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Euler simulation of stochastic differential equations (SDEs)

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

```
[Paths, Times, Z] = simByEuler(MDL, NPERIODS, 'Name1',
Value1, 'Name2', Value2, ...)
```

All classes in the SDE Class Hierarchy.

This method simulates any vector-valued SDE of the form

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

*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.

`MDL` | Stochastic differential equation object created with the `sdeddo` constructor. |

`NPERIODS` | Positive scalar integer number of simulation periods. The value
of `NPERIODS` determines the number of rows of the
simulated output series. |

Specify optional inputs as matching parameter name/value pairs as follows:

Specify the parameter name as a character vector, followed by its corresponding value.

You can specify parameter name/value pairs in any order.

Parameter names are case insensitive.

You can specify unambiguous partial character vector matches.

Valid parameter names are:

`NTRIALS` | Positive scalar integer number of simulated trials (sample
paths) of `NPERIODS` observations each. If you do
not specify a value for this argument, the default is `1` ,
indicating a single path of correlated state variables. |

`DeltaTime` | Scalar or `NPERIODS` -by-`1` column
vector of positive time increments between observations. `DeltaTime` represents
the familiar dt found in stochastic differential
equations, and determines the times at which the simulated paths of
the output state variables are reported. If you do not specify a value
for this argument, the default is `1` . |

`NSTEPS` | Positive scalar integer number of intermediate time steps
within each time increment If
you do not specify a value for |

`Antithetic` | Scalar logical flag that indicates whether When Odd trials `(1,3,5,...)` correspond to the primary Gaussian paths.Even trials `(2,4,6,...)` are the matching antithetic paths of each pair derived by negating the Gaussian draws of the corresponding primary (odd) trial.
If you specify |

`Z` | Direct specification of the dependent random noise process
used to generate the Brownian motion vector (Wiener process) that
drives the simulation. Specify this argument as a function, or as
an A real-valued scalar observation time *t*.An `NVARS` -by-`1` state vector*X*._{t}
If you do not specify a value for |

`StorePaths` | Scalar logical flag that indicates how the output array If |

`Processes` | Function or cell array of functions that indicates a sequence of end-of-period processes or state vector adjustments of the form $${X}_{t}=P(t,{X}_{t})$$
If
you specify more than one processing function, If
you do not specify a processing function, |

`Paths` | `(NPERIODS + 1)` -by-`NVARS` -by-`NTRIALS` three-dimensional
time series array, consisting of simulated paths of correlated state
variables. For a given trial, each row of `Paths` is
the transpose of the state vector X_{t} at
time t. When the input flag `StorePaths` = `FALSE` , `simByEuler` returns `Paths` as
an empty matrix. |

`Times` | `(NPERIODS + 1)` -by-`1` column
vector of observation times associated with the simulated paths. Each
element of `Times` is associated with the corresponding
row of `Paths` . |

`Z` | `(NPERIODS * NSTEPS)` -by-`NBROWNS` -by-`NTRIALS` three-dimensional
time series array of dependent random variates used to generate the
Brownian motion vector (Wiener processes) that drive the simulation. |

Implementing Multidimensional Equity Market Models, Implementation 5: Using the simByEuler Method

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).

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.

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

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

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

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

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

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