In: Finance
Suppose that, in each period, the cost of a security either goes up by a factor of u = 2 or down by a factor d = 1/2. Assume the initial price of the security is $100 and that the interest rate r is 0.
a). Compute the risk neutral probabilities p (price moves up) and q = 1−p (price moves down) for this model
b). Assuming the strike price of a European call option on this
security is $150, compute the possible payoffs of the call option
given that the option expires in two periods. It may help to sketch
a diagram of the possible security price movement over two
periods.
c). What is the expected value of the payoff of the call option?
d). What should the no-arbitrage price of the call option be?
a). For a two period model, we have:
p = (e^(r*t) - d)/(u-d) where r (interest rate) = 0; t (time covered in 1 step) = 0.5
p = (e^(0*0.5)-0.5)/(2-0.5) = 0.3333 (or 33.33%)
q = 1-p= 1-0.3333 = 0.6667 (or 66.67%)
b). Stock price movement:
T = 0: S0 = 100
T = 1: S1+ = S0*u = 100*2 = 200
S1- = 100*d = 100*0.5 = 50
T = 2: S2++ = S1+*u = 200*2 = 400
S2 = S1+*d = 200*0.5 = 100
S2-- = S1-*d = 50*0.5 = 25
Possible payoff at expiry:
P2++ = max(S2++ - strike price, 0) = max(400-150, 0) = 250
P2 = max(S2 - strike price, 0) = max(100-150,0) = 0
P2-- = max(S2-- - strike price,0) = max(25-150,0) = 0
Payoffs at expiry can be 250 or 0.
c). Expected value of the payoff = (P2++*p) = 250*0.3333 = 83.33
d). No-arbitrage price will be the expected value of the payoff, discounted back one more period to T = 0. Discount factor = 1 since e^(r*0.5) = e^(0*0.5) = 1
No arbitrage price = 83.33*p = 83.33*0.3333 = 27.78