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3rd Conference on Financial Mathematics & Applications.
30, 31 January 2013, Semnan University, Semnan, Iran, pp. xxx-xxx
Jump-Diffusions Stochastic Differential Equations: Simulation
and Financial Applications
∗
B. †Kafash
‡
A. Delavarkhalafi†
M.Hasani‡
∗, ,
Faculty of Mathematics, Yazd University, Yazd, Iran.
Emam Javad Higher Education Institute, yazd, Iran.
E-mail: [email protected],
E-mail: [email protected], hasani [email protected]
∗
Abstract
The modelling of many real life phenomena for which either the parameter estimation is
difficult, or which are subject to random noisy perturbation, is often carried out by using
stochastic differential equations (SDEs). In this paper, we introduce jump-diffusion stochastic differential equations and express it’s applications. Also simulate Merton jump-diffusion
model via Matlab software.
Keywords and phrases: Jump-diffusion stochastic differential equations, Simulation, Merton jump-diffusion model
1. Introduction and Preliminaries
Stochastic process models play a prominent role in a range of application areas, including
economics and finance. In mathematical modelling, if we use stochastic systems then we will
assume that the system follows a probabilistic rule and the future behaviour of the system
will not be known for sure.
Suppose W (t) is a standard Brownian motion, and f and g are some given functions. The
solution of stochastic differential equation, dX(t) = f (t, X(t))dt + g(t, X(t))dW (t), is continuous almost every where, if exists. To have a solution of stochastic differential equation
with jump, we should add a jump part to the above equation. The Poisson process P (t)
and Brownian motion W (t) are the two fundamental examples in the theory of continuous
time stochastic processes. The Wiener and Poisson processes form the tools of a toolbox to
create jump-diffusion process models.
2. Jump-Diffusion Rules and SDEs
Wiener diffusion and simple Poisson jump processes provide an introduction to elementary
stochastic initial value problem in continuous time for the simple jump-diffusion state process:
dX(t) = f (X(t), t)dt + g(X(t), t)dW (t) + h(X(t), t)dP (t), X(0) = X0 .
(2.1)
With a set of continuous coefficient functions {f, g, h}, possibly nonlinear in the state X(t).
1
2
2.1. Chain Rule.
Ito stochastic chain rule for jump-diffusions with simple Poisson jumps is
Let F (x, t) be twice continuously differentiable in x and once in t, then [1].
1
dF (X(t), t) =
Ft + f Fx + g 2 Fxx (X(t), t)dt + (gFx )(X(t), t)dW (t)
2
+ (F (X(t) + h(X(t), t), t) − F (X(t), t))dP (t).
(2.2)
2.2. Linear Jump-Diffusion SDEs.
Let the linear diffusion and jump SDEs be combined into a single SDE:
dX(t) = X(t)(µ(t)dt + σ(t)dW (t) + ν(t)dP (t)),
(2.3)
X(t0 ) = x0 > 0 with probability one (this is for simplicity, but only x0 6= 0 is sufficient),
where the set of coefficients {µ(t), σ(t), ν(t), λ(t)} are assumed to be bounded and integrable,
with ν(t) > 1 (otherwise, positivity of X(t) cannot be maintained) and σ(t) > 0 (for consistency with the interpretation as a standard deviation coefficient of the process). The
logarithmic transformation of the state process Y (t) = ln(X(t)) the state from the right
hand side of the SDE using the jump-diffusion chain rule (2.2) and the first two logarithmic
derivatives, so
dY (t) = (µ(t) − σ 2 (t)/2)dt + σ(t)dW (t) + ln(1 + ν(t))dP (t).
(2.4)
SDE (2.4) is a linear combination of the deterministic, diffusion and jump processes with
deterministic time dependent coefficients, so it can be immediately but formally integrated
to yield,
Z t
Y (t) = y0 +
(µ(s) − σ 2 (s)/2)ds + σ(s)dW (s) + ln(1 + ν(s))dP (s)),
(2.5)
t0
where y0 = ln(x0 ), recalling that it has been assumed that x0 > 0. Inverting logarithmic
state Y (t) back to the original state
X(t) = exp(Y (t))
leads to
X(t) = x0 × exp
Z
t
t0
(µ(s)σ 2 (s)/2)ds + σ(s)dW (s) + ln(1 + ν(s))dP (s)) ,
(2.6)
2.3. Linear Jump-Diffusion SDEs with Constant Coefficients:
For the special case of constant rate coefficients, µ(t) = µ0 , σ(t) = σ0 , ν(t) = ν0 and
λ(t) = λ0 , also setting t0 = 0, leads to the SDE:
dX(t) = X(t)(µ0 dt + σ0 dW (t) + ν0 dP (t)),
(2.7)
X(t0 ) = x0 > 0 with probability one with solution:
X(t)
=
x0 exp((µ0 − σ02 /2)t + σ0 W (t) + ln(1 + ν0 )P (t)
=
x0 (1 + ν0 )P (t) exp((µ0 − σ02 /2)t + σ0 W (t)),
(2.8)
3
Linear Jump−Diffusion Simulations
1.4
X(t) Sample 1
X(t) Sample 2
X(t) Sample 3
X(t) Sample 4
X(t), Jump−Diffusion State
1.2
1
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t, Time
Figure 1. four Sample path of Merton jump-diffusion model
3. Simulating Stochastic Differential Equations: Jump-Diffusion Models
We now briefly discuss how to approximately simulate certain types of jump-diffusion
processes when exact simulation is impossible. We discretize time and utilize an Eulertype scheme. Let P (t) be a Poisson process, W (t) a standard Brownian motion and Y =
{Y1 , Y2 , · · · } a sequence of IID random variables. We assume P (t), W (t) and Y are all independent of one another. Now, consider a jump-diffusion model (2.1). X(t) might represent
the time t value of an underlying security or relevant state variable. We can approximately
simulate a path of X(t) on [0, T ] in a number of ways. We now describe one such way:
1: Define an initial grid 0, h, 2h, · · · , mh = T .
2: Since the Poisson process, P (t), is independent of W (t), we can imagine that we first
simulate the jump times of the process in [0; T ]. Let these times be τ1 , τ2 · · · τP (T )
noting of course that P (T ) will vary from sample path to sample path.
3: We now create a combined time grid, 0 = t0 , t1 , · · · , tM = T consisting of the
original mh + 1 grid points as well as the P (T ) jump times. We therefore have
M = mh + 1 + P (T ) .
4: We then approximately simulate X(t) at points on the combined grid.
If we want to generate n sample paths then we repeat steps 1 to 4 a total of n times, noting
that we obtain a different combined grid for each sample path.
3.1. Merton jump-diffusion model.
The unique solution of Merton jump-diffusion model [3], dX(t) = X(t− )(µdt + σdW (t) +
Q t
1
dP (t)) is X(t) = X(0) exp((µ − σ 2 )t + σW (t)) P
i=1 ξi . Where compound Poisson, P (t) =
2
P Pt
i=1 (ξi − 1) and ξi ’s are IID random variables. The results of simulation of equation above
equation with µ = 0.05, σ = 0.2, π = −0.2, λ = 10, X0 = 1 and T = 1 using Matlab is
shown in Figure 1.
References
[1] Hanson, F. B. , Applied Stochastic Processes and Control for Jump-Diffusions: Modeling, Analysis and Computation, University of Illinois Chicago, Illinois, USA, (2007).
[2] Oksendal, B., Stochastic Differential Equations, Springer-Verlag, Berlin, (2003).
[3] Platen, E., Bruti Liberati, N.,Numerical solutions of stochastic differential equations with jumps, Sydney, Springer, (2010).