NUMERICAL METHODS OF QUANTITATIVE FINANCE - University Of Technology Sydney
NUMERICAL METHODS OF QUANTITATIVE FINANCE Eckhard Platen Finance Discipline Group and School of Mathematical Sciences University of Technology, Sydney Platen, E. & Bruti-Liberati, N.: Numerical Solution of SDEs with Jumps in Finance Springer, Applications of Mathematics (2010). Platen, E. & Heath, D.: A Benchmark Approach to Quantitative Finance Springer Finance, 700 pp., 199 illus., Hardcover, ISBN-10 3-540-26212-1 (2010). Kloeden, P. & Platen, E.: Numerical Solution of Stochastic Differential Equations Springer, Applications of Mathematics, 600 pp., Hardcover, ISBN-3-540-54062-8 (1999).
Contents 1 Stochastic Expansions 2 Introduction to Scenario Simulation 3 Monte Carlo Simulation of SDEs 133 4 Numerical Stability 224 5 Variance Reduction Techniques 262 6 Trees and Markov Chain Approximations 312 7 Partial Differential Equation Methods 392 References 476 c ⃝ Copyright E. Platen 1 BA to QF 26 Chap. ?
1 Stochastic Expansions Stochastic Taylor Expansions Deterministic Taylor Formula ordinary differential equation d dt Xt a(Xt ) integral form t Xt Xt0 a(Xs ) ds t0 c ⃝ Copyright E. Platen NS of SDEs Chap. 1 1
deterministic chain rule d dt f (Xt ) a(Xt ) x f (Xt ) operator L a c ⃝ Copyright E. Platen NS of SDEs x Chap. 1 2
integral equation t f (Xt ) f (Xt0 ) L f (Xs ) ds t0 special case f (x) x Lf a, LLf (L)2 f La, . . . apply to f a c ⃝ Copyright E. Platen NS of SDEs Chap. 1 3
t Xt Xt0 ( a(Xt0 ) t0 ) s L a(Xz ) dz ds t0 t Xt0 a(Xt0 ) t s ds t0 L a(Xz ) dz ds t0 t0 nontrivial Taylor expansion c ⃝ Copyright E. Platen NS of SDEs Chap. 1 4
apply f La t t0 t s t0 z (L)2 a(Xu ) du dz ds t0 Copyright E. Platen s dz ds R1 t0 R1 c ⃝ ds L a(Xt0 ) Xt Xt0 a(Xt0 ) remainder term t t0 t0 NS of SDEs Chap. 1 5
continuing classical deterministic Taylor formula f (Xt ) f (Xt0 ) r (t t0 )ℓ l! l 1 t ··· t0 (L)ℓ f (Xt0 ) s2 (L)r 1 f (Xs1 ) ds1 . . . dsr 1 t0 for t [t0 , T ] and r N c ⃝ Copyright E. Platen NS of SDEs Chap. 1 6
Wagner-Platen Expansion Wagner & Pl. (1978), Pl. (1982), Pl. & Wagner (1982) and Kloeden & Pl. (1992a) SDE t a(Xs ) ds Xt Xt0 t0 c ⃝ Copyright E. Platen t NS of SDEs b(Xs ) dWs t0 Chap. 1 7
Itô formula f (Xt ) f (Xt0 ) t ( a(Xs ) t0 x f (Xs ) t b(Xs ) t0 x b2 (Xs ) ) f (Xs ) ds 2 2 x f (Xs ) dWs t 0 L1 f (Xs ) dWs L f (Xs ) ds t0 Copyright E. Platen 2 t f (Xt0 ) c ⃝ 1 t0 NS of SDEs Chap. 1 8
operators L0 a x 1 2 b2 2 x2 and L1 b f (x) x c ⃝ Copyright E. Platen x L0 f a and L1 f b NS of SDEs Chap. 1 9
apply Itô formula to f a and f b Xt Xt0 t( a(Xt0 ) t0 s L0 a(Xz ) dz t0 t0 s t ds b(Xt0 ) t0 Copyright E. Platen ) L1 b(Xz ) dWz dWs t0 t Xt0 a(Xt0 ) s L0 b(Xz ) dz t0 c ⃝ L1 a(Xz ) dWz ds t0 t( b(Xt0 ) ) s dWs R2 t0 NS of SDEs Chap. 1 10
remainder term t s s L1 a(Xz ) dWz ds 0 L a(Xz ) dz ds R2 t0 t0 t0 t t s L0 b(Xz ) dz dWs t0 Copyright E. Platen t0 s c ⃝ t t0 L1 b(Xz ) dWz dWs t0 NS of SDEs Chap. 1 t0 11
apply Itô formula to f L1 b t Xt Xt0 a(Xt0 ) ds b(Xt0 ) t0 t s L b(Xt0 ) dWz dWs R3 t0 Copyright E. Platen NS of SDEs dWs t0 1 c ⃝ t t0 Chap. 1 12
remainder term t R3 t0 s t s L0 a(Xz ) dz ds t0 L1 a(Xz ) dWz ds t0 t t0 s L0 b(Xz ) dz dWs t0 t t0 s z L0 L1 b(Xu ) du dWz dWs t0 t t0 s t0 z L1 L1 b(Xu ) dWu dWz dWs t0 t0 t0 example for Wagner-Platen expansion c ⃝ Copyright E. Platen NS of SDEs Chap. 1 13
multiple Itô integrals t ds t t0 , t0 t dWs Wt Wt0 , t0 t s dWz dWs t0 t0 1( 2 (Wt Wt0 )2 (t t0 ) ) remainder term R3 consisting of next following multiple Itô integrals with nonconstant integrands c ⃝ Copyright E. Platen NS of SDEs Chap. 1 14
Example d m 1 for function f (t, x) x, times ϱ 0, τ t hierarchical set drift A {α Mm : ℓ(α) 3} a(t, x) a(x) diffusion coefficient b(t, x) b(x) dXt a (Xt ) dt b(Xt ) dWt Wagner-Platen expansion in the form: c ⃝ Copyright E. Platen NS of SDEs Chap. 1 15
( Xt X0 a I(0) b I(1) a a′ 1 2 ) b2 a′′ I(0,0) ( ) 1 a b′ b2 b′′ I(0,1) b a′ I(1,0) b b′ I(1,1) 2 [ ( ) 1 1 2 ′′ ′ 2 ′ ′′ 2 ′′′ a a a (a ) b b a b a b (a a′′′ 3 a′ a′′ 2 2 ] ( ′ 2 ) ) 1 (b ) b b′′ a′′ 2 b b′ a′′′ b4 a(4) I(0,0,0) 4 [ ( ) 1 1 2 ( ′′ ′ ′ ′ ′′ ′ ′′ 2 ′′′ a a b ab bb b b b b a b 2 a′ b′′ 2 2 ] ) ) ( 1 a b′′′ (b′ )2 b b′′ b′′ 2 b b′ b′′′ b2 b(4) I(0,0,1) 2 c ⃝ Copyright E. Platen NS of SDEs Chap. 1 16
[ ′ ′ ′′ a (b a b a ) 1 2 ] b2 (b′′ a′ 2 b′ a′′ b a′′′ ) I(0,1,0) [ ] ) ( ′ 2 1 2 ′′ ′ ′′ a (b ) b b b (b b 2 b b′′ b b′′′ ) I(0,1,1) 2 ( ) 1 2 ′′′ ′′ ′ 2 ′ ′′ b a a (a ) b b a b a I(1,0,0) 2 ( ) 1 b a b′′ a′ b′ b b′ b′′ b2 b′′′ I(1,0,1) 2 ) ( ′ 2 ′ ′ ′′ ′′ b (a b a b) I(1,1,0) b (b ) b b I(1,1,1) R6 c ⃝ Copyright E. Platen NS of SDEs Chap. 1 17
Examples for Wagner Platen Expansions Vasicek interest rate model drt γ (r̄ rt ) dt β dWt for t [0, T ], r0 0 c ⃝ Copyright E. Platen NS of SDEs Chap. 1 18
hierarchical set A {α M1 : ℓ(α) 3} rt r0 γ (r̄ r0 ) t β Wt γ 2 (r̄ r0 ) t βγ Ws ds γ 3 (r̄ r0 ) 0 βγ t Copyright E. Platen 2 t3 6 s2 2 Ws1 ds1 ds2 R6 0 c ⃝ t2 0 NS of SDEs Chap. 1 19
Black-Scholes dynamics dSt St (a dt σ dWt ) for t [0, T ], S0 0 Wagner-Platen expansion hierarchical set A {α M1 : ℓ(α) 3} c ⃝ Copyright E. Platen NS of SDEs Chap. 1 20
is given by St ( 2 t S 0 1 a t σ Wt a2 a σ (I(0,1) I(1,0) ) 2 3 ( ) 1 t σ2 (Wt )2 t a3 2 6 a2 σ (I(0,0,1) I(0,1,0) I(1,0,0) ) a σ 2 (I(0,1,1) I(1,0,1) I(1,1,0) ) ) ( ) 1 σ3 (Wt )3 3 t Wt 6 R6 c ⃝ Copyright E. Platen NS of SDEs Chap. 1 21
relationship t Wt Wt t c ⃝ 1( 2 t2 2 ) (Wt ) t Copyright E. Platen 2 I(0) I(1) I(0,1) I(1,0) I(1) I(0,0) I(0,0,1) I(0,1,0) I(1,0,0) I(0) I(1,1) I(0,1,1) I(1,0,1) I(1,1,0) NS of SDEs Chap. 1 22
Wagner-Platen expansion ( St S0 1 a t σ Wt a2 t2 2 a σ t Wt 3 2 ) σ2 ( t t (Wt )2 t a3 a2 σ Wt 2 6 2 ) ( ) 1 a σ2 (Wt )2 t σ 3 (Wt )3 3 t Wt R6 2 6 t( c ⃝ Copyright E. Platen ) NS of SDEs Chap. 1 23
squared Bessel process dXt ν dt 2 Xt dWt for t [0, T ] with X0 0 hierarchical set A {α M1 : ℓ(α) 2} c ⃝ Copyright E. Platen NS of SDEs Chap. 1 24
Wagner-Platen expansion Xt X0 (ν 1) t 2 X0 Wt ν 1 X0 c ⃝ Copyright E. Platen t s dWs (Wt )2 R̃ 0 NS of SDEs Chap. 1 25
2 Introduction to Scenario Simulation Discrete Time Approximation 0 τ0 τ 1 · · · τ n · · · τ N T one-dimensional SDE dXt a(t, Xt ) dt b(t, Xt ) dWt c ⃝ Copyright E. Platen NS of SDEs Chap. 2 26
Euler scheme Euler-Maruyama scheme Maruyama (1955) ( Yn 1 Yn a(τn , Yn ) (τn 1 τn ) b(τn , Yn ) Wτn 1 Wτn Y0 X0 , Yn Yτn n τn 1 τn c ⃝ Copyright E. Platen NS of SDEs Chap. 2 27 )
maximum step size max n {0,1,.,N 1} n equidistant time discretization τn n n c ⃝ Copyright E. Platen T N NS of SDEs Chap. 2 28
Euler approximation recursively computed random increments Wn Wτn 1 Wτn for n {0, 1, . . . , N 1} Wiener process W {Wt t [0, T ]} Gaussian distributed mean E ( Wn ) 0 variance c ⃝ Copyright E. Platen ( ) 2 E ( Wn ) n NS of SDEs Chap. 2 29
Gaussian pseudo-random numbers generated abbreviation f f (τn , Yn ) Euler scheme Yn 1 Yn a n b Wn , discrete time approximations c ⃝ Copyright E. Platen NS of SDEs Chap. 2 30
Simulating Geometric Brownian Motion illustrate scenario simulation standard market model as geometric Brownian motions dXt a Xt dt b Xt dWt for t [0, T ], X0 0 drift coefficient a(t, x) a x diffusion coefficient b(t, x) b x c ⃝ Copyright E. Platen NS of SDEs Chap. 2 31
appreciation rate volatility a b ̸ 0 explicit solution (( ) ) 1 Xt X0 exp a b2 t b Wt 2 Wiener process W {Wt , t [0, T ]} c ⃝ Copyright E. Platen NS of SDEs Chap. 2 32
Scenario Simulation simulate trajectory of Euler approximation 1. initial value Y0 X0 2. proceed recursively Yn 1 Yn a Yn n b Yn Wn for n {0, 1, . . . , N 1} Wn Wτn 1 Wτn c ⃝ Copyright E. Platen NS of SDEs Chap. 2 33
comparison with explicit solution at time τn (( Xτn X0 exp a 1 2 ) b2 τn b n ) Wi 1 i 1 for n {0, 1, . . . , N 1} Euler approximation mathematically another new object different negative ? alternative ways ? c ⃝ Copyright E. Platen NS of SDEs Chap. 2 34
10 8 6 4 2 0 0 0.2 0.4 0.6 0.8 1 Figure 2.1: Euler approximations for 0.25 and 0.0625 and exact solution for Black-Scholes SDE. c ⃝ Copyright E. Platen NS of SDEs Chap. 2 35
Strong Approximation not specified a criterion for classification scenario simulation approximates paths testing of calibration methods statistical estimators filtering c ⃝ Copyright E. Platen NS of SDEs Chap. 2 36
Monte-Carlo simulation approximates probabilities functionals simulates expectations moments prices of contingent claims or risk measures as Value at Risk c ⃝ Copyright E. Platen NS of SDEs Chap. 2 37
Order of Strong Convergence absolute error criterion ε( ) E ( XT YT ) A discrete time approximation Y converges strongly with order γ 0 at time T if there exists a positive constant C, which does not depend on , and a δ0 0 such that ( ) ε( ) E XT YT C γ for each (0, δ0 ). c ⃝ Copyright E. Platen NS of SDEs Chap. 2 38
Strong Taylor Schemes use Wagner-Platen expansions appropriate truncation operators 0 L t d a k xk k 1 0 L c ⃝ Copyright E. Platen t d m 1 2 NS of SDEs b k,ℓ 1 j 1 d a k 1 Chap. 2 k k,j b ℓ,j xk xℓ xk 39
and j j L L d b k,j k 1 xk for j {1, 2, . . . , m}, where ak ak c ⃝ Copyright E. Platen NS of SDEs m 1 2 Lj bk,j j 1 Chap. 2 40
multiple Itô integrals τn 1 I(j1 ,.,jℓ ) s2 . τn τn dWsj11 . . . dWsjℓℓ multiple Stratonovich integrals τn 1 J(j1 ,.,jℓ ) s2 . τn τn dWsj11 . . . dWsjℓℓ for j1 , . . . , jℓ {0, 1, . . . , m}, ℓ {1, 2, . . .} and n {0, 1, . . .} Wt0 t for all t [0, T ] c ⃝ Copyright E. Platen NS of SDEs Chap. 2 41
Itô SDE dXt a(t, Xt ) dt m bj (t, Xt ) dWtj j 1 equivalent Stratonovich SDE dXt a(t, Xt ) dt m bj (t, Xt ) dWtj j 1 c ⃝ Copyright E. Platen NS of SDEs Chap. 2 42
Euler Scheme simplest strong Taylor approximation order of strong convergence γ 0.5 d m 1 Euler scheme Yn 1 Yn a b W τn 1 τn W Wn Wτn 1 Wτn N (0, ) independent Gaussian distributed c ⃝ Copyright E. Platen NS of SDEs Chap. 2 43
multi-dimensional Euler scheme m 1 and d {1, 2, . . .} kth component k Yn 1 Ynk ak bk W for k {1, 2, . . . , d} a (a1 , . . ., ad ) and b (b1 , . . ., bd ) c ⃝ Copyright E. Platen NS of SDEs Chap. 2 44
general multi-dimensional Euler scheme d, m {1, 2, . . .} k Yn 1 Ynk ak m bk,j W j j 1 W j Wτjn 1 Wτjn N (0, ) independent Gaussian distributed W j1 and W j2 b [bk,j ]d,m k,j 1 independent for j1 ̸ j2 d m-matrix truncated Wagner-Platen expansion c ⃝ Copyright E. Platen NS of SDEs Chap. 2 45
Theorem 2.1 such that Suppose that we have initial values X0 and Y0 Y0 ( E X0 and ( E X0 2 2 Y0 ) ) 12 1 K1 2 . Furthermore, assume the Lipschitz condition a(t, x) a(t, y) b(t, x) b(t, y) K2 x y , the linear growth condition a(t, x) b(t, x) K3 (1 x ) c ⃝ Copyright E. Platen NS of SDEs Chap. 2 46
and a(s, x) a(t, x) b(s, x) b(t, x) K4 (1 x ) s t 1 2 for all s, t [0, T ] and x, y ℜd , where the constants K1 , . . ., K4 do not depend on . Then the Euler approximation Y converges with strong order γ 0.5, that is we have the estimate ) ( 1 K5 2 , E XT YT where the constant K5 does not depend on . c ⃝ Copyright E. Platen NS of SDEs Chap. 2 47
A Simulation Example Black-Scholes SDE dXt µ Xt dt σ Xt dWt exact solution {( XT X0 exp c ⃝ Copyright E. Platen µ NS of SDEs σ2 2 Chap. 2 ) } T σ WT 48
absolute error ε( ) E( XT YN ) default parameters: X0 1, µ 0.06, σ 0.2 and T 1 5000 simulations fitted line ln(ε( )) 3.86933 0.46739 ln( ) c ⃝ Copyright E. Platen NS of SDEs Chap. 2 49
Log -Strong Error -4 -4.5 -5 -5.5 -6 -4 -3 -2 -1 0 Log -dt Figure 2.2: Log-log plot of the absolute error ε( ) for an Euler Scheme against ln( ). c ⃝ Copyright E. Platen NS of SDEs Chap. 2 50
Milstein Scheme Milstein (1974) Milstein scheme d m 1 Yn 1 Yn a b W 1 2 { } 2 b b ( W ) ′ order γ 1.0 of strong convergence c ⃝ Copyright E. Platen NS of SDEs Chap. 2 51
multi-dimensional Milstein scheme m 1 and d {1, 2, . . .} ( k Yn 1 Ynk ak k b W d ℓ 1 c ⃝ Copyright E. Platen NS of SDEs Chap. 2 b ℓ b k xℓ ) 1{ 2 ( W ) 2 } 52
general multi-dimensional Milstein scheme d, m {1, 2, . . .} kth component k Yn 1 Ynk ak m bk,j W j m j 1 j1 ,j2 1 m m Lj1 bk,j2 I(j1 ,j2 ) Itô integrals I(j1 ,j2 ) alternatively k Ynk ak Yn 1 bk,j W j j 1 c ⃝ Copyright E. Platen NS of SDEs Lj1 bk,j2 J(j1 ,j2 ) j1 ,j2 1 Chap. 2 53
Stratonovich integrals J(j1 ,j2 ) j1 ̸ j2 j1 , j2 {1, 2, . . . , m} τn 1 s1 J(j1 ,j2 ) I(j1 ,j2 ) τn τn dWsj21 dWsj12 cannot be simply expressed by W j1 and W j2 I(j1 ,j1 ) } 1{ j1 2 ( W ) 2 and J(j1 ,j1 ) 1( 2 W ) j1 2 for j1 {1, 2, . . . , m} c ⃝ Copyright E. Platen NS of SDEs Chap. 2 54
Commutative Noise commutativity condition Lj1 bk,j2 Lj2 bk,j1 for all j1 , j2 {1, 2, . . . , m}, k {1, 2, . . . , d} and (t, x) [0, T ] ℜd satisfied for Black-Scholes SDEs additive noise single Wiener process c ⃝ Copyright E. Platen NS of SDEs Chap. 2 55
Milstein scheme under commutative noise k Yn 1 Ynk ak m bk,j W j j 1 1 2 m Lj1 bk,j2 W j1 W j2 j1 ,j2 1 for k {1, 2, . . . , d} no double Wiener integrals c ⃝ Copyright E. Platen NS of SDEs Chap. 2 56
A Black-Scholes Example correlated Black-Scholes dynamics d m 2 first risky security dXt1 ] [ 2 2 2 1 1 1 r dt θ (θ dt dWt ) θ (θ dt dWt ) [( ) ( ) 1 Xt1 r (θ 1 )2 (θ 2 )2 dt 2 ] Xt1 θ 1 dWt1 θ 2 dWt2 c ⃝ Copyright E. Platen NS of SDEs Chap. 2 57
second risky security dXt2 [ ] 1 1,1 1 1 r dt (θ σ ) (θ dt dWt ) [( ( )) 1 1 1 Xt2 r (θ 1 σ 1,1 ) θ σ 1,1 dt 2 2 ] Xt2 (θ 1 σ 1,1 ) dWt1 c ⃝ Copyright E. Platen NS of SDEs Chap. 2 58
L1 b1,2 2 bk,1 k 1 2 b k,2 xk xk k 1 b1,2 θ 1 Xt1 θ 2 b1,1 L2 b1,1 and 1 L b 2,2 2 k 1 c ⃝ b k,1 xk 2,2 b 0 2 k 1 commutativity condition satisfied Milstein scheme : Copyright E. Platen NS of SDEs Chap. 2 b k,2 xk b2,1 L2 b2,1 59
1 Yn 1 Yn1 ( ) 1( 1 2 1 2 2 Yn r (θ ) (θ ) θ 1 W 1 θ 2 W 2 2 1 2 (θ 1 )2 ( W 1 )2 (( 2 Yn 1 Yn2 Yn2 r 1 2 1 2 ) (θ 2 )2 ( W 2 )2 θ 1 θ 2 W 1 W 2 ) (θ 1 σ 1,1 ) (θ 1 σ 1,1 ) (θ 1 σ 1,1 ) W 1 1 2 ) (θ 1 σ 1,1 )2 ( W 1 )2 ( may produce negative trajectories ) c ⃝ Copyright E. Platen NS of SDEs Chap. 2 60
A Square Root Process Example square root process of dimension ν dXt X0 1 η ν 4 ( η 1 η ) Xt dt Xt dWt1 for ν 2 Milstein Yn 1 Yn c ⃝ Copyright E. Platen ν 4 ( η 1 η ) Yn ) 1( 2 Yn W W 4 NS of SDEs Chap. 2 61
40 35 30 25 20 15 10 5 0 0 20 40 60 80 100 Figure 2.3: Square root process simulated by the Milstein scheme. c ⃝ Copyright E. Platen NS of SDEs Chap. 2 62
Convergence Theorem ( 2 E X0 ( E c ⃝ Copyright E. Platen X0 ) 2 Y0 NS of SDEs ) 12 K1 Chap. 2 1 2 63
a(t, x) a(t, y) c ⃝ K2 x y bj1 (t, x) bj1 (t, y) K2 x y Lj1 bj2 (t, x) Lj1 bj2 (t, y) K2 x y a(t, x) Lj a(t, x) K3 (1 x ) bj1 (t, x) Lj bj2 (t, x) K3 (1 x ) Lj Lj1 bj2 (t, x) K3 (1 x ) Copyright E. Platen NS of SDEs Chap. 2 64
and a(s, x) a(t, x) b (s, x) b (t, x) j1 j1 Lj1 bj2 (s, x) Lj1 bj2 (t, x) 1 K4 (1 x ) s t 2 K4 (1 x ) s t 1 2 1 K4 (1 x ) s t 2 for all s, t [0, T ], x, y ℜd , j {0, . . ., m} and j1 , j2 {1, 2, . . . , m}. Then the Milstein scheme converges with strong order γ 1.0 ( ) E XT YT K5 . Kloeden & Pl. (1992b). c ⃝ Copyright E. Platen NS of SDEs Chap. 2 65
A Simulation Study Log -Strong Error -5 -6 -7 -8 -9 -4 -3 -2 -1 0 Log -dt Figure 2.4: Log-log plot of the absolute error against log-step size for a Milstein scheme. c ⃝ Copyright E. Platen NS of SDEs Chap. 2 66
linear regression ln(ε( )) 4.91021 0.95 ln( ) c ⃝ Copyright E. Platen NS of SDEs Chap. 2 67
Order 1.5 Strong Taylor Scheme simulation tasks that require more accurate schemes extreme log-returns including further multiple stochastic integrals order 1.5 strong Taylor scheme d m 1 c ⃝ Copyright E. Platen NS of SDEs Chap. 2 68
Yn 1 Yn a b W 1 2 { } 2 b b ( W ) ′ ( ) 1 1 a′ b Z a a′ b2 a′′ 2 2 2 ( ) 1 2 ′′ ′ a b b b { W Z} 2 { } ( ) 1 1 ′′ ′ 2 b b b (b ) ( W )2 W 2 3 c ⃝ Copyright E. Platen NS of SDEs Chap. 2 69
double integral τn 1 s2 Z I(1,0) dWs1 ds2 τn τn Gaussian E( Z) 0 2 E(( Z) ) 1 3 3 1 2 E( Z W ) 2 c ⃝ Copyright E. Platen NS of SDEs Chap. 2 70
two independent N (0, 1) U1 and U2 W U1 ( ) 1 3 Z 2 U1 U2 2 3 1 , triple Wiener integral I(1,1,1) 1 2 { 1 3 } ( W 1 )2 W 1 scaled monic Hermite polynomial c ⃝ Copyright E. Platen NS of SDEs Chap. 2 71
multi-dimensional order 1.5 strong Taylor scheme d {1, 2, . . .} and m 1 k Yn 1 Ynk ak bk W 1 2 L1 bk {( W )2 } L1 ak Z L b { W Z} 0 c ⃝ Copyright E. Platen 1 2 k { 1 1 L L b NS of SDEs k 1 3 Chap. 2 1 2 L0 a k 2 } ( W )2 W 72
general multi-dimensional order 1.5 strong Taylor scheme d, m {1, 2, . . .} k Yn 1 Ynk k a m 1 2 L0 a k 2 (bk,j W j L0 bk,j I(0,j) Lj ak I(j,0) ) j 1 m m Lj1 bk,j2 I(j1 ,j2 ) j1 ,j2 1 Lj1 Lj2 bk,j3 I(j1 ,j2 ,j3 ) j1 ,j2 ,j3 1 in case of additive noise simplifies considerably strong order γ 1.5 c ⃝ Copyright E. Platen NS of SDEs Chap. 2 73
Approximate Multiple Stochastic Integrals Let ξj , ζj,1 , . . . , ζj,p , ηj,1 , . . . , ηj,p , µj,p and ϕj,p be independent N (0, 1) for j, j1 , j2 , j3 {1, 2, . . . , m} and some p {1, 2, . . .} set I(j) W ξj , j with aj,0 c ⃝ Copyright E. Platen I(j,0) p 2 1 π r 1 r NS of SDEs ζj,r 1 2 ( ) ξj aj,0 2 ϱp µj,p Chap. 2 74
where ϱp 1 12 1 2π 2 I(0,j) W I(j,0) , j I(j,j,j) p I(j 1 ,j2 ) c ⃝ 1 2 Copyright E. Platen 1 { 2 ξj1 ξj2 1 3 1 2 p 1 r 1 I(j,j) r2 } 1{ j 2 ( W ) 2 } ( W j )2 W j (ξj1 aj2 ,0 ξj2 aj1 ,0 ) Apj1 ,j2 NS of SDEs Chap. 2 75
Order 2.0 Strong Taylor Scheme use Stratonovich-Taylor expansion d m 1 Yn 1 Yn a b W 1 2 1 2! b b′ ( W )2 b a′ Z a a′ 2 a b′ { W Z} 1 3! ′ ′ 3 b (b b ) ( W ) ′ 1 4! ( )′ ′ ′ b b (b b ) ′ ( W )4 ′ a (b b′ ) J(0,1,1) b (a b′ ) J(1,0,1) b (b a′ ) J(1,1,0) c ⃝ Copyright E. Platen NS of SDEs Chap. 2 76
Approximate Multiple Stratonovich Integrals W Z p J(1,0,1) p J(0,1,1) p J(1,1,0) c ⃝ 1 3! 1 3! Copyright E. Platen 1 3! p J(1,0) 2 ζ12 2 ζ12 1 2π 1 2 2 ζ1 4 1 4 p J(1) 1 2 ζ1 ( a21,0 3 2 ζ1 b1 a21,0 1 2π NS of SDEs 2 ) ζ1 a1,0 1 π 3 p 2 ζ1 b1 2 B1,1 1 p B1,1 4 3 2 ζ1 b1 Chap. 2 1 4 3 p 2 a1,0 ζ1 2 C1,1 3 2 p a1,0 ζ1 2 C1,1 77
with a1,0 p 1 1 2 ξ1,r 2 ϱp µ1,p π r r 1 1 ϱp b1 p 1 2 p B1,1 Copyright E. Platen r2 r 1 αp c ⃝ 12 π2 180 1 4π 2 p 1 1 2π 2 r 1 η1,r r 1 2π 2 r2 NS of SDEs αp ϕ1,p p 1 1 p 1 ( r2 r 1 r4 2 2 ξ1,r η1,r Chap. 2 ) 78
and p C1,1 1 2π 2 p r,l 1 r̸ l ( r r2 l2 1 l ξ1,r ξ1,ℓ l r ) η1,r η1,ℓ ζ1 , ξ1,r η1,r , µ1,p and ϕ1,p N (0, 1) i.i.d. c ⃝ Copyright E. Platen NS of SDEs Chap. 2 79
Multi-dimensional Order 2.0 Strong Taylor Scheme m 1 k Yn 1 Ynk k k a b W 1 2 1 2! L1 bk ( W )2 L1 ak Z L0 ak 2 L0 bk { W Z} 1 3! 1 1 k 3 L L b ( W ) 1 4! L1 L1 L1 bk ( W )4 L0 L1 bk J(0,1,1) L1 L0 bk J(1,0,1) L1 L1 ak J(1,1,0) c ⃝ Copyright E. Platen NS of SDEs Chap. 2 80
general multi-dimensional order 2.0 strong Taylor scheme k Yn 1 Ynk k a m ( 1 2 L0 a k 2 0 k,j bk,j W j L b j J(0,j) L ak J(j,0) ) j 1 ( m Lj1 bk,j2 J(j1 ,j2 ) L0 Lj1 bk,j2 J(0,j1 ,j2 ) j1 ,j2 1 ) Lj1 L0 bk,j2 J(j1 ,0,j2 ) Lj1 Lj2 ak J(j1 ,j2 ,0) m Lj1 Lj2 bk,j3 J(j1 ,j2 ,j3 ) j1 ,j2 ,j3 1 m Lj1 Lj2 Lj3 bk,j4 J(j1 ,j2 ,j3 ,j4 ) j1 ,j2 ,j3 ,j4 1 c ⃝ Copyright E. Platen NS of SDEs Chap. 2 81
Derivative Free Strong Schemes similar to Runge-Kutta schemes for ODEs Explicit Order 1.0 Strong Schemes Pl. (1984) d m 1 Yn 1 }{ } 1 { 2 b(τn , Ῡn ) b ( W ) Yn a b W 2 with supporting value Ῡn Yn a b c ⃝ Copyright E. Platen NS of SDEs Chap. 2 82
multi-dimensional Platen scheme m 1 k Yn 1 Ynk ak bk W )( ) 1 ( k k 2 b (τn , Ῡn ) b ( W ) 2 with the vector supporting value Ῡn Yn a b c ⃝ Copyright E. Platen NS of SDEs Chap. 2 83
general multi-dimensional case k Yn 1 Ynk ak m bk,j W j j 1 1 m ( b ( k,j2 ) j1 τn , Ῡn bk,j2 I(j1 ,j2 ) j1 ,j2 1 with Ῡjn ) Yn a b j for j {1, 2, . . .} c ⃝ Copyright E. Platen NS of SDEs Chap. 2 84
commutative noise k Yn 1 Ynk ak m 1 ( 2 ) ( ) bk,j τn , Ῡn bk,j W j j 1 with Ῡn Yn a m bj W j j 1 strong order γ 1.0 c ⃝ Copyright E. Platen NS of SDEs Chap. 2 85
ln(ε( )) 4.71638 0.946112 ln( ) Log -Strong Error -5 -6 -7 -8 -9 -4 -3 -2 -1 0 Log -dt Figure 2.5: Log-log plot of the absolute error for a Platen scheme. c ⃝ Copyright E. Platen NS of SDEs Chap. 2 86
two stage Runge-Kutta method γ 1.0 m d 1 Burrage (1998) ( Yn 1 Yn a(Yn ) 3 a(Ȳn ) 1 ( with Ȳn Yn c ⃝ Copyright E. Platen 4 2 3 ) 4 ) b(Yn ) 3 b(Ȳn ) W (a(Yn ) b(Yn ) W ) NS of SDEs Chap. 2 87
Explicit Order 1.5 Strong Schemes Pl. (1984) d m 1 with Ῡ Yn a b and Φ̄ Ῡ b(Ῡ ) c ⃝ Copyright E. Platen NS of SDEs Chap. 2 88
Yn 1 c ⃝ ) 1 ( Yn b W a(Ῡ ) a(Ῡ ) Z 2 ) 1( a(Ῡ ) 2a a(Ῡ ) 4 )( ) 1 ( b(Ῡ ) b(Ῡ ) ( W )2 4 )( ) 1 ( b(Ῡ ) 2b b(Ῡ ) W Z 2 ) 1 ( b(Φ̄ ) b(Φ̄ ) b(Ῡ ) b(Ῡ ) 4 ( ) 1 ( W )2 W 3 Copyright E. Platen NS of SDEs Chap. 2 89
order 1.5 Platen scheme k Yn 1 Ynk ak m bk,j W j j 1 m m ( ( ) ( )) 1 j j bk,j2 Ῡ 1 bk,j2 Ῡ 1 I(j1 ,j2 ) 2 j2 0 j1 1 1 2 1 2 m m ( Copyright E. Platen ( Ῡj 1 ) 2b k,j2 b k,j2 ( Ῡj 1 )) I(0,j2 ) j2 0 j1 1 ( m j1 ,j2 ,j3 1 bk,j3 c ⃝ b k,j2 ( Ῡj 1 bk,j3 ) ( Φ̄j 1 ,j2 bk,j3 NS of SDEs ( ) bk,j3 Ῡj 1 Chap. 2 ( Φ̄j 1 ,j2 ) )) I(j1 ,j2 ,j3 ) 90
with Ῡj Yn 1 m a b and Φ̄j 1 ,j2 Ῡj 1 bj2 ( j ) Ῡj 1 where we interpret bk,0 as ak c ⃝ Copyright E. Platen NS of SDEs Chap. 2 91
A Simulation Study -4 Error -6 Euler Log -Strong -8 Milstein -10 R -K -12 1.5 Taylor -4 -3 -2 -1 0 Log -dt Figure 2.6: Log-log plot of the absolute error for various strong schemes. c ⃝ Copyright E. Platen NS of SDEs Chap. 2 92
Method CPU Time Euler 3.5 Seconds Milstein 4.1 Seconds 1.0 Strong Platen 4.1 Seconds 1.5 Strong Taylor 4.4 Seconds Table 1: Computational times for various strong methods. 500000 time steps c ⃝ Copyright E. Platen NS of SDEs Chap. 2 93
Numerical Stability ability of a scheme to control the propagation of initial and roundoff errors numerical stability has higher priority than a potentially high strong order c ⃝ Copyright E. Platen NS of SDEs Chap. 2 94
Deterministic A-Stability one-step method Yn 1 Yn Ψ(τn , Yn , Yn 1 , ) ordinary differential equation dx dt a(t, x) a(t, x) satisfies Lipschitz condition c ⃝ Copyright E. Platen NS of SDEs Chap. 2 95
numerically stable if there exist positive constants 0 and M such that Yn Ỹn M Y0 Ỹ0 n {0, 1, . . . , N }, 0 and any two solutions Y , Ỹ corresponding to the initial values Y0 Ỹ0 , respectively asymptotically numerically stable if there exist 0 and M such that lim Yn Ỹn M Y0 Ỹ0 n for any two Y , Ỹ c ⃝ Copyright E. Platen NS of SDEs Chap. 2 96
test equation dxt dt λ xt with λ ℜ c ⃝ Copyright E. Platen NS of SDEs Chap. 2 97
numerical scheme in recursive form Yn 1 G(λ ) Yn A-stability region: those λ ℜ for which G(λ ) 1 c ⃝ Copyright E. Platen NS of SDEs Chap. 2 98
explicit Euler scheme Yn 1 Yn a(tn , Yn ) Yn 1 (1 λ ) Yn A-stability region open unit interval centered at 1 and ending at 0 since G(λ ) 1 λ c ⃝ Copyright E. Platen NS of SDEs Chap. 2 99
1 0.5 -2 -1 0 -0.5 -1 Figure 2.7: Region of A-stability for the deterministic Euler scheme. c ⃝ Copyright E. Platen NS of SDEs Chap. 2 100
implicit Euler scheme Yn 1 Yn a(tn 1 , Yn 1 ) Yn 1 Yn λ Yn 1 (1 λ ) Yn 1 Yn G(λ ) G(λ ) 1 1 λ 1 1 λ exterior of an interval with center at 1 beginning at 0 scheme is called A-stable if it covers at least the left half axis c ⃝ Copyright E. Platen NS of SDEs Chap. 2 101
Stochastic A-Stability test equation dXt λ Xt dt dWt λ ℜ c ⃝ Copyright E. Platen NS of SDEs Chap. 2 102
discrete time approximation Yn 1 GA (λ ) Yn Zn , Zn does not depend on Y0 , Y1 , . . . , Yn , Yn 1 c ⃝ Copyright E. Platen NS of SDEs Chap. 2 103
A-stability region real numbers λ for which GA (λ ) 1 If A-stability region covers left half of the real axis c ⃝ then A-stable Copyright E. Platen NS of SDEs Chap. 2 104
Drift Implicit Euler Scheme drift implicit Euler scheme strong order γ 0.5 d m 1 Yn 1 Yn a (τn 1 , Yn 1 ) b W c ⃝ Copyright E. Platen NS of SDEs Chap. 2 105
family of drift implicit Euler schemes Yn 1 Yn {α a (τn 1 , Yn 1 ) (1 α) a} b W degree of implicitness c ⃝ α [0, 1] for α 0 explicit Euler scheme for α 1 drift implicit Euler scheme Copyright E. Platen NS of SDEs Chap. 2 106
general multi-dimensional family of drift implicit Euler schemes k Yn 1 Ynk ( αk a (τn 1 , Yn 1 ) (1 αk ) a m k k ) bk,j W j j 1 αk [0, 1], k {1, 2, . . . , d} c ⃝ Copyright E. Platen NS of SDEs Chap. 2 107
for test equation ( ) Yn 1 Yn α λ Yn 1 (1 α) λ Yn Wn Yn 1 GA (λ ) Yn Wn (1 α λ ) 1 transfer function GA (λ ) (1 α λ ) 1 (1 (1 α) λ ) c ⃝ Copyright E. Platen NS of SDEs Chap. 2 108
Ȳn corresponding discrete time approximation that starts at initial value Ȳ0 Yn 1 Ȳn 1 GA (λ ) (Yn Ȳn ) (GA (λ ))n (Y0 Ȳ0 ) as long as GA (λ ) 1 the initial error (Y0 Ȳ0 ) is decreased for α c ⃝ 1 2 ( λ 2 1 2α ) ,0 A-stable Copyright E. Platen NS of SDEs Chap. 2 109
Drift Implicit Milstein Scheme d m 1 Itô version Yn 1 Yn a (τn 1 , Yn 1 ) b W c ⃝ Copyright E. Platen NS of SDEs Chap. 2 1 2 ′ ( b b ( W ) 2 ) 110
Stratonovich version Yn 1 Yn a (τn 1 , Yn 1 ) b W adjusted Stratonovich drift 1 2 b b′ ( W )2 a a 21 b b′ both schemes are different c ⃝ Copyright E. Platen NS of SDEs Chap. 2 111
multi-dimensional case with d m {1, 2, . . .} and commutative noise { } k k k k Yn 1 Yn αk a (τn 1 , Yn 1 ) (1 αk ) a m bk,j W j j 1 1 2 m j1 k,j2 L b { } j1 j2 W W 1{j1 j2 } j1 ,j2 1 and k Yn 1 Ynk { } k k αk a (τn 1 , Yn 1 ) (1 αk ) a m bk,j W j j 1 c ⃝ Copyright E. Platen NS of SDEs m 1 2 Lj1 bk,j2 W j1 W j2 j1 ,j2 1 Chap. 2 112
general drift implicit Milstein scheme Itô version ( ) k Yn 1 Ynk αk ak (τn 1 , Yn 1 ) (1 αk ) ak m bk,j W j j 1 c ⃝ Copyright E. Platen m Lj1 bk,j2 I(j1 ,j2 ) j1 ,j2 1 NS of SDEs Chap. 2 113
Stratonovich version ( ) k Yn 1 Ynk αk ak (τn 1 , Yn 1 ) (1 αk ) ak m m bk,j W j j 1 Lj1 bk,j2 J(j1 ,j2 ) j1 ,j2 1 αk [0, 1] for k {1,. . ., d} multiple stochastic integrals I(j1 ,j2 ) and J(j1 ,j2 ) approximated c ⃝ Copyright E. Platen NS of SDEs Chap. 2 114
Drift Implicit Order 1.0 Strong Runge-Kutta Scheme d m 1 Yn 1 Yn a (τn 1 , Yn 1 ) b W ) )( ) 1 ( ( 2 b τn , Ῡn b ( W ) 2 with supporting value Ῡ Yn a b c ⃝ Copyright E. Platen NS of SDEs Chap. 2 115
family of drift implicit order 1.0 strong Runge-Kutta schemes k Yn 1 Ynk ( αk a (τn 1 , Yn 1 ) (1 αk ) a m k ) bk,j W j j 1 1 m ( b ( k,j2 ) ) τn , Ῡjn1 bk,j2 I(j1 ,j2 ) j1 ,j2 1 with Ῡjn Yn a b j for j {1, 2, . . . , m} αk [0, 1] for k {1, 2, . . . , d} c ⃝ Copyright E. Platen NS of SDEs Chap. 2 116
for commutative noise ( ) k Yn 1 Ynk αk ak (τn 1 , Yn 1 ) (1 αk ) ak m 1 ( 2 b k,j ) ) ( k,j τn , Ψ̄n b W j j 1 with Ψ̄n Yn a m bj W j j 1 αk [0, 1] for k {1, 2, . . . , d} c ⃝ Copyright E. Platen NS of SDEs Chap. 2 117
Alternative Implicit Methods implicit methods are important overcome a range of numerical instabilities the above strong schemes do not provide implicit diffusion terms important limitation drift implicit methods are well adapted for small noise and additive noise for relatively large multiplicative noise implicit diffusion terms seem unavoidable c ⃝ Copyright E. Platen NS of SDEs Chap. 2 118
illustration with multiplicative noise dXt σ Xt dWt explicit strong methods have large errors fo
FINANCE Eckhard Platen Finance Discipline Group and School of Mathematical Sciences University of Technology, Sydney Platen, E. & Bruti-Liberati, N.: Numerical Solution of SDEs with Jumps in Finance Springer, Applications of Mathematics (2010). Platen, E. & Heath, D.: A Benchmark Approach to Quantitative Finance
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