Question

A company is currently trading at $100 per share. After examining the stock, determined that in each 3 month period, its price will either increase to 25% or decrease by 20%. The interest rare is 3% every 3 months.

a) A six month Europrean call option on this company has an exercise price of $90. What is the value of this call option?

b) What is the value of a six month European put option on this company with an exercise price of $90?

c) Verify that put-call parity holds.

d) Now suppose that the company pays a dividend equal to $25 in three months. What is the value of a six month American call option on the company with an exercise price of $90? Would an individual ever want to exercise this option early?

Answer 1

ANSWER:

a)

u = 1.25 , d =0.8

The stock lattice is

		156.25
	125.00	100.00
100.00	80.00	64.00
t=0	t=1	t=2

risk neutral probability p= (1.03-0.8)/(1.25-0.8) =0.5111

So, the European call option lattice is

		66.25
	37.62	10.00
21.02	4.96	0.00
t=0	t=1	t=2

So, the price of the European call option is $21.02

b)

The European put option lattice is

		0.00
	0.00	0.00
5.86	12.34	26.00
t=0	t=1	t=2

So, the price of the European put option is $5.86

c)

From put call parity ,

c+K/(1+r)^t = p+S

Here, c+K/(1+r)^t = 21.02+ 90/1.03^2 =105.86

and p+S= 5.86+100 = 105.86

So, the prices follow put-call parity

d)

in case of dividends, the stock price after 3 months will experience a decrease of $25 ex-dividend and the next 3 months price will be accordingly set. The only time early exercise is profitable is before the dividend

the stock lattice (ex-dividend) price is

		125.00
	100.00	80.00
100.00		68.75
	55.00	44.00
t=0	t=1	t=2

The European Option lattice is

		35.00
	17.37	0.00
8.62		0.00
	0.00	0.00
t=0	t=1	t=2

So, if the option is held till maturity, the value of the option is $8.62, however, if the option is exercised just before the dividend date, the value of the option is $35 at t=1 when stock price is $125 (cum dividend) and $100 (ex-dividend)

Similarly at t=1, when stock price is $80 (cum dividend) and $55 (ex-dividend), the value of exercising is 0

So, The American option lattice is as shown below

		35.00
	35.00	0.00
17.37		0.00
	0.00	0.00
t=0	t=1	t=2

Value of the american call option is $17.37 and early exercise is optimal at t=1 if the stock price reaches $125 (cum dividend) and $100 (ex-dividend)

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