(Initial blurb (1996)) This course is divided into four parts. First, we will cover fundamentals on Ito Calculus and Arbitrage Pricing Theory, emphasizing the use of trinomial trees, or finite-difference schemes, for pricing derivative securities (options, forwards, callable bonds, etc.) Next, we will move into the area of transaction costs due to liquidity constraints. We will analyze how low liquidity in trading the underlying asset affects implied volatility and dynamic hedging. Third, we will consider different ways of of modeling markets with uncertain volatility. We will cover stochastic volatility models, auto-regressive models as well as recently derived nonlinear pricing models. We will emphasize, in particular, worst-case scenario risk-management techniques, and managing volatility risk with options. Finally, we will cover more practical risk-management techniques and guidelines such as the ``Value at Risk'' outlined in the J.P. Morgan technical document on measuring financial risk.
(Spring 1998) Mathematics of Finance II borrows some of the same basic material from the original Risk-management course but emphasizes mostly (i) continous time finance and (ii) modeling the term structure of interest rates. We do a little bit of ``taxonomy'' of models as well as some conceptual thinking about the significance and applications of term-structure models. Topics tend to change from one year to the next.
3. Ito processes, continuous-time martingales and Girsanov's Theorem (revised)
4. Continuous-time finance: an introduction
5. Valuation of derivative securities
6. Uncertain Volatility Model & worst-case scenario pricing
7. Trinomial trees and finite-difference schemes
#2 Gauss-Markov processes
#3 Term-structure Modeling
Extra problem: digital
barrier and volalility term-structure
Homework # 2