SDE Models

Introduction

Most models and utilities available with Monte Carlo Simulation of SDEs are represented as MATLAB^® objects. Therefore, this documentation often uses the terms model and object interchangeably.

However, although all models are represented as objects, not all objects represent models. In particular, drift, diffusion objects are used in model specification, but neither of these types of objects in and of themselves makes up a complete model. Usually, you do not need to create drift, diffusion objects directly, so you do not need to differentiate between objects and models. It is important, however, to understand the distinction between these terms.

In many of the following examples, most model parameters are evaluated or invoked like any MATLAB function. Although it is helpful to examine and access model parameters as you would data structures, think of these parameters as functions that perform actions.

Creating SDE Objects

• Creating Objects

• Displaying Objects

• Assigning and Referencing Object Parameters

• Creating and Evaluating Models

• Specifying SDE Simulation Parameters

Creating Objects

For examples and more information on creating SDE objects, see:

• Available SDE Models

• Simulating Equity Prices

• Simulating Interest Rates

Displaying Objects

• Objects display like traditional MATLAB data structures.

• Displayed object parameters appear as nouns that begin with capital letters. In contrast, parameters such as simulate and interpolate appear as verbs that begin with lowercase letters, which indicate tasks to perform.

Assigning and Referencing Object Parameters

• Objects support referencing similar to data structures. For example, statements like the following are valid:

  A = obj.A 

• Objects support complete parameter assignment similar to data structures. For example, statements like the following are valid:

  obj.A = 3

• Objects do not support partial parameter assignment as data structures do. Therefore, statements like the following are invalid:

  obj.A(i,j) = 0.3   

Creating and Evaluating Models

• You can create objects of any model class only if enough information is available to determine unambiguously the dimensionality of the model. Because each object offers unique input interfaces, some models require additional information to resolve model dimensionality.

• You need only enter required input parameters in placeholder format, where a given input argument is associated with a specific position in an argument list. You can enter optional inputs in any order as parameter name-value pairs, where the name of a given parameter appears in single quotation marks and precedes its corresponding value.

• Association of dynamic (time-variable) behavior with function evaluation, where time and state (t,X_t) are passed to a common, published interface, is pervasive throughout the SDE class system. You can use this function evaluation approach to model or construct powerful analytics. For a simple example, see Specify Univariate Geometric Brownian Motion Model.

Specifying SDE Simulation Parameters

The SDE engine allows the simulation of generalized multivariate stochastic processes, and provides a flexible and powerful simulation architecture. The framework also provides you with utilities and model classes that offer various parametric specifications and interfaces. The architecture is fully multidimensional in both the state vector and the Brownian motion, and offers both linear and mean-reverting drift-rate specifications.

You can specify most parameters as MATLAB arrays or as functions accessible by a common interface, that supports general dynamic/nonlinear relationships common in SDE simulation. Specifically, you can simulate correlated paths of any number of state variables driven by a vector-valued Brownian motion of arbitrary dimensionality. This simulation approximates the underlying multivariate continuous-time process using a vector-valued stochastic difference equation.

Consider the following general stochastic differential equation:

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

(1)

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.

The drift and diffusion rates, F and G, respectively, are general functions of a real-valued scalar sample time t and state vector X_t. Also, static (non-time-variable) coefficients are simply a special case of the more general dynamic (time-variable) situation, just as a function can be a trivial constant; for example, f(t,X_t) = 4. The SDE in Equation 1 is useful in implementing derived classes that impose additional structure on the drift and diffusion-rate functions.

Specifying User-Defined Functions as Model Parameters.  Several examples in this documentation emphasize the evaluation of object parameters as functions accessible by a common interface. In fact, you can evaluate object parameters by passing to them time and state, regardless of whether the underlying user-specified parameter is a function. However, it is helpful to compare the behavior of object parameters that are specified as functions to that of user-specified noise and end-of-period processing functions.

Model parameters that are specified as functions are evaluated in the same way as user-specified random number (noise) generation functions. (For more information, see Evaluating Different Types of Functions.) Model parameters that are specified as functions are inputs to remove object constructors. User-specified noise and processing functions are optional inputs to simulation methods.

Because class constructors offer unique interfaces, and simulation methods of any given model have different implementation details, models often call parameter functions for validation purposes a different number of times, or in a different order, during object creation, simulation, and interpolation.

Therefore, although parameter functions, user-specified noise generation functions, and end-of-period processing functions all share the interface and are validated at the same initial time and state (obj.StartTime and obj.StartState), parameter functions are not guaranteed to be invoked only once before simulation as noise generation and end-of-period processing functions are. In fact, parameter functions might not even be invoked the same number of times during a given Monte Carlo simulation process.

In most applications in which you specify parameters as functions, they are simple, deterministic functions of time and/or state. There is no need to count periods, count trials, or otherwise accumulate information or synchronize time.

However, if parameter functions require more sophisticated bookkeeping, the correct way to determine when a simulation has begun (or equivalently, to determine when model validation is complete) is to determine when the input time and/or state differs from the initial time and state (obj.StartTime and obj.StartState, respectively). Because the input time is a known scalar, detecting a change from the initial time is likely the best choice in most situations. This is a general mechanism that you can apply to any type of user-defined function.

Evaluating Different Types of Functions.  It is useful to compare the evaluation rules of user-specified noise generation functions to those of end-of-period processing functions. These functions have the following in common:

• They both share the same general interface, returning a column vector of appropriate length when evaluated at the current time and state:

  $$X_t=f(t,X_t)$$

  $$z_t=Z(t,X_t)$$

• Before simulation, the simulation method itself calls each function once to validate the size of the output at the initial time and state, obj.StartTime, and obj.StartState, respectively.

• During simulation, the simulation method calls each function the same number of times: NPeriods * NSteps.

However, there is an important distinction regarding the timing between these two types of functions. It is most clearly drawn directly from the generic SDE model:

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

This equation is expressed in continuous time, but the simulation methods approximate the model in discrete time as:

$$X_{t+\Delta t}=X_t+F(t,X_t)\Delta t+G(t,X_t)\sqrt{\Delta t}Z(t,X_t)$$

where Δt > 0 is a small (and not necessarily equal) period or time increment into the future. This equation is often referred to as a Euler approximation, a simulation technique that provides a discrete-time approximation of a continuous-time stochastic process. All functions on the rightmost side are evaluated at the current time and state (t, X_t).

In other words, over the next small time increment, the simulation evolves the state vector based only on information available at the current time and state. In this sense, you can think of the noise function as a beginning-of-period function, or as a function evaluated from the left. This is also true for any user-supplied drift or diffusion function.

In contrast, user-specified end-of-period processing functions are applied only at the end of each simulation period or time increment. For more information about processing functions, see Price European Stock Options Using Monte Carlo Simulation.

Therefore, all simulation methods evaluate noise generation functions as:

$$z_t=Z(t,X_t)$$

for t = t₀, t₀ + Δt, t₀ + 2Δt, ..., T – Δt.

Yet simulation methods evaluate end-of-period processing functions as:

$$X_t=f(t,X_t)$$

for t = t₀ + Δt, t₀ + 2Δt, ..., T.

where t₀ and T are the initial time (taken from the object) and the terminal time (derived from inputs to the simulation method), respectively. These evaluations occur on all sample paths. Therefore, during simulation, noise functions are never evaluated at the final (terminal) time, and end-of-period processing functions are never evaluated at the initial (starting) time.

Drift and Diffusion

For example, an SDE with a linear drift rate has the form:

$$F(t,X_t)=A(t)+B(t)X_t$$

(2)

where A is an NVars-by-1 vector-valued function and B is an NVars-by-NVars matrix-valued function.

As an alternative, consider a drift-rate specification expressed in mean-reverting form:

$$F(t,X_t)=S(t)[L(t)-X_t]$$

(3)

where S is an NVars-by-NVars matrix-valued function of mean reversion speeds (that is, rates of mean reversion), and L is an NVars-by-1 vector-valued function of mean reversion levels (that is, long run average level).

Similarly, consider the following diffusion-rate specification:

$$G(t,X_t)=D(t,X_t^{\alpha (t)})V(t)$$

(4)

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 also an NVars-by-1 vector-valued function. V is an NVars-by-NBrowns matrix-valued function of instantaneous volatility rates. Each row of V corresponds to a particular state variable, and each column corresponds to a particular Brownian source of uncertainty. V associates the exposure of state variables with sources of risk.

The parametric specifications for the drift and diffusion-rate functions associate parametric restrictions with familiar models derived from the general SDE class, and provide coverage for many models.

The class system and hierarchy of the SDE engine use industry-standard terminology to provide simplified interfaces for many models by placing user-transparent restrictions on drift and diffusion specifications. This design allows you to mix and match existing models, and customize drift-rate or diffusion-rate functions.

Available SDE Models

For example, the following models are special cases of the general SDE model.

SDE Models

Model Name	Specification
Brownian Motion (BM)	$$d{X}_t=A(t)dt+V(t)d{W}_t$$
Geometric Brownian Motion (GBM)	$$d{X}_t=B(t)X_{t}dt+V(t)X_{t}d{W}_t$$
Constant Elasticity of Variance (CEV)	$$d{X}_t=B(t)X_{t}dt+V(t)X_t^{\alpha (t)}d{W}_t$$
Cox-Ingersoll-Ross (CIR)	$$d{X}_t=S(t)(L(t)-X_t)dt+V(t)X_t^{\frac{1}{2}}d{W}_t$$
Hull-White/Vasicek (HWV)	$$d{X}_t=S(t)(L(t)-X_t)dt+V(t)d{W}_t$$
Heston	$$d{X}_{1t}=B(t)X_{1t}dt+\sqrt{X_{2t}}{X}_{1t}d{W}_{1t}$$ $$d{X}_{2t}=S(t)[L(t)-X_{2t}]dt+V(t)\sqrt{X_{2t}}d{W}_{2t}$$
Merton	$$d{X}_t=B(t,X_t)X_{t}dt+D(t,X_t)V(t,x_t)d{W}_t+Y(t,X_t)X_{t}d{N}_t$$
Bates	Bates models are bivariate composite models. Each Bates model consists of two coupled univariate models: A geometric Brownian motion (gbm) model with a stochastic volatility function. $$d{X}_{1t}=B(t)X_{1t}dt+\sqrt{X_{2t}}{X}_{1t}d{W}_{1t}+Y(t)X_{1t}d{N}_t$$ A Cox-Ingersoll-Ross (cir) square root diffusion model. $$d{X}_{2t}=S(t)[L(t)-X_{2t}]dt+V(t)\sqrt{X_{2t}}d{W}_{2t}$$
rvm	The rough volatility model (RVM) model allows you to simulate any vector-valued RVM of the form: $$\begin{array}{l}d{X}_{1t}=B(t)X_{1t}dt+X_{2t}{X}_{1t}d{W}_{1t}\\ X_{2t}=\xi e^{Y_t}\\ Y_t={\displaystyle {\int }_{-\infty }^{t}g(t-s){\nu }_{s}d{W}_s}\end{array}$$
roughbergomi	The rough Bergomi volatility model allows you to simulate any vector-valued RVM of the form: $$\begin{array}{l}d{X}_{1t}=B(t)X_{1t}dt+\sqrt{X_{2t}}{X}_{1t}d{W}_{1t}\\ X_{2t}={\xi }_0(t)e^{\eta \sqrt{2a+1}{Y}_t^{\alpha }-\frac{{\eta }^2}{2}{t}^{2\alpha +1}}\\ Y_t={\displaystyle {\int }_0^t{(t-s)}^{\alpha }d{W}_s}\end{array}$$
roughheston	The rough Heston volatility model allows you to simulate any vector-valued RVM of the form: $$\begin{array}{l}d{X}_t=B(t)X_{t}dt+\sqrt{Y_t}{X}_{t}d{W}_t\\ Y_t=Y_0+\frac{1}{\Gamma (\alpha +1)}{\displaystyle {\int }_0^t{(t-s)}^{\alpha }\kappa (\theta -Y_{s)}ds+\frac{1}{\Gamma (\alpha +1)}{\displaystyle {\int }_0^t{(t-s)}^{\alpha }\sigma }\sqrt{Y_s}d{W}_s}\end{array}$$

SDE Simulation and Interpolation Methods

The sde class provides default simulation and interpolation methods for all derived classes:

• simulate: High-level wrapper around the user-specified simulation method stored in the Simulation property

• simByEuler: Default Euler approximation simulation method

• interpolate: Stochastic interpolation method (that is, Brownian bridge)

You can use the simBySolution function to simulate approximate solutions of diagonal-drift processes for the following classes:

• gbm supports simBySolution

• hwv supports simBySolution

• merton supports simBySolution

You can use the simBySolution function with the name-value arguments for MonteCarloMethod and QuasiSequence to simulates approximate solutions of quasi-Monte Carlo simulations for the following classes:

• gbm supports simBySolution

• hwv supports simBySolution

• merton supports simBySolution

In addition, you can also use:

• simByTransition with a cir object to approximates a continuous-time Cox-Ingersoll-Ross (CIR) model by an approximation of the transition density function.

• simByTransition with a bates object to approximates a continuous-time Bates model by an approximation of the transition density function.

• simByTransition with a heston object to approximates a continuous-time Heston model by an approximation of the transition density function.

• simByQuadExp with a heston, bates, or cir object to generate sample paths by using a Quadratic-Exponential discretization scheme.

• simByMilstein with a heston, bates, cir, sde, or merton object to process sample paths by Milstein approximation.

• simByMilstein2 with a heston, bates, cir, sde, or merton object to process sample paths by second order Milstein approximation.

• simByHybrid with a rvm or roughbergomi object to process sample paths by hybrid approximation.

• simByEuler with a rvm, roughheston, or roughbergomi object to process sample paths by Euler approximation.

SDE Classes and Associated Simulation Methods

The following table lists the Financial Toolbox™ SDE classes and their supported simulation methods. For more information on models for the supported classes, see Available SDE Models.

SDE Class Names	Supported Simulation Methods
sde	simByEuler, simulate
bm	simByEuler, simByMilstein, simByMilstein2, simulate
gbm	simByEuler, simBySolution, simByMilstein, simByMilstein2, simulate
merton	simBySolution, simByMilstein, simByMilstein2, simulate
bates	simByTransition, simByQuadExp, simByMilstein, simByMilstein2, simulate
hwv	simBySolution, simByMilstein, simByMilstein2, simulate
heston	simByTransition, simByQuadExp, simByMilstein, simByMilstein2, simulate
cev	simByEuler, simByMilstein, simByMilstein2, simulate
cir	simByEuler, simByTransition, simByQuadExp, simByMilstein, simByMilstein2, simulate
sdeddo	simByEuler, simByMilstein, simByMilstein2, simulate
sdeld	simByEuler, simByMilstein, simByMilstein2, simulate
sdemrd	simByEuler, simByMilstein, simByMilstein2, simulate
rvm	simByEuler, simByHybrid
roughbergomi	simByEuler, simByHybrid
roughheston	simByEuler