Simulate Bates sample paths by Euler approximation

`[`

simulates `Paths`

,`Times`

,`Z`

,`N`

] = simByEuler(`MDL`

,`NPeriods`

)`NTrials`

sample paths of Bates bivariate models driven
by `NBrowns`

Brownian motion sources of risk and
`NJumps`

compound Poisson processes representing the arrivals
of important events over `NPeriods`

consecutive observation
periods. The simulation approximates continuous-time stochastic processes by the
Euler approach.

Bates models are bivariate composite models. Each Bates model consists of two coupled univariate models.

One model is a geometric Brownian motion (

`gbm`

) model with a stochastic volatility function and jumps.$$d{X}_{1t}=B(t){X}_{1t}dt+\sqrt{{X}_{2t}}{X}_{1t}d{W}_{1t}+Y(t){X}_{1t}d{N}_{t}$$

This model usually corresponds to a price process whose volatility (variance rate) is governed by the second univariate model.

The other model is a Cox-Ingersoll-Ross (

`cir`

) square root diffusion model.$$d{X}_{2t}=S(t)[L(t)-{X}_{2t}]dt+V(t)\sqrt{{X}_{2t}}d{W}_{2t}$$

This model describes the evolution of the variance rate of the coupled Bates price process.

This simulation engine provides a discrete-time approximation of the underlying
generalized continuous-time process. The simulation is derived directly from the
stochastic differential equation of motion. Thus, the discrete-time process approaches
the true continuous-time process only as `DeltaTimes`

approaches
zero.

[1] Deelstra, Griselda, and Freddy Delbaen. “Convergence of Discretized
Stochastic (Interest Rate) Processes with Stochastic Drift Term.”
*Applied Stochastic Models and Data Analysis.* 14, no. 1, 1998,
pp. 77–84.

[2] Higham, Desmond, and Xuerong Mao. “Convergence of Monte Carlo Simulations
Involving the Mean-Reverting Square Root Process.” *The Journal of
Computational Finance* 8, no. 3, (2005): 35–61.

[3] Lord, Roger, Remmert Koekkoek, and Dick Van Dijk. “A Comparison of Biased
Simulation Schemes for Stochastic Volatility Models.” *Quantitative
Finance* 10, no. 2 (February 2010): 177–94.

- Implementing Multidimensional Equity Market Models, Implementation 5: Using the simByEuler Method
- 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