binary options

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Th probability of an asset price being above the strike price at the matrutiy of an option is N(d2). Hull has used this to value cash or nothing call option...but uses N(d1) for an asset or nothing call option...

The only difference between these two options is what the holder would receive (either a fixed cash or the asset's price).If that's the case..why the probability measure changing from N(d2) to N(d1)..

Kindly advise

 

[David Harper CFA FRM]

Hi vinoth,

I can't seem to find an intuition for it (intuitively, i struggle with the issue that the discontinuity--the high possibility of zero payout isn't manifesting), however please note we can justify (as is often the case) with the synthetic equivalence that "a regular European call option is equivalent to a long position in an asset-or-nothing call and a short position in a cash-or-nothing call where the cash payoff in the cash-or-nothing call equals the strike price" (Hull). This statement (imo) is intuitive. Then it follows:
if c = S*exp(-qT)*N(d1) - Q*exp(-rT)*N(d2); the asset-or-nothing must be the first term.

In longhand:
As c = S*exp(-qT)*N(d1) - Q*exp(-rT)*N(d2)
And Since a Euro call (c) = long asset-or-nothing + short cash-or-nothing;
c = (asset or nothing) - (cash-or-nothing);
(asset-or-nothing) = c + cash-or-nothing;
asset-or-nothing = [S*exp(-qT)*N(d1) - Q*exp(-rT)*N(d2)] + [Q*exp(-rT)*N(d2)] = S*exp(-qT)*N(d1)

I cannot tell you that i intuitively see why this price reduces to discounted stock multiplied by delta [N(d1)] but i hope that helps, Thanks! David