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simBySolution

Class: gbm

Simulate approximate solution of diagonal-drift GBM processes

Syntax

[Paths,Times,Z] = simBySolution(MDL,NPERIODS)
[Paths,Times,Z] = simBySolution(MDL,NPERIODS,Name,Value)

Description

[Paths,Times,Z] = simBySolution(MDL,NPERIODS) simulates approximate solution of diagonal-drift for geometric Brownian motion (GBM) processes.

[Paths,Times,Z] = simBySolution(MDL,NPERIODS,Name,Value) simulates approximate solution of diagonal-drift for GBM processes with additional options specified by one or more Name,Value pair arguments.

The simBySolution method simulates NTRIALS sample paths of NVARS correlated state variables, driven by NBROWNS Brownian motion sources of risk over NPERIODS consecutive observation periods, approximating continuous-time GBM short-rate models by an approximation of the closed-form solution.

Consider a separable, vector-valued GBM model of the form:

dXt=μ(t)Xtdt+D(t,Xt)V(t)dWt

where:

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

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

  • V is an NVARS-by-NBROWNS instantaneous volatility rate matrix.

  • dWt is an NBROWNS-by-1 Brownian motion vector.

The simBySolution method simulates the state vector Xt using an approximation of the closed-form solution of diagonal-drift models.

When evaluating the expressions, simBySolution assumes that all model parameters are piecewise-constant over each simulation period.

In general, this is not the exact solution to the models, because the probability distributions of the simulated and true state vectors are identical only for piecewise-constant parameters.

When parameters are piecewise-constant over each observation period, the simulated process is exact for the observation times at which Xt is sampled.

Input Arguments

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Geometric Brownian motion (GBM) model, specified as a gbm object that is created using the gbm constructor.

Data Types: struct

Number of simulation periods, specified as a positive scalar integer. The value of this argument determines the number of rows of the simulated output series.

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.

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Number of simulated trials (sample paths), specified as positive scalar integer of NPERIODS observations each. If you do not specify a value for this argument, the default is 1, indicating a single path of correlated state variables.

Data Types: double

Time increments between observations, specified as scalar or NPERIODS-by-1 column vector of positive values. DeltaTime represents the familiar dt found in stochastic differential equations, and determines the times at which simBySolution reports the simulated paths of the output state variables. If you do not specify a value for this argument, the default is 1.

Data Types: double

Number of intermediate time steps within each time increment dt (defined as DeltaTime), specified positive scalar integer. simBySolution partitions each time increment dt into NSTEPS subintervals of length dt/NSTEPS, and refines the simulation by evaluating the simulated state vector at NSTEPS - 1 intermediate points. Although simBySolution does not report the output state vector at these intermediate points, the refinement improves accuracy by allowing the simulation to more closely approximate the underlying continuous-time process. If you do not specify a value for NSTEPS, the default is 1, indicating no intermediate evaluation.

Data Types: double

Flag that indicates whether antithetic sampling is used to generate the Gaussian random variates that drive the Brownian motion vector (Wiener processes), specified using a scalar logical with values 0 or 1. When Antithetic is TRUE (logical 1), simBySolution performs sampling such that all primary and antithetic paths are simulated and stored in successive matching pairs:

  • Odd trials (1,3,5,...) correspond to the primary Gaussian paths

  • Even trials (2,4,6,...) are the matching antithetic paths of each pair derived by negating the Gaussian draws of the corresponding primary (odd) trial.

If you specify Antithetic to be any value other than TRUE, simBySolution assumes that it is FALSE (logical 0) by default, and does not perform antithetic sampling. When you specify an input noise process (see Z), simBySolution ignores the value of Antithetic.

Data Types: logical

Direct specification of the dependent random noise process used to generate the Brownian motion vector (Wiener process) that drives the simulation, specified as an (NPERIODS * NSTEPS)-by-NBROWNS-by-NTRIALS array of dependent random variates. If you specify Z as a function, it must return an NBROWNS-by-1 column vector, and you must call it with two inputs:

  • A real-valued scalar observation time t.

  • An NVARS-by-1 state vector Xt.

Data Types: double

Flag that indicates how output array Paths is stored, specified as a scalar logical with values 0 or 1. If StorePaths is TRUE (the default value) or is unspecified, simBySolution returns Paths as a three-dimensional time series array.

If StorePaths is FALSE (logical 0), simBySolution returns the Paths output array as an empty matrix.

Data Types: logical

Function or cell array of functions indicating a sequence of end-of-period processes or state vector adjustments of the form

Xt=P(t,Xt)

specified as function or cell array of functions.

simBySolution applies processing functions at the end of each observation period. These functions must accept the current observation time t and the current state vector Xt, and return a state vector that may be an adjustment to the input state. If you specify more than one processing function, simBySolution invokes the functions in the order in which they appear in the cell array. You can use this argument to specify boundary conditions, prevent negative prices, accumulate statistics, plot graphs, and more.

Data Types: double

Output Arguments

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Simulated paths of correlated state variables, returned as a (NPERIODS + 1)-by-NVARS-by-NTRIALS three-dimensional time series array. For a given trial, each row of Paths is the transpose of the state vector Xt at time t.

When the input flag StorePaths = FALSE, simBySolution returns Paths as an empty matrix.

Observation times associated with simulated paths, returned as a (NPERIODS + 1)-by-1 column vector. Each element of Times is associated with the corresponding row of Paths.

Array of dependent random variates used to generate the Brownian motion vector, returned as a (NPERIODS * NSTEPS)-by-NBROWNS-by-NTRIALS three-dimensional time series array.

Examples

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Use GBM simulation methods. Separable GBM models have two specific simulation methods:

  • An overloaded Euler simulation method, designed for optimal performance.

  • A simBySolution method that provides an approximate solution of the underlying stochastic differential equation, designed for accuracy.

Load the Data_GlobalIdx2 data set and specify the SDE model as in Representing Market Models Using SDE Objects, and the GBM model as in Representing Market Models Using SDELD, CEV, and GBM Objects.

load Data_GlobalIdx2
prices  = [Dataset.TSX Dataset.CAC Dataset.DAX ...
    Dataset.NIK Dataset.FTSE Dataset.SP];

returns =  tick2ret(prices);

nVariables  = size(returns,2);
expReturn   = mean(returns);
sigma       = std(returns);
correlation = corrcoef(returns);
t           = 0;
X           = 100;
X           = X(ones(nVariables,1));

F = @(t,X) diag(expReturn)* X;
G = @(t,X) diag(X) * diag(sigma);

SDE = sde(F, G, 'Correlation', ...
    correlation, 'StartState', X);

GBM = gbm(diag(expReturn),diag(sigma), 'Correlation', ...
    correlation, 'StartState', X);

To illustrate the performance benefit of the overloaded Euler approximation method, increase the number of trials to 10000.

nPeriods = 249;      % # of simulated observations
dt       =   1;      % time increment = 1 day
rng(142857,'twister')
[X,T] = simulate(GBM, nPeriods, 'DeltaTime', dt, ...
    'nTrials', 10000);

whos X
  Name        Size                     Bytes  Class     Attributes

  X         250x6x10000            120000000  double              

Using this sample size, examine the terminal distribution of Canada's TSX Composite to verify qualitatively the lognormal character of the data.

histogram(squeeze(X(end,1,:)), 30), xlabel('Price'), ylabel('Frequency')
title('Histogram of Prices after One Year: Canada (TSX Composite)')

Simulate 10 trials of the solution and plot the first trial:

rng('default')
[S,T] = simulate(SDE, nPeriods, 'DeltaTime', dt, 'nTrials', 10);
rng('default')
[X,T] = simBySolution(GBM, nPeriods,...
    'DeltaTime', dt, 'nTrials', 10);
subplot(2,1,1)
plot(T, S(:,:,1)), xlabel('Trading Day'),ylabel('Price')
title('1st Path of Multi-Dim Market Model:Euler Approximation')
subplot(2,1,2)
plot(T, X(:,:,1)), xlabel('Trading Day'),ylabel('Price')
title('1st Path of Multi-Dim Market Model:Analytic Solution')

In this example, all parameters are constants, and simBySolution does indeed sample the exact solution. The details of a single index for any given trial show that the price paths of the Euler approximation and the exact solution are close, but not identical.

The following plot illustrates the difference between the two methods:

subplot(1,1,1)
plot(T, S(:,1,1) - X(:,1,1), 'blue'), grid('on')
xlabel('Trading Day'), ylabel('Price Difference')
title('Euler Approx Minus Exact Solution:Canada(TSX Composite)')

The simByEuler Euler approximation literally evaluates the stochastic differential equation directly from the equation of motion, for some suitable value of the dt time increment. This simple approximation suffers from discretization error. This error can be attributed to the discrepancy between the choice of the dt time increment and what in theory is a continuous-time parameter.

The discrete-time approximation improves as DeltaTime approaches zero. The Euler method is often the least accurate and most general method available. All models shipped in the simulation suite have this method.

In contrast, the simBySolution method provides a more accurate description of the underlying model. This method simulates the price paths by an approximation of the closed-form solution of separable models. Specifically, it applies a Euler approach to a transformed process, which in general is not the exact solution to this GBM model. This is because the probability distributions of the simulated and true state vectors are identical only for piecewise constant parameters.

When all model parameters are piecewise constant over each observation period, the simulated process is exact for the observation times at which the state vector is sampled. Since all parameters are constants in this example, simBySolution does indeed sample the exact solution.

For an example of how to use simBySolution to optimize the accuracy of solutions, see Optimizing Accuracy: About Solution Precision and Error.

Algorithms

  • The input argument Z allows you to directly specify the noise generation process. This process takes precedence over the Correlation parameter of the sde object and the value of the Antithetic input flag. If you do not specify a value for Z, simBySolution generates correlated Gaussian variates, with or without antithetic sampling as requested.

  • Gaussian diffusion models, such as HWV, allow negative states. By default, simBySolution does nothing to prevent negative states, nor does it guarantee that the model be strictly mean-reverting. Thus, the model may exhibit erratic or explosive growth.

  • The end-of-period Processes argument allows you to terminate a given trial early. At the end of each time step, simBySolution tests the state vector Xt for an all-NaN condition. Thus, to signal an early termination of a given trial, all elements of the state vector Xt must be NaN. This test enables a user-defined Processes function to signal early termination of a trial, and offers significant performance benefits in some situations (for example, pricing down-and-out barrier options).

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.

Introduced in R2008a

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