Contents

• .
• Acknowledgements
• .
• 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 $\boldsymbol{\rho = 0}$
  • 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 $\rho$ 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 ...