Let St be the dollar price of a stock at t years from today. Consider an agreeme

Question

Let St be the dollar price of a stock at t years from today. Consider an agreement (contract) between two parties, the “holder” and the “seller”, according to which the seller pays the holder $1 at time t if St > K, where K is some positive constant. If St K, no transaction occurs.

(a) Suppose that St is a lognormal random variable, i.e., ln (St/S0) N (µ (^ 2/ 2) t, ^2 t) , where S0 is the price of the stock today, µ and are positive numbers. Obtain an expression for P(St > K) that involves only S0, K, µ, , t and the cdf of the standard normal distribution.

(b) Compute this probability when S0 = 60, K = 70, µ = 0.3, = 0.2, t = 0.5.

Explanation / Answer

At the time when the contract is written, we don't knowST, we can only guess at it, or, more

formally, assign a probability distribution to it. A widely used model (which underlies the Black-Scholes analysis) is that stock prices arelognormally distributed. That is, there are constantsvand such that thelogarithmofST=S0(the stock price at timeTdivided by that at time zero,usually called thereturn) is normally distributed with meanvand variance2. In symbols

b.St/S0) N (µ (^ 2/ 2) t, ^2 t)

St=s0[N(0.3-(0.2^2/2)0.5,0.2^2(0.5)]

Related Questions

Navigate

• Browse (All)
• Browse L
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.