Question

An asset’s price is $29.00 and its volatility is 29% per year. A European put on the asset has an exercise price of $30.00, eighteen months until expiration, and the risk-free interest rate is 3.8% per year. According to a two-period binomial option pricing model, what is the option’s value?

A) $3.07
	B) $3.54
	C) $2.53
	D) $4.78
	E) $3.48

Answer 1

E. $3.48

Size of the upmove factor U =

t is the size of each step in the binomial model

Size of the upmove factor U = e^(0.29*((9/12)^0.5)) = 1.2855

Size of the downmove factor D = 1/U = 1/1.2855= 0.778

Probability of up-move =

Probability of up-move = (e^(0.038*0.75)-0.778) / (1.2855- 0.778) = 0.4944

Probability of down-move = 1- 0.4944 = 0.5056

Since, this is a European option, option can only be exercised at maturity

Hence, accordingly to the probable stock prices at maturity, the stock is in the money at node 4, 5 and 6.

The payoff at node 4 = 30 - 29.0034 = 0.9966

The payoff at node 5 = 30 - 29.0034 = 0.9966

The payoff at node 6= 30 - 17.553= 12.447

Value of the payoff at t=0 = Probability of payoff * PV of payoff

Probability of ending up at node 4 = 0.4944*0.5056

PV of node 4 payoff = 0.9966*e^(-0.038*1.5)

Value of the node 4 payoff at t(0) = 0.4944*0.5056* 0.9966*e^(-0.038*1.5) = $0.235316

Probability of ending up at node 5= 0.5056*0.4944

PV of node 5 payoff = 0.9966*e^(-0.038*1.5)

Value of the node 5 payoff at t(0) = 0.5056*0.4944* 0.9966*e^(-0.038*1.5) = $0.235316

Probability of ending up at node 6 = 0.5056*0.5056

PV of node 6 payoff = 12.447*e^(-0.038*1.5)

Value of the node 6 payoff at t(0) = 0.5056*0.5056*12.447*e^(-0.038*1.5) = $3.0055

Total expected value at t(0) = Put option's value = Value of the node 4 payoff at t(0) + Value of the node 5 payoff at t(0) +Value of the node 6 payoff at t(0)

Put option's value = $0.235316+ $0.235316+ $3.0055

Put option's value = $3.48