Construct Heston model
heston = heston(Return, Speed, Level, Volatility)
heston = heston(Return, Speed, Level, Volatility,
'Name1', Value1, 'Name2', Value2, ...)
This constructor creates and displays heston
objects,
which derive from thesdeddo
(SDE
from drift and diffusion objects) class. Use heston
objects
to simulate sample paths of two state variables. Each state variable
is driven by a single Brownian motion source of risk over NPERIODS
consecutive
observation periods, approximating continuoustime stochastic volatility
processes.
Heston models are bivariate composite models. Each Heston model consists of two coupled univariate models:
A geometric Brownian motion (gbm
) model with a stochastic volatility function.
$$d{X}_{1t}=B(t){X}_{1t}dt+\sqrt{{X}_{2t}}{X}_{1t}d{W}_{1t}$$
This model usually corresponds to a price process whose volatility (variance rate) is governed by the second univariate model.
A CoxIngersollRoss (cir
) square root diffusion model.
$$d{X}_{2t}=S(t)[L(t){X}_{2t}]dt+V(t)\sqrt{{X}_{2t}}d{W}_{2t}$$
This model describes the evolution of the variance rate of the coupled GBM price process.
Specify required input parameters as one of the following types:
A MATLAB^{®} array. Specifying an array indicates a static (nontimevarying) parametric specification. This array fully captures all implementation details, which are clearly associated with a parametric form.
A MATLAB function. Specifying a function provides indirect support for virtually any static, dynamic, linear, or nonlinear model. This parameter is supported via an interface, because all implementation details are hidden and fully encapsulated by the function.
You can specify combinations of array and function input parameters as needed.
Moreover, a parameter is identified as a deterministic function
of time if the function accepts a scalar time t
as
its only input argument. Otherwise, a parameter is assumed to be a
function of time t and state X(t) and
is invoked with both input arguments.
The required input parameters are:
Return  If you specify

Speed  If you specify

Level  If you specify

Volatility  If you specify

Although the constructor does not enforce restrictions on the signs of any of these input arguments, each argument is specified as a positive value.
Specify optional input arguments as variablelength lists of
matching parameter name/value pairs: 'Name1'
, Value1
, 'Name2'
, Value2
,
... and so on. The following rules apply when specifying parametername
pairs:
Specify the parameter name as a character vector, followed by its corresponding parameter value.
You can specify parameter name/value pairs in any order.
Parameter names are case insensitive.
You can specify unambiguous partial character vector matches.
The following table lists valid parameter names.
StartTime  Scalar starting time of the first observation, applied to all
state variables. If you do not specify a value for StartTime ,
the default is 0 . 
StartState  Scalar, If If If If you do not specify
a value for 
Correlation  Correlation between Gaussian random variates drawn to
generate the Brownian motion vector (Wiener processes). Specify A As a deterministic function
of time, If you do not specify a value for 
Simulation  A userdefined simulation function or SDE simulation method.
If you do not specify a value for Simulation , the
default method is simulation by Euler approximation (simByEuler ). 
heston  Object of class

See Creating Heston Stochastic Volatility Models.
AitSahalia, Y. “Testing ContinuousTime Models of the Spot Interest Rate.” The Review of Financial Studies, Spring 1996, Vol. 9, No. 2, pp. 385–426.
AitSahalia, 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, SpringerVerlag, 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: ContinuousTime Models. New York: SpringerVerlag, 2004.