We will derive a two-state put option value in this problem. Data: S0 = 180; X =

Question

We will derive a two-state put option value in this problem. Data: S0 = 180; X = 190; 1 + r = 1.1. The two possibilities for ST are 210 and 110. a. The range of S is 100 while that of P is 20 across the two states. What is the hedge ratio of the call? Hedge ratio .18 b. Calculate the value of a call option on the stock with an exercise price of 190. (Do not use continuous compounding to calculate the present value of X in this example, because the interest rate is quoted as an effective per-period rate.) Call value $

Explanation / Answer

(a)

S0 = 180, X = 190, r = 0.1 , Su = 210, Sd = 110

Cu = max(Su-X, 0) = max(210-190,0) = 20

Cd = max(Sd-X, 0) = max(110-190,0) = 0

hedge ratio(H) = (Cu-Cd)/(Su-Sd) = (20-0)/(210-110) = 0.2

(b)

u = Su/S0 = 210/180 = 7/6

d = Sd/S0 = 110/180 = 11/18

C0 = (1+r-d)/(u-d)*Cu/(1+r) + (u-1-r)/(u-d)*Cd/(1+r)

= (1+0.1-11/18)/(7/6-11/18)*20/(1+0.1)

= 16

The answer is $16

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