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Contents
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Acknowledgements
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Contents
Overview
Derivatives and Options
Securities
Common Stocks
Bonds and Fixed Income Securities
Derivatives
Options
The Practical Uses of Options
Speculation
Hedging
The Principle of no Arbitrage
The Equivalence of European and American Calls on a Stock that does not Pay Dividends
Put-Call Parity
Stochastic Processes : An Elementary Introduction
A One-Dimensional Random Walk
Multi-Dimensional Random Walks
The Wiener Process (Brownian Motion)
Properties of the Wiener Process
Scale Invariance
Expectation Values
White Noise
Martingales
The Langevin Equation
The Black-Scholes Equation
The Assumptions Behind the Black-Scholes Equation
Self-financing, Replicating Hedging Strategies
Risk-Neutral Valuation
Ito's Lemma
The Black-Scholes Equation
Solution of the Black-Scholes Equation
Solution Using the Principle of Risk-Neutral Valuation
Solution Using the Quantum Mechanical Formalism
Stochastic Volatility
The Stochastic Process Followed by the Volatility
The Merton-Garman Equation
The ``Straightforward'' Solution when
A Quantum Mechanical Formulation of the Problem
The Hamiltonian for the Problem
The Lagrangian for the Problem
The Action for the Problem
Merton's Theorem
An Extension to Merton's Theorem
Risk-Neutrality
Brownian Motion on a Riemmanian Manifold
The Algorithm
A Short, Quick Reminder of the Monte Carlo Method
Our Monte Carlo Method for this Problem
A Comparison of Our Algorithm with Standard Monte Carlo Techniques
Checks on the Accuracy of the Program
The Results
The Effect of
on Option Prices
The Effect of Mean Reversion on the Option Price
Calibration with Market Data
Conclusion
Gaussian Integrals
One-Dimensional Gaussian Integrals
Higher-Dimensional Gaussian Integrals
The General Program
The Specialised Program for Call Options
Bibliography
About this document ...
Marakani Srikant 2000-08-15