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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:

`$d{X}_{t}=\mu \left(t\right){X}_{t}dt+D\left(t,{X}_{t}\right)V\left(t\right)d{W}_{t}$`

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

`${X}_{t}=P\left(t,{X}_{t}\right)$`
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.