# 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.

expand all

### 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.