Homework Set 5 Math 456 Topics in Financial Mathematics Prof. Wickerhauser Read Chapters 9 and 10 of the textbook, "Binomial Models In Finance" by John van der Hoek and Robert J. Elliott NOTE: When asked to produce a spreadsheet, you may instead implement the model in Octave or another system. For full credit you must translate the algorithm into a computer program and produce output from several examples. Include your code so that the grader can see and reproduce your work. Do the following exercises from the textbook Chapter 9.6, p.126: Exercise 9.3. Derive Eq.9.1 from the Black-Scholes formula. You may use the Call-Put Parity formula to derive Eq.9.2. Likewise, derive Eq.9.8, Eq.9.9, Eq.9.13, Eq.9.14, Eq.9.18, and Eq.9.19. NOTE: the definitions of d1 and d2 are on p.248 of the textbook, along with the Black-Scholes formula for European-style Call options, Eq.A.17. Additional references on Black-Scholes formulas are linked from the class website. HINT 1: Octave functions normcdf(d1) and normpdf(d1) compute N(d1) and N'(d1), respectively. HINT 2: prove and use the identity [S*N'(d1)-K*exp(-r*T)*N'(d2)]=0 to simplify. Also, N(X)-1=-N(-X) for every X. Exercise 9.4. Either create a spreadsheet or use Octave programs. HINTS: Use h=1 in the centered difference formula for Gamma. Expect a poor result that is even worse with smaller h. Use h=0.1 in the centered difference formulas for Delta and Theta. Use h=0.01 for Vega/Kappa. Use h=0.001 for Rho. Do the following exercises from the textbook Chapter 10.4, p.134: Exercise 10.10. The unstated spot price S(0,0) should be \$80. Exercise 10.11. The last strict inequality ">" is actually ">=" since both sides may be zero for some (n,j). Exercise 10.12 There is a typo in the example. To get the claimed results, the (unstated) strike price should be K=\$80 as in Example 10.8, and the risk-free return should be R=exp(0.10) rather than exp(0.01) as stated in the textbook. Thanks to Mr Eric Tang for solving this puzzle. Exercise 10.13 There is a typo in this example as well. To get the claimed results, the risk-free return should be R=exp(0.10) rather than exp(0.01) as stated in the textbook. Thanks again to Mr Eric Tang for solving this puzzle as well as for Example 10.7. Exercise 10.14 There is no strict equality here in general, despite the textbook's claim, since for some (n,j) it is possible that both sides are zero.