1. Math4143 W08, HM Zhu 2. Lattice Methods 2.3 Dealing with Options on Dividend-paying Stocks (Hull, Sec. 17.3, page 401)

2. Math4143, W08, HM ZHU 2 With Continuous Dividend Yields

3. Math4143, W08, HM ZHU 3 With Continuous Dividend Yields

4. Math4143, W08, HM ZHU 4 With Continuous Dividend Yields

5. Math4143, W08, HM ZHU 5 With Discrete Dividend Yields

6. Math4143, W08, HM ZHU 6 With Dollar Dividend

7. Math4143, W08, HM ZHU 7 A better procedure: –Draw the tree for the stock price less the present value of the dividends –Create a new tree for the stock price by adding the present value of the dividends at each node This ensures that the tree recombines and makes assumptions similar to those when the Black-Scholes model is used With Dollar Dividend

8. Math4143, W08, HM ZHU 8 With Dollar Dividend

9. Example 1: American put option on a stock: S 0 = 52; K = 50; D = $2.06 r =10%;   = 40%; T = 5 months;  = 3.5 months  t = 1 month

10. Math4143, W08, HM ZHU 10 Example 1: Tree model for S * +$2.05

11. Math4143 W08, HM Zhu 2. Lattice Methods 2.4 Comments and Extensions of the model

12. Math4143, W08, HM ZHU 12 Trees for Options on Indices, Currencies and Futures Contracts (Hull, Sec. 11.9, Sec. 17.2) As with Black-Scholes: –For options on stock indices, replace the continuous dividend yield D 0 with the dividend yield on the index –For options on a foreign currency, D 0 equals the foreign risk-free rate r f –For options on futures contracts D 0 = r

13. Math4143, W08, HM ZHU 13 Time Dependent Interest Rate and Dividend Yield (page 409) Making interest rate r or dividend yield D a function of time does not affect the geometry of the tree. The probabilities on the tree become functions of time Discounting factor becomes a function of time as well

14. Math4143, W08, HM ZHU 14 Time Dependent Volatility (page 409) Changing  at each time step does affect the geometry of the tree. (The probabilities on the tree become functions of time) Or we can make  a function of time by making the lengths of the time steps inversely proportional to the variance rate.

15. Math4143, W08, HM ZHU 15 Trinomial Tree (see Technical Note 9, www.rotman.utoronto.ca/~hull) SS Sd Su pupu pmpm pdpd

16. Math4143, W08, HM ZHU 16 Adaptive Mesh Model This is a way of grafting a high resolution tree on to a low resolution tree We need high resolution in the region of the tree close to the strike price and option maturity Numerically efficient over a binomial or trinomial tree Figlewski and Gao, “The adaptive mesh model: a new approach to efficient option pricing”, J. of Financial Ecomonics, 53:313-351 (1999)

17. Math4143, W08, HM ZHU 17 Further Reading 1.L. Clewlow and C. Strickland. Implementing Derivatives Models. Wiley, Chichester, West Sussex, England, 1998 (relationship between finite differences and trinomial trees) 2.D J Higham. Nine Ways to Implement the Binomial Method for Option Valuation in Matlab. SIAM review, 44:661-677, 2002 (issues of implementing binomial trees) 3.J C Hull. Option, Futures, and Other Derivatives. Prentice Hall. (classic reference) 4.G. Levy. Computational Finance. Numerical Methods for Pricing Financial Instruments. Elsevier Butterworth- Heinemann, oxford, 2004 (implied lattices and efficient implementations)