Looking at the formula, it appears to me that the normal distribution is modeling a distribution of prices. In fact, this is explicitly stated here - http://bradley.bradley.edu/~arr/bsm/pg04.html - in assumption six: "returns on the underlying stock are normally distributed".
So the distribution is the model for the price of the underlying. On Tue, Apr 17, 2012 at 4:31 PM, KM Chakahwata <kmchakahw...@first-derivative.com> wrote: > sorry, I meant "expected option price", as of valuation date. for call, this > would be Expected(S-K), which is the same as Expected(S)-K > > enjoy > ken > > -----Original Message----- > From: programming-boun...@jsoftware.com > [mailto:programming-boun...@jsoftware.com] On Behalf Of KM Chakahwata > Sent: 17 April 2012 20:39 > To: 'Programming forum' > Subject: Re: [Jprogramming] black-scholes and levy distribution > > the special form of Black-Scholes equation is based on the root stochastic > differential equation that, through a series of so-called risk-neutral > arguments leads to stock (or underlying) prices being lognormal. then we > just find the present value (using risk free rate) of the expected stock > price at expiration -- where the expectation is with respect to the > lognormal distribution -- or normal when appropriate transformations are > made from lognormal to normal (for mean and variance). > > now, given this, i dont know whether one can simply replace the cummulative > distribution function "N" in BS formula with the appropriate Levy > equivalent. i dont know that much about Levy, but i would be very pleasantly > surprised if this is indeed the case -- but somehow i doubt it. i suspect > you may have to go back to first principles and actually integrate the > expectation equation somehow, assuming the risk neutral arguments still > apply. > > anyone know for sure? > > enjoy > ken > > -----Original Message----- > From: programming-boun...@jsoftware.com > [mailto:programming-boun...@jsoftware.com] On Behalf Of Devon McCormick > Sent: 17 April 2012 19:51 > To: Programming forum > Subject: Re: [Jprogramming] black-scholes and levy distribution > > Raul - > > I've thought about this. Basically, you want to replace "N" (normal > distribution) with "L" (Levy distribution) in the formula. This would > require changing the "cnd" verb in McDonnel's essay to a "cld" > (cumulative Levy distribution) verb, which gets right to the heart of > the most complex piece of that code. > > I guess you've figured out how to write the Levy distribution verb? > This is the crucial thing to get right - here's some graphs of it: > http://en.wikipedia.org/wiki/L%C3%A9vy_distribution . > > What is "levychar"? Do you have some examples of using these verbs? > > Regards, > > Devon > > On Tue, Apr 17, 2012 at 2:04 PM, Raul Miller <rauldmil...@gmail.com> wrote: >> I was reading > http://triplehelixblog.com/2012/04/fractal-finance-a-rogue-mathematician%E2% > 80%99s-search-for-answers/ >> and then I was reading wikipedia's writeup on the levy distribution >> (http://en.wikipedia.org/wiki/L%C3%A9vy_distribution) and then I was >> poking around on jsoftware.com to find an implementation of erfc >> >> That gets me to here: >> >> require 'stats/distribs' >> erfc=: erfc_pdistribs_ >> >> NB. m: location parameter (domain: y > m) >> NB. n: scale parameter >> levypdf=:2 :0 >> (%: n%o.2) * ^@(n % 2 * m - ]) % 1.5 ^~ m -~ ] >> ) >> >> levydist=:2 :0 >> erfc@%:@(n % 2 * m -~ ]) >> ) >> >> NB. no graph of this one -- it's complex -- not sure how to detect >> stupid mistakes >> levychar=:2 :0 >> ^@((0j1*m)&* - 0j_2 %:@*n*]) >> ) >> >> But I noticed this writeup on black-scholes: >> http://www.jsoftware.com/papers/play193.htm and I got to wondering how >> that would be rephrased if it used the assumptions that lead to the >> levy distribution. >> >> Does anyone know how to approach this problem? >> >> Thanks, >> >> -- >> Raul >> ---------------------------------------------------------------------- >> For information about J forums see http://www.jsoftware.com/forums.htm > > > > -- > Devon McCormick, CFA > ^me^ at acm. > org is my > preferred e-mail > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm -- Devon McCormick, CFA ^me^ at acm. org is my preferred e-mail ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
