Exact Sampling of Jump-Diffusion Processes

Transcription

1 1 Exact Sampling of Jump-Diffusion Processes and Dmitry Smelov Management Science & Engineering Stanford University

2 Exact Sampling of Jump-Diffusion Processes 2 Jump-Diffusion Processes Ubiquitous in finance and economics Price models: equity, commodity, rates, energy, FX Default timing models (jump component) We develop a method for the exact sampling of a jump-diffusion process with state-dependent drift, volatility, jump intensity, and jump size Leads to unbiased simulation estimators of derivative prices, risk measures, and other quantities The method extends an innovative acceptance/rejection scheme developed by Beskos & Roberts (2005, AAP) and Chen (2009) for diffusions

3 Exact Sampling of Jump-Diffusion Processes 3 Jump-Diffusion Fix a filtered probability space (Ω, F, F, P) For suitable functions µ and σ, consider the SDE dx t = µ(x t )dt + σ(x t )dw t + dj t where W is a standard Brownian motion and J is a jump process J t = n N t c(x T n, Z n ) N is a counting process with event times (T n ) and intensity λ t = Λ(X t ) for a suitable function Λ (Z n ) is a sequence of mark variables valued in E c : D X E D X determines the jump magnitudes J is self-exciting; dependence between jump sizes and frequency

4 Exact Sampling of Jump-Diffusion Processes 4 Jump-Diffusion For a suitable function f and a horizon T we wish to calculate E { f(x T, (J t ) t T ) } Price of a derivative written on X T Price of a credit derivative written on J Alternative approaches Analytical solutions: rare; e.g. Merton and Kou models Semi-analytical transform approaches: AJDs, LQJDs, Lévy PIDE approaches: fewer restrictions Monte Carlo simulation: perhaps the widest scope

5 Exact Sampling of Jump-Diffusion Processes 5 Discretizing the Jump-Diffusion Standard approach to simulating X X is approximated on a discrete-time grid Euler or higher order scheme for diffusion component Thinning or time-scaling scheme for jump times T n Simulation estimator is biased Magnitude of the bias? Confidence intervals? Convergence of scheme? Allocation of computational budget?

6 Exact Sampling of Jump-Diffusion Processes 6 Discretizing the Jump-Diffusion Time-scaling for jumps: T n d = inf{t : t 0 Λ(X s)ds E E n } Intensity Loss Time Time S 6 (ω) S 5 (ω) S 4 (ω) S 3 (ω) S 2 (ω) S 1 (ω) Compensator T 1 (ω) T 2 (ω) T 3 (ω) T 4 (ω) T 5 (ω) T 6 (ω) 0

7 Exact Sampling of Jump-Diffusion Processes 7 Exact Sampling We provide an exact sampling scheme for X that avoids discretization entirely, and leads to unbiased simulation estimators First step: transform X into a unit-diffusion SDE Lamperti transform F (x) = x X 0 1 σ(u) du Then Y t = F (X t ) solves dy t = α(y t )dt + dw t + dj Y t where α(y) = µ(f 1 (y)) σ(f 1 (y)) 1 2 σ (F 1 (y)) Jt Y = (Y T n, Z n ) n N t for (y, z) = F (F 1 (y) + c(f 1 (y), z)) y

8 Exact Sampling of Jump-Diffusion Processes 8 Exact Sampling Assumptions 1. σ(x) is bounded away from 0 2. µ(x) and Λ(x) are continuously differentiable and σ(x) is twice continuously differentiable. 3. Conditions on α(y) and c(x, z) guaranteeing that Y does not reach the boundaries in finite time (known).

9 Exact Sampling of Jump-Diffusion Processes 9 Acceptance/Rejection Scheme The exact method uses an A/R scheme Suppose we want to sample from a density f(y) and there is another density g(y) and a constant c > 0 such that A/R scheme 1. Draw a sample Y from g c f(y) g(y) 1 2. Draw a Bernoulli variable I with success probability c f(y ) g(y ) 3. Accept Y as a sample from f if I = 1

10 Exact Sampling of Jump-Diffusion Processes 10 A/R for Continuous Y Beskos & Roberts (2005, AAP) We wish to sample Y T using the A/R scheme Under Novikov and additional boundedness conditions, [ ( f YT (y) T ] E exp φ(w s )ds) W T = y =: H(y) g(y) where φ = (α + α 2 )/2 and g(y) is a proposal density 0 For the A/R step, note that H(y) = P(M T = 0 W T = y) where M is a doubly-stochastic Poisson process with intensity φ(w s ) The Bernoulli indicator {I = 1} = {M T = 0} can be generated by sampling M T given W T = Y with Y drawn from g If φ(x) π, then this can be done by thinning a Poisson process with intensity π (requires sampling from BB)

11 Exact Sampling of Jump-Diffusion Processes 11 A/R for Continuous Y Localization: Chen (2009)

12 Exact Sampling of Jump-Diffusion Processes 12 A/R for Continuous Y Generating the first piece of Y Target exit time ζ 1 = inf{t 0 : Y t Y 0 L} for L > 0 Proposal exit time τ = inf{t 0 : W t L} The LR between (ζ 1, Y 1 Y 0 ) and (τ, W τ ) is proportional to [ ( τ ) ] E exp φ(y 0 + W s )ds τ, W τ 0 Because of the continuity assumptions, φ(y 0 + W s ) is bounded and thinning can always be used in the acceptance test Need to sample from Brownian meander The density of τ is known and can be sampled from using the method of Burq & Jones (2006)

13 Exact Sampling of Jump-Diffusion Processes 13 A/R for Jump-Diffusion Y Introducing jumps

14 Exact Sampling of Jump-Diffusion Processes 14 A/R for Jump-Diffusion Y Generating the first piece of Y Sample τ as before using a bound L. Suppose τ T. Sample candidate jump times (σ 1,..., σ n ) of Y during [0, τ] from a Poisson process with intensity π Λ(F 1 (Y 0 + y)), y [ L, L] Sample (W σ1,..., W σn, W τ ) from a Brownian meander Perform acceptance tests for the σ i by drawing Bernoulli variables with success probabilities Λ(F 1 (Y 0 + W σi ))/π Perform acceptance test for (ζ 1, Y σ1 Y 0,..., Y σ Y 0 ) given the k proposal (τ, W σ1,..., W σk ), where σ k is the first candidate jump time of Y accepted in the previous step If the skeleton is accepted, draw mark Z 1 and compute Y T1 = Y σ + (Y k σ, Z 1 ) k

15 Exact Sampling of Jump-Diffusion Processes 15 A/R for Jump-Diffusion Y Likelihood ratio The LR for the last acceptance test is proportional to [ ( σk e A(Y 0+W σk ) E exp φ(y 0 + W u )du) τ, W,..., W σ1 σ k where A(x) = x 0 α(u)du and φ = (α + α 2 )/2 0 Generate Bernoulli indicator by generating the jump times of a doubly-stochastic Poisson process with intensity φ(y 0 + W u ) Thinning applies Sample from Brownian meander ]

16 Exact Sampling of Jump-Diffusion Processes 16 Numerical examples Jump-extended CEV model of Carr & Linetsky (2006): dx t = (r + Λ(X t ))X t dt + σ(x t )dw t + dj t where X 0 > 0 and for a > 0, b 0, c 1 2 and β < 0 Λ(x) = b + ca 2 x 2β is the jump intensity σ(x) = ax β+1 is the volatility c(x, z) = xz for z (0, 1) is the jump size The firm defaults at the first jump time T 1 of J The default intensity λ = Λ(X) is unbounded Violates boundedness hypothesis of thinning scheme for jumps (Glasserman & Merener (2003), Casella & Roberts (2010)) Convergence order of discretization scheme unknown

17 Exact Sampling of Jump-Diffusion Processes 17 Numerical examples We are interested in X during [0, T T 1 ] for some T > 0 The target functional takes the form Examples f(x T, (J t ) t T ) = h 1 (X T )1 {JT =0} + h 2 (X T )1 {JT 0} Probability of survival to T : h 1 (x) = 1 and h 2 (x) = 0 European put with strike K and maturity T : h 1 (x) = e rt (K x) + and h 2 (x) = Ke rt Carr & Linetsky (2006) provide analytical solutions to these and other quantities

18 Exact Sampling of Jump-Diffusion Processes 18 Numerical examples We estimate the price of a European put on X We consider the RMSE = Bias 2 + SE 2 for The exact method, for which the bias is 0 The discretization method (Euler plus time-scaling for jumps) The number of time steps is equal to the square root of the number of trials (Duffie & Glynn (1995)) The bias is estimated using 10 million trials Matlab implementation (favors discretization)

19 Exact Sampling of Jump-Diffusion Processes 19 Numerical examples X 0 = 50, β = 1, r = 0.05, a = 50/4, b = 0, c = 0.5, T = 1, strike 5, analytical value Method Trials Steps Value Bias SE RMSE Time (sec) Exact 10K N/A Exact 20K N/A Exact 40K N/A Exact 100K N/A Exact 500K N/A Euler 10K Euler 20K Euler 40K Euler 100K Euler 500K

20 Exact Sampling of Jump-Diffusion Processes 20 Numerical Examples Convergence of RMSEs (log-log plot), strike K = 5

21 Exact Sampling of Jump-Diffusion Processes 21 Numerical examples X 0 = 50, β = 1, r = 0.05, a = 50/4, b = 0, c = 0.5, T = 1, strike 50, analytical value Method Trials Steps Value Bias SE RMSE Time (sec) Exact 10K N/A Exact 20K N/A Exact 40K N/A Exact 100K N/A Exact 500K N/A Euler 10K Euler 20K Euler 40K Euler 100K Euler 500K

22 Exact Sampling of Jump-Diffusion Processes 22 Numerical Examples Convergence of RMSEs (log-log plot), strike K = 50

23 Exact Sampling of Jump-Diffusion Processes 23 Extensions The target functional can take the form E{f((X t ) t S, (J t ) t T )} for a discrete set S of times t [0, T ] Treatment of certain path-dependent payoffs The intensity can take the form λ t = Λ(X t, J t, t)

24 Exact Sampling of Jump-Diffusion Processes 24 Conclusions We develop a method for the exact sampling of a one-dimensional jump-diffusion process with state-dependent drift, volatility, jump intensity and jump size Only mild conditions on the coefficients are required Numerical experiments indicate the advantages of the method over a conventional discretization scheme Future research Efficiency: choice of localization bound Extension to multiple dimensions: stochastic volatility with state-dependent jumps