Financial Mathematics

AY2019/2020 Semester 2

This course aims at developing quantitative skills for the pricing and hedging of financial derivatives, using stochastic calculus and partial differential equations. It will enable you to design both discrete and continuous-time financial pricing models by combining the power of analytical and probabilistic methods. This is a level 4 course and no finance prerequisite is required. Course Content: 1 Discrete-Time Martingales 1.1 Filtrations and Conditional Expectations 1.2 Martingales - Definition and Properties . 1.3 Stopping Times 1.4 Ruin Probabilities 1.5 Mean Game Duration 2 Assets, Portfolios, and Arbitrage 2.1 Definitions and Notation 2.2 Portfolio Allocation and Short Selling 2.3 Arbitrage 2.4 Risk-Neutral Measures 2.5 Hedging of Contingent Claims 2.6 Market Completeness 2.7 Example 3 Discrete-Time Model 3.1 Discrete-Time Compounding 3.2 Stochastic Processes 3.3 Portfolio Strategies and Arbitrage 3.4 Contingent Claims 3.5 Martingales and Conditional Expectation 3.6 Market Completeness and Risk-Neutral Measures 3.7 The Cox-Ross-Rubinstein (CRR) Market Model 4 Pricing and hedging in discrete time 4.1 Pricing of Contingent Claims 4.2 Pricing of Vanilla Options in the CRR Model 4.3 Hedging of Contingent Claims 4.4 Hedging of Vanilla Options in the CRR model 4.5 Hedging of Exotic Options in the CRR Model 4.6 Convergence of the CRR Model 5 Brownian Motion and Stochastic Calculus 5.1 Brownian Motion 5.2 Constructions of Brownian Motion 5.3 Wiener Stochastic Integral 5.4 Ito Stochastic Integral 5.5 Stochastic Calculus 5.6 Geometric Brownian Motion 5.7 Stochastic Differential Equations 6 The Black-Scholes PDE 6.1 Continuous-Time Market Model 6.2 Self-Financing Portfolio Strategies 6.3 Arbitrage and Risk-Neutral Measures 6.4 Market Completeness 6.5 The Black-Scholes Formula 6.6 The Heat Equation 6.7 Solution of the Black-Scholes PDE 7 Martingale Approach to Pricing and Hedging 7.1 Martingale Property of the Ito Integral 7.2 Risk-neutral Measures 7.3 Change of Measure and the Girsanov Theorem 7.4 Pricing by the Martingale Method 7.5 Hedging Strategies 8 Estimation of Volatility 8.1 Historical Volatility 8.2 Implied Volatility 8.3 Local Volatility 8.4 The VIX ? Volatility Index 8.5 Stochastic Volatility 8.6 Volatility Derivatives 9 Basic Numerical Methods 9.1 Discretized Heat Equation 9.2 Discretized Black-Scholes PDE 9.3 Euler Discretization 9.4 Milshtein Discretization