GPU acceleration in early-exercise option valuation ´ Alvaro Leitao and Cornelis W. Oosterlee Financial Mathematics and Supercomputing A Coru˜ na - September 26, 2018 ´ A. Leitao & Kees Oosterlee SGBM on GPU A Coru˜ na - September 26, 2018 1 / 36
Motivation Efficient valuation of early-exercise options. Novel method: combination of successful previous ideas. Originally introduced by Jain and Oosterlee in 2013. Multi-dimensional early-exercise option contracts. Increase the dimensionality. The technique becomes very expensive. Solution: parallelization of the method. GPU computing (GPGPU). ´ A. Leitao & Kees Oosterlee SGBM on GPU A Coru˜ na - September 26, 2018 2 / 36
Outline Definitions 1 Basket Bermudan Options 2 Stochastic Grid Bundling Method 3 Parallel GPU Implementation 4 Results 5 Conclusions 6 ´ A. Leitao & Kees Oosterlee SGBM on GPU A Coru˜ na - September 26, 2018 3 / 36
Definitions Option A contract that offers the buyer the right, but not the obligation, to buy (call) or sell (put) a financial asset at an agreed-upon price (the strike price) during a certain period of time or on a specific date (exercise date). Investopedia. Option price The fair value to enter in the option contract. In other words, the (discounted) expected value of the contract. V t = D t E [ f ( S t )] where f is the payoff function, S the underlying asset, t the exercise time and D t the discount factor. ´ A. Leitao & Kees Oosterlee SGBM on GPU A Coru˜ na - September 26, 2018 4 / 36
Definitions (II) Pricing techniques Stochastic process, S t , governing by a SDE. Simulation: Monte Carlo method. PDEs: Feynman-Kac theorem. Fourier inversion techniques: Characteristic function. Types of options - Exercise time European: End of the contract, t = T . American: Anytime, t ∈ [0 , T ]. Bermudan: Some predefined times, t ∈ { t 1 , . . . , t M } Many others: Asian, barrier, . . . ´ A. Leitao & Kees Oosterlee SGBM on GPU A Coru˜ na - September 26, 2018 5 / 36
Definitions (III) Early-exercise option price American: V t = sup D t E [ f ( S t )] . t ∈ [0 , T ] Bermudan: V t = sup D t E [ f ( S t )] . t ∈{ t 1 ,..., t M } Pricing early-exercise options PDEs: Hamilton-Jacobi-Bellman equation. Fourier inversion techniques: low dimensions. Simulation: ◮ Least-squares method (LSM), Longstaff and Schwartz. ◮ Stochastic Grid Bundling method (SGBM) [JO15]. ´ A. Leitao & Kees Oosterlee SGBM on GPU A Coru˜ na - September 26, 2018 6 / 36
Basket Bermudan Options Right to exercise at a set of times: t ∈ { t 0 = 0 , . . . , t m , . . . , t M = T } . d -dimensional underlying process: S t = ( S 1 t , . . . , S d t ) ∈ R d . Driven by a system of SDE in the form: d S 1 t = µ 1 ( S t ) d t + σ 1 ( S t ) d W 1 t , d S 2 t = µ 2 ( S t ) d t + σ 2 ( S t ) d W 2 t , . . . d S d t = µ d ( S t ) d t + σ d ( S t ) d W d t , where W δ t , δ = 1 , 2 , . . . , d , are correlated standard Brownian motions. t and W j The instantaneous correlation coefficient between W i t is ρ i , j . ´ A. Leitao & Kees Oosterlee SGBM on GPU A Coru˜ na - September 26, 2018 7 / 36
Basket Bermudan Options (II) Intrinsic value of the option: h t := h ( S t ). The value of the option at the terminal time T : V T ( S T ) = f ( S T ) = max( h ( S T ) , 0) . The conditional continuation value Q t m , i.e. the discounted expected payoff at time t m : � � Q t m ( S t m ) = D t m E V t m +1 ( S t m +1 ) | S t m . The Bermudan option value at time t m and state S t m : V t m ( S t m ) = f ( S T ) = max( h ( S t m ) , Q t m ( S t m )) . Value of the option at the initial state S t 0 , i.e. V t 0 ( S t 0 ). ´ A. Leitao & Kees Oosterlee SGBM on GPU A Coru˜ na - September 26, 2018 8 / 36
Basket Bermudan options scheme Figure: d-dimensional Bermudan option ´ A. Leitao & Kees Oosterlee SGBM on GPU A Coru˜ na - September 26, 2018 9 / 36
Stochastic Grid Bundling Method Dynamic programming approach. Simulation and regression-based method. Forward in time: Monte Carlo simulation. Backward in time: Early-exercise policy computation. Step I: Generation of stochastic grid points { S t 0 ( n ) , . . . , S t M ( n ) } , n = 1 , . . . , N . Step II: Option value at terminal time t M = T V t M ( S t M ) = max( h ( S t M ) , 0) . ´ A. Leitao & Kees Oosterlee SGBM on GPU A Coru˜ na - September 26, 2018 10 / 36
Stochastic Grid Bundling Method (II) Backward in time, t m , m ≤ M , : Step III: Bundling into ν non-overlapping sets or partitions B t m − 1 (1) , . . . , B t m − 1 ( ν ) Step IV: Parameterizing the option values Z ( S t m , α β t m ) ≈ V t m ( S t m ) . Step V: Computing the continuation and option values at t m − 1 Q t m − 1 ( S t m − 1 ( n )) = E [ Z ( S t m , α β � t m ) | S t m − 1 ( n )] . The option value is then given by: V t m − 1 ( S t m − 1 ( n )) = max( h ( S t m − 1 ( n )) , � � Q t m − 1 ( S t m − 1 ( n ))) . ´ A. Leitao & Kees Oosterlee SGBM on GPU A Coru˜ na - September 26, 2018 11 / 36
Bundling Original: Iterative process (K-means clustering). Problems: Too expensive (time and memory) and distribution. New technique: Equal-partitioning. Efficient for parallelization. Two stages: sorting and splitting. SORT SPLIT Figure: Equal partitioning scheme ´ A. Leitao & Kees Oosterlee SGBM on GPU A Coru˜ na - September 26, 2018 12 / 36
Parametrizing the option value Basis functions φ 1 , φ 2 , . . . , φ K . � � S t m , α β In our case, Z depends on S t m only through φ k ( S t m ): t m � � K � S t m , α β α β Z = t m ( k ) φ k ( S t m ) . t m k =1 Computation of α β α β t m (or � t m ) by least squares regression. The α β t m determines the early-exercise policy. The continuation value: �� K � � � � α β Q t m − 1 ( S t m − 1 ( n )) = D t m − 1 E t m ( k ) φ k ( S t m ) | S t m − 1 � k =1 K � � � α β = D t m − 1 � t m ( k ) E φ k ( S t m ) | S t m − 1 . k =1 ´ A. Leitao & Kees Oosterlee SGBM on GPU A Coru˜ na - September 26, 2018 13 / 36
Basis functions � � Choosing φ k : the expectations E φ k ( S t m ) | S t m − 1 should be easy to calculate. The intrinsic value of the option, h ( · ) , is usually an important and useful basis function. For example: ◮ Geometric basket Bermudan: � d � 1 � d S δ h ( S t ) = t δ =1 ◮ Arithmetic basket Bermudan: d � h ( S t ) = 1 S δ t m d δ =1 For S t following a GBM: expectations analytically available. ´ A. Leitao & Kees Oosterlee SGBM on GPU A Coru˜ na - September 26, 2018 14 / 36
Estimating the option value SGBM has been developed as duality-based method . Provide two estimators (confidence interval). Direct estimator (high-biased estimation): � �� � � � � , � V t m − 1 ( S t m − 1 ( n )) = max h S t m − 1 ( n ) Q t m − 1 S t m − 1 ( n ) , N � V t 0 ( S t 0 )] = 1 E [ � � V t 0 ( S t 0 ( n )) . N n =1 Path estimator (low-biased estimation): τ ∗ ( S ( n )) = min { t m : h ( S t m ( n )) ≥ � � Q t m ( S t m ( n )) , m = 1 , . . . , M } , � � v ( n ) = h S � , τ ∗ ( S ( n )) N L � 1 V t 0 ( S t 0 ) = lim v ( n ) . N L N L n =1 ´ A. Leitao & Kees Oosterlee SGBM on GPU A Coru˜ na - September 26, 2018 15 / 36
SGBM - schematic algorithm Data: S t 0 , X , µ δ , σ δ , ρ i , j , T , N , M Pre-Bundling (only in k-means case). Generation of the grid points (Monte Carlo). Step I. Option value at terminal time t = M . Step II. for Time t = ( M − 1) . . . 1 do Bundling. Step III. for Bundle β = 1 . . . ν do Exercise policy (Regression). Step IV. Continuation value. Step V. Direct estimator. Step V. Generation of the grid points (Monte Carlo). Step I. Option value at terminal time t = M . Step II. for Time t = ( M − 1) . . . 1 do Bundling. Step III. for Bundle β = 1 . . . ν do Continuation value. Step V. Path estimator. Step V. ´ A. Leitao & Kees Oosterlee SGBM on GPU A Coru˜ na - September 26, 2018 16 / 36
Continuation value computation: new approach More generally applicable. More involved models or options. First discretize, then derive the discrete characteristic function. S 1 tm +1 = S 1 W 1 tm + µ 1 ( S tm )∆ t + σ 1 ( S tm )∆ ˜ tm +1 , S 2 tm +1 = S 2 W 1 W 2 tm + µ 2 ( S tm )∆ t + ρ 1 , 2 σ 2 ( S tm )∆ ˜ tm +1 + L 2 , 2 σ 2 ( S tm )∆ ˜ tm +1 , · · · S d tm +1 = S d W 1 W 2 W d tm + µ d ( S tm )∆ t + ρ 1 , d σ d ( S tm )∆ ˜ tm +1 + L 2 , d σ d ( S tm )∆ ˜ tm +1 + · · · + L d , d σ d ( S tm )∆ ˜ tm +1 , By definition, the d -variate discrete characteristic function: d iu j S j � | S tm � u 1 , u 2 , . . . , u d | S tm � = E exp ψ S tm +1 tm +1 j =1 j d S j W k � � L k , j ∆ ˜ | S tm = E exp iu j tm + µ j ( S tm )∆ t + σ j ( S tm ) tm +1 j =1 k =1 d d d � � � S j � � iu j L k , j σ j ( S tm )∆ ˜ W k · = exp iu j tm + µ j ( S tm )∆ t E exp tm +1 j =1 k =1 j = k d d d � � � S j � � · , = exp iu j tm + µ j ( S tm )∆ t ψ N (0 , ∆ t ) u j L k , j σ j ( S tm ) j =1 k =1 j = k ´ A. Leitao & Kees Oosterlee SGBM on GPU A Coru˜ na - September 26, 2018 17 / 36
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