Math-580.Ê Risk Management I:Ê Stocks and Derivative Securities.

The course concerns financial markets, and deals, in particular, with such notions as

á Evolution of stock prices;

á Prices for options, forwards, etc;

á Trading or investment strategies involving options and other securities.Ê

The core of the course concerns some modern mathematical finance models, Êbut we also talk a lot about ãreal situationsä. In a certain sense, the course answers the questions:

á How can prices for stocks change?

á How to trade them in an optimal way?

á What can one do to stabilize her/his profit?

The prerequisite is ordinary calculus (not complicated, but a student should be able, for example, to differentiate simple functions, and to know what the number e is), and an introductory course of Probability Theory (say, Stat-550 or Stat-551a are more than enough).

ÊÊÊÊ The course is self-contained; in particular Math-581, or Math-544 are NOT needed.

List of Topics.

1. ÊIntroduction. Different types of random assets. Underlying assets and derivatives. Forwards and futures. Options. Other derivatives. Types of traders.

2. The simplest static model. State prices, and non-arbitrage pricing. Option prices.

3. Futures markets. Hedging using futures. Forward and futures prices.

4. Options, bounds on prices, put-call parity.

5. Back to basic notions: price and equilibrium. Random assets. What is price and equilibrium in this case?

6. Dominance and arbitrage. State prices and arbitrage. Risk-neutral pricing.

7. Binomial trees, risk-neutral evaluation in this case.Ê

8. The notion of martingale. Martingale transformation. Fair game. Back to pricing for the binomial tree.

9. Brownian motion. Geometrical Brownian motion.

10. The stock price process.Ê

11. The Black-Scholes formula. Derivation based on the arbitrage theorem.

12. Ito's lemma.Ê

13. The option price process. The PDE for the option price and its solution (again the Black-Scholes formula).

14. Some variations of the Black-Scholes formula.

15. Trading: spreads and combinations.Ê

16. Interest rate derivatives and models of the yield curve.

17. Some notions of the management of market risk. Alternative models.

References:Ê

1. Hull, John C.Ê Options, Futures and Other Derivatives. 4thÊ edition, 2000, Prentice-Hall.

2. Financial Economics,Ê Ed. Panjer, 1998, The Actuarial Foundation.

3. Stampfli J., Goodman V.ÊÊ The Mathematic of Finance: Modelling and Hedging, 2001. Books/Cole.

4. Ingersol, J.E.ÊÊ Theory of Financial Decision Making, 1987, Rowman & Littlefield Publishers.