# simulate

Simulate multivariate stochastic differential equations (SDEs) for
`SDE`

, `BM`

, `GBM`

,
`CEV`

, `CIR`

, `HWV`

,
`Heston`

, `SDEDDO`

, `SDELD`

,
`SDEMRD`

, `Merton`

, or `Bates`

models

## Description

`[`

adds optional inputs for `Paths`

,`Times`

,`Z`

] = simulate(___,`Optional`

,`Scheme`

)`Optional`

and
`Scheme`

.

The `Optional`

input argument for `simulate`

accepts any variable-length list of input arguments that the simulation method or
function referenced by the `SDE.Simulation`

parameter requires or
accepts. It passes this input list directly to the appropriate SDE simulation method
or user-defined simulation function.

The optional input `Scheme`

lets you specify an approximation
scheme used to simulate the sample paths and you can use this optional input with or
without an `Optional`

input argument.

## Examples

## Input Arguments

## Output Arguments

## More About

## Algorithms

This function simulates any vector-valued SDE of the form:

$$d{X}_{t}=F(t,{X}_{t})dt+G(t,{X}_{t})d{W}_{t}$$ | (1) |

*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.

## References

[1] Ait-Sahalia, Y. “Testing Continuous-Time Models of the Spot Interest
Rate.” *The Review of Financial Studies*, Spring 1996, Vol.
9, No. 2, pp. 385–426.

[2] Ait-Sahalia, Y. “Transition Densities for Interest Rate and Other
Nonlinear Diffusions.” *The Journal of Finance*, Vol. 54,
No. 4, August 1999.

[3] Glasserman, P. *Monte Carlo Methods in Financial
Engineering.* New York, Springer-Verlag, 2004.

[4] Hull, J. C. *Options, Futures, and Other Derivatives*, 5th
ed. Englewood Cliffs, NJ: Prentice Hall, 2002.

[5] Johnson, N. L., S. Kotz, and N. Balakrishnan. *Continuous Univariate
Distributions.* Vol. 2, 2nd ed. New York, John Wiley & Sons,
1995.

[6] Shreve, S. E. *Stochastic Calculus for Finance II: Continuous-Time
Models.* New York: Springer-Verlag, 2004.

## Version History

**Introduced in R2008a**

## See Also

`simByEuler`

| `simBySolution`

| `simBySolution`

| `sde`

| `merton`

| `bates`

| `bm`

| `gbm`

| `sdeddo`

| `sdeld`

| `cev`

| `cir`

| `heston`

| `hwv`

| `sdemrd`