Strong Taylor Schemes for Stochastic Volatility
Ito and Stratonovich Stochastic Calculus
Ito-Stratonovich drift conversion
Strong Numerical Schemes for SDE
Milstein scheme for commutative noise
Approximations of Volatility Models
General 2D Milstein scheme for stochastic volatility models
Approximations of the Double Integral
Subdivision (Kloeden - IC = 0)
Simulation of the Double Integral
Formulae derivation for Heston Volatility
Derivation of the 2D Milstein Scheme
Strong Taylor Schemes for Stochastic Volatility
Ito-Stratonovich drift conversion
The Stratonovich SDE:
f or i = 1, ..., N ; j = 1, ..., M
with the same solutions as the N-dimensional Ito SDE with an M-dimensional Wiener process:
has a drift coefficient that is defined given a component-wise by:
(7)
whereas a given a is defined component-wise by:
(8)
These are called the drift correction formulas. Note that the diffusion coef-
ficients are the same in both the Ito and Stratonovich SDEs.
Prof. Klaus Schmitz
Strong Taylor Schemes for Stochastic Volatility
This method requires formulas that are not always easy or possible to find. In this document, we present the corresponding approximations for both Euler and Milstein schemes for the usual Geometric Brownian Motion and the stochastic volatility models. Also, we present five methods of how we can simulate the double integrals for the 2 dimensional Milstein approximation.
By Prof. Klaus Erich Schmitz Abe