Numerical Solution of Stochastic Differential Equations with Jumps in Finance

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 Solutions of Stochastic Differential Equations Springer, Applications of Mathematics 23 (1992,1995,1999).

Pl, E. & Heath, D.: A Benchmark Approach to Quantitative Finance, Springer Finance (2006).

Bruti-Liberati, N. & Pl, E.: Numerical Solutions of Stochastic Differential Equations with Jumps Springer, Applications of Mathematics (2008).

1 23

Springer Finance

A Benchmark Approach to Quantitative Finance

1 A Benchmark

Approach to Quantitative Finance

S F

Platen · Heath

Eckhard Platen David Heath

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Springer Finance

E. Platen · D. Heath

The benchmark approach provides a general framework for financial market modeling, which extends beyond the standard risk neutral pricing theory.

It allows for a unified treatment of portfolio optimization, derivative pricing, integrated risk management and insurance risk modeling. The existence of an equivalent risk neutral pricing measure is not required. Instead, it leads to pricing formulae with respect to the real world probability measure. This yields important modeling freedom which turns out to be necessary for the derivation of realistic, parsimonious market models.

The first part of the book describes the necessary tools from probability theory, statistics, stochastic calculus and the theory of stochastic differential equations with jumps. The second part is devoted to financial modeling under the bench- mark approach. Various quantitative methods for the fair pricing and hedging of derivatives are explained. The general framework is used to provide an under- standing of the nature of stochastic volatility.

The book is intended for a wide audience that includes quantitative analysts, postgraduate students and practitioners in finance, economics and insurance.

It aims to be a self-contained, accessible but mathematically rigorous introduction to quantitative finance for readers that have a reasonable mathematical or quanti- tative background. Finally, the book should stimulate interest in the benchmark approach by describing some of its power and wide applicability.

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ISBN 3-540-26212-1

›

Jump-Diffusion Multi-Factor Models

Bj¨ork, Kabanov & Runggaldier (1997)

• continuous time

• Markovian

• explicit transition densities in special cases

• benchmark framework

• discrete time approximations

• suitable for simulation

• Markov chain approximations

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

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

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

Continuous and Event Driven Risk

• Wiener processes W^k, k ∈ {1, 2, . . ., m}

• counting processes p^k intensity h^k

jump martingale q^k

dW_t^m+k = dq_t^k = dp^k_t − h^kt dt

h^k_t
⁻¹₂ k ∈ {1, 2, . . . , d−m}

W_t = (W_t¹, . . . , W_t^m, q_t¹, . . . , q_t^d−m)^⊤

Primary Security Accounts

dS_t^j = S_t−^j a^j_t dt +

d

X

k=1

b^j,k_t dW_t^k

!

Assumption 1

b^j,k_t ≥ − q

h^k−m_t k ∈ {m + 1, . . . , d}.

Assumption 2

Generalized volatility matrix b_t = [b^j,k_t ]^d_j,k=1 invertible.

• market price of risk

θ_t = (θ_t¹, . . . , θ_t^d)^⊤ = b⁻_t ¹ [at − r^t 1]

• primary security account

dS_t^j = S_t−^j r_t dt +

d

X

k=1

b^j,k_t (θ_t^k dt + dW_t^k)

!

• portfolio

dS_t^δ =

d

X

j=0

δ_t^j dS_t^j

• fraction

π_δ,t^j = δ_t^j S_t^j S_t^δ

• portfolio

dS_t^δ = S_t−^δ n

r_t dt + π^⊤_δ,t− b_t (θt dt + dWt)o

Assumption 3

q

h^k−m_t > θ_t^k

• generalized GOP volatility

c^k_t =







θ_t^k for k ∈ {1, 2, . . . , m}

θ^k_t

1−θt^k (h^k−mt )⁻¹² for k ∈ {m + 1, . . . , d}

• GOP fractions

π_δ∗,t = (π_δ¹

∗,t, . . . , π_δ^d

∗,t)^⊤ = 

c^⊤_t b⁻_t ¹^⊤

• Growth Optimal Portfolio dS_t^δ∗ = S_t−^δ∗ 

r_t dt + c^⊤_t (θt dt + dWt)

• optimal growth rate g_t^δ∗ = rt + 1

2

m

X

k=1

(θ_t^k)²

−

d

X

k=m+1

h^k−m_t



ln



1 + θ_t^k q

h^k−m_t − θt^k



 + θ_t^k q

h^k−m_t





• benchmarked portfolio

Sˆ_t^δ = S_t^δ S_t^δ∗

Theorem 4 Any nonnegative benchmarked portfolio ˆS^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 (Loewenstein & Willard (2000))

Multi-Factor Model

model mainly:

• benchmarked primary security accounts

Sˆ_t^j = S_t^j S_t^δ∗ j ∈ {0, 1, . . . , d}

supermartingales, often SDE driftless, local martingales, sometimes martingales

savings account

S_t⁰ = exp

Z t

0

r_s ds



=⇒ GOP

S_t^δ∗ = S_t⁰ Sˆ_t⁰

=⇒ stock

S_t^j = ˆS_t^j S_t^δ∗

additionally dividend rates foreign interest rates

Example

Black-Scholes Type Market

d ˆS_t^j = − ˆS_t−^j

d

X

k=1

σ_t^j,k dW_t^k

h^j_t, σ_t^j,k, r_t

Examples

• Merton jump-diffusion model

dX_t = Xt− (µ dt + σ dWt + dpt) ,

⇓

X_t = X₀ e^{(µ−12σ2)t+σWt}

Nt

Y

i=1

ξ_i

• Bates model

dS_t = St−

α dt + p

V_t dW_t^S + dpt

 dV_t = ξ(η − V^t) dt + θp

V_t dW_t^V

0 0.5 1 1.5 2 2.5 3

0 5 10 15 20

time

Figure 1: Simulated benchmarked primary security accounts.

0 1 2 3 4 5 6 7 8 9 10

0 5 10 15 20

time

Figure 2: Simulated primary security accounts.

0.5 1 1.5 2 2.5 3 3.5 4 4.5

0 5 10 15 20

time

GOP EWI

Figure 3: Simulated GOP and EWI for d = 50.

0.5 1 1.5 2 2.5 3 3.5 4 4.5

0 5 10 15 20

time

GOP index

Figure 4: Simulated accumulation index and GOP.

Diversification

• diversified portfolios

π_δ,t^j    ≤

K₂ d^12+K1

Theorem 5

In a regular market any diversified portfolio is an approximate GOP.

Pl. (2005)

• robust characterization

• similar to Central Limit Theorem

• model independent

0 10 20 30 40 50 60

0 5 9 14 18 23 27 32

Figure 5: Benchmarked primary security accounts.

0 50 100 150 200 250 300 350 400 450

0 5 9 14 18 23 27 32

Figure 6: Primary security accounts under the MMM.

0 10 20 30 40 50 60 70 80 90 100

0 5 9 14 18 23 27 32

EWI GOP

Figure 7: GOP and EWI.

0 10 20 30 40 50 60 70 80 90 100

0 5 9 14 18 23 27 32

Market index GOP

Figure 8: GOP and market index.

• fair security

benchmarked security (A, P )-martingale ⇐⇒ fair

• minimal replicating portfolio

fair nonnegative portfolio S^δ with S_τ^δ = Hτ

=⇒ minimal nonnegative replicating portfolio

• fair pricing formula

V_Hτ(t) = S_t^δ∗ E

H_τ S_τ^δ∗

    A^t



No need for equivalent risk neutral probability measure!

Fair Hedging

• fair portfolio S_t^δ

• benchmarked fair portfolio

Sˆ_t^δ = E

H_τ S_τ^δ∗

    A^t



• martingale representation H_τ

S_τ^δ∗ = E

H_τ S_τ^δ∗

    A^t

 +

d

X

k=1

Z τ

t

x^k_Hτ(s) dW_s^k + MHτ(t)

M_Hτ-(A, P )-martingale (pooled) E M_Hτ, W^k

t
= 0 F¨ollmer & Schweizer (1991)

No need for equivalent risk neutral probability measure!

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

• intensity of jump process

– regular schemes =⇒ high intensity

– jump-adapted schemes =⇒ low intensity

SDE with Jumps

dX

= a(t, X

)dt + b(t, X

)dW

+ c(t −, X

) dp

X₀ ∈ ℜ^d

• p^t = Nt: Poisson process, intensity λ < ∞

• p^t = PNt

i=1(ξi − 1): compound Poisson, ξ_i i.i.d r.v.

• Poisson random measure Z

E

c(t−, X^t−, v) pφ(dv × dt)

• {(τⁱ, ξ_i), i = 1, 2, . . . , N_T}

Numerical Schemes

• time discretization

t_n = n∆

• discrete time approximation

Y_n^∆₊₁ = Y_n^∆ + a(Y_n^∆)∆ + b(Y_n^∆)∆Wn + c(Y_n^∆)∆pn

Strong Convergence

• Applications: scenario analysis, filtering and hedge simulation

• strong order γ if

ε_s(∆) = r

E  

X_T − YN^∆

2

≤ K ∆^γ

Weak Convergence

• Applications: derivative pricing, utilities, risk measures

• weak order β if

ε_w(∆) = |E(g(X^T)) − E(g(YN^∆))| ≤ K∆^β

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

• Gardo `n (2004) =⇒ γ ∈ {0.5, 1, . . .} strong schemes

• Higham & Kloeden (2005) =⇒ implicit Euler scheme

• Bruti-Liberati & Pl (2005) =⇒ γ ∈ {0.5, 1, . . .}

explicit, implicit, derivative-free, predictor-corrector

Euler Scheme

• Euler scheme

Y_n+1 = Y_n + a(Y_n)∆ + b(Y_n)∆W_n + c(Y_n)∆p_n where

∆W_n ∼ N (0, ∆) and ∆p_n = N_tn+1−N^tn ∼ P oiss(λ ∆)

• γ = 0.5

Strong Taylor Scheme

Wagner-Platen expansion =⇒

Yn+1 = Yn + a(Yn)∆ + b(Yn)∆Wn + c(Yn)∆pn + b(Yn)b^′(Yn) I_(1,1) + b(Yn) c^′(Yn) I_(1,−1) + {b(Yⁿ + c(Yn)) − b(Yⁿ)} I^(−1,1)

+{c (Yⁿ + c(Yn)) − c(Yⁿ)} I^(−1,−1) with

I_(1,1) = ¹₂{(∆Wⁿ)² − ∆}, I_(−1,−1) = 1

2{(∆pⁿ)² − ∆pⁿ} I_(1,−1) = P^N(tn+1)

i=N (tn)+1 Wτi − ∆pⁿ Wtn, I_(−1,1) = ∆pn ∆Wn − I^(1,−1)

• simulation jump times τⁱ : W_τi =⇒ I_(1,−1) and I_(−1,1)

• Computational effort heavily dependent on intensity λ

Derivative-Free Strong Schemes

avoid computation of derivatives

⇓

order 1.0 derivative-free strong scheme

Implicit Strong Schemes

wide stability regions

⇓

implicit Euler scheme

order 1.0 implicit strong Taylor scheme

Predictor-Corrector Euler Scheme

• corrector

Y_n+1 = Yn + 

θ a¯_η( ¯Y_n+1) + (1 − θ) ¯a^η(Yn)

∆n

+ 

η b( ¯Y_n+1) + (1 − η) b(Yⁿ)

∆Wn +

p(tn+1)

X

i=p(tⁿ)+1

c (ξi)

¯

a_η = a − η b b^′

• predictor

Y¯_n+1 = Y_n + a(Y_n) ∆_n + b(Y_n) ∆W_n +

p(tⁿ₊₁)

X

i=p(tⁿ)+1

c (ξ_i)

θ, η ∈ [0, 1] degree of implicitness

Jump-Adapted Time Discretization

t₀ t₁ t₂ t₃ = T

regular

τ₁ r

τ₂

r

jump times

t₀ t₁ t₂ t₃ t₄ t₅ = T

r r

jump-adapted

Jump-Adapted Strong Approximations

jump-adapted time discretisation

⇓

jump times included in time discretisation

• jump-adapted Euler scheme

Y_tn+1− = Ytn + a(Ytn)∆tn + b(Ytn)∆Wtn

and

Y_tn+1 = Ytn+1− + c(Ytn+1−) ∆pn

• γ = 0.5

Merton SDE : µ = 0.05, σ = 0.2, ψ = −0.2, λ = 10, X⁰ = 1, T = 1

0 0.2 0.4 0.6 0.8 1

T

0 0.2 0.4 0.6 0.8 1

X

Figure 9: Plot of a jump-diffusion path.

0 0.2 0.4 0.6 0.8 1

T

-0.00125 -0.001 -0.00075 -0.0005 -0.00025 0 0.00025 0.0005

Error

Figure 10: Plot of the strong error for Euler(red) and 1.0 Taylor(blue)

Merton SDE : µ = −0.05, σ = 0.1, λ = 1, X⁰ = 1, T = 0.5

-10 -8 -6 -4 -2 0

Log₂dt

-25 -20 -15 -10

Log 2Error

15TaylorJA 1TaylorJA 1Taylor EulerJA Euler

Figure 11: Log-log plot of strong error versus time step size.

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, extrapo- lation and simplified schemes

• Kubilius & Pl (2002) and Glasserman & Merener (2003)

=⇒ jump-adapted Euler with weaker assumptions on coefficients

• Bruti-Liberati & Pl (?) =⇒ jump-adapted order β ∈ {1, 2 . . .}

derivative-free, implicit and predictor-corrector schemes

Simplified Euler Scheme

• Euler scheme =⇒ β = 1

• simplified Euler scheme

Y_n+1 = Yn + a(Yn)∆ + b(Yn)∆ ˆW_n + c(Yn) ( ˆξ_n − 1)∆ˆp_n

• if ∆ ˆW_n and ∆ ˆp_n match the first 3 moments of ∆Wn and ∆pn up to an O(∆²) error =⇒ β = 1

•

P(∆ ˜W_n = ±√

∆) = 1 2

Jump-Adapted Taylor Approximations

• jump-adapted Euler scheme =⇒ β = 1

• jump-adapted order 2 weak Taylor scheme

Y_tn+1⁻ = Ytn + a∆tn + b∆Wtn + b b^′ 2

(∆Wtn)² − ∆^tn

+ a^′ b ∆Ztn

+ 1 2



a a^′ + 1

2a^{′ ′}b²



∆²_tn +



a b^′ + 1

2b^′′ b²



{∆W^tn ∆tn− ∆Z^tn} and

Y_tn+1 = Ytn+1− + c(Ytn+1−) ∆pn

• β = 2

Predictor-Corrector Schemes

• predictor-corrector =⇒ stability and efficiency

• jump-adapted predictor-corrector Euler scheme Y_tn+1− = Ytn + 1

2

na( ¯Y_tn+1−) + ao

∆tn + b∆Wtn

with predictor

Y¯_tn+1− = Ytn + a∆tn + b∆Wtn

• β = 1

-5 -4 -3 -2 -1

Log₂dt

-2 -1 0 1 2 3

Log2Error

PredCorrJA ImplEulerJA EulerJA

Figure 12: Log-log plot of weak error versus time step size.

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

Conclusions

• low intensity =⇒ jump-adapted higher order predictor-corrector

• high intensity =⇒ regular schemes

• distinction between strong and weak predictor-corrector schemes

References

Bj¨ork, T., Y. Kabanov, & W. Runggaldier (1997). Bond market structure in the presence of marked point processes. Math. Finance 7, 211–239.

Bruti-Liberati, N. & E. Platen (2005). On the strong approximation of jump-diffusion processes. Technical report, University of Technology, Sydney. QFRC Research Paper 157.

Delbaen, F. & W. Schachermayer (2006). The Mathematics of Arbitrage. Springer Finance. Springer.

F¨ollmer, 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.

Gardo`n, A. (2004). The order of approximations for solutions of Itˆo-type stochastic differential equations with jumps. Stochastic Analysis and Applications 22(3), 679–699.

Glasserman, P. & N. Merener (2003). Numerical solution of jump-diffusion LIBOR market models. Fi- nance Stoch. 7(1), 1–27.

Higham, D. & P. 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 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.

Loewenstein, M. & G. A. Willard (2000). Local martingales, arbitrage, and viability: Free snacks and cheap thrills. Econometric Theory 16(1), 135–161.

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 Itˆo equa- tions. 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 pro- cesses. Math. Nachr. 151, 233–239.

Platen, E. (1982). An approximation method for a class of Itˆo processes with jump component. Liet. Mat.

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Platen, E. (2005). Diversified portfolios with jumps in a benchmark framework. Asia-Pacific Financial Markets 11(1), 1–22.

Platen, E. & W. J. Runggaldier (2005). A benchmark approach to filtering in finance. Asia-Pacific Finan- cial Markets 11(1), 79–105.

Platen, E. & W. J. Runggaldier (2007). A benchmark approach to portfolio optimization under partial information. Technical report, University of Technology, Sydney. QFRC Research Paper 191.