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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Summary:

• 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