Numerical Solution of Stochastic Differential Equations with Jumps in Finance Eckhard Platen School of Finance and Economics and School of Mathematical Sciences University of Technology, Sydney Kloeden, P.E.&Pl, E.: Numerical Solution of Stochastic Differential Equations Springer, Applications of Mathematics 23 (1992,1995,1999). Pl, E.&Heath, D.: A Benchmark Approach to Quantitative Finance, Springer Finance (2010). Pl, E.&Bruti-Niberati, N.: Numerical Solution of SDEs with Jumps in Finance, Springer, Stochastic Modelling and Applied Probability 64 (2010).
Jump-Diffusion Multi-Factor Models Björk, Kabanov & Runggaldier (1997) Øksendal & Sulem (2005) Markovian explicit transition densities in special cases benchmark framework discrete time approximations suitable for simulation Markov chain approximations c Copyright E. Platen SDE Jump 1
Pathwise Approximations: scenario simulation of entire markets testing statistical techniques on simulated trajectories filtering hidden state variables Pl. & Runggaldier (2005, 2007) hedge simulation dynamic financial analysis extreme value simulation stress testing = higher order strong schemes predictor-corrector methods c Copyright E. Platen SDE Jump 2
Probability Approximations: derivative prices sensitivities expected utilities portfolio selection risk measures long term risk management = Monte Carlo simulation, higher order weak schemes, predictor-corrector, variance reduction, Quasi Monte Carlo, or Markov chain approximations, lattice methods c Copyright E. Platen SDE Jump 3
Essential Requirements: parsimonious models respect no-arbitrage in discrete time approximation numerically stable methods efficient methods for high-dimensional models higher order schemes, predictor-corrector c Copyright E. Platen SDE Jump 4
Continuous and Event Driven Risk Wiener processes W k, k {1,2,..., m} counting processes p k intensity h k jump martingaleq k dw m+k t = dq k t = ( dp k t hk t dt)( h k t ) 1 2 k {1,2,...,d m} W t = (W 1 t,...,wm t,q 1 t,...,qd m t ) c Copyright E. Platen SDE Jump 5
Primary Security Accounts Assumption 1 ds j t = S j t ( b j,k t a j t dt + d k=1 h k m t b j,k t dw k t ) k {m + 1,...,d}. Assumption 2 Generalized volatility matrixb t = [b j,k t ] d j,k=1 invertible. c Copyright E. Platen SDE Jump 6
market price of risk θ t = (θ 1 t,...,θd t ) = b 1 t [a t r t 1] primary security account ds j t = S j t ( r t dt + d k=1 b j,k t (θ k t dt + dwk t ) ) portfolio ds δ t = d j=0 δ j t ds j t c Copyright E. Platen SDE Jump 7
fraction π j δ,t = δj t S j t S δ t portfolio ds δ t = Sδ t { } r t dt + π δ,t b t(θ t dt + dw t ) c Copyright E. Platen SDE Jump 8
Assumption 3 h k m t > θ k t generalized GOP volatility c k t = θ k t for k {1,2,...,m} θ k t 1 θ k t (hk m t ) 1 2 for k {m + 1,...,d} GOP fractions π δ,t = (π 1 δ,t,...,πd δ,t ) = ( c t b 1 t ) c Copyright E. Platen SDE Jump 9
Growth Optimal Portfolio ( ) ds δ t = S δ t r t dt + c t (θ tdt + dw t ) optimal growth rate m g δ t = r t + 1 2 d k=1 k=m+1 (θ k t )2 h k m t ln 1 + θ k t h k m t θ k t + θk t h k m t c Copyright E. Platen SDE Jump 10
benchmarked portfolio Ŝ δ t = Sδ t S δ t Theorem 4 Any nonnegative benchmarked portfolio Ŝ δ is an (A, P)-supermartingale. = no strong arbitrage but there may exist: free lunch with vanishing risk (Delbaen & Schachermayer (2006)) free snacks or cheap thrills (?)) c Copyright E. Platen SDE Jump 11
Multi-Factor Model model mainly: benchmarked primary security accounts j {0,1,...,d} Ŝ j t = Sj t S δ t supermartingales, often SDE driftless, local martingales, sometimes martingales c Copyright E. Platen SDE Jump 12
savings account S 0 t = exp { t 0 } r s ds = GOP S δ t = S0 t Ŝ 0 t = stock S j t = Ŝ j t S δ t additionally dividend rates foreign interest rates c Copyright E. Platen SDE Jump 13
Example Black-Scholes Type Market dŝ j t = Ŝ j t d k=1 σ j,k t dw k t h j t, σ j,k t, r t c Copyright E. Platen SDE Jump 14
Examples Merton jump-diffusion model dx t = X t (µdt + σdw t + dp t ), N t X t = X 0 e (µ 1 2 σ2 )t+σw t i=1 ξ i Bates model ds t = S t ( αdt + V t dw S t + dp t ) dv t = ξ(η V t )dt + θ V t dw V t c Copyright E. Platen SDE Jump 15
3 2.5 2 1.5 1 0.5 0 0 5 10 15 20 time Figure 1: Simulated benchmarked primary security accounts. c Copyright E. Platen SDE Jump 16
10 9 8 7 6 5 4 3 2 1 0 0 5 10 15 20 time Figure 2: Simulated primary security accounts. c Copyright E. Platen SDE Jump 17
4.5 GOP EWI 4 3.5 3 2.5 2 1.5 1 0.5 0 5 10 15 20 time Figure 3: Simulated GOP and EWI ford = 50. c Copyright E. Platen SDE Jump 18
4.5 GOP index 4 3.5 3 2.5 2 1.5 1 0.5 0 5 10 15 20 time Figure 4: Simulated accumulation index and GOP. c Copyright E. Platen SDE Jump 19
Diversification diversified portfolios π j δ,t K 2 d 1 2 +K 1 c Copyright E. Platen SDE Jump 20
Theorem 5 In a regular market any diversified portfolio is an approximate GOP. Pl. (2005) robust characterization similar to Central Limit Theorem model independent c Copyright E. Platen SDE Jump 21
60 50 40 30 20 10 0 0 5 9 14 18 23 27 32 Figure 5: Benchmarked primary security accounts. c Copyright E. Platen SDE Jump 22
450 400 350 300 250 200 150 100 50 0 0 5 9 14 18 23 27 32 Figure 6: Primary security accounts under the MMM. c Copyright E. Platen SDE Jump 23
100 90 EWI GOP 80 70 60 50 40 30 20 10 0 0 5 9 14 18 23 27 32 Figure 7: GOP and EWI. c Copyright E. Platen SDE Jump 24
100 90 Market index GOP 80 70 60 50 40 30 20 10 0 0 5 9 14 18 23 27 32 Figure 8: GOP and market index. c Copyright E. Platen SDE Jump 25
fair security benchmarked security (A,P)-martingale fair minimal replicating portfolio fair nonnegative portfolio S δ with S δ τ = H τ = minimal nonnegative replicating portfolio fair pricing formula V Hτ (t) = S δ t E ( Hτ S δ τ ) A t No need for equivalent risk neutral probability measure! c Copyright E. Platen SDE Jump 26
Fair Hedging fair portfolio S δ t benchmarked fair portfolio martingale representation ( ) H τ Hτ = E A t + S δ τ S δ τ Ŝ δ t = E ( Hτ S δ τ d k=1 τ t ) A t x k H τ (s)dw k s + M H τ (t) M Hτ -(A,P)-martingale (pooled) E ([ M Hτ,W k] t) = 0 Föllmer & Schweizer (1991), Pl. & Du (2011) No need for equivalent risk neutral probability measure! c Copyright E. Platen SDE Jump 27
Simulation of SDEs with Jumps strong schemes (paths) Taylor explicit derivative-free implicit balanced implicit predictor-corrector weak schemes (probabilities) Taylor simplified explicit derivative-free implicit, predictor-corrector c Copyright E. Platen SDE Jump 28
intensity of jump process regular schemes = high intensity jump-adapted schemes = low intensity c Copyright E. Platen SDE Jump 29
SDE with Jumps dx t = a(t,x t )dt + b(t,x t )dw t + c(t,x t )dp t X 0 R d p t = N t : Poisson process, intensity λ < p t = N t i=1 (ξ i 1): compound Poisson, ξ i i.i.d r.v. Poisson random measure c(t,x t,v)p φ (dv dt) {(τ i,ξ i ),i = 1,2,...,N T } E c Copyright E. Platen SDE Jump 30
Numerical Schemes time discretization t n = n discrete time approximation Y n+1 = Y n + a(y n ) + b(y n ) W n + c(y n ) p n c Copyright E. Platen SDE Jump 31
Strong Convergence Applications: scenario analysis, filtering and hedge simulation strong order γ if ε s ( ) = E( XT Y N 2) K γ c Copyright E. Platen SDE Jump 32
Weak Convergence Applications: derivative pricing, utilities, risk measures weak order β if ε w ( ) = E(g(X T )) E(g(Y N )) K β c Copyright E. Platen SDE Jump 33
Literature on Strong Schemes with Jumps Pl (1982), Mikulevicius&Pl (1988) = γ {0.5, 1,...} Taylor schemes and jump-adapted Maghsoodi (1996, 1998) = strong schemes γ 1.5 Jacod & Protter (1998) = Euler scheme for semimartingales Gardoǹ (2004) = γ {0.5, 1,...} strong schemes Higham & Kloeden (2005) = implicit Euler scheme Bruti-Liberati & Pl (2007) = γ {0.5,1,...} explicit, implicit, derivative-free, predictor-corrector c Copyright E. Platen SDE Jump 34
Euler Scheme Euler scheme where Y n+1 = Y n + a(y n ) + b(y n ) W n + c(y n ) p n W n N(0, ) and p n = N tn+1 N tn Poiss(λ ) γ = 0.5 c Copyright E. Platen SDE Jump 35
Strong Taylor Scheme Wagner-Platen expansion = Y n+1 = Y n + a(y n ) + b(y n ) W n + c(y n ) p n + b(y n )b (Y n )I (1,1) +b(y n )c (Y n )I (1, 1) + {b(y n + c(y n )) b(y n )}I ( 1,1) +{c(y n + c(y n )) c(y n )}I ( 1, 1) with I (1,1) = 1 {( W 2 n) 2 }, I ( 1, 1) = 1 2 {( p n) 2 p n } I (1, 1) = N(t n+1 ) i=n(t n )+1 W τ i p n W tn, I ( 1,1) = p n W n I (1, 1) simulation jump times τ i : W τi = I (1, 1) and I ( 1,1) Computational effort heavily dependent on intensity λ c Copyright E. Platen SDE Jump 36
Derivative-Free Strong Schemes avoid computation of derivatives order1.0 derivative-free strong scheme c Copyright E. Platen SDE Jump 37
Implicit Strong Schemes wide stability regions implicit Euler scheme order1.0 implicit strong Taylor scheme c Copyright E. Platen SDE Jump 38
Predictor-Corrector Euler Scheme corrector Y n+1 = Y n + ( ) θā η (Ȳ n+1 ) + (1 θ)ā η (Y n ) n + ā η = a ηbb ( ) ηb(ȳ n+1 ) + (1 η)b(y n ) W n + p(t n+1 ) i=p(t n )+1 c(ξ i ) predictor Ȳ n+1 = Y n + a(y n ) n + b(y n ) W n + p(t n+1 ) i=p(t n )+1 c(ξ i ) θ, η [0, 1] degree of implicitness c Copyright E. Platen SDE Jump 39
Jump-Adapted Time Discretization t 0 t 1 t 2 t 3 = T regular τ 1 τ 2 jump times jump-adapted t 0 t 1 t 2 t 3 t 4 t 5 = T c Copyright E. Platen SDE Jump 40
Jump-Adapted Strong Approximations jump-adapted time discretisation jump times included in time discretisation jump-adapted Euler scheme and Y tn+1 = Y tn + a(y tn ) tn + b(y tn ) W tn Y tn+1 = Y tn+1 + c(y tn+1 ) p n γ = 0.5 c Copyright E. Platen SDE Jump 41
Merton SDE :µ = 0.05, σ = 0.2, ψ = 0.2, λ = 10, X 0 = 1, T = 1 1 0.8 0.6 X 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 T Figure 9: Plot of a jump-diffusion path. c Copyright E. Platen SDE Jump 42
0.0005 0.00025 0 Error -0.00025-0.0005-0.00075-0.001-0.00125 0 0.2 0.4 0.6 0.8 1 T Figure 10: Plot of the strong error for Euler(red) and 1.0 Taylor(blue) scheme. c Copyright E. Platen SDE Jump 43
Merton SDE :µ = 0.05, σ = 0.1, λ = 1, X 0 = 1, T = 0.5-10 Log 2 Error -15-20 -25 Euler EulerJA 1Taylor 1TaylorJA 15TaylorJA -10-8 -6-4 -2 0 Log 2 dt Figure 11: Log-log plot of strong error versus time step size. c Copyright E. Platen SDE Jump 44
Literature on Weak Schemes with Jumps Mikulevicius & Pl (1991) = jump-adapted order β {1, 2...} weak schemes Liu & Li (2000) = order β {1,2...} weak Taylor, extrapolation and simplified schemes Kubilius&Pl (2002) and Glasserman & Merener (2003) = jump-adapted Euler with weaker assumptions on coefficients Bruti-Liberati&Pl (2006) = jump-adapted orderβ {1,2...} derivative-free, implicit and predictor-corrector schemes c Copyright E. Platen SDE Jump 45
Simplified Euler Scheme Euler scheme = β = 1 simplified Euler scheme Y n+1 = Y n + a(y n ) + b(y n ) Ŵ n + c(y n )(ˆξ n 1) ˆp n if Ŵ n and ˆp n match the first 3 moments of W n and p n up to an O( 2 ) error = β = 1 P( W n = ± ) = 1 2 c Copyright E. Platen SDE Jump 46
Jump-Adapted Taylor Approximations jump-adapted Euler scheme = β = 1 jump-adapted order 2 weak Taylor scheme Y tn+1 = Y tn + a tn + b W tn + bb ) (( W tn ) 2 tn + a b Z tn 2 + 1 (aa + 12 ) 2 a b 2 2 t n + (ab + 12 ) b b 2 { W tn tn Z tn } and Y tn+1 = Y tn+1 + c(y tn+1 ) p n β = 2 c Copyright E. Platen SDE Jump 47
Predictor-Corrector Schemes predictor-corrector = stability and efficiency jump-adapted predictor-corrector Euler scheme Y tn+1 = Y tn + 1 2 { } a(ȳ tn+1 ) + a tn + b W tn with predictor Ȳ tn+1 = Y tn + a tn + b W tn β = 1 c Copyright E. Platen SDE Jump 48
3 EulerJA 2 ImplEulerJA PredCorrJA Log 2 Error 1 0-1 -2-5 -4-3 -2-1 Log 2 dt Figure 12: Log-log plot of weak error versus time step size. c Copyright E. Platen SDE Jump 49
Regular Approximations higher order schemes : time, Wiener and Poisson multiple integrals random jump size difficult to handle higher order schemes: computational effort dependent on intensity c Copyright E. Platen SDE Jump 50
Conclusions low intensity = jump-adapted higher order predictor-corrector high intensity = regular schemes distinction between strong and weak predictor-corrector schemes c Copyright E. Platen SDE Jump 51
References Björk, T., Y. Kabanov, & W. J. Runggaldier (1997). Bond market structure in the presence of marked point processes. Math. Finance 7, 211 239. Bruti-Liberati, N. & E. Platen (2006). On weak predictor-corrector schemes for jump-diffusion processes in finance. Technical report, University of Technology, Sydney. QFRC Research Paper 179. Bruti-Liberati, N. & E. Platen (2007). Strong approximations of stochastic differential equations with jumps. J. Comput. Appl. Math. 205(2), 982 1001. Delbaen, F. & W. Schachermayer (2006). The Mathematics of Arbitrage. Springer Finance. Springer. Föllmer, H. & M. Schweizer (1991). Hedging of contingent claims under incomplete information. In M. H. A. Davis and R. J. Elliott (Eds.), Applied Stochastic Analysis, Volume 5 of Stochastics Monogr., pp. 389 414. Gordon and Breach, London/New York. Gardoǹ, A. (2004). The order of approximations for solutions of Itô-type stochastic differential equations with jumps. Stochastic Anal. Appl. 22(3), 679 699. Glasserman, P. & N. Merener (2003). Numerical solution of jump-diffusion LIBOR market models. Finance Stoch. 7(1), 1 27. Higham, D. J. & P. E. Kloeden (2005). Numerical methods for nonlinear stochastic differential equations with jumps. Numer. Math. 110(1), 101 119. Jacod, J. & P. Protter (1998). Asymptotic error distribution for the Euler method for stochastic differential equations. Ann. Probab. 26(1), 267 307. Kubilius, K. & E. Platen (2002). Rate of weak convergence of the Euler approximation for diffusion c Copyright E. Platen SDE Jump 52
processes with jumps. Monte Carlo Methods Appl. 8(1), 83 96. Liu, X. Q. & C. W. Li (2000). Weak approximation and extrapolations of stochastic differential equations with jumps. SIAM J. Numer. Anal. 37(6), 1747 1767. Maghsoodi, Y. (1996). Mean-square efficient numerical solution of jump-diffusion stochastic differential equations. SANKHYA A 58(1), 25 47. Maghsoodi, Y. (1998). Exact solutions and doubly efficient approximations of jump-diffusion Itô equations. Stochastic Anal. Appl. 16(6), 1049 1072. Mikulevicius, R. & E. Platen (1988). Time discrete Taylor approximations for Ito processes with jump component. Math. Nachr. 138, 93 104. Mikulevicius, R. & E. Platen (1991). Rate of convergence of the Euler approximation for diffusion processes. Math. Nachr. 151, 233 239. Øksendal, B. & A. Sulem (2005). Applied stochastic control of jump-duffusions. Universitext. Springer. Platen, E. (1982). An approximation method for a class of Itô processes with jump component. Liet. Mat. Rink. 22(2), 124 136. Platen, E. (2005). Diversified portfolios with jumps in a benchmark framework. Asia-Pacific Financial Markets 11(1), 1 22. Platen, E. & K. Du (2011). Benchmarked risk minimization for jump diffusion markets. UTS Working Paper. Platen, E. & W. J. Runggaldier (2005). A benchmark approach to filtering in finance. Asia-Pacific Financial Markets 11(1), 79 105. Platen, E. & W. J. Runggaldier (2007). A benchmark approach to portfolio optimization under partial c Copyright E. Platen SDE Jump 53
information. Asia-Pacific Financial Markets 14(1-2), 25 43. c Copyright E. Platen SDE Jump 54