Question

Consider a European call with an exercise price of 50 on a stock priced at 60. The stock can go

up by 15% or down by 20% each of the two binomial periods. The risk-free rate is 10%.

Determine the price of the option today. Then construct a risk-free hedge for a long stock and a

short option. At each point in the binomial tree, show the composition and value of the hedge

portfolio. For period 1 (that is h), demonstrate that the return is the same as the risk-free rate.

(Assume 1,000 calls).

Answer 1

K = 50 , S0 = 60 , u=1.15 d=0.8 r= 0.1

time 0 1 2 payoff

69 79.35 29.35

60 55.2 5.2

  48   

38.4 0

q = exp(0.1)-0.8/(1.15-0.8) = 0.8719170

1-q = 0.1280830

V0 = price of european call option

= {29.35*0.8919170^2+ 2*5.2*0.8719170*0.1280830}*exp(-0.1*2)

= 21.90

Vo portdolio at time zeo which is replicating portfolio

Vo=share + cash bonds

V1 = value of portfolio at time 1

*share * u + = 19

*share*d + = 0

= 19/21

21.90 = 19/21*60 + cash bounds units

cash bounds = - 32.38

V1 value of portfolio at time 1

V2 =