CM2-15 Practice Question 15.2(i)

[yuli2513]

The question is on page 25 and the answer is given on page 32-33.

I do not fully understand how the share bonus and cash bonuses are calculated using the Black-Scholes formula, to be specific, for the share benefit, the first term of the Black-Scholes formula for a European call option is kept, whereas for the cash benefit, the second term of the Black-Scholes formula for a European call option is kept.

May I please ask what's the logic behind this? What does each term in the formula actually mean? The answer gave some explanation but I still do not fully understand it.

Thanks a lot for helping out!

 

[Steve Hales]

When a call option is exercised, two things happen: the investors pays the strike price (but only if the share price is greater than the strike price) and the investor receives the share (but, again, only if the share price is greater than the strike price).
The two terms of the Black-Scholes pricing formula quantify the values of these two legs of the transaction.
These two transactions can be isolated and considered separately. To prove this you can use the general risk-neutral pricing formula to value the options which pay out one unit of cash or one share depending on the final share price.
Hope that helps.

 

[yuli2513]

When a call option is exercised, two things happen: the investors pays the strike price (but only if the share price is greater than the strike price) and the investor receives the share (but, again, only if the share price is greater than the strike price).
The two terms of the Black-Scholes pricing formula quantify the values of these two legs of the transaction.
These two transactions can be isolated and considered separately. To prove this you can use the general risk-neutral pricing formula to value the options which pay out one unit of cash or one share depending on the final share price.
Hope that helps.

Hi Steve,

Thanks a lot for the answer. I understand it now.

 

[Bill SD]

When a call option is exercised, two things happen: the investors pays the strike price (but only if the share price is greater than the strike price) and the investor receives the share (but, again, only if the share price is greater than the strike price).
The two terms of the Black-Scholes pricing formula quantify the values of these two legs of the transaction.

Thanks Steve -have 2 follow-up questions to this historic answer.

Q1: Your answer helps me intuitively understand the discounting, St and K terms in the Black-scholes formula (which is the Garman-Kohlhagen with q=0). But still unsure what d1 and d2 intuitively represent?

Q2: In the practice Question which is the subject of this thread [Q15.2(i) pgs 32-33 solutions] why do we treat the future share price £8.59 (=£7.81*1.1) as the strike price rather than saying the strike price =0 (since the bonus scheme is free to managers) and the future share price (St) =8.59?

Similarly, why do we use only the second term (representing the strike price) to value the £10,000 cash bonus? According to your explanation this term represents what the investor pays but here the cash is free!

 

[Steve Hales]

Hi
Q1: d1 and d2 don't have any financial significance on their own. They're just intermediate calculations without any interpretation.

Q2: Think of the "strike price" as the location where the payout diagram bends - it's the point where the option moves from being in- to out-of-the-money. Whether or not the investor actually has to pay that is a separate issue. Usually (as in the case of the call and the put) those two concepts coincide, but sometimes we're just interested in the shape of the payoff diagram.