Math 586 - Computational Financial

1. Introduction (Reading Hull 5th ed. Chapter 1)

   Two Examples of financial derivatives

   • Currency Risk - forward contract, European call option
   • Equity-Linked Note or CD Lehman Brothers Report

2. What are financial derivatives; (history and disasters), underlying assets; exchange traded and OTC derivatives (WSJ article on exchanges); stocks and stock options.

3. European call and put options, intrinsic value, time value, payoff and profit and loss diagrams, moneyness, covered call, trading options (option strategies guide), interest rates and compounding. (Reading Hull Chapter 7, 9), ( WSJ article Options and Greed ), Interesting derivative info site NumaWeb, Options Industry Council.

4. Continuous compounding; pricing equation for future value of a money market account as an IVP; pricing equation for the present value of an asset as a FVP; pricing ZCB and coupon bonds; pricing an annuity using a pricing equation and constructing solution.

   Reading from Albanese - Intro. to Chapter 1 and 1.1

5. pricing and arbitrage: simple contract; one-step binomial tree; pricing approaches - risk-free portfolio, risk-neutral probabilities, replication, class notes -Hull ex. (Reading - Baxter and Rennie - Chapter 1, 2.1; Hull - 10.1, 10.2)

6. continued discussing one step binomial pricing model; forward contracts on investment assets - determining the delivery or forward price and value using an arbitrage argument; put-call parity using arbitrage argument.

7. Simple bounds on European call option, binary options (put-call parity); very brief review of probability.

   Reading from Albanese - 1.2 and 1.3

8. more probability; modeling random behavior of stock prices; daily return and relationship to normal; stochastic difference equation; simulation.

9. changing observation interval; instantaneous parameters; motivation for continuous limit; random walk - path description.

10. Review of random walk stochastic difference equation and solution; distribution approach - forward equation and solution; re-scaling to obtain continuous limit W(t) - Brownian motion; diffusion equation and solution.

11. very brief discussion of stochastic integrals, SDE formulation and relationship to stochastic integrals, "solving" SDE using the stochastic integral formulation.

    Reading from Albanese - Chapter 7 and 1.4, 1.5

12. strong solution of SDE, density function solution, Bachelier's stock model, naive pricing formula.

13. introduced Ito's lemma, constructed strong solution of a stock price SDE, derived the lognormal density.

    Reading from Albanese - 1.6

14. diffusion processes and stochastic differential equations J. Chang's notes on diffusion

15. Forward and backward Kolmogorov equations; special problems involving BKE; derivation of Black-Scholes eqn using risk-free portfolio.

16. derived solution of Black-Scholes equation for some special payoff functions (derivatives); price of forward contract on nondividend asset and asset with continuous yield.

17. Continued derivation of Black-Scholes formula; Black-Scholes formula on asset with continuous yield.

18. Risk-neutral approach to Black-Scholes formula

    Reading from Albanese - 1.6 Read again very carefully

    Compare with derivations in class.
19. Derivation using forward price; Black's formula

20. multi-asset models, correlated noise; pricing equation for multi-asset models

21. Chooser option (see Real Choosers ) Binary options

    Reading from Albanese - 1.7 and 1.8

22. static hedges - butterfly spread option

23. static hedge for general payoff in terms of call, puts and combination of puts and calls

24. dynamic hedging; the Greeks

25. delta hedge; delta-gamma hedge

26. Extensions of the Black-Scholes problem - risk-free rate is deterministic function of time, i.e. r(t), numeraire, martingale pricing.

27. Models in which underlying asset pays discrete dividends; Model 1 - discrete dividend yield - exact solution

28. Model 2 - discrete cash dividend; paper of Hong, Haug, and Lewis (need reference) - discussion of problems; solution assuming adjusted price satisfies geometric Brownian motion.

29. Importance of volatility; historical volatility forcasting; ARCH and GARCH models; implied volatility. Robert Engle's nice article in J. Econ. Perspectives

30. American options; optionality; bounds and relationship to European versions; linear complementary problem; optimal exercise boundary; conditions at optimal boundary

31. Perpetual American put; exact solution; determining optimal boundary by maximizing option value

32. More perpetual options - calls with or without continuous yield

33. American option solutions and approximations; Barone-Adesi; RGW; compound options

34. more RGW; bivariate normal

35. Integral equation representation using early exercise premium; reduction for perpetual option

36. Begin numerical methods; binomial tree - review early material; recombining trees; multiplicative price tree.

37. multiperiod trees; forward induction for asset price; backward induction for derivative price.

38. schemes for selecting parameters: Wilmott's exact formulas - u*d = 1 and equal probability; CRR formulas

39. Local volatility formulas - Chriss

40. Modification of binomial tree models to include dividends

©2005 Charles Tier