counterren
New Member
For cash or nothing, we use Nd1 and for asset or nothing, we use Nd2. I thought both option will exercise if end price>strike? Should both be ND2 which is the probability end price>strike?
cash or nothing and asset or nothing (why N(d1) and N(d2))
- Thread starter counterren
- Start date
Hi counterren,
It is true that N(d2) is risk-neutral probability option will expire in the money. Consequently, the intuition of a binary (digital) cash-or-nothing call = Q*exp(-rT)*N(d2) is pretty direct and elegant. I don't think i've ever been able to directly intuit the asset or nothing call, based on its use of N(d1) which is the option delta.
But it must be true, due to: a Euro call (c) = long asset-or-nothing + short cash-or-nothing
... the asset-or-nothing value is a consequence of its synthetic equivalent --> Euro call + cash-or-nothing
See discussion http://forum.bionicturtle.com/threa...t-or-nothing-binary-on-shares.3309/#post-9030
and here is the referenced spreadsheet which "proves:" https://public.sheet.zoho.com/public/btzoho/0517-binary
i.e.,
Since 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 hope that helps, thanks, David
It is true that N(d2) is risk-neutral probability option will expire in the money. Consequently, the intuition of a binary (digital) cash-or-nothing call = Q*exp(-rT)*N(d2) is pretty direct and elegant. I don't think i've ever been able to directly intuit the asset or nothing call, based on its use of N(d1) which is the option delta.
But it must be true, due to: a Euro call (c) = long asset-or-nothing + short cash-or-nothing
... the asset-or-nothing value is a consequence of its synthetic equivalent --> Euro call + cash-or-nothing
See discussion http://forum.bionicturtle.com/threa...t-or-nothing-binary-on-shares.3309/#post-9030
and here is the referenced spreadsheet which "proves:" https://public.sheet.zoho.com/public/btzoho/0517-binary
i.e.,
Since 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 hope that helps, thanks, David
Hi David,
In one of the FRM study materials, I came across a statement reflecting that "an asset-or-nothing call pays the value of the stock when the contract is initiated if the stock price ends up above the strike price at expiration", which I believe is not accurate since it would mean that the asset or nothing is a cash or nothing where Q=S0.
I appreciate your feedback on the above.
In one of the FRM study materials, I came across a statement reflecting that "an asset-or-nothing call pays the value of the stock when the contract is initiated if the stock price ends up above the strike price at expiration", which I believe is not accurate since it would mean that the asset or nothing is a cash or nothing where Q=S0.
I appreciate your feedback on the above.
Hi @patrickbs Yes, I do agree with you: if the binary (aka, digital) call option pays Q = S(0) then it is effectively a cash-or-nothing call option. An asset-or-nothing call pays the asset price at the (future) time of exercise, which is unknown (unknowable) at contract initiation (aka, purchase or grant). I do see how somebody can make this mistake: the present value of an asset-or-nothing call, per Hull, is given by S(0)*exp(-qT)*N(d1).
This formula might lead some readers to infer that the payoff is S(0). However, in the risk-neutral world, the present value of the expected future asset price is the current asset price; i.e. E[S(t)]*exp(-rT) = S(0); so the fact that S(0) values the option as a function of the current asset price is a reflection of the idea that the payoff is the future S(t). I somewhat describe this valuation mechanic (albeit indirectly via synthesis of the digital option) in this recent YouTube video, I hope that's hopeful:
This formula might lead some readers to infer that the payoff is S(0). However, in the risk-neutral world, the present value of the expected future asset price is the current asset price; i.e. E[S(t)]*exp(-rT) = S(0); so the fact that S(0) values the option as a function of the current asset price is a reflection of the idea that the payoff is the future S(t). I somewhat describe this valuation mechanic (albeit indirectly via synthesis of the digital option) in this recent YouTube video, I hope that's hopeful:
Similar threads
