Pricing Early-exercise options

Transcription

1 Pricing Early-exercise options GPU Acceleration of SGBM method Delft University of Technology - Centrum Wiskunde & Informatica Álvaro Leitao Rodríguez and Cornelis W. Oosterlee Lausanne - December 4, 2016 A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28

2 Outline 1 Definitions 2 Basket Bermudan Options 3 Stochastic Grid Bundling Method 4 Parallel GPU Implementation 5 Results 6 Conclusions A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28

3 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 (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28

4 Definitions - cont. Pricing techniques Stochastic process, S t. Simulation: Monte Carlo method. PDEs: Feynman-Kac theorem. Types of options - Exercise time European: End of the contract, t = T. American: Anytime, t [0, T ]. Bermudan: Some predefined times, t {t1,..., t M } Many others: Asian, barrier,... A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28

5 Definitions - cont. Early-exercise option price American: Bermudan: V t = V t = Pricing early-exercise options sup D t E [f (S t )]. t [0,T ] sup D t E [f (S t )]. t {t1,...,t M } PDEs: Hamilton-Jacobi-Bellman equation. Simulation: Least-squares method (LSM), Longstaff and Schwartz. Stochastic Grid Bundling method (SGBM) [JO15]. A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28

6 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. 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 tm, i.e. the discounted expected payoff at time t m : Q tm (S tm ) = D tm E [ V tm+1 (S tm+1 ) S tm ]. The Bermudan option value at time t m and state S tm : V tm (S tm ) = f (S T ) = max(h(s tm ), Q tm (S tm )). Value of the option at the initial state S t0, i.e. V t0 (S t0 ). A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28

7 Basket Bermudan options scheme Figure: d-dimensional Bermudan option A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28

8 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 t0 (n),..., S tm (n)}, n = 1,..., N. Step II: Option value at terminal time t M = T V tm (S tm ) = max(h(s tm ), 0). A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28

9 Stochastic Grid Bundling Method (II) Backward in time, t m, m M,: Step III: Bundling into ν non-overlapping sets or partitions B tm 1 (1),..., B tm 1 (ν) Step IV: Parameterizing the option values Z(S tm, α β t m ) V tm (S tm ). Step V: Computing the continuation and option values at t m 1 Q tm 1 (S tm 1 (n)) = E[Z(S tm, α β t m ) S tm 1 (n)]. The option value is then given by: V tm 1 (S tm 1 (n)) = max(h(s tm 1 (n)), Q tm 1 (S tm 1 (n))). A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28

10 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 (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28

11 Parametrizing the option value Basis functions φ 1, φ 2,..)., φ K. In our case, Z (S tm, αt β m depends on S tm only through φ k (S tm ): ) K Z (S tm, αt β m = αt β m (k)φ k (S tm ). k=1 Computation of αt β m (or α t β m ) by least squares regression. The αt β m determines the early-exercise policy. The continuation value: [( K ) ] Q tm 1 (S tm 1 (n)) = D tm 1 E α t β m (k)φ k (S tm ) S tm 1 k=1 K = D tm 1 α t β m (k)e [ ] φ k (S tm ) S tm 1. k=1 A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28

12 Basis functions Choosing φ k : the expectations E [ φ k (S tm ) S tm 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: h(s t ) = Arithmetic basket Bermudan: ( d δ=1 S δ t d ) 1 d h(s t ) = 1 d δ=1 S δ t m For S t following a GBM: expectations analytically available. A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28

13 Estimating the option value SGBM has been developed as duality-based method. Provide two estimators (confidence interval). Direct estimator (high-biased estimation): ( V tm 1 (S tm 1 (n)) = max h ( S tm 1 (n) ), Q ( tm 1 Stm 1 (n) )), E[ V t0 (S t0 )] = 1 N N V t0 (S t0 (n)). n=1 Path estimator (low-biased estimation): τ (S(n)) = min{t m : h (S tm (n)) Q tm (S tm (n)), m = 1,..., M}, v(n) = h ( S τ (S(n))), 1 V t0 (S t0 ) = lim NL N L N L v(n). n=1 A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28

14 Parallel SGBM on GPU NVIDIA CUDA platform. Parallel strategy: two parallelization stages: Forward: Monte Carlo simulation. Backward: Bundles Oportunity of parallelization. Novelty in early-exercise option pricing methods. Two implementations K-means vs. Equal-partitioning: K-means: sequential parts. K-means: transfers between CPU and GPU cannot be avoided. K-means: all data need to be stored. K-means: Load-balancing. Equal-partitioning: fully parallelizable. Equal-partitioning: No transfers. Equal-partitioning: efficient memory use. A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28

15 Parallel SGBM on GPU - Forward in time One GPU thread per Monte Carlo simulation. Random numbers on the fly : curand library. Compute intermediate results: Expectations. Intrinsic value of the option. Equal-partitioning: sorting criterion calculations. Intermediate results in the registers: fast memory access. Original bundling: all the data still necessary. A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28

16 Parallel SGBM on GPU - Forward in time Figure: SGBM Monte Carlo A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28

17 Parallel SGBM on GPU - Backward in time One parallelization stage per exercise time step. Sort w.r.t bundles: efficient memory access. Parallelization in bundles. Each bundle calculations (option value and early-exercise policy) in parallel. All GPU threads collaborate in order to compute the continuation value. A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28

18 Parallel SGBM on GPU - Backward in time Figure: SGBM backward stage A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28

19 Parallel SGBM on GPU - Backward in time Figure: SGBM backward stage A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28

20 Results Accelerator Island system of Cartesius Supercomputer. Intel Xeon E v2. NVIDIA Tesla K40m. C-compiler: GCC CUDA version: 5.5. Geometric and arithmetic basket Bermudan put options: S t0 = (40,..., 40) R d, X = 40, r t = 0.06, σ = (0.2,..., 0.2) R d, ρ ij = 0.25, T = 1 and M = 10. Basis functions: K = 3. Multi-dimensional Geometric Brownian Motion. Euler discretization, δt = T /M. A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28

21 Equal-partitioning: convergence test d Reference price 5d Direct estimator 5d Path estimator 10d Reference price 10d Direct estimator 10d Path estimator 15d Reference price 15d Direct estimator 15d Path estimator d Direct estimator 5d Path estimator 10d Direct estimator 10d Path estimator 15d Direct estimator 15d Path estimator Vt0 (St0) 1.3 Vt0 (St0) Bundles ν (a) Geometric basket put option Bundles ν (b) Arithmetic basket put option Figure: Convergence with equal-partitioning bundling technique. Test configuration: N = 2 18 and t = T /M. A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28

22 Speedup Geometric basket Bermudan option k-means equal-partitioning d = 5 d = 10 d = 15 d = 5 d = 10 d = 15 C CUDA Speedup Arithmetic basket Bermudan option k-means equal-partitioning d = 5 d = 10 d = 15 d = 5 d = 10 d = 15 C CUDA Speedup Table: SGBM total time (s) for the C and CUDA versions. Test configuration: N = 2 22, t = T /M and ν = A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28

23 Speedup - High dimensions Geometric basket Bermudan option ν = 2 10 ν = 2 14 d = 30 d = 40 d = 50 d = 30 d = 40 d = 50 C CUDA Speedup Arithmetic basket Bermudan option ν = 2 10 ν = 2 14 d = 30 d = 40 d = 50 d = 30 d = 40 d = 50 C CUDA Speedup Table: SGBM total time (s) for a high-dimensional problem with equal-partitioning. Test configuration: N = 2 20 and t = T /M. A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28

24 Conclusions Efficient parallel GPU implementation. Extend the SGBM s applicability: Increasing dimensionality. New bundling technique. Future work: Explore the new CUDA 7 features: cusolver (QR factorization). CVA calculations. A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28

25 References Shashi Jain and Cornelis W. Oosterlee. The Stochastic Grid Bundling Method: Efficient pricing of Bermudan options and their Greeks. Applied Mathematics and Computation, 269: , Álvaro Leitao and Cornelis W. Oosterlee. GPU Acceleration of the Stochastic Grid Bundling Method for Early-Exercise options. International Journal of Computer Mathematics, 92(12): , A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28

26 Acknowledgements Thank you for your attention A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28

27 Appendix Geo. basket Bermudan option - Basis functions: φ k (S tm ) = ( ( ) k 1 d St δ m ) 1 d, k = 1,..., K, δ=1 The expectation can directly be computed as: E [ φ k (S tm ) S tm 1 (n) ] = ( P tm 1 (n)e ( ) ) k 1 µ+ (k 1) σ2 t 2, where, ( d P tm 1 (n) = St δ m 1 (n) δ=1 ) 1 d ( d, µ = 1 d δ=1 r q δ σ2 δ 2 ), σ 2 = 1 d 2 2 d d Cpq 2. p=1 q=1 A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28

28 Appendix Arith. basket Bermudan option - Basis functions: φ k (S tm ) = ( 1 d ) k 1 d St δ m, k = 1,..., K., δ=1 The summation can be expressed as a linear combination of the products: ( d ) k St δ m = δ=1 k 1 +k 2 + +k d =k ( k k 1, k 2,..., k d ) 1 δ d ( S δ t m ) kδ, And the expression for Geometric basket option can be applied. A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU Lausanne - December 4, / 28