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,
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
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
For examples and more information on creating SDE objects, see:
Objects display like traditional MATLAB data structures.
Displayed object parameters appear as nouns that begin with capital letters. In contrast, parameters such as
interpolateappear 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,Xt) 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 Example: Univariate GBM Models.
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:
X is an NVars-by-
1state vector of process variables (for example, short rates or equity prices) to simulate.
W is an NBrowns-by-
1Brownian motion vector.
F is an NVars-by-
1vector-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 Xt. 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,Xt) = 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.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
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
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:
Before simulation, the simulation method itself calls each function once to validate the size of the output at the initial time and state,
During simulation, the simulation method calls each function the same number of times:
However, there is an important distinction regarding the
timing between these two types of functions. It is most clearly drawn
directly from the generic
This equation is expressed in continuous time, but the simulation methods approximate the model in discrete time as:
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, Xt).
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 Pricing Equity Options.
Therefore, all simulation methods evaluate noise generation functions as:
for t = t0, t0 + Δt, t0 + 2Δt, ..., T – Δt.
Yet simulation methods evaluate end-of-period processing functions as:
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:
where A is an NVars-by-
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:
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:
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
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.
For example, the following models are special cases of the general SDE model.
|Brownian Motion (BM)|
|Geometric Brownian Motion (GBM)|
|Constant Elasticity of Variance (CEV)|
Bates models are bivariate composite models. Each Bates model consists of two coupled univariate models:
SDE Simulation and Interpolation Methods
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
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:
You can use the
simBySolution function with the name-value
QuasiSequence to simulates approximate solutions of
quasi-Monte Carlo simulations for the following classes:
In addition, you can also use:
cirobject to approximates a continuous-time Cox-Ingersoll-Ross (CIR) model by an approximation of the transition density function.
batesobject to approximates a continuous-time Bates model by an approximation of the transition density function.
hestonobject to approximates a continuous-time Heston model by an approximation of the transition density function.
cirobject to generate sample paths by using a Quadratic-Exponential discretization scheme.