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.