The purpose of this document is to introduce implied, local and stochastic volatility, to review evidence of non-constant volatility, and to consider the implications for option pricing of alternative random or stochastic volatility models. We focus on continuous time diffusion models for the volatility, but we also briefly discuss certain classes of discrete time models, such as ARV or ARCH.

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

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Download Matlab version: Local_vs_Dupire.m

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Summary:

•Introduction

•Implied Volatility - Ito's Lemma

•Taylor Series Expansion

•Applying Ito to the Hedging Portfolio

•Risk-Neutralization and No-Arbitrage

•Implied Volatility (Smiles and Skews)

•Local Volatility

•Stochastic Volatility

•Coupled SDEs for Stochastic Volatility

•Risk-Neutralization and No-Arbitrage

•The Heston Model

•Exact Solution for Heston Volatility

•Ornstein-Uhlenbeck Process

•Formulae derivation for Heston Volatility

•The fundamental solution

•References

•Books Related