Pricing of cross-currency interest rate derivatives on Graphics Processing Units

Transcription

1 Pricing of cross-currency interest rate derivatives on Graphics Processing Units Duy Minh Dang Department of Computer Science University of Toronto Toronto, Canada Joint work with Christina Christara and Ken Jackson Workshop on Parallel and Distributed Computing in Finance IEEE International Parallel & Distributed Processing Symposium Atlanta, USA, April 19 23, /18

2 Outline 1 Power Reverse Dual Currency (PRDC) swaps 2 The model and the associated PDE 3 GPU-based parallel numerical methods 4 Numerical results 5 Summary and future work 2/18

3 Power Reverse Dual Currency (PRDC) swaps PRDC swaps Long-dated swaps ( 30 years) Two currencies: domestic and foreign (unit zero-coupon bond prices P d and P f ) PRDC coupons in exchange for domestic LIBOR payments (funding leg) Two parties: the issuer (pays PRDC coupons) and the investor (pays LIBOR) PRDC coupon and LIBOR rates are applied on the domestic currency principal N d Tenor structure: T 0 < T 1 <... < T β 1 < T β, ν α ν(t α 1,T α) = T α T α 1 At each of the times T α, α = 1,...,β 1, the issuer 1 P d (T α 1,T α) receives ν αn d L d (T α 1,T α), where L d (T α 1,T α) = ν(t α 1,T α)p d (T α 1,T α) pays PRDC coupon amount ν αn d C α, where the coupon rate C α has the structure ( ( s(t ) ) α) C α = min max c f c d,b f,b c f α s(t α) : the spot FX-rate at time T α f α: scaling factor, usually is set to the forward FX rate F(0,T α) = P f(0,t α) P d (0,T α) s(0) c d,c f : domestic and foreign coupon rates; b f,b c : a cap and a floor In the standard case (b f = 0 and b c = ), C α is a call option on the spot FX rate C α = h αmax(s(t α) k α,0), h α = c f f α,k α = fαc d c f 3/18

4 Power Reverse Dual Currency (PRDC) swaps time T α 4/18 Bermudan cancelable PRDC swaps The issuer has the right to cancel the underlying swap at any of the times {T α} β 1 α=1 after the occurrence of any exchange of fund flows scheduled on that date. Observation: terminating a swap at T α is the same as i. continuing the underlying swap, and ii. entering into the offsetting swap at T α the issuer has a long position in an associated offsetting Bermudan swaption Pricing framework: dividing the pricing of a Bermudan cancelable PRDC swap into i. the pricing of the underlying PRDC swap (a vanilla PRDC swap), and ii. the pricing of the associated offsetting Bermudan swaption Notations uα c (t) and uf α (t): value at time t of the coupon and the LIBOR part scheduled after T α, respectively uα h (t): value at time t of the offsetting Bermudan swaption that has only the dates {T α+1,...,t β 1 } as exercise opportunities u e α(t): value at time t of all fund flows in the offsetting swap scheduled after T α u h β 1 (T β 1) = u e β 1 (T β 1) = 0 Note: u h α(t α) is the hold value and u e α(t α) is the exercise value of the option at

5 Power Reverse Dual Currency (PRDC) swaps Backward pricing algorithm T 0 T 1 T β 3 T β 2 T β 1 T β 5/18

6 Power Reverse Dual Currency (PRDC) swaps Backward pricing algorithm T 0 T 1 T β 3 T β 2 T β 1 T β u c β 2 (T β 2 ) GPU1 N d C β 1 u h β 2 (T β 2 ) GPU2 0 5/18

7 Power Reverse Dual Currency (PRDC) swaps Backward pricing algorithm T 0 T 1 T β 3 T β 2 T β 1 T β u c β 2 (T β 2 ) GPU1 N d C β 1 u h β 2 (T β 2 ) GPU2 0 uc β 3 (T β 3 ) u h β 3 (T β 3 ) GPU1 GPU2 N d C β 2 + u c β 2 (T β 2 ) ( max u β 2 e (T β 2 ),u h ) β 2 (T β 2 ) } {{ } (u β 2 c ) (T β 2 )+uf β 2 (T β 2 ) 5/18

8 Power Reverse Dual Currency (PRDC) swaps Backward pricing algorithm T 0 T 1 T β 3 T β 2 T β 1 T β u c β 2 (T β 2 ) GPU1 N d C β 1 u h β 2 (T β 2 ) GPU2 0 uc β 3 (T β 3 ) u h β 3 (T β 3 ) GPU1 GPU2 N d C β 2 + u c β 2 (T β 2 ) ( max u β 2 e (T β 2 ),u h ) β 2 (T β 2 ) } {{ } (u β 2 c ) (T β 2 )+uf β 2 (T β 2 ) 5/18

9 Power Reverse Dual Currency (PRDC) swaps Backward pricing algorithm T 0 T 1 T β 3 T β 2 T β 1 T β u c β 2 (T β 2 ) GPU1 N d C β 1 u h β 2 (T β 2 ) GPU2 0 uc β 3 (T β 3 ) u h β 3 (T β 3 ) GPU1 GPU2 N d C β 2 + u c β 2 (T β 2 ) ( max u β 2 e (T β 2 ),u h ) β 2 (T β 2 ) } {{ } (u β 2 c ) (T β 2 )+uf β 2 (T β 2 ) uc 0 (T 0 ) u h 0 (T 0 ) GPU1 GPU2 N d C 1 + u c 2 (T 1 ) ( max u 1 e (T 1 ),u h ) 1 (T 1 ) } {{ } (u 2 c ) (T 2 )+uf 2 (T 2 ) u f α(t α): obtained by the fixed notional method, not by solving a PDE Price of the underlying PRDC swap: u f 0(T 0)+u c 0(T 0) Price of the Bermudan cancelable PRDC swap: (u f 0(T 0)+u c 0(T 0))+u h 0(T 0) 5/18

10 The model and the associated PDE The pricing model Consider the following model under domestic risk neutral measure ds(t) s(t) =(r d(t) r f (t))dt+γ(t,s(t))dw s(t), dr d (t)=(θ d (t) κ d (t)r d (t))dt +σ d (t)dw d (t), dr f (t)=(θ f (t) κ f (t)r f (t) ρ fs (t)σ f (t)γ(t,s(t)))dt +σ f (t)dw f (t), r i(t),i = d,f: domestic and foreign interest rates with mean reversion rate and volatility functions κ i(t) and σ i(t) s(t): the spot FX rate (units domestic currency per one unit foreign currency) W d (t),w f (t), and W s(t) are correlated Brownian motions with dw d (t)dw s(t) = ρ ds dt, dw f (t)dw s(t) = ρ fs dt, dw d (t)dw f (t) = ρ df dt ( s(t) ) ς(t) 1 Local volatility function γ(t, s(t)) = ξ(t) L(t) - ξ(t): relative volatility function - ς(t): constant elasticity of variance (CEV) parameter - L(t): scaling constant (e.g. the forward FX rate F(0,t)) 6/18

11 The model and the associated PDE The 3-D pricing PDE Let u = u(s,r d,r f,t) be the value of a security at time t, with a terminal payoff measurable with respect to the σ-algebra at maturity time T end and without intermediate payments. On R 3 + [T start,t end ), u satisfies the PDE u t u +Lu t +(r d r f )s u s + ( θ d (t) κ d (t)r d ) u r d + ( ) u θ f (t) κ f (t)r f ρ fs σ f (t)γ(t,s(t)) r f γ2 (t,s(t))s 2 2 u s σ2 d(t) 2 u + 1 rd 2 2 σ2 f(t) 2 u rf 2 +ρ ds σ d (t)γ(t,s(t))s 2 u r d s +ρ fs σ f (t)γ(t,s(t))s 2 u r f s +ρ dfσ d (t)σ f (t) 2 u r d r f r d u = 0 Derivation: Multi-dimensional Itô s formula Boundary conditions: Dirichlet-type stopped process boundary conditions Backward PDE: the change of variable τ = T end t Difficulties: High-dimensionality, cross-derivative terms 7/18

12 GPU-based parallel numerical methods Discretization Space: Second-order central finite differences on uniform mesh Time: ADI technique based on Hundsdorfer and Verwer (HV) approach u m : the vector of approximate values A m 0 : matrix of all mixed derivatives terms; Am i,i = 1,...,3: matrices of the second-order spatial derivative in the s-, r d -, and r s- directions, respectively gi m,i = 0,...,3 : vectors obtained from the boundary conditions A m = 3 i=0 Am i ; g m = 3 i=0 gm i Timestepping HV scheme from time t m 1 to t m: Phase 1: (I 1 2 τam i ) } {{ } Phase 2: Â m i v 0 = u m 1 + τ(a m 1 u m 1 +g m 1 ), v i = v i τam 1 i u m τ(gm i g m 1, i = 1,2,3, i ) } {{ } v i ṽ 0 = v τ(am v 3 A m 1 u m 1 )+ 1 2 τ(gm g m 1 ), (I 1 2 τam i )ṽ i = ṽ i τam i v 3, i = 1,2,3, u m = ṽ 3. 8/18

13 GPU-based parallel numerical methods Parallel algorithm overview Focus on the parallelism within one timestep via a parallelization of the HV scheme With respect to the CUDA implementation, the two phases of the HV scheme are essentially the same. Hence, we focus on describing the parallelization of the first phase. Main steps of Phase 1: Step a.1: computes the matrices A m i, i = 0,1,2,3, the matrices Âm i, i = 1,2,3, the products A m i u m 1, i = 0,1,2,3, and the vector v 0 ; Step a.2: computes v 1 and solves Âm 1 v 1 = v 1 ; Step a.3: computes v 2 and solves Âm 2 v 2 = v 2 ; Step a.4: computes v 3 and solves Âm 3 v 3 = v 3 ; Steps a.2, a.3, and a.4 are inherently parallelizable (block-diagonal, with tridiagonal blocks) Step a.1, on the other hand, the computation of the products A m i u m 1 is more difficult to parallelize efficiently. 9/18

14 GPU-based parallel numerical methods Phase 1 - Step a.1: Overview Grid partitioning/assignment of gridpoints to threads computational grid of size n p q is partitioned into 3-D blocks of size n b p b q, each of which can be viewed as consisting of q 2-D blocks, referred to as tiles, of size n b p b. A grid of ceil(n/n b ) ceil(p/p b ) threadblocks is invoked, each of which consists of an n b p b array of threads. Each threadblock does a q-iteration loop, processing an n b p b tile at each iteration, i.e. each thread does a q-iteration loop, processing one gridpoint at each iteration n b = 4 p b = 2 q = 10 p = 8 n = 8 r f s r d Computation details of a threadblock at each iteration loads from the global memory to its shared memory the components of u m 1 corresponding to a tile, and the associated halo values; computes the respective rows of matrices A m i and Âm i, and respective entries of A m i u m 1 and v 0 copies new rows and new values from the shared memory to the global memory 10/18

15 GPU-based parallel numerical methods Phase 1 - Step a.1: Computation of v 0 During the kth iteration, each threadblock 1. loads from the global memory into its shared memory the old data (vector u m 1 ) corresponding to the (k +1)st tile, and the associated halos (in the s- and r d -directions), if any, 2. computes and stores new values for the kth tile using data of the (k 1)st, kth and (k +1)st tiles, and of the associated halos, if any, 3. copies the newly computed data of the kth tile from the shared memory to the global memory, and frees the shared memory locations taken by the data of the (k 1)st tile, and associated halos, if any, so that they can be used in the next iteration. s North r d halo data West South Figure: An example of n b p b = 8 8 tiles with halos. East Memory coalescing: fully coalesced loading for interior data of a tile and halos along the s-direction (North and South), but not for halos along the r d -direction (East and West) 11/18

16 GPU-based parallel numerical methods Phase 1 - Steps a.2/a.3/a.4: Tridiagonal solves Motivated by the block structure of the tridiagonal matrices Âm i = I 1 2 τam i Based on the parallelism arising from independent tridiagonal solutions, rather than the parallelism within each one When solved in one direction, the data are partitioned with respect to the other two Assign each tridiagonal system to one of the threads Example: (I 1 2 τam 1 ) } {{ } Â m 1 v 1 = v τam 1 1 u m τ(gm 1 g m 1 1 ) } {{ } v 1 i. Partition Âm 1 and v 1 into pq independent n n tridiagonal systems ii. Assign each tridiagonal system to one of pq threads. iii. Use multiple 2-D threadblocks of identical size r t c t, i.e. a 2-D grid of threadblocks of size ceil( p r t ) ceil( q c t ) is invoked. Memory coalescence: fully achieved for the tridiagonal solves in the r d - and r f - directions, but not in the s-direction. Could be improved by renumbering gridpoints between steps of the first phase., 12/18

17 Numerical results Market Data Two economies: Japan (domestic) and US (foreign) Initial spot FX rate: s(0) = 105 Interest rate curves, volatility parameters, correlations: P d (0,T) = exp( 0.02 T) σ d (t) = 0.7% κ d (t) = 0.0% ρ df = 25% ρ ds = 15% P f (0,T) = exp( 0.05 T) σ f (t) = 1.2% κ f (t) = 5.0% ρ fs = 15% Local volatility function: period period (years) (ξ(t)) (ς(t)) (years) (ξ(t)) (ς(t)) (0 0.5] 9.03% -200% (7 10] 13.30% -24% (0.5 1] 8.87% -172% (10 15] 18.18% 10% (1 3] 8.42% -115% (15 20] 16.73% 38% (3 5] 8.99% -65% (20 25] 13.51% 38% (5 7] 10.18% -50% (25 30] 13.51% 38% Truncated computational domain: {(s,r d,r f ) [0,S] [0,R d ] [0,R f ]} {[0,305] [0,0.06] [0,0.15]} 13/18

18 Numerical results Specification Bermudan cancelable PRDC swaps Principal: N d (JPY); Settlement/Maturity dates: 23 Apr. 2010/23 Nov Details: paying annual PRDC coupon, receiving JPY LIBOR Year coupon funding (FX options) leg 1 s(1) max(c f F(0,1) c d,0)n d L d (0,1)N d s(29) max(c f F(0,29) c d,0)n d L d (28,29)N d Leverage level level low medium high c f 4.5% 6.25% 9.00% c d 2.25% 4.36% 8.10% The payer has the right to cancel the swap on each of {T α} β 1 α=1, β = 30 (years) Architectures Xeon running at 2.0GHz host system with a NVIDIA Tesla S870 (four Tesla C870 GPUs, 16 multi-processors, each containing 8 processors running at 1.35GHz, and 16 KB of shared memory) The tile sizes are chosen to be n b p b 16 4 (for Step a.1), and r t c t 16 4 (for Steps a.2, a.3, a.4), which appears to be optimal on Tesla C /18

19 Numerical results Prices and convergence underlying swap cancelable swap leverage m n p q value change ratio value change ratio (t) (s) (r d ) (r f ) (%) (%) low e e e e e e medium e e e e e e high e e e e e e Computed prices and convergence results for the underlying swap and cancelable swap with the FX skew model 15/18

20 Numerical results Parallel speedup underlying swap (one Tesla C870) m n p q value CPU GPU speed (t) (s) (r d ) (r f ) (%) time (s.) time (s.) up cancelable swap (two Tesla C870) m n p q value CPU GPU speed (t) (s) (r d ) (r f ) (%) time (s.) time (s.) up Computed prices and timing results for the underlying swap and cancelable swap for the low-leverage case 16/18

21 Summary and future work Summary and future work Summary GPU-based algorithm for pricing exotic cross-currency interest rate derivatives under a FX local volatility skew model via a PDE approach, with strong emphasis on Bermudan cancelable PRDC swaps The parallel algorithm is based on i. partitioning the pricing of cancelable PRDC swaps into two entirely independent pricing subproblems in each period of the tenor structure ii. efficient parallelization on GPUs of the HV ADI scheme at each timestep for the efficient solution of each of these subproblems Results indicate speedup of 44 with two Tesla C870, for the cancellable swap. Ongoing projects Exotic features: knockout, FX-TARN (interesting) GPU-based parallel methods for pricing multi-asset American options (penalty + ADI) Future work Numerical methods: non-uniform/adaptive grids, higher-order ADI schemes Modeling: stochastic models/regime switch for the volatility of the spot FX rate, multi-factor models for the short rates Parallelization: extension to multi-gpu platforms 17/18

22 Summary and future work Thank you! 1 D. M. Dang, C. C. Christara, K. R. Jackson and A. Lakhany (2009) A PDE pricing framework for cross-currency interest rate derivatives Available at 2 D. M. Dang (2009) Pricing of cross-currency interest rate derivatives on Graphics Processing Units Available at 3 D. M. Dang, C. C. Christara and K. R. Jackson (2010) GPU pricing of exotic cross-currency interest rate derivatives with a foreign exchange volatility skew model Available at 4 D. M. Dang, C. C. Christara and K. R. Jackson (2010) Parallel implementation on GPUs of ADI finite difference methods for parabolic PDEs with applications in finance Available at More at 18/18