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Klaus Erich Schmitz Abe:

Introduction to Implied, Local and Stochastic Volatility

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/

Download Matlab version: Local_vs_Dupire.m

Download Excel version: Local_vs_Dupire.xls

Download Matlab version: vol_test.m



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


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