Question

3) Show that the beta of a European call option, ?? , must be equal to:

?? = ???

Where ?? is the CAPM equity beta and ? is the option Greek relating option prices and percentage changes in stock prices.

4) Show that it is never optimal to exercise an American call option on a non-dividend paying stock prior to expiration. Why is this not the case for American puts on nondividend-paying stocks?

(SHOW ALL WORK and EXPLAIN each, Please)

Answer 1

Answer 3

The beta of a call is given by Black and Scholes

.........................................1

...............................2

..................................................3

From 1 & 3

..............................4

From 2

...................................................5

From equation 4 & 5

Answer 4

Consider an American Call Option that pays dividend D1, D2, D3....Dn at time t1, t2, t3....tn.

Consider the possibility of an early exercise just prior to final dividend date. If the option is exercised the investor receives

S(tn)-X

If the option is not exercised, the stock price drops to S(tn) – Dn and the value od the option is then greater than

S(tn) – Dn –Xe^-r(T-tn)

It follows that if

S(tn) – Dn -Xe^-r(T-tn) >=S(tn) –X

That is

Dn<=X(1-e^-r(T-tn)

Hence it cannot be optimal to exercise an American option before the date of dividend payment.