Simulate approximate solution of diagonal-drift GBM processes
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:
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
function 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.
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.
gbm
| simByEuler
| simBySolution
| simulate