Simulate approximate solution of diagonal-drift GBM processes
Simulating Equity Markets Using GBM Simulation Functions
Use GBM simulation functions. Separable GBM models have two specific simulation functions:
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 function (
simulate), increase the number of trials to
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 functions:
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)')
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 function is often the least accurate and most general method available. All models shipped in the simulation suite have the
In contrast, the
simBySolution function provides a more accurate description of the underlying model. This function 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.
MDL — Geometric Brownian motion (GBM) model
Geometric Brownian motion (GBM) model, specified as a
gbm object that is created using
NPeriods — Number of simulation periods
Number of simulation periods, specified as a positive scalar integer. The
NPeriods determines the number of rows of the
simulated output series.
comma-separated pairs of
the argument name and
Value is the corresponding value.
Name must appear inside quotes. You can specify several name and value
pair arguments in any order as
[Paths,Times,Z] = simBySolution(GBM,NPeriods,'DeltaTime',dt,'NTrials',10)
NTrials — Simulated trials (sample paths) of
NPERIODS observations each
1 (single path of correlated state
variables) (default) | positive integer
Simulated trials (sample paths) of
observations each, specified as the comma-separated pair consisting of
'NTrials' and a positive scalar integer.
DeltaTimes — Positive time increments between observations
1 (default) | scalar | column vector
Positive time increments between observations, specified as the
comma-separated pair consisting of
'DeltaTimes' and a
scalar or a
DeltaTime represents the familiar
dt found in stochastic differential equations,
and determines the times at which the simulated paths of the output
state variables are reported.
NSteps — Number of intermediate time steps within each time increment dt (specified as
1 (indicating no intermediate
evaluation) (default) | positive integer
Number of intermediate time steps within each time increment
dt (specified as
specified as the comma-separated pair consisting of
'NSteps' and a positive scalar integer.
simBySolution function partitions each time
increment dt into
subintervals of length dt/
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
Antithetic — Flag to indicate whether
simBySolution uses antithetic sampling to generate the Gaussian random variates
False (no antithetic
sampling) (default) | logical with values
Flag to indicate whether
antithetic sampling to generate the Gaussian random variates that drive
the Brownian motion vector (Wiener processes), specified as the
comma-separated pair consisting of
'Antithetic' and a
scalar logical flag with a value of
When you specify
simBySolution performs sampling such that all
primary and antithetic paths are simulated and stored in successive
(1,3,5,...)correspond to the primary Gaussian paths.
(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 an input noise process (see
the value of
Z — Direct specification of the dependent random noise process used to generate the Brownian motion vector
generates correlated Gaussian variates based on the
Correlation member of the
object (default) | function | three-dimensional array of dependent random variates
Direct specification of the dependent random noise process used to
generate the Brownian motion vector (Wiener process) that drives the
simulation, specified as the comma-separated pair consisting of
'Z' and a function or as an
three-dimensional array of dependent random variates.
The input argument
Z allows you to directly specify
the noise generation process. This process takes precedence over the
Correlation parameter of the input
gbm object and the value of
Antithetic input flag.
If you specify
Z as a function, it must return
1 column vector,
and you must call it with two inputs:
A real-valued scalar observation time t.
1state vector Xt.
StorePaths — Flag that indicates how the output array
Paths is stored and returned
(default) | logical with values
Flag that indicates how the output array
stored and returned, specified as the comma-separated pair consisting of
'StorePaths' and a scalar logical flag with a
default value) or is unspecified,
Paths as a three-dimensional time series
simBySolution returns the
Paths output array as an empty matrix.
Processes — Sequence of end-of-period processes or state vector adjustments of the form
simBySolution makes no adjustments
and performs no processing (default) | function | cell array of functions
Sequence of end-of-period processes or state vector adjustments of the
form, specified as the comma-separated pair consisting of
'Processes' and a function or cell array of
functions of the form
simBySolution function runs processing
functions at each interpolation time. They must accept the current
interpolation time t, and the current state vector
Xt, and return a state
vector that may be an adjustment to the input state.
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
return a state vector that may be an adjustment to the input
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
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).
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.
Paths — Simulated paths of correlated state variables
Simulated paths of correlated state variables, returned as a
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
Paths as an
Times — Observation times associated with the simulated paths
Observation times associated with the simulated paths, returned as a
(NPERIODS + 1)-by-
1 column vector.
Each element of
Times is associated with the
corresponding row of
Z — Dependent random variates used to generate the Brownian motion vector
Dependent random variates used to generate the Brownian motion vector
(Wiener processes) that drive the simulation, returned as a
three-dimensional time series array.
Simulation methods allow you to specify a popular variance reduction technique called antithetic sampling.
This technique attempts to replace one sequence of random observations with another of the same expected value, but smaller variance. In a typical Monte Carlo simulation, each sample path is independent and represents an independent trial. However, antithetic sampling generates sample paths in pairs. The first path of the pair is referred to as the primary path, and the second as the antithetic path. Any given pair is independent of any other pair, but the two paths within each pair are highly correlated. Antithetic sampling literature often recommends averaging the discounted payoffs of each pair, effectively halving the number of Monte Carlo trials.
This technique attempts to reduce variance by inducing negative dependence between paired input samples, ideally resulting in negative dependence between paired output samples. The greater the extent of negative dependence, the more effective antithetic sampling is.
simBySolution function simulates
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
Consider a separable, vector-valued GBM model of the form:
Xt is an
1state vector of process variables.
μ is an
NVARSgeneralized expected instantaneous rate of return matrix.
V is an
NBROWNSinstantaneous volatility rate matrix.
dWt is an
1Brownian motion vector.
simBySolution function 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.
Gaussian diffusion models, such as
hwv, allow negative states. By default,
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.
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 Aït-Sahalia, Yacine. “Transition Densities for Interest Rate and Other Nonlinear Diffusions.” The Journal of Finance 54, no. 4 (Aug. 1999): 1361–95.
 Glasserman, Paul. Monte Carlo Methods in Financial Engineering. New York: Springer-Verlag, 2004.
 Hull, John C. Options, Futures and Other Derivatives. 7th ed, Prentice Hall, 2009.
 Johnson, Norman Lloyd, Samuel Kotz, and Narayanaswamy Balakrishnan. Continuous Univariate Distributions. 2nd ed. Wiley Series in Probability and Mathematical Statistics. New York: Wiley, 1995.
 Shreve, Steven E. Stochastic Calculus for Finance. New York: Springer-Verlag, 2004.