# simBySolution

Simulate approximate solution of diagonal-drift GBM processes

## Description

## Examples

## Input Arguments

## Output Arguments

## More About

## 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*is an_{t}`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*is an_{t}`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* 9, no. 2 ( Apr. 1996): 385–426.

[2] Aït-Sahalia, Yacine.
“Transition Densities for Interest Rate and Other Nonlinear Diffusions.” *The
Journal of Finance* 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.

## See Also

`simByEuler`

| `simulate`

| `gbm`

| `simBySolution`

**Introduced in R2008a**