F70CF - Continuous-Time Finance

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Syllabus

1. Introduction to stochastic calculus (1.1 1. Theory of Martingales in continuous time, 1.2 2. Brownian motion; definitions and properties, 1.3 3. Brownian motion as the limit of a binomial random-walk process, 1.4 4. Introduction to stochastic integration, stochastic differential equations and Ito’s formula, 1.5 5. Geometric Brownian motion; the Ornstein-Uhlenbeck process, 1.6 6. Introduction to Girsanov’s theorem and the martingale representation theorem)

2. Derivative pricing in the Black-Scholes model (2.1 1. The Black-Scholes model, 2.2 2. Derivatives pricing using the Black-Scholes model using the martingale and PDE approaches to pricing, 2.3 3. Extensions to foreign currencies and dividend-paying stocks, 2.4 4. Portfolio risk management using the Greeks)

3. Interest rate models and credit risk (3.1 1. Introduction to interest rate models, 3.2 2. Introduction to credit risk models)

Learning outcomes

By the end of the course, students should be able to do the following:

• apply their knowledge of Brownian motion to analyse properties of continuous-time stochastic processes defined as functions of Brownian motion.
• apply Ito’s formula to evaluate stochastic differentials of Ito processes.
• utilise martingale measures and their connection with arbitrage free/complete markets to analyse financial models.
• derive the Black-Scholes formula and the Black-Scholes PDE.
• price contingent claims (in particular European style options and forward contracts) in the Black-Scholes model.
• apply the Greeks in portfolio risk management.
• utilise the relationship between forward interest rates, spot rates and zero coupon bond prices, to price bonds in the Vasicek and CIR models.
• apply the Merton model and the 2-state model for credit risk, to price simple corporate bonds and calculate credit spreads.

Further details

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SCQF Level: 10

Credits: 15