Question

Assume that the current stock price (S0) is £42, and that it either increases at a rate of 10% (u = 1:10) or it decreases at a rate of 5% (d = 0:95) over a period of three months. Further assume a European call option written on the stock, with a strike price (K) equal to £40 and a time-to-maturity (T) equal to six months. The risk-free interest is 10% pa.

(a) Draw a binomial tree showing the possible evolutions of the stock price over the next six months (i.e., until the maturity time of the option).
(b) Compute the payoff of the European call option at maturity for each possible stock price, and indicate this payoff in the graph.

(c) Compute the risk-neutral probability of an upward move and that of a downward move in the stock price. Do these differ across the binomial tree (i.e., are these different at time t = 0 compared to, for example, time t = 2)? Why or why not?
(d) Using the risk-neutral probabilities, compute the value of the European call option.

(e) Would your answer to question (d) be different if the call option had not been European but instead American? Why or why not?

Answer 1

Part (a)

Please see the diagram above. The binomial tree is shown. Stock price at each node is show in yellow color. Prices have been calculated using the following rule:

Su = u x S0; Sd = d x S0; Suu = u x Su; Sud = u x Sd = d x Su; Sdd = d x Sd; S0 = 42

Part (b)

the payoff of the European call option at maturity for each possible stock price has been shown in blue colored cells. The payoff is governed by = max (S - K, 0) = max (S - 40, 0)

Part (c)

The risk-neutral probability of an upward move = p = (e^rt - d) / (u - d) = (e^{10% x 3/12} - 0.95) / (1.1 - 0.95) = 50.21%
and that of a downward move in the stock price = 1 - p = 1 - 50.21% = 49.79%

They don't differ across the binomial tree (i.e., are these different at time t = 0 compared to, for example, time t = 2). They don't differ because:

• Risk free rate is same at each time period
• Width of each time interval is constant
• up factor and down factor, u & d are constant

Part (d)

Value of the European call option = PV of Sum of (Call value x probability) at each node = e^-rT[p² x Cuu + 2p(1 - p)Cud+ (1 - p)²Cdd] = e^{-10% x 6/12}[0.2521052 x 10.82 + 0.4999912 x 3.89 + 0.2479036 x 0] =  4.44

Part (e)

Let's check if early exercise made sense at node Su and Sd. At Sd, payoff on exercise = max (Sd - K, 0) = max (39.90 - 40, 0) = 0. So, the early exercise would not have made sense. At Su, payoff = max (Su - K, 0) = max (46.20 - 40, 0) = 6.20.

Expected PV of Cuu and Cud at the node Su = e^-rt[p x Cuu + (1 - p) x Cud]

= e^{-10% x 3/12}[50.21% x 10.82 + 49.79% x 3.89]

= 7.19 > payoff due to an early exercise.

Hence, the option holder will not exercise early.

In that case, the answer to question (d) will not be different if the call option had not been European but instead American, because the early exercise doesn't make sense for the option holder.