Put call parity

[higaurav]

Hi David,

Q1 Put call parity - Practice question (Par 4 difficulty). If the options are instead American-style (paying dividends), when is it optimal to exercise?

- Is it same for both put and call options (i.e before stock goes ex dividend). Because after dividend, stock price should go down and put option holder should exercise after that as S < X and value of put option will increase. Pls suggest.


Q2 If the stock pays a dividend of 2%, how do we adjust the Black-Scholes-Merton to account for the dividends? What is the intuition?

- Can we incorporate this in the BS formula itself like - N(d1) = (log (s/x)+ ( Rf - d + var/2)*t)/vol*sqrt(t) .. Here d is the dividend yield. Although the result is different from N(d1)* exp(-d *t). Infact, when I look your excel sheet given in the solution of Option Gamma - Practice Question (Par 4 difficulty)..you have used the former approach in the formula although dividend there is zero.

Also, in case of N(d1)* exp(-d *t), how to incorporate this for N(d2)? .. look for your guidance.

Thnks
OM

 

[David Harper CFA FRM]

Hi Om

Re optimal exercise, here is the graphic from the Market A episode.

http://learn.bionicturtle.com/images/forum/optimal_exercise.png

Note the put and the call are not deemed the same here. Hull gives the logic, but maybe one way to think about this is: the expected return of the stock is positive (i.e., without systemetic risk, E(r) on stock is riskless rate). So, if the put is "in the money," the *expectation* is only that, if we wait, as (S) increases, the put can only lose value. I am really simplifying/distorting what Hull says about this, but the key difference, in this context, between an in-the-money call and put is that, given E[growth in stock price] > 0, we might expect the intrinsic value of a call to increase, but not true for a put.

Re dividend. If the dividend is continuous, yes, absolutely the adjustment is just as you have it for d1. I do tend to use this general version (Rf - d + var/2) because it works in both cases; i.e., if no dividend, it is the same Black-Scholes. If the dividend is continuous, you just have to remember there are two adjustments. One, within the d1 as you have, and two, to again discount the stock price:
c = S*N(d1*)*EXP[(-d)(T)] - (Strike)*N(d2*)*EXP[(-r)(T)]

d2 with a dividend (d2*) is the same: d1 - volatility*SQRT[T]

David