Let A(0) = 100, A(1) = 105, A(2) = 110.25 and suppose that stockprices can follo

Question

Let A(0) = 100, A(1) = 105, A(2) = 110.25 and suppose that stockprices can follow four possible
scenarios:
Scenario     S(0)    S(1)      S(2)
w1             80       88        100
w2             80       88         81
w3            80       74         79
w4             80       74         68
(a) (10 points) Draw the tree of the stock prices.
(b) (10 points) Find a risk-neutral probability, if possible.
(c) (10 points) Apply the Fundamental Theorem of Asset Pricing tofind the time 0 and 1 prices
of a European put option with strike price $95 maturing after twosteps.

Explanation / Answer

Tree: _________100 ____88 _________81 80 _________79 ____74 _________68 Risk-Free Rate: rf = 105/100 - 1 = 5% = 110.25/105 - 1 Risk Neutral Probability: I'll find the risk neutral probability for one branch. I'llleave the rest to you (the probabilities will likely be equal inmost problems, but don't have to be). You want to find the p_up such that the expected rate of return onthe stock is the risk free rate. Remember: This is anARTIFICIAL CONSTRUCT and IS NOT the actual probability of thestock's price going up. So, looking at the top right node... ____100 88 ____81 p_up*(100-88)/88 + (1-p_up)*(81-88)/88 = .05 Solving, we get p_up = .6 Option Pricing Via Synthetics: Again, I'll do it for the upper right node and let you do therest. Payout of a put = max(K - S, 0) So the payout in the upper branch is 0 and in the lower branch it's4. We now need to replicate these cash flows using a bonds and astock. Solve the simultaneous equ's. These tell you how much of eachasset you have to "buy" to replicate the cash flows. 100*S + 1.05B = 0 81*S + 1.05B = 14 Solving, we get S = -.737 and B = 70.175. In general, for allputs, S will be negative and B will be positive. For allcalls, S will be positive and B will be negative. This makessense as a put is similar to shorting (make money when the stockprice goes down) and a call is similar to longing (make money whenthe stock price goes up). Finally, to compute the cost of the put, multiply S and B by theirrespective price at that time. S*88 + B*1 = 5.33. Now, to do the others, work backwards.

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