Economics 4113, Spring 2013, University of Minnesota. Instructor: David Rahman. Homework 4 Due date: Wednesday, April 24, in class. Please substantiate your answers. Question 1 (American Put). Consider a stock price S that follows a geometric Brownian motion: dS = rSdt + !Sdz, where r > 0 is the interest rate and ! is the volatility parameter. An investor holds an American put option with infinite maturity, i.e., the right to sell the stock anytime at a prespecified price K. By buying back the stock at market price S, the investor’s payo? from exercising this option is ?(S) = K – S. Let F(S) be the value of the option as a function of the stock price. The asset return equation for F is given byrF(S) = max{K – S,E[dF]/dt} Let S? be the cuto? value of S below which it is optimal to exercise the option. 1. Use Ito’s Lemma to show that the value of the option F must satisfy 12!2S2F00(S) + rSF0(S) – rF(S) = 0 (?) in the continuation region, i.e., whenever S > S?. 2. Derive the two possible values of ” for which the function F(S) = AS! satisfies (?). (Hint: Plug F(S) into (?), derive the fundamental quadratic, and calculate its roots.) Show that one such root is ” = 1. What is the value of the other root? 3. Show that F(S) = A1S!1 + A2S, where “1 6= 1 is a root from part (b), solves (?). For a put option, it must be the case that limS!1 F(S) = 0, i.e., it is never worth buying back an infinitely expensive stock. Use this condition to show that A2 = 0, hence F(S) = A1S!1 . 4. Use value matching (F(S?) = K – S?) and smooth pasting (F0(S?) = -1) to find A1, S? and F in terms of the problem’s parameters. 5. What can you say about how S? depends on the parameters? 1
Question 2 (European Call ). Consider a stock price S that follows a geometric Brownian motion: dS = rSdt + !Sdz, 1. Derive the law of motion for ln S using Ito’s Lemma. Find ?t and “t such that ln ST – ln St ? N(?t, “t), where ? is the mean of ln ST – ln St given St and “t the variance. 2. A European call option…