gbm

Geometric Brownian motion (GBM) model

Description

Creates and displays a geometric Brownian motion model (GBM), which derives from the cev (constant elasticity of variance) class.

Geometric Brownian motion (GBM) models allow you to simulate sample paths of NVars state variables driven by NBrowns Brownian motion sources of risk over NPeriods consecutive observation periods, approximating continuous-time GBM stochastic processes. Specifically, this model allows the simulation of vector-valued GBM processes of the form

$$d{X}_t=\mu (t)X_{t}dt+D(t,X_t)V(t)d{W}_t$$

where:

• X_t is an NVars-by-1 state vector of process variables.

• μ is an NVars-by-NVars generalized expected instantaneous rate of return matrix.

• D is an NVars-by-NVars diagonal matrix, where each element along the main diagonal is the corresponding element of the state vector X_t.

• V is an NVars-by-NBrowns instantaneous volatility rate matrix.

• dW_t is an NBrowns-by-1 Brownian motion vector.

Creation

Syntax

GBM = gbm(Return,Sigma)

GBM = gbm(___,Name,Value)

Description

GBM = gbm(Return,Sigma) creates a default GBM object.

Specify the required input parameters as one of the following types:

• A MATLAB^® array. Specifying an array indicates a static (non-time-varying) 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.

Note

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.

GBM = gbm(___,Name,Value) creates a GBM 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 GBM object has the following Properties:

• StartTime — Initial observation time

• StartState — Initial state at StartTime

• Correlation — Access function for the Correlation input, 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

• Simulation — A simulation function or method

• Return — Access function for the input argument Return, callable as a function of time and state

• Sigma — Access function for the input argument Sigma, callable as a function of time and state

Input Arguments

Return — Return represents the parameter μ

array or deterministic function of time or deterministic function of time and state

Return represents the parameter μ, specified as an array or deterministic function of time.

If you specify Return as an array, it must be an NVars-by-NVars matrix representing the expected (mean) instantaneous rate of return.

As a deterministic function of time, when Return is called with a real-valued scalar time t as its only input, Return must produce an NVars-by-NVars matrix. If you specify Return as a function of time and state, it must return an NVars-by-NVars matrix when invoked with two inputs:

• A real-valued scalar observation time t.

• An NVars-by-1 state vector X_t.

Data Types: double | function_handle

Sigma — Sigma represents the parameter V

array or deterministic function of time or deterministic function of time and state

Sigma represents the parameter V, specified as an array or a deterministic function of time.

If you specify Sigma as an array, it must be an NVars-by-NBrowns matrix of instantaneous volatility rates or as a deterministic function of time. In this case, each row of Sigma corresponds to a particular state variable. Each column corresponds to a particular Brownian source of uncertainty, and associates the magnitude of the exposure of state variables with sources of uncertainty.

As a deterministic function of time, when Sigma is called with a real-valued scalar time t as its only input, Sigma must produce an NVars-by-NBrowns matrix. If you specify Sigma as a function of time and state, it must return an NVars-by-NBrowns matrix of volatility rates when invoked with two inputs:

• A real-valued scalar observation time t.

• An NVars-by-1 state vector X_t.

Although the gbm object enforces no restrictions on the sign of Sigma volatilities, they are specified as positive values.

Data Types: double | function_handle

Properties

StartTime — Starting time of first observation, applied to all state variables

0 (default) | scalar

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

Data Types: double

StartState — Initial values of state variables

1 (default) | scalar, column vector, or matrix

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

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

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

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

Data Types: double

Correlation — Correlation between Gaussian random variates drawn to generate the Brownian motion vector (Wiener processes)

NBrowns-by-NBrowns identity matrix representing independent Gaussian processes (default) | positive semidefinite matrix | deterministic function

Correlation between Gaussian random variates drawn to generate the Brownian motion vector (Wiener processes), specified as an NBrowns-by-NBrowns positive semidefinite matrix, or as a deterministic function C(t) that accepts the current time t and returns an NBrowns-by-NBrowns positive semidefinite correlation matrix. If Correlation is not a symmetric positive semidefinite matrix, use nearcorr to create a positive semidefinite matrix for a correlation matrix.

A Correlation matrix represents a static condition.

As a deterministic function of time, Correlation allows you to specify a dynamic correlation structure.

Data Types: double

Simulation — User-defined simulation function or SDE simulation method

simulation by Euler approximation (simByEuler) (default) | function | SDE simulation method

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

Data Types: function_handle

Drift — Drift rate component of continuous-time stochastic differential equations (SDEs)

value stored from drift-rate function (default) | drift object or function accessible by (t, X_t)

This property is read-only.

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

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 drift) of the form:

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

where:

• A is an NVars-by-1 vector-valued function accessible using the (t, X_t) interface.

• B is an NVars-by-NVars matrix-valued function accessible using the (t, X_t) interface.

The displayed parameters for a drift object are:

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

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

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

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, X_t) interface. However, you can specify the components A and B as functions that adhere to the common (t, X_t) interface, or as MATLAB arrays of appropriate dimension.

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

Data Types: struct | double

Diffusion — Diffusion rate component of continuous-time stochastic differential equations (SDEs)

value stored from diffusion-rate function (default) | diffusion object or functions accessible by (t, X_t)

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

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 diffusion):

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

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, X_t) interface.

The diffusion object's displayed parameters are:

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

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

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

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, X_t) interface. However, you can specify the components A and B as functions that adhere to the common (t, X_t) interface, or as MATLAB arrays of appropriate dimension.

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

Data Types: struct | double

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
simBySolution	Simulate approximate solution of diagonal-drift GBM processes
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

Create a gbm Object

Create a univariate gbm object to represent the model: $d{X}_t=0.25X_{t}dt+0.3X_{t}d{W}_t$.

obj = gbm(0.25, 0.3)  % (B = Return, Sigma)

obj = 
   Class GBM: Generalized Geometric Brownian Motion
   ------------------------------------------------
     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
         Return: 0.25
          Sigma: 0.3

gbm objects display the parameter B as the more familiar Return

Compute Price of European Option Using Monte Carlo Simulation with GBM Object

This example shows the workflow to compute the price of a European option using Monte Carlo simulation with a gbm object.

Set up the parameters for the Geometric Brownian Motion (GBM) model and the European option.

% Parameters for the GBM model and option
S0 = 100; % Initial stock price
K = 110; % Strike price
T = 1; % Time to maturity in years
r = 0.05; % Risk-free interest rate
sigma = 0.20; % Volatility
nTrials = 10000; % Number of Monte Carlo trials
nPeriods = 1; % Number of periods (for one year, this can be set to 1)

Create a gbm object.

% Create GBM object
gbmobj = gbm(r,sigma,'StartState',S0);

Use simulate to simulate the end-of-year stock prices using the GBM model (gbm) over nTrials trials.

% Simulate stock prices at maturity
[Paths, ~, ~] = simulate(gbmobj,nPeriods,'nTrials',nTrials,'DeltaTime',T);

% Extract the final prices for all trials
ST = Paths(end, :, 1);

Calculate the payoff for European call and put options based on the simulated prices.

% Calculate payoffs for call and put options
callPayoff = max(ST - K, 0)

callPayoff = 
5.7533

putPayoff = max(K - ST, 0)

putPayoff = 
0

Discount these payoffs back to the present value and then average the payoff values to estimate the option prices.

% Discount payoffs back to present value and average
callPrice = exp(-r * T) * mean(callPayoff)

callPrice = 
5.4727

putPrice = exp(-r * T) * mean(putPayoff)

putPrice = 
0

Display the estimated prices for both the call and put options.

% Display results
fprintf('European Call Option Price: %.4f\n', callPrice);

European Call Option Price: 5.4727

fprintf('European Put Option Price: %.4f\n', putPrice);

European Put Option Price: 0.0000

More About

Instance Hierarchy

There are inheritance relationships among the SDE classes.

The following figure illustrates the inheritance relationships.

For more information, see SDE Models, SDE Classes and Associated Simulation Methods, and SDE Class Hierarchy.

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, gbm 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

R2023b: Added simByMilstein2 method

Use the simByMilstein2 method to approximate a numerical solution of a stochastic differential equation.

R2023a: Added simByMilstein method

Use the simByMilstein method to approximate a numerical solution of a stochastic differential equation.