { X t 0 , . . . , X t M } , obtained by a discretization scheme X t m ( n ) = F m − 1 ( X t m − 1 ( n ) , Z t m − 1 ( n ) , θ ) , where n = 1 , . . . , N is the index of the path. Z t m − 1 is a d- dimensional standard normal random vector. F m − 1 is a transformation from R d to R d . The method starts by computing the option value at terminal time as V t M ( X t M ) = max( h ( X t M ) , 0) . The following SGBM components are performed for each time step, t m , m ≤ M , moving backwards in time, starting from t M . S. Jain & A. Leitao & Kees Oosterlee Rolling Adjoints May 16, 2018 6 / 31

SGBM - Bundling The grid points at t m − 1 are bundled into B t m − 1 (1) , . . . , B t m − 1 ( ν ) non-overlapping sets or partitions. Several bundling techniques can be employed, ◮ Equal-partitioning ◮ k-means clustering algorithm ◮ recursive bifurcation ◮ recursive bifurcation of a reduced state space t m − 1 : N [1 , N β ] �→ N [1 , N ] , is defined which maps ordered A mapping I β indices of paths in a bundle B t m − 1 ( β ) to the original path indices, where N β := |B t m − 1 ( β ) | is the cardinality of the β -th bundle, β = 1 , . . . , ν . S. Jain & A. Leitao & Kees Oosterlee Rolling Adjoints May 16, 2018 7 / 31

SGBM - Regression Regress-later approach within each bundle B t m − 1 ( β ) , β = 1 , . . . , ν . G : R d × R K �→ R , which assigns values A parameterized value function ˜ ˜ G ( X t m , α β t m ) to states X t m , is introduced. The aim is to choose, for each t m and β , a vector α β t m so that � � ˜ X t m , α β = V t m ( X t m ) . G t m The option value is approximated as a linear combination of a finite number of orthonormal basis functions φ k as � � K � V t m ( X t m ) ≈ � X t m , α β α β G := t m ( k ) φ k ( X t m ) . t m k =1 The α β t m weights are approximated using a least squares regression by � ��� 2 N β � � �� � � K � � I β α β I β argmin t m − 1 ( n ) − � t m ( k ) φ k t m − 1 ( n ) V t m X t m X t m . α β � n =1 k =1 tm S. Jain & A. Leitao & Kees Oosterlee Rolling Adjoints May 16, 2018 8 / 31

SGBM - Continuation and option values The continuation values for X t m − 1 ( n ) ∈ B t m − 1 ( β ), n = 1 , . . . , N , β = 1 , . . . , ν, are approximated by � � � � � � � � X t m , α β X t m − 1 ( n ) = E | X t m − 1 ( n ) Q t m − 1 G . t m Exploiting the linearity of the expectation operator, it is written as K � � � � α β Q t m − 1 ( X t m − 1 ( n )) = � t m ( k ) E φ k ( X t m ) | X t m − 1 ( n ) . k =1 The vector of basis functions φ k should ideally be chosen such that � � the expectations E φ k ( X t m ) | X t m − 1 are known in closed-form, or have analytic approximations. The option value at each exercise time is then given by � �� � � � � � � , � V t m − 1 X t m − 1 ( n ) = max h X t m − 1 ( n ) Q t m − 1 X t m − 1 ( n ) . S. Jain & A. Leitao & Kees Oosterlee Rolling Adjoints May 16, 2018 9 / 31

Sensitivities along the paths with SGBM Naturally, we follow a backward iteration, starting at maturity, where the sensitivities are again trivial to calculate. We focus on two main sensitivities of interest: ∂ V tm − 1 ( X tm − 1 ) ◮ With respect to X t m − 1 , i.e. . ∂ X tm − 1 ∂ V tm − 1 ( X tm − 1 ) ◮ With respect to the model parameters, . ∂θ α β The method requires the derivatives of the regression coefficients, � t m . Assuming minimal smoothness of the option value function V , � � V t m ( X t m ) � �� � ∂ � V t m ( X t m ) �� � � � ∂ � � E � X t m − 1 = E � X t m − 1 . ∂θ B t m ∂θ B t m S. Jain & A. Leitao & Kees Oosterlee Rolling Adjoints May 16, 2018 10 / 31

Delta along the paths Delta is the sensitivity of the option value at t m − 1 w.r.t. X t m − 1 , � � � � ∂ V t m − 1 ( X t m − 1 ) ∂ h X t m − 1 = 1 Q tm − 1 < h ( X tm − 1 ) ∂ X t m − 1 ∂ X t m − 1 � � � � ∂ Q t m − 1 X t m − 1 + 1 Q tm − 1 ≥ h ( X tm − 1 ) . ∂ X t m − 1 The derivative of the immediate payoff, h , is usually easy to compute. The computation of the sensitivity of the continuation value function � K � ∂ � � � � Q t m − 1 ( X t m − 1 ( n )) ∂ α β = � t m ( k ) E φ k ( X t m ) | X t m − 1 ( n ) ∂ X t m − 1 ∂ X t m − 1 k =1 � K � α β � � ∂ � t m ( k ) = φ k ( X t m ) | X t m − 1 ( n ) E ∂ X t m − 1 k =1 �� � ∂ α β + � t m ( k ) E φ k ( X t m ) | X t m − 1 ( n ) . ∂ X t m − 1 S. Jain & A. Leitao & Kees Oosterlee Rolling Adjoints May 16, 2018 11 / 31

Delta along the paths � � ∂ φ k ( X t m ) | X t m − 1 ( n ) is readily computed. ∂ X tm − 1 E The derivative of the regression coefficients is the difficult part. Let us first define matrix A β t m as   φ 1 ( X t m ( I β φ 2 ( X t m ( I β φ K ( X t m ( I β t m − 1 (1))) t m − 1 (1))) t m − 1 (1))) . . .    φ 1 ( X t m ( I β φ 2 ( X t m ( I β φ K ( X t m ( I β  t m − 1 (2))) t m − 1 (2))) t m − 1 (2))) . . .     A β t m :=   . . . , ...   . . .   . . .     φ 1 ( X t m ( I β φ 2 ( X t m ( I β φ K ( X t m I β t m − 1 ( N β ))) t m − 1 ( N β ))) . . . t m − 1 (( N β ))) where X t m ( I β t m − 1 (1)) , . . . , X t m ( I β t m − 1 ( N β )) are the states of the paths in bundle B t m − 1 ( β ) . The corresponding vector of option values for these paths   V t m ( X t m ( I β � t m − 1 (1)))   V t m ( X t m ( I β � t m − 1 (2)))   V β   t m := . .   .  .  V t m ( X t m ( I β � t m − 1 ( N β ))) S. Jain & A. Leitao & Kees Oosterlee Rolling Adjoints May 16, 2018 12 / 31

Delta along the paths The least squares coefficients computation can be written as: ⊤ A β ⊤ ) V β α β t m = ( A β t m ) − 1 ( A β � t m . t m t m The derivative of the regression coefficients is then given by ⊤ A β ∂α β ∂ ( A β t m ) − 1 ⊤ ) V β t m t m ( A β = t m t m ∂ X t m − 1 ∂ X t m − 1 ⊤ t m ) − 1 ∂ A β ⊤ A β ( A β t m V β + t m t m ∂ X t m − 1 ⊤ ) ∂ V β ⊤ A β ( A β t m ) − 1 ( A β t m + , t m t m ∂ X t m − 1 The derivative of the matrix inverse can be further expanded as � � ⊤ A β ⊤ ∂ ( A β t m ) − 1 ∂ A β ⊤ ∂ A β ⊤ A β ⊤ A β t m t m t m = − ( A β t m ) − 1 A β t m + A β ( A β t m ) − 1 t m t m t m ∂ X t m − 1 ∂ X t m − 1 ∂ X t m − 1 S. Jain & A. Leitao & Kees Oosterlee Rolling Adjoints May 16, 2018 13 / 31