The BENCHOP project The BENCHmarking project in Option Pricing A. - PowerPoint PPT Presentation

The BENCHOP project The BENCHmarking project in Option Pricing ´ A. Leitao, S. Jain and C. W. Oosterlee Delft University of Technology - Centrum Wiskunde & Informatica Reading group, September 8, 2017 Reading group, September 8, 2017 1 / ´ Alvaro Leitao (CWI & TUDelft) The BENCHOP project 28

Introduction 1 Problems formulation 2 Our contribution 3 Numerical results 4 Reading group, September 8, 2017 2 / ´ Alvaro Leitao (CWI & TUDelft) The BENCHOP project 28

The BENCHOP project The purpose and aim of BENCHOP is to provide sets of benchmark problems. Facilitating comparison and evaluation of different methods. Expecting that future papers in the financial field will compare method performances with the methods in BENCHOP. Contributing to a more uniform comparison and understanding of different methods’ pros and cons. Results published in a journal articles. This is the second edition. The results of the first edition can be found in [vSHL + 15]. Reading group, September 8, 2017 3 / ´ Alvaro Leitao (CWI & TUDelft) The BENCHOP project 28

Aspects 2nd edition Implementation should be in Matlab. Preferable, use of high-performance features: parallel computing toolbox. ◮ parfor. ◮ GPU array. Two categories: ◮ Basket options. ◮ Stochastic and local volatility. Benchmark: Error (accuracy) in the solution as a function of CPU (GPU) time. Reading group, September 8, 2017 4 / ´ Alvaro Leitao (CWI & TUDelft) The BENCHOP project 28

Basket options - Problem formulation Underlying prices modelled by a multidimensional Merton model: � � d S i ( t ) e J i ( t ) − 1 S i ( t ) = ( r − λκ i ) d t + d B i ( t ) + d P ( t ) . d B i ( t ) , i = 1 , . . . , d is a multidimensional Brownian motion with covariance matrix Σ B ij = σ B i ( S i , t ) σ B j ( S j , t ) ρ B ij . P ( t ) is a Poisson process with the arrival rate λ . J i ( t ) , i = 1 , . . . , d follows a multivariate normal distribution with mean values µ J i and covariance matrix Σ J ij = σ J i ( S i , t ) σ J j ( S j , t ) ρ J ij . The expected jump of the i th component is   � � d � i + 1 e J i ( t ) − 1  µ J σ J i σ J j ρ J  − 1 . κ i = E = exp ij 2 j =1 When λ = 0 and σ i constant: multi Black-Scholes model. Reading group, September 8, 2017 5 / ´ Alvaro Leitao (CWI & TUDelft) The BENCHOP project 28

Basket options - Problems For all the problems: Price u . For some problems also: ∆ = ∂ u ∂ S i and V = ∂ u ∂σ i 1 European spread option g ( S ) = max { S 1 − S 2 − K , 0 } , with settings: GBM, S i = 100, r = 0 . 03, T = 1, ρ = 0 . 5 and K = 5. Two problems: constant volatility ( σ i = 0 . 15) or given by the function σ i ( S i , t ) = 0 . 15 + 0 . 15(0 . 5 + 2 t ) ( S i / 100 − 1 . 2) 2 ( S i / 100) 2 + 1 . 44 . Reading group, September 8, 2017 6 / ´ Alvaro Leitao (CWI & TUDelft) The BENCHOP project 28

Basket options - Problems 2 American put on the minimum of two assets g ( S ) = max { K − min { S 1 , S 2 } , 0 } , with settings: S i = 40, r = 0 . 05, σ i = 0 . 3 T = 0 . 5, ρ = 0 . 5 and K = 40. Two problems: without jumps (Black-Scholes) or with jumps ( µ J i = − 0 . 5, σ J i = 0 . 4, ρ J ij = 0 . 5 and λ = 0 . 4). 3 Arithmetic basket options on 3 and 10 assets � � d � K − 1 g ( S ) = max S i , 0 , d i =1 with settings: GBM, S i = 40, r = 0 . 06, σ i = 0 . 2, T = 1 and K = 40. Four problems: European/American and low constant correlation ( ρ = 0 . 25), European/American high variable correlations ( ρ ij = 0 . 9 | i − j | ). Reading group, September 8, 2017 7 / ´ Alvaro Leitao (CWI & TUDelft) The BENCHOP project 28

Basket options - Problems 4 European arithmetic basket options on four assets � � � d K − 1 g ( S ) = max S i , 0 , d i =1 with settings: GBM, S i = 40, r = 0 . 06, σ i = 0 . 3, T = 1 and K = 40. Correlation matrix:   1 0 . 3 0 . 4 0 . 5   0 . 3 1 0 . 2 0 . 25   ρ =  .  0 . 4 0 . 2 1 0 . 3 0 . 5 0 . 25 0 . 3 1 Reading group, September 8, 2017 8 / ´ Alvaro Leitao (CWI & TUDelft) The BENCHOP project 28

Basket options - Problems 5 European/American arithmetic basket options on five assets � � d � g ( S ) = max K − w i S i , 0 , i =1 with settings: GBM, S i = 1, r = 0 . 05, σ = [0 . 518 , 0 . 648 , 0 . 623 , 0 . 570 , 0 . 530], w = [0 . 381 , 0 . 065 , 0 . 057 , 0 . 270 , 0 . 227], T = 1 and K = 1. Correlation matrix:   1 0 . 79 0 . 82 0 . 91 0 . 84   0 . 79 1 0 . 73 0 . 80 0 . 76     ρ = 0 . 82 0 . 73 1 0 . 77 0 . 72 .     0 . 91 0 . 80 0 . 77 1 0 . 90 0 . 84 0 . 76 0 . 72 0 . 90 1 Reading group, September 8, 2017 9 / ´ Alvaro Leitao (CWI & TUDelft) The BENCHOP project 28

Stochastic and local volatility - Problems European call options. Three prices: in-the-money, at-the-money and out-the-money. 1 SABR model The formal definition of the SABR model reads d S ( t ) = σ ( t ) S β ( t ) d W S ( t ) , S (0) = S 0 exp ( rT ) , d σ ( t ) = ασ ( t ) d W σ ( t ) , σ (0) = σ 0 , where S ( t ) = ¯ S ( t ) exp ( r ( T − t )). Correlation between the Brownian motions, ρ . Two parameter sets: T = 2, r = 0 . 0, S 0 = 0 . 5, σ 0 = 0 . 5, α = 0 . 4, β = 0 . 5, ρ = 0. T = 10, r = 0 . 0, S 0 = 0 . 07, σ 0 = 0 . 4, α = 0 . 8, β = 0 . 5, ρ = − 0 . 6. European call option payoff (max( S ( T ) − K i ( T ) , 0)) with √ K i ( T ) = S (0) exp(0 . 1 × T × δ i ) , δ i = − 1 . 5 , − 1 . 0 , − 0 . 5 , 0 . 0 , 0 . 5 , 1 . 0 , 1 . 5 . Reading group, September 8, 2017 10 / ´ Alvaro Leitao (CWI & TUDelft) The BENCHOP project 28

Stochastic and local volatility - Problems 2 Quadratic local stochastic volatility model � d S ( t ) = rS ( t ) d t + V ( t ) f ( S ( t )) d W S ( t ) , � d V ( t ) = κ ( η − V ( t )) d t + σ V ( t ) d W V ( t ) , 2 α s 2 + β s + γ . with f ( s ) = 1 3 Heston-Hull-White model � d S ( t ) = R ( t ) S ( t ) d t + V ( t ) S ( t ) d W S ( t ) , � d V ( t ) = κ ( η − V ( t )) d t + σ 1 V ( t ) d W V ( t ) , d R ( t ) = a ( b − V ( t )) d t + σ 2 d W R ( t ) . Reading group, September 8, 2017 11 / ´ Alvaro Leitao (CWI & TUDelft) The BENCHOP project 28

Our contribution We propose Monte Carlo-based methods. For Basket options: Stochastic Grid Bundling method (SGBM). For SABR model: ◮ The mSABR simulation scheme [LGO17]. ◮ Multi Level Monte Carlo, MLMC, to exploit parallel features. Reading group, September 8, 2017 12 / ´ Alvaro Leitao (CWI & TUDelft) The BENCHOP project 28

Stochastic Grid Bundling Method Early-exercise pricing method [JO15]. 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) . Reading group, September 8, 2017 13 / ´ Alvaro Leitao (CWI & TUDelft) The BENCHOP project 28

Stochastic Grid Bundling Method 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 ))) . Reading group, September 8, 2017 14 / ´ Alvaro Leitao (CWI & TUDelft) The BENCHOP project 28

Stochastic Grid Bundling Method 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 = t m ( k ) φ k ( S t m ) . Z 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 Reading group, September 8, 2017 15 / ´ Alvaro Leitao (CWI & TUDelft) The BENCHOP project 28

Stochastic Grid Bundling Method � � 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. Reading group, September 8, 2017 16 / ´ Alvaro Leitao (CWI & TUDelft) The BENCHOP project 28