# sdemrd

SDE with Mean-Reverting Drift (`SDEMRD`

) model

## Description

Creates and displays a SDE object whose drift rate is expressed in
mean-reverting drift-rate form and which derives from the `sdeddo`

class (SDE from drift and diffusion objects).

Use `sdemrd`

objects to simulate of sample paths of
`NVars`

state variables expressed in mean-reverting drift-rate
form, and provide a parametric alternative to the linear drift form (see `sdeld`

). These state variables are driven by `NBrowns`

Brownian motion sources of risk over `NPeriods`

consecutive observation
periods, approximating continuous-time stochastic processes with mean-reverting
drift-rate functions.

The `sdemrd`

object allows you to simulate any vector-valued SDEMRD
of the form:

$$d{X}_{t}=S(t)[L(t)-{X}_{t}]dt+D(t,{X}_{t}^{\alpha (t)})V(t)d{W}_{t}$$

where:

*X*is an_{t}`NVars`

-by-`1`

state vector of process variables.*S*is an`NVars`

-by-`NVars`

matrix of mean reversion speeds.*L*is an`NVars`

-by-`1`

vector of mean reversion levels.*D*is an`NVars`

-by-`NVars`

diagonal matrix, where each element along the main diagonal is the corresponding element of the state vector raised to the corresponding power of*α*.*V*is an`NVars`

-by-`NBrowns`

instantaneous volatility rate matrix.*dW*is an_{t}`NBrowns`

-by-`1`

Brownian motion vector.

## Creation

### Description

creates a `SDEMRD`

= sdemrd(___,`Name,Value`

)`SDEMRD`

object with additional options specified
by one or more `Name,Value`

pair arguments.

`Name`

is a property name and `Value`

is
its corresponding value. `Name`

must appear inside single
quotes (`''`

). You can specify several name-value pair
arguments in any order as
`Name1,Value1,…,NameN,ValueN`

.

The `SDELD`

object has the following displayed Properties:

`StartTime`

— Initial observation time`StartState`

— Initial state at time`StartTime`

`Correlation`

— Access function for the`Correlation`

input argument, callable as a function of time`Drift`

— Composite drift-rate function, callable as a function of time and state`Diffusion`

— Composite diffusion-rate function, callable as a function of time and state`Speed`

— Access function for the input argument`Speed`

, callable as a function of time and state`Level`

— Access function for the input argument`Level`

, callable as a function of time and state`Alpha`

— Access function for the input argument`Alpha`

, callable as a function of time and state`Sigma`

— Access function for the input argument`Sigma`

, callable as a function of time and state`Simulation`

— A simulation function or method

### Input Arguments

## Properties

## Object Functions

`interpolate` | Brownian interpolation of stochastic differential equations (SDEs) for
`SDE` , `BM` , `GBM` ,
`CEV` , `CIR` , `HWV` ,
`Heston` , `SDEDDO` , `SDELD` , or
`SDEMRD` models |

`simulate` | Simulate multivariate stochastic differential equations (SDEs) for
`SDE` , `BM` , `GBM` ,
`CEV` , `CIR` , `HWV` ,
`Heston` , `SDEDDO` , `SDELD` ,
`SDEMRD` , `Merton` , or `Bates`
models |

`simByEuler` | Euler simulation of stochastic differential equations (SDEs) for
`SDE` , `BM` , `GBM` ,
`CEV` , `CIR` , `HWV` ,
`Heston` , `SDEDDO` , `SDELD` , or
`SDEMRD` models |

`simByMilstein` | Simulate diagonal diffusion for `BM` , `GBM` ,
`CEV` , `HWV` , `SDEDDO` ,
`SDELD` , or `SDEMRD` sample paths by Milstein
approximation |

`simByMilstein2` | Simulate `BM` , `GBM` , `CEV` ,
`HWV` , `SDEDDO` , `SDELD` ,
`SDEMRD` process sample paths by second order Milstein
approximation |

## Examples

## More About

## Algorithms

When you specify the required input parameters as arrays, they are associated with a specific parametric form. By contrast, when you specify either required input parameter as a function, you can customize virtually any specification.

Accessing the output parameters with no inputs simply returns the original input specification. Thus, when you invoke these parameters with no inputs, they behave like simple properties and allow you to test the data type (double vs. function, or equivalently, static vs. dynamic) of the original input specification. This is useful for validating and designing methods.

When you invoke these parameters with inputs, they behave like functions, giving the
impression of dynamic behavior. The parameters accept the observation time
*t* and a state vector
*X _{t}*, and return an array of appropriate
dimension. Even if you originally specified an input as an array,

`sdemrd`

treats it as a static function of time and state, by that
means guaranteeing that all parameters are accessible by the same interface.## References

[1] Aït-Sahalia, Yacine. “Testing
Continuous-Time Models of the Spot Interest Rate.” *Review of Financial
Studies*, vol. 9, no. 2, Apr. 1996, pp. 385–426.

[2] Aït-Sahalia, Yacine.
“Transition Densities for Interest Rate and Other Nonlinear Diffusions.” *The
Journal of Finance*, vol. 54, no. 4, Aug. 1999, pp.
1361–95.

[3] Glasserman, Paul.
*Monte Carlo Methods in Financial Engineering*. Springer,
2004.

[4] Hull, John.
*Options, Futures and Other Derivatives*. 7th ed, Prentice
Hall, 2009.

[5] Johnson, Norman Lloyd, et al.
*Continuous Univariate Distributions*. 2nd ed, Wiley,
1994.

[6] Shreve, Steven E.
*Stochastic Calculus for Finance*. Springer,
2004.

## Version History

**Introduced in R2008a**

## See Also

`drift`

| `diffusion`

| `sdeddo`

| `simByEuler`

| `nearcorr`

### Topics

- Creating Stochastic Differential Equations from Mean-Reverting Drift (SDEMRD) Models
- Simulating Equity Prices
- Simulating Interest Rates
- Stratified Sampling
- Price American Basket Options Using Standard Monte Carlo and Quasi-Monte Carlo Simulation
- Base SDE Models
- Drift and Diffusion Models
- Linear Drift Models
- Parametric Models
- SDEs
- SDE Models
- SDE Class Hierarchy
- Quasi-Monte Carlo Simulation
- Performance Considerations