Question

Problem 4 (6 pts) In a simple market one-period binomial tree model, we know that the simple compounding annual effective interest rate is given by i=8%. The stock price satisfies S(0)=65 and

S(1)={ 80, with probabilit p,
{ 40, with probabilit 1−p,
where 0

(a) Let us consider a Put option written on this stock with the strike price 60 and the maturity time T=1. Find the initial price P(0) of this Put option.
(b) Let us consider a Call option written on the Put option in part (a) with the strike price K=10, i.e., the holder of this Call option has the right to get the payoff of the Put option in part (a) by paying the strike price K=10. Find the initial price C(0) of this "call on put" option by constructing replication portfolio using stock and bank account.

Answer 1

(a) To find the initial price P(0) of the Put option, we can use the risk-neutral valuation approach. First, we need to calculate the risk-neutral probability p*, which is given by:

p* = (1 + i - d) / (u - d)

where i is the annual effective interest rate, u is the up factor of the stock price, and d is the down factor of the stock price. In this case, i = 8%, u = 1.25, and d = 0.5, so we have:

p* = (1 + 0.08 - 0.5) / (1.25 - 0.5) = 0.6

Next, we can calculate the put option prices at the two nodes of the tree at time T=1. At S(1)=80, the put option has a payoff of 0 because the stock price is higher than the strike price of 60. At S(1)=40, the put option has a payoff of 20 because the stock price is lower than the strike price by 20. Therefore, we have:

P(1,80) = 0

P(1,40) = 20

Using the risk-neutral valuation formula, we can then calculate the initial price P(0) of the put option as:

P(0) = e^{-iT} [p*P(1,uS(0)) + (1-p*)P(1,dS(0))]

where T is the time to maturity, S(0) is the current stock price, and uS(0) and dS(0) are the possible stock prices at time T=1. Plugging in the numbers, we get:

P(0) = e^{-0.08} [0.6*P(1,80) + 0.4*P(1,40)]

= e^{-0.08} [0.6*0 + 0.4*20]

= 6.707

Therefore, the initial price of the Put option is $6.707.

(b) To find the price of the Call option written on the Put option in part (a), we can use a similar approach. The Call option has a payoff of C(T) = max(P(T) - K, 0), where K is the strike price of 60. Using the same values for p*, P(1,80), and P(1,40), we can calculate the value of the Call option at time T=1 as:

C(1,80) = max(0 - 60, 0) = 0

C(1,40) = max(20 - 60, 0) = 0

Using the risk-neutral valuation formula, we can then calculate the initial price C(0) of the Call option as:

C(0) = e^{-iT} [p*C(1,uP(0)) + (1-p*)C(1,dP(0))]