Euler simulation of stochastic differential equations (SDEs)
[
specifies options using one or more name-value pair arguments in addition to the
input arguments in the previous syntax.Paths
,Times
,Z
] = simByEuler(___,Name,Value
)
This function simulates any vector-valued SDE of the form
where:
X is an
NVARS-by-1
state vector of process
variables (for example, short rates or equity prices) to simulate.
W is an
NBROWNS-by-1
Brownian motion
vector.
F is an
NVARS-by-1
vector-valued
drift-rate function.
G is an NVARS-by-NBROWNS matrix-valued diffusion-rate function.
simByEuler
simulates NTRIALS
sample
paths of NVARS
correlated state variables driven by
NBROWNS
Brownian motion sources of risk over
NPERIODS
consecutive observation periods, using the Euler
approach to approximate continuous-time stochastic processes.
This simulation engine provides a discrete-time approximation of the
underlying generalized continuous-time process. The simulation is derived
directly from the stochastic differential equation of motion. Thus, the
discrete-time process approaches the true continuous-time process only as
DeltaTime
approaches zero.
The input argument Z
allows you to directly specify the
noise-generation process. This process takes precedence over the
Correlation
parameter of the sde
object and the value of the
Antithetic
input flag. If you do not specify a value
for Z
, simByEuler
generates correlated
Gaussian variates, with or without antithetic sampling as requested.
The end-of-period Processes
argument allows you to
terminate a given trial early. At the end of each time step,
simByEuler
tests the state vector
Xt for an
all-NaN
condition. Thus, to signal an early termination
of a given trial, all elements of the state vector
Xt must be
NaN
. This test enables a user-defined
Processes
function to signal early termination of a
trial, and offers significant performance benefits in some situations (for
example, pricing down-and-out barrier options).
[1] Deelstra, G. and F. Delbaen. “Convergence of discretized stochastic (interest rate) processes with stochastic drift term.” Applied Stochastic Models and Data Analysis., 1998, Vol. 14, Number 1, pp. 77–84.
[2] Higham, D. and X. Mao. “Convergence of Monte Carlo simulations involving the mean-reverting square root process.” Journal of Computational Finance., 2005, Vol. 8, Number 3, pp. 35–61.
[3] Lord, R., R. Koekkoek, and D. Van Dijk. “A comparison of biased simulation schemes for stochastic volatility models.” Quantitative Finance., 2010, Vol. 10, Number 2, pp. 177–194.
|
bm
| cev
| cir
| gbm
| heston
| hwv
| sde
| sdeddo
| sdeld
| sdemrd
| simBySolution
| simByTransition
| simulate