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Strong Taylor Schemes for Stochastic Volatility

Introduction

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)

Fourier Lévy formulae

Exact Fourier Lévy formulae

Real Variance formulae

Simulation of the Double Integral

Conclusions and Observations

Ornstein-Uhlenbeck Process

Formulae derivation for Heston Volatility

The fundamental solution

Derivation of the 2D Milstein Scheme

Numerical Data of the Double Integral

References

Books Related

Klaus Erich Schmitz Abe:

Strong Taylor Schemes for Stochastic Volatility

Keywords:Modelli Econometrici; Stochastic

We can price any financial instrument using Monte Carlo and the payoff. Another method of pricing is using the exact solution for its corresponding SDEs. 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.

Simulations, observations, and conclusions between all methods are given at the end of the document showing the big advantage of using better approximations to price derivative instruments. The future plans are to calculate and investigate more about Taylor 1.5 and Stratonovich in order to make a good comparison (advantages and disadvantages) between the use of all of them.

By Prof. Klaus Erich Schmitz Abe http://www.maths.ox.ac.uk/~schmitz/


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