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# simByEuler

Euler simulation of stochastic differential equations (SDEs)

## Synopsis

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

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

## Classes

All classes in the SDE Class Hierarchy.

## Description

This method simulates any vector-valued SDE of the form

`$d{X}_{t}=F\left(t,{X}_{t}\right)dt+G\left(t,{X}_{t}\right)d{W}_{t}$`
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.

## Input Arguments

 `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.

## Optional Input Arguments

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 dt (specified as `DeltaTime`). The `simByEuler` method partitions each time increment dt into `NSTEPS` subintervals of length dt/`NSTEPS`, and refines the simulation by evaluating the simulated state vector at ```NSTEPS − 1``` intermediate points. Although `simByEuler` does not report the output state vector at these intermediate points, the refinement improves accuracy by allowing the simulation to more closely approximate the underlying continuous-time process. If you do not specify a value for `NSTEPS`, the default is `1`, indicating no intermediate evaluation. `Antithetic` Scalar logical flag that indicates whether `simByEuler` uses antithetic sampling to generate the Gaussian random variates that drive the Brownian motion vector (Wiener processes). When `Antithetic` is `TRUE` (logical `1`), `simByEuler` performs sampling such that all primary and antithetic paths are simulated and stored in successive matching pairs: 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 `Antithetic` to be any value other than `TRUE`, `simByEuler` assumes that it is `FALSE` (logical `0`) by default, and does not perform antithetic sampling. When you specify an input noise process (see `Z`), `simByEuler` ignores the value of `Antithetic`. `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 `(NPERIODS * NSTEPS)`-by-`NBROWNS`-by-`NTRIALS` three-dimensional array of dependent random variates. If you specify `Z` as a function, it must return an `NBROWNS`-by-`1` column vector, and you must call it with two inputs: A real-valued scalar observation time t.An `NVARS`-by-`1` state vector Xt.If you do not specify a value for `Z`, `simByEuler` generates correlated Gaussian variates based on the `Correlation` member of the `SDE` object. `StorePaths` Scalar logical flag that indicates how the output array `Paths` is stored and returned to the caller. If `StorePaths` is `TRUE` (the default value) or is unspecified, `simByEuler` returns `Paths` as a three-dimensional time series array. If `StorePaths` is `FALSE` (logical `0`), `simByEuler` returns the `Paths` output array as an empty matrix. `Processes` Function or cell array of functions that indicates a sequence of end-of-period processes or state vector adjustments of the form `${X}_{t}=P\left(t,{X}_{t}\right)$``simByEuler` applies processing functions at the end of each observation period. These functions must accept the current observation time t and the current state vector Xt, and return a state vector that may be an adjustment to the input state. If you specify more than one processing function, `simByEuler` invokes the functions in the order in which they appear in the cell array. You can use this argument to specify boundary conditions, prevent negative prices, accumulate statistics, plot graphs, and more.If you do not specify a processing function, `simByEuler` makes no adjustments and performs no processing.

## Output Arguments

 `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 Xt 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.

## Examples

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

## Algorithms

• 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).

## References

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.