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# simBySolution

Simulate approximate solution of diagonal-drift HWV and GBM processes

## Synopsis

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

[Paths, Times, Z] = simBySolution(MDL, NPERIODS, 'Name1', Value1,'Name2', Value2, ...)

## Description

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 Hull-White/Vasicek (HWV) and geometric Brownian motion (GBM) short-rate models by an approximation of the closed-form solution.

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

 $d{X}_{t}=S\left(t\right)\left[L\left(t\right)-{X}_{t}\right]dt+V\left(t\right)d{W}_{t}$ (18-12)

where:

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

• S is an NVARS-by-NVARS matrix of mean reversion speeds (the rate of mean reversion).

• L is an NVARS-by-1 vector of mean reversion levels (long-run mean or level).

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

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

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

 $d{X}_{t}=\mu \left(t\right){X}_{t}dt+D\left(t,{X}_{t}\right)V\left(t\right)d{W}_{t}$ (18-13)

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

 MDL Hull-White/Vasicek (HWV) or geometric Brownian motion (GBM) model. NPERIODS Positive scalar integer number of simulation periods. The value of this argument determines the number of rows of the simulated output series.

## Optional Input Arguments

Specify optional input arguments as variable-length lists of matching parameter name/value pairs: 'Name1', Value1, 'Name2', Value2, ... and so on. The following rules apply when specifying parameter-name pairs:

• Specify the parameter name as a character string, followed by its corresponding parameter value.

• You can specify parameter name/value pairs in any order.

• Parameter names are case insensitive.

• You can specify unambiguous partial string matches.

Valid parameter names are:

 NTRIALS Positive scalar integer number of simulated trials (sample paths) 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. DeltaTime Scalar or NPERIODS-by-1 column vector of positive time increments between observations. 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. NSTEPS Positive scalar integer number of intermediate time steps within each time increment dt (specified as DeltaTime). 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. Antithetic Scalar logical flag that indicates whether antithetic sampling is used to generate the Gaussian random variates that drive the Brownian motion vector (Wiener processes). 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. Z Direct specification of the dependent random noise process used to generate the Brownian motion vector (Wiener process) that drives the simulation. Specify this argument as a function, or 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.If you do not specify a value for Z, simBySolution generates correlated Gaussian variates based on the Correlation member of the SDE object. StorePaths Scalar logical flag that indicates how simBySolution stores the output array Paths and returns it to the caller. 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. Processes Function or cell array of functions that indicates a sequence of end-of-period processes or state vector adjustments of the form ${X}_{t}=P\left(t,{X}_{t}\right)$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.If you do not specify a processing function, simBySolution makes no adjustments and performs no processing.

## Output Arguments

 Paths (NPERIODS + 1)-by-NVARS-by-NTRIALS three-dimensional time series array, consisting of simulated paths of correlated state variables. 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. Times (NPERIODS + 1)-by-1 column vector of observation times associated with the simulated paths. Each element of Times is associated with the corresponding row of Paths. Z (NPERIODS * NSTEPS)-by-NBROWNS-by-NTRIALS three-dimensional time series array of dependent random variates used to generate the Brownian motion vector (Wiener processes) that drive the simulation.

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### 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.