Consider a three period binomial model (one initial date and two steps) as in cl
ID: 1121001 • Letter: C
Question
Consider a three period binomial model (one initial date and two steps) as in class. Let the initial stock price be $100. Let the upward increments be u = 1.2 and the downward be d = .9. Thus, the up-up state should have Suu = u u S0 = 1.2 1.2 100 = 144 and so on. The risk free rate is ten percent per period. The call option with strike price of $100 was priced in class.
a. Price the put option with strike price of $100 using replicating portfolios. Does put-call parity hold (remember the interest accrues in discrete steps)? Explain.
b. Suppose we had an Asian call option whose payoff is max{SK, 0} where S is the average stock price, with strike K = 100. So, if the state is up-up the payoff would be (100 + 120 + 144)/3 100 = 21.33. Use replicating portfolios to price this option.
Explanation / Answer
Here we have u = 4/3
And d = ¾
The initial price of the stock S0 = 10
Risk free rate rf = 0
For a one-period model (S0 -> S1), we have t=1
the state prices ?u = initial price * u
= S0 * u = 10 * 4/3 = 13.33
and ?d = initial price * d
= S0 * d = 10 * 3/4 = 7.5
the risk neutral probability qu = (e^(rf*t) – d)/ (u - d)
= (e^(0*1) – 3/4) / (4/3 -3/4)
= (1 -3/4) /(4/3 -3/4)
= 0.4286
and the risk neutral probability qd = (u - e^(rf*t) / (u - d)
= (4/3 - e^(0*1)) / (4/3 -3/4)
= (4/3 - 1) /(4/3 -3/4)
= 0.5714