TonyC
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| Nuclear Energy Trader |
| Total Posts: 1048 |
| Joined: May 2004 |
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having a brain fart, i seem to remember that there was a simple relation twixt the price of eurpean options and the distribution of the asset price.
something like "the derivative of this black-scholes input with respect to that black-scholes input gives the probabliity" [or was it the cumulative probablility till that point?] . . . is it:
- derivative of the strike with respect to option's price?
or
- derivative of option's price with respect to strike?
or some other simple thing like that?
[i'm trying to determine the distribution implied by the observed smile] |
flaneur/boulevardier/remittance man/energy trader |
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baghead
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Gusak
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| Joined: Jul 2004 |
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if you have option prices for all strikes (say, through an interpolated implied vol function vol(K) ), you can extract implied probability (for asset distr. at time T =options' maturity) in the following form:
d^2[Call(T,vol(K),K)] /dK^2 = discountFactor(T) * p(K),
where p(K) is the prob. density function of the asset at time T. Which 'means' that
Prob( K<S(T)<K+dK) = p(K)*dK
So indeed, forgetting the disc. gactor it is second derivative of the option price wrt strike
You can see that this is true very easily:
the idea here is that the second derivative of the options price with respect to strike a limiting case of a very very tight butterfly. And such a tight butterfly will only pay out when spot ends up in a very tight interval given by the butterflies strikes.
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Gusak
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| Total Posts: 210 |
| Joined: Jul 2004 |
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ok, and if you are after the probability distrubution function, like Prob (S<K), then
it's as simple as the undiscounted digital price D(K),
or just a tight call spread.. |
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Martingale
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| NP House Mouse |
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| Joined: Jun 2004 |
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Another way to see this is
C(S,K) = E{ max (S_T-K, 0)} = \int f(S,S_T)max (S_T -K,0) dS_T
Take derivative twice with respect to K on both side, notice that second derivative of max (S_T-K, 0) with respect to K is just the Dirac-Delta funtion, so the RHS becomes the density f. |
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Graeme
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| This is the argument of Breeden and Litzenberger, who I think were the first to realise this result. The first few pages of their paper are actually well worth reading. |
Graeme West |
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TonyC
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| Nuclear Energy Trader |
| Total Posts: 1048 |
| Joined: May 2004 |
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thanks to all, and
> This is the argument of Breeden and Litzenberger, who I think were the first to realise this result.
BINGO! thats what my cobwebbed mind was remembering, some guy whose name began with an "L" . . .
> The first few pages of their paper are actually well worth reading.
what paper was that again? |
flaneur/boulevardier/remittance man/energy trader |
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M1kz
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| Total Posts: 121 |
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Given what Martingale wrote below, I guess the paper is their classic one, i.e.:
Breeden, Douglas T & Litzenberger, Robert H, 1978.
"Prices of State-contingent Claims Implicit in Option Prices," Journal of Business, University of Chicago Press, vol. 51(4), pages 621-51.
I got a pdf-version available, email me if you want it. |
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jungle
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| Chief Rhythm OfficerCSD LLC |
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apine
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| Total Posts: 963 |
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M1kz,
if possible pls post the article. i was going to email you, but thought others might want it, too. |
People can't stand two things - randomness and responsibility. -- Art Cashin |
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AVt
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| Total Posts: 720 |
| Joined: Jun 2004 |
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May be it is worth to remember, that the RND is not the asset distribution
because then one would get all prices from history (besides scaling). |
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jungle
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| Chief Rhythm OfficerCSD LLC |
| Total Posts: 2939 |
| Joined: Jul 2004 |
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Attached File: JoB 51.pdf
breeden and litzenberger attached. |
think about shoes, inhale...okay, think about cool grey fabrics, exhale....better? |
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apine
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| thank you. |
People can't stand two things - randomness and responsibility. -- Art Cashin |
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