Question

Consider a European call with an exercise price of $50 on a stock priced at $60. The stock price can go up by 15 percent or down by 20 percent in each of two binomial periods. The risk free rate is 10 percent. (Show ALL your workings.)

(a) Calculate the stock price sequence.

(b) Determine the possible prices of the call at expiration.

(c) Find the possible prices of the call at the end of the first period.

(d) What is the current price of the call?

(e) Construct an example to show that the hedge ratio works with 1,000 units of the option. You need to show your strategy with ONLY ONE hedge ratio.

Answer 1

(a) Stock Price Sequence

Su = 60*1.15 = 69

Su² = 79.35

.Sud = 55.2

Sd = 48

Sd² = 38.4

(b) Price of the call at expiration :-

Fu² = 29.35

.Fud = 5.2

Fd² = 0

(c) Fu =((p*Fu²)+(1-p)*Fud)/1.10

((29.35*0.7142)+(5.2*0.2857))/1.10

= 20.40674

Fd = ((p*Fud)+(1-p)*Fd²)/1.10

=((0.7142*5.2)+(0.2857*0))/1.1

= 3.3762

(d) Value of Option :-

((p*Fu)+(1-p)*Fd)/1.10

=((0.7142*20.40674)+(0.2857*3.3762))/1.10

= 14.12643

Working Note:-

(Value of P = (1.10-u)/(u-d) = (1.10-0.85)/(1.20-0.85) = 0.25/0.35 = 0.7142