Consider the put-call parity relationship for European call and put options on a stock that pays discrete dividends only. Assume the initial stock price is S0, the continuously compounded risk-free interest rate is r, the options expire at time T, the stock price at time T is denoted ST and there are two discrete dividends paid between time 0 and time T at times t1 and t2, 0 < t1 < t2 < T with amounts denoted dt1 and dt2 respectively. Write the put-call parity formula and explain why it is true

Question

Damion Park

Mathematics

14 June, 12:58

Consider the put-call parity relationship for European call and put options on a stock that pays discrete dividends only. Assume the initial stock price is S0, the continuously compounded risk-free interest rate is r, the options expire at time T, the stock price at time T is denoted ST and there are two discrete dividends paid between time 0 and time T at times t1 and t2, 0 < t1 < t2 < T with amounts denoted dt1 and dt2 respectively. Write the put-call parity formula and explain why it is true

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Leroy Tapia · Answer 1

Answer and explanation:

The solution is determined using the explained formula below

C₀ + X * e - r * t = P₀ + S₀

S₀ = Stock price today

X = Strike price

C₀ = European call option premium

P₀ = European put option premium

T = Time to expiration

r = Risk-free rate of return

another thing which we have to take into consideration is the impact of dividend on put-call parity.

Since interest is a cost to an investor who borrows funds to purchase stock and benefit to the investor who shorts the stock or securities by investing the funds.

Here we will examine how the Put-Call parity equation would be adjusted if the stock pays a dividend. Also, we assume that dividend which is paid during the life of the option is known.

Here, the equation would be adjusted with the present value of the dividend. And along with the call option premium, the total amount to be invested by the investor is cash equivalent to the present value of a zero-coupon bond (which is equivalent to the strike price) and the present value of the dividend. Here, we are making an adjustment in the fiduciary call strategy. The adjusted equation would be

C₀ + (D + X * e - r * t) = P₀ + S₀ where,

D = Present value of dividends

We can adjust the dividends in another way also which will yield the same value. The only basic difference between these two ways is while in the first one we have added the amount of the dividend in strike price, in the other one we have adjusted the dividends amount directly from the stock.

P₀ = C₀ + X * e - r * t - S₀ - (S₀ * e - r * t),