Question

Suppose a stock currently trades at a price of 150. The stock price can go up 33% or down 15%.The risk free rate is 4.5%

1. use a one period binomial model to calculate the price of a put option with exercise price of 150.

2. Suppose the put price is currently 14, show how to execute an arbitrage transaxtion that will earn more than the risk free rate . use 10,000 put options.

3. Suppose the put price is currently 11, show how to execute an arbitragr transaction that will earn more than the risk free rate . use 10,000 put options

Answer 1

1.

S₀=150

S₊=150*1.33=199.5

u=199.5/150=1.33

S_-=150*0.85=127.5

d=127.5/150=0.85

p=(e^r-d)/(u-d)=(e^0.045-0.85)/(1.33-0.85)=40.839%

(1-p)=1-40.839%=59.161%

Put possibilities at maturity,

P₊ = MAX(K-S₊,0) = MAX(150-199.5,0) = 0

P_- = MAX(K-S_-,0) = MAX(150-127.5,0) = 22.5

P_t=0 = {[p * P₊] + [(1-p) * P_-] }*e^-r = [(0.40839*0)+(0.59161*22.5)]*e^-4.5%=$12.255

Excel calculations:

We can calculate delta = [P₍₊₎ - P_(-)] /[S₍₊₎ - S_(-)] = -22.5/(199.5-127.5) = -0.3125

2.

*MM refers tomoney market.

Thus we, see if stock goes up (to 199.5) or down (to 127.5), we always cash in $13331.96 despite zero investment initially. Hence we get riskless profit (arbitrage).

3.

Thus we, see if stock goes up (to 199.5) or down (to 127.5), we always cash in $7588.60 despite zero investment initially. Hence we get riskless profit (arbitrage).