Question

Consider a stock currently priced at $25 with an expected volatility of 25% p.a. A dividend of $2.50 is expected in 1.5 months. You are working for a bank and one of your clients wants to buy a call option on the stock with a maturity of three months and a strike of $24.

a) The client has asked you to quantify what the additional cost will be if they have the right to exercise the call option early. Use a three-step binomial model and assume a constant risk-free rate of 4% p.a. continuously compounded.

b) If the client buys an option that allows for early exercise, based on the pricing model, when would you advise them to exercise this right?

Answer 1

Answer to question (a):

Actually there are two type of options American option which can be exercised on or before expiry date.

Another one is Europien option which can be exercised only on the expiry date.

It depends on type of option that we can exercise it before expiry date or not. No additional cost it requires here to exercise it prior.

Note: To avoid dishonour the contract in case of options only option premium is additionally given . And it is given irrespectively the option is exercised or not.

Computation of theoritical value of call option as per continuous compounding .

Risk free rate 4%

Dividend $2.5 for 1.5 months period.

Spot price = $25

Strike price $24

value of option = ?

Value of option=  (s -d)e ^{r x t}

s= spot price

d= Dividend

r= Risk free rate

t= Time / Period

e= Exponential Function. And its value 207183 remain same always.

Solution:

Here forst present value of dividend will be calculated.

Present value of dividend = Divident amount / e ^{r x t = 2.5 / 2.7183 x.04 x 1.5/12}

= 2.5 / 2.7183 .04x1.5/12 = 2.5/1.005 = $2.4875 or 2.49 after rounding off.

(2.7183 is value of exponential function. after solution it will give value 1.005)

Value of option =   (s -d)e ^{r x t = (25 - 2.49) 2.7183 x.04x3/12 = 22.51 x 1.01005 = $ 22.736}

(2.7183 is value of exponential function. after solution it will give value 1.01005)

theritical value of option will be = $ 22.736

Answer to question (b)

It is suggested to the investor not to exercise this option because secutiy is cheaper in sopt market & value of call option os more.

Kindly see uploded images for further clerification.