Simulate approximate solution of diagonal-drift HWV processes

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

`[Paths,Times,Z] = simBySolution(___,Name,Value)`

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) by an
approximation of the closed-form solution.

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

$$d{X}_{t}=S(t)[L(t)-{X}_{t}]dt+V(t)d{W}_{t}$$

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.

The `simBySolution`

method 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.

[1] 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.

[2] Ait-Sahalia, Y. “Transition Densities for Interest Rate and Other Nonlinear
Diffusions.” *The Journal of Finance*, Vol. 54, No. 4, August
1999.

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

[4] Hull, J. C. *Options, Futures, and Other Derivatives*, 5th ed.
Englewood Cliffs, NJ: Prentice Hall, 2002.

[5] Johnson, N. L., S. Kotz, and N. Balakrishnan. *Continuous Univariate
Distributions.* Vol. 2, 2nd ed. New York, John Wiley & Sons,
1995.

[6] Shreve, S. E. *Stochastic Calculus for Finance II: Continuous-Time
Models.* New York: Springer-Verlag, 2004.

`hwv`

| `simByEuler`

| `simBySolution`

| `simulate`

- Creating Hull-White/Vasicek (HWV) Gaussian Diffusion Models
- Simulating Equity Prices
- Simulating Interest Rates
- Stratified Sampling
- Pricing American Basket Options by Monte Carlo Simulation
- Base SDE Models
- Drift and Diffusion Models
- Linear Drift Models
- Parametric Models
- SDEs
- SDE Models
- SDE Class Hierarchy
- Performance Considerations