Consider the following binomial model which gives the price of a stock at differ

Question

Consider the following binomial model which gives the price of a stock at different points in time:

The return in the ’up’ state (u) is 10%, the return in the down state (d) is -9.09%, and the risk-free interest rate (r) is 4%. We are interested in pricing American options that expire at time 2.

a) Find the risk-neutral probability (p) of an ’up’ state. Briefly interpret this measure.

b) At each node of the binomial tree compute the no-arbitrage price of a call option with an exercise price (X) of 95.

c) Derive the hedge ratio (h) between the stock and the call option at the nodes at time 0 and time 1. Briefly explain why h increases when the stock price goes up.

d) Repeat parts (b) and (c) for a put option with an exercise price of 105.

Explanation / Answer

a) Rf = 4% Current Price= 100 Time to maturity = 2 years Cu = 110 PV of future cash flow = 105.77 110 = Max((110-100,0)*p)+Max((90.91-100,0)*1-p) 110= 10*p+0*(1-p) 110= 10*p p= 11.00% c) Hedge Ratio = (Cu-Cd)/(u*S0-d*S0) Value of option at Cu= 10 (110-100) Value of option at Cd= 0 Cu = 110 Cd = 90.91 Hedge Ratio (H)= 0.52 For every call option written by the market maker, 0.52 shares of stock must be held to hedge away risk. As the stock price goes up, the value of option also increases. So the risk increase and to hedge this increased risk, higher number of shares are required. So H increases with increase in stock price.

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