Euler simulation of stochastic differential equations (SDEs)

`[Paths,Times,Z] = simByEuler(MDL,NPeriods)`

`[Paths,Times,Z] = simByEuler(___,Name,Value)`

`[`

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

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

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*X*for an all-_{t}`NaN`

condition. Thus, to signal an early termination of a given trial, all elements of the state vector*X*must be_{t}`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`

- Implementing Multidimensional Equity Market Models, Implementation 5: Using the simByEuler Method
- Simulating Equity Prices
- Simulating Interest Rates
- Stratified Sampling
- Pricing American Basket Options by Monte Carlo Simulation
- Base SDE Models
- Drift and Diffusion Models
- Linear Drift Models
- Parametric Models
- SDEs
- SDE Models
- SDE Class Hierarchy
- Performance Considerations