## Transcribed Text

6. Consider a CRR model with T = 2, S0 = $100, S1 = $200 or S1 = $50
(the same model as in Exercise 2). Now consider an American put option
with strike price K = $120. Assume that the risk free interest rate is r = 0.1.
(a) Use a binary tree to compute the arbitrage free initial price of the
American put option.
(b) Determine an explicit superhedging strategy 0* for this option.
(c) Suppose that you can buy the American put option at time zero for
$1 less than its arbitrage free price. Explicitly describe a strategy
that yields an arbitrage opportunity for a buyer of the American
put option.
(d) Try to automate the arbitrage free pricing of an American put
option in a computer program where T, S0, u, d and K are variables.
4. Let T : S
{0, 1, 2, , T be a random variable on a finite sample space S2, and let {Fit= =
0, 1, 2,
,
I} be the filtration generated by the stock prices in a binomial model, that is,
Ft = 0 (So, S1 ,
St) for t = 0, 1, , T. (The result actually holds for any filtration.)
(a) Show that if T is an (Fi}-stopping time, then for every t = 0,1, , T, {T t} E Ft.
(b) Show that if every t = 0,1, T, {T t} E Ft, then T is an {Fi}-stopping time (that is,
the converse of part a is true).

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction
of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice.
Unethical use is strictly forbidden.