simBySolution

Simulate approximate solution of diagonal-drift GBM processes

Syntax

[Paths,Times,Z] = simBySolution(MDL,NPeriods)

[Paths,Times,Z] = simBySolution(___,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(___,Name,Value) adds optional name-value pair arguments.

You can perform quasi-Monte Carlo simulations using the name-value arguments for MonteCarloMethod, QuasiSequence and BrownianMotionMethod. For more information, see Quasi-Monte Carlo Simulation.

Examples

Simulating Equity Markets Using GBM Simulation Functions

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

• An overloaded Euler simulation function (simulate), designed for optimal performance.

• A simBySolution function 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 Represent Market Models Using SDE Models, and the GBM model as in Represent 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 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 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)')

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 function is often the least accurate and most general method available. All models shipped in the simulation suite have the simByEuler function.

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.

Simulating Equity Markets Using GBM Model with Quasi-Monte Carlo Simulation

This example shows how to use simBySolution with a GBM model to perform a quasi-Monte Carlo simulation. Quasi-Monte Carlo simulation is a Monte Carlo simulation that uses quasi-random sequences instead pseudo random numbers.

Load the Data_GlobalIdx2 data set and specify the GBM model as in Represent 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);
X           = 100;
X           = X(ones(nVariables,1));

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

Perform a quasi-Monte Carlo simulation by using simBySolution with the optional name-value arguments for 'MonteCarloMethod','QuasiSequence', and 'BrownianMotionMethod'.

[paths,time,z] = simBySolution(GBM, 10,'ntrials',4096,'MonteCarloMethod','quasi','QuasiSequence','sobol','BrownianMotionMethod','brownian-bridge');

Calculate Price for European Call Option Using Monte Carlo Simulation with GBM Object

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

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

% Parameters for the GBM model and option
S0 = 80; % Initial stock price
K = 40; % 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 simBySolution to simulate the end-of-year stock prices using the GBM model (gbm) over nTrials trials.

% Simulate stock prices at maturity
FuturePrices = simBySolution(gbmobj,nPeriods,'nTrials',nTrials,'DeltaTime',T);

Calculate the payoff for the European call option based on the simulated prices.

% Calculate payoffs for the call option
Payoffs = max(FuturePrices(:, end) - K, 0);

Discount the payoff back to the present value and then average the payoff value to estimate the option price.

% Discount payoff back to present value and then average payoff value
DiscountedPayoffs = exp(-r * T) * Payoffs;

OptionPrice = mean(DiscountedPayoffs);

Display the estimated price for the call option.

% Display results
disp(['Estimated Europen option price using Monte Carlo simulation: ', num2str(OptionPrice)]);

Estimated Europen option price using Monte Carlo simulation: 40.0231

Input Arguments

MDL — Geometric Brownian motion (GBM) model

gbm object

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

Data Types: object

NPeriods — Number of simulation periods

positive integer

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

Data Types: double

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: [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 NPERIODS observations each, specified as the comma-separated pair consisting of 'NTrials' and a positive scalar integer.

Data Types: double

DeltaTime — Positive time increments between observations

1 (default) | scalar | column vector

Positive time increments between observations, specified as the comma-separated pair consisting of 'DeltaTime' and a scalar or a NPERIODS-by-1 column vector.

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.

Data Types: double

NSteps — Number of intermediate time steps within each time increment dt (specified as DeltaTime)

1 (indicating no intermediate evaluation) (default) | positive integer

Number of intermediate time steps within each time increment dt (specified as DeltaTime), specified as the comma-separated pair consisting of 'NSteps' and a positive scalar integer.

The simBySolution function 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.

Data Types: double

Antithetic — Flag to indicate whether simBySolution uses antithetic sampling to generate Gaussian random variates

False (no antithetic sampling) (default) | logical with values True or False

Flag to indicate whether simBySolution uses 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 True or False.

When you specify True, 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.

Note

If you specify an input noise process (see Z), simBySolution ignores the value of Antithetic.

Data Types: logical

MonteCarloMethod — Monte Carlo method to simulate stochastic processes

"standard" (default) | string with values "standard", "quasi", or "randomized-quasi" | character vector with values 'standard', 'quasi', or 'randomized-quasi'

Monte Carlo method to simulate stochastic processes, specified as the comma-separated pair consisting of 'MonteCarloMethod' and a string or character vector with one of the following values:

• "standard" — Monte Carlo using pseudo random numbers

• "quasi" — Quasi-Monte Carlo using low-discrepancy sequences

• "randomized-quasi" — Randomized quasi-Monte Carlo

Note

If you specify an input noise process (see Z), simBySolution ignores the value of MonteCarloMethod.

Data Types: string | char

QuasiSequence — Low discrepancy sequence to drive stochastic processes

"sobol" (default) | string with value "sobol" | character vector with value 'sobol'

Low discrepancy sequence to drive the stochastic processes, specified as the comma-separated pair consisting of 'QuasiSequence' and a string or character vector with the following value:

• "sobol" — Quasi-random low-discrepancy sequences that use a base of two to form successively finer uniform partitions of the unit interval and then reorder the coordinates in each dimension

Note

• If MonteCarloMethod option is not specified or specified as"standard", QuasiSequence is ignored.

• If you specify an input noise process (see Z), simBySolution ignores the value of QuasiSequence.

Data Types: string | char

BrownianMotionMethod — Brownian motion construction method

"standard" (default) | string with value "brownian-bridge" or "principal-components" | character vector with value 'brownian-bridge' or 'principal-components'

Brownian motion construction method, specified as the comma-separated pair consisting of 'BrownianMotionMethod' and a string or character vector with one of the following values:

• "standard" — The Brownian motion path is found by taking the cumulative sum of the Gaussian variates.

• "brownian-bridge" — The last step of the Brownian motion path is calculated first, followed by any order between steps until all steps have been determined.

• "principal-components" — The Brownian motion path is calculated by minimizing the approximation error.

Note

If an input noise process is specified using the Z input argument, BrownianMotionMethod is ignored.

The starting point for a Monte Carlo simulation is the construction of a Brownian motion sample path (or Wiener path). Such paths are built from a set of independent Gaussian variates, using either standard discretization, Brownian-bridge construction, or principal components construction.

Both standard discretization and Brownian-bridge construction share the same variance and, therefore, the same resulting convergence when used with the MonteCarloMethod using pseudo random numbers. However, the performance differs between the two when the MonteCarloMethod option "quasi" is introduced, with faster convergence for the "brownian-bridge" construction option and the fastest convergence for the "principal-components" construction option.

Data Types: string | char

Z — Direct specification of the dependent random noise process used to generate Brownian motion vector

generates correlated Gaussian variates based on the Correlation member of the SDE 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 (NPERIODS * NSTEPS)-by-NBROWNS-by-NTRIALS 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 the Antithetic input flag.

Note

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 X_t.

Data Types: double | function

StorePaths — Flag that indicates how the output array Paths is stored and returned

True (default) | logical with values True or False

Flag that indicates how the output array Paths is stored and returned, specified as the comma-separated pair consisting of 'StorePaths' and a scalar logical flag with a value of True or False.

• 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

Processes — Sequence of end-of-period processes or state vector adjustments

simBySolution makes no adjustments and performs no processing (default) | function | cell array of functions

Sequence of end-of-period processes or state vector adjustments, specified as the comma-separated pair consisting of 'Processes' and a function or cell array of functions of the form

$$X_t=P(t,X_t)$$

The simBySolution function runs processing functions at each interpolation time. They must accept the current interpolation time t, and the current state vector X_t, 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 X_t, and return a state vector that may be an adjustment to the input state.

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 X_t for an all-NaN condition. Thus, to signal an early termination of a given trial, all elements of the state vector X_t 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.

Data Types: cell | function

Output Arguments

Paths — Simulated paths of correlated state variables

array

Simulated paths of correlated state variables, returned as an (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 X_t at time t. When the input flag StorePaths = False, simBySolution returns Paths as an empty matrix.

Times — Observation times associated with simulated paths

column vector

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

Z — Dependent random variates used to generate Brownian motion vector

array

Dependent random variates used to generate the Brownian motion vector (Wiener processes) that drive the simulation, returned as an (NPERIODS * NSTEPS)-by-NBROWNS-by-NTRIALS three-dimensional time series array.

More About

Antithetic Sampling

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.

Algorithms

The simBySolution function 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:

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

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.

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

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

The simBySolution function simulates the state vector X_t 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 X_t is sampled.

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.

References

[1] Aït-Sahalia, Yacine. “Testing Continuous-Time Models of the Spot Interest Rate.” Review of Financial Studies, Vol. 9, No. 2 ( Apr. 1996): 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): 1361–95.

[3] Glasserman, Paul. Monte Carlo Methods in Financial Engineering, New York: Springer-Verlag, 2004.

[4] Hull, John C. Options, Futures and Other Derivatives, 7th ed, Prentice Hall, 2009.

[5] Johnson, Norman Lloyd, Samuel Kotz, and Narayanaswamy Balakrishnan. Continuous Univariate Distributions, 2nd ed. Wiley Series in Probability and Mathematical Statistics. New York: Wiley, 1995.

[6] Shreve, Steven E. Stochastic Calculus for Finance, New York: Springer-Verlag, 2004.

Version History

Introduced in R2008a

R2022b: Perform Brownian bridge and principal components construction

Perform Brownian bridge and principal components construction using the name-value argument BrownianMotionMethod.

R2022a: Perform Quasi-Monte Carlo simulation

Perform Quasi-Monte Carlo simulation using the name-value arguments MonteCarloMethod and QuasiSequence.