Documentation

This is machine translation

Translated by
Mouseover text to see original. Click the button below to return to the English verison of the page.

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

sde class

Superclasses:

Stochastic Differential Equation (SDE) model

Description

The `sde` constructor creates and displays general stochastic differential equation (`SDE`) models from user-defined drift and diffusion rate functions. Use `sde` objects to simulate sample paths of `NVARS` state variables driven by `NBROWNS` Brownian motion sources of risk over `NPERIODS` consecutive observation periods, approximating continuous-time stochastic processes.

An `sde` object enables you to simulate 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:

• Xt is an `NVARS`-by-`1` state vector of process variables.

• dWt 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.

Construction

`SDE = sde(DriftRate,DiffusionRate)` constructs a default `sde` object.

`SDE = sde(DriftRate,DiffusionRate,Name,Value)` constructs a `sde` 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`.

For more information on constructing a `sde` object, see `sde`.

Input Arguments

expand all

`DriftRate` is a user-defined drift-rate function and represents the parameter F, specified as a vector or object of class `drift`.

`DriftRate` is a function that returns an `NVARS`-by-`1` drift-rate vector when called with two inputs:

• A real-valued scalar observation time t.

• An `NVARS`-by-`1` state vector Xt.

Alternatively, `DriftRate` can also be an object of class `drift` that encapsulates the drift-rate specification. In this case, however, `sde` uses only the `Rate` parameter of the object. For more information on the `drift` object, see `drift`.

Data Types: `double`

`DiffusionRate` is a user-defined drift-rate function and represents the parameter G, specified as a matrix or object of class `diffusion`.

`DiffusionRate` is a function that returns an `NVARS`-by-`NBROWNS` diffusion-rate matrix when called with two inputs:

• A real-valued scalar observation time t.

• An `NVARS`-by-`1` state vector Xt.

Alternatively, `DiffusionRate` can also be an object of class `diffusion` that encapsulates the diffusion-rate specification. In this case, however, `sde` uses only the `Rate` parameter of the object. For more information on the `diffusion` object, see `diffusion`.

Data Types: `double`

Name-Value Pair Arguments

Specify optional comma-separated pairs of `Name,Value` arguments. `Name` is the argument name and `Value` is the corresponding value. `Name` must appear inside single quotes (`' '`). You can specify several name and value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.

For more information on using optional name-value arguments, see `sde`.

Properties

expand all

Drift rate component of continuous-time stochastic differential equations (SDEs), specified as a drift object or function accessible by (t, Xt.

The drift rate specification supports the simulation of sample paths of `NVARS` state variables driven by `NBROWNS` Brownian motion sources of risk over `NPERIODS` consecutive observation periods, approximating continuous-time stochastic processes.

The `drift` class allows you to create drift-rate objects (using the`drift` constructor) of the form:

`$F\left(t,{X}_{t}\right)=A\left(t\right)+B\left(t\right){X}_{t}$`
where:

• `A` is an `NVARS`-by-`1` vector-valued function accessible using the (t, Xt) interface.

• `B` is an `NVARS`-by-`NVARS` matrix-valued function accessible using the (t, Xt) interface.

The `drift` object's displayed parameters are:

• `Rate`: The drift-rate function, F(t,Xt)

• `A`: The intercept term, A(t,Xt), of F(t,Xt)

• `B`: The first order term, B(t,Xt), of F(t,Xt)

`A` and `B` enable you to query the original inputs. The function stored in `Rate` fully encapsulates the combined effect of `A` and `B`.

When specified as MATLAB® double arrays, the inputs `A` and `B` are clearly associated with a linear drift rate parametric form. However, specifying either `A` or `B` as a function allows you to customize virtually any drift rate specification.

Note

You can express `drift` and `diffusion` classes in the most general form to emphasize the functional (t, Xt) interface. However, you can specify the components `A` and `B` as functions that adhere to the common (t, Xt) interface, or as MATLAB arrays of appropriate dimension.

Example: `F = drift(0, 0.1) % Drift rate function F(t,X)`

Attributes:

 `SetAccess` `private` `GetAccess` `public`

Data Types: `struct` | `double`

Diffusion rate component of continuous-time stochastic differential equations (SDEs), specified as a drift object or function accessible by (t, Xt.

The diffusion rate specification supports the simulation of sample paths of `NVARS` state variables driven by `NBROWNS` Brownian motion sources of risk over `NPERIODS` consecutive observation periods, approximating continuous-time stochastic processes.

The `diffusion` class allows you to create diffusion-rate objects (using the constructor `diffusion` constructor):

`$G\left(t,{X}_{t}\right)=D\left(t,{X}_{t}^{\alpha \left(t\right)}\right)V\left(t\right)$`
where:

• `D` is an `NVARS`-by-`NVARS` diagonal matrix-valued function.

• Each diagonal element of `D` is the corresponding element of the state vector raised to the corresponding element of an exponent `Alpha`, which is an `NVARS`-by-`1` vector-valued function.

• `V` is an `NVARS`-by-`NBROWNS` matrix-valued volatility rate function `Sigma`.

• `Alpha` and `Sigma` are also accessible using the (t, Xt) interface.

The `diffusion` object's displayed parameters are:

• `Rate`: The diffusion-rate function, G(t,Xt).

• `Alpha`: The state vector exponent, which determines the format of D(t,Xt) of G(t,Xt).

• `Sigma`: The volatility rate, V(t,Xt), of G(t,Xt).

`Alpha` and `Sigma` enable you to query the original inputs. (The combined effect of the individual `Alpha` and `Sigma` parameters is fully encapsulated by the function stored in `Rate`.) The `Rate` functions are the calculation engines for the `drift` and `diffusion` objects, and are the only parameters required for simulation.

Note

You can express `drift` and `diffusion` classes in the most general form to emphasize the functional (t, Xt) interface. However, you can specify the components `A` and `B` as functions that adhere to the common (t, Xt) interface, or as MATLAB arrays of appropriate dimension.

Example: ```G = diffusion(1, 0.3) % Diffusion rate function G(t,X) ```

Attributes:

 `SetAccess` `private` `GetAccess` `public`

Data Types: `struct` | `double`

Starting time of first observation, applied to all state variables, specified as a scalar

Attributes:

 `SetAccess` `public` `GetAccess` `public`

Data Types: `double`

Initial values of state variables, specified as a scalar, column vector, or matrix.

If `StartState` is a scalar, the `gbm` constructor applies the same initial value to all state variables on all trials.

If `StartState` is a column vector, the`gbm` constructor applies a unique initial value to each state variable on all trials.

If `StartState` is a matrix, the `gbm` constructor applies a unique initial value to each state variable on each trial.

Attributes:

 `SetAccess` `public` `GetAccess` `public`

Data Types: `double`

User-defined simulation function or SDE simulation method, specified as a function or SDE simulation method.

Attributes:

 `SetAccess` `public` `GetAccess` `public`

Data Types: `function_handle`

Methods

The following methods are from the `sde` class.

`interpolate`

`simulate`

`simByEuler`

Instance Hierarchy

The following figure illustrates the inheritance relationships among SDE classes.

For more information, see SDE Class Hierarchy.

Copy Semantics

Value. To learn how value classes affect copy operations, see Copying Objects (MATLAB).

Examples

expand all

Construct an `SDE` object `obj` to represent a univariate geometric Brownian Motion model of the form: .

Create drift and diffusion functions that are accessible by the common interface:

```F = @(t,X) 0.1 * X; G = @(t,X) 0.3 * X;```

Pass the functions to the `sde` constructor to create an object `obj` of class `sde`:

`obj = sde(F, G) % dX = F(t,X)dt + G(t,X)dW`
```obj = Class SDE: Stochastic Differential Equation ------------------------------------------- Dimensions: State = 1, Brownian = 1 ------------------------------------------- StartTime: 0 StartState: 1 Correlation: 1 Drift: drift rate function F(t,X(t)) Diffusion: diffusion rate function G(t,X(t)) Simulation: simulation method/function simByEuler ```

`obj` displays like a MATLAB® structure, with the following information:

• The object's class

• A brief description of the object

• A summary of the dimensionality of the model

The object's displayed parameters are as follows:

• `StartTime`: The initial observation time (real-valued scalar)

• `StartState`: The initial state vector (`NVARS`-by-1 column vector)

• `Correlation`: The correlation structure between Brownian process

• `Drift`: The drift-rate function

• `Diffusion`: The diffusion-rate function

• `Simulation`: The simulation method or function.

Of these displayed parameters, only `Drift` and `Diffusion` are required inputs.

The only exception to the (, ) evaluation interface is `Correlation`. Specifically, when you enter `Correlation` as a function, the SDE engine assumes that it is a deterministic function of time, . This restriction on `Correlation` as a deterministic function of time allows Cholesky factors to be computed and stored before the formal simulation. This inconsistency dramatically improves run-time performance for dynamic correlation structures. If `Correlation` is stochastic, you can also include it within the simulation architecture as part of a more general random number generation function.

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 Xt, and return an array of appropriate dimension. Even if you originally specified an input as an array, `sde` treats it as a static function of time and state, by that means guaranteeing that all parameters are accessible by the same interface.

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.