Due Monday 11/20/2017Fin Finance 421 (Towner) Homework 6 Problem 2: Binomial Opt

Question

Due Monday 11/20/2017Fin Finance 421 (Towner) Homework 6 Problem 2: Binomial Options Pricing (5 points) Imagine a case where there were only two possible stock prices in one month: $100 and $80, occurring with probabilities 60% and 40% respectively, and the current stock price was $91; 60% /. S100 $91 40% $80 We will build up to finding the no-arbitrage price of a call option with strike K 95 that expires exactly one month from now. (a) What will the call payoff be in case that ST -$100? What about Sr 880? We'l call the first number Ou and the second Oa (b) What portfolio of risk-free bonds with face value B and shares of the underlying stock will replicate the call option? B=

Explanation / Answer

a:

Ct = max (St – K, 0) where K = 95

Cu = 100 – 95 = 5, Cd = 0

Risk-neutral probability: p = (R – d)/(u – d)

0.6 = (R – 80/91)/(100/91 – 80/91) = (91R – 80)/20

91R – 80 = 12

Risk-neutral growth rate: R = 92/91

Using risk-neutral probability, the call value is

C = [p*Cu + (1 – p)*Cd]/R

C = (0.6*5 + 0.4*0)/(92/91) = 2.97

Call payoff = Cu = 5 when S = 100

Call payoff = Cd = 0 when S = 80

b:

= (Cu – Cd)/(Su – Sd) = (5 – 0)/(100 – 80) = 5/20 = 0.25

PV of amount invested on bond = (Cu – Su )/R

Cu – Su ) = 5 – 100*0.25 = 5 – 25 = -10

Negative sign shows borrowed

FV of amount invested on bond = Su - Cu = $10

B = $10

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