American option

[Unusualskill]

Hi,

I would like to ask about GARP Assigned Reading- Hull, Chapter 15

When a stock pays a divided , D , at time n.
At the last dividend date before expiration, t_n, the exercised value of the option is: S(t_n) - X
If the call option is unexercised and the dividend is paid, its unexercised value is : S(t_n)-D_n -X^(-r(T-t_n))

Why investor will only exercise when S(t_n) - X > S(t_n) - D_n - X^(-r(T-t_n)) ?

And why the closer the option is to expiration and the larger the dividend, the more optimal early exercise will become?

Appreciate your help!

Thank you

 

[David Harper CFA FRM]

Hi @Unusualskill I just answered a question similar to this, can you please look here at https://forum.bionicturtle.com/threads/lower-bounds-on-dividend-paying-options.10667/#post-52014
The pure theory idea is actually pretty straightforward: if you hold an american call option, you face a trade off:

• exercise now, which earns you the underlying share and therefore the dividend, so this is a gain of the PV(dividends)
• do not exercise, which has value because the strike price is fixed. So if the strike is K, but you exercise in T years (e.g., 0.25 years for three months), then the present value of that future exercise is K*exp(-r*T). So deferring exercise by itself actually gains you K - K*exp(-rT) = K*[1 - exp(-rt)]

Ergo it is advisable to exercise the option if PV(dividends) > K*[1 - exp(-rt)]; i.e., if the dividends earned are moe PV profitable than the strike price savings. I hope that's a helpful start!

 

[Unusualskill]

@David Harper CFA FRM Thank you, you are great !