Question

Calculate the value of a put option for both continuous and discrete dividend yields( one payment). What is the put-call parity relation in the latter case? Do the dividends increase or decrease the value of the put? Why?

Answer 1

1) The vale of European put option in case continuous risk free rate & dividend yield is as follows:

= Initial stock price - present value of dividends , K = strike price., r = risk free rate, is the volatility and T is time to expiry of option.

Present value of dividends if yield is continuous = , where r is the risk free rate and is time difference between ex-dividend date and expiry date.

Present value of dividends if yield is discrete =

In case the dividend yield is discrete, then one needs to convert the rate into continuous compounding rate by using the formula ->

where is the continuous compounding rate and is the rate compounded at m frequency.

For e.g. to convert annual compounding rate to continuous compounding rate, replace m =1 in the equation above. For semi-annual compounding rate, replace m = 2. So, if in this problem, dividend yield is given at rate on per annum basis, use m =1.

2 ) Put call parity in case where the dividend yield is discrete is given by ->

where, c= value of call

D = present value of dividends which is calculated as where   is the time difference between dividend date and expiry date.

p = value of put , k =strike price, is initial stock price and r is risk free rate.

3) Dividends leads to increase in value of put option as evident from the put call parity equation mentioned above.

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