tag:blogger.com,1999:blog-7905515.post2952171509901352950..comments2024-03-14T11:09:32.759-05:00Comments on Falkenblog: The Easiest Way to Derive Black-ScholesEric Falkensteinhttp://www.blogger.com/profile/07243687157322033496noreply@blogger.comBlogger6125tag:blogger.com,1999:blog-7905515.post-5039797845481742542013-02-12T18:18:14.611-06:002013-02-12T18:18:14.611-06:00Eric,
I never cared for the risk neutral version ...Eric,<br /><br />I never cared for the risk neutral version of this proof. If you assume log-normal with mean mu then you can use it to get the expected value of both the stock and call option in terms of mu:<br /><br />stock = f(mu)<br />call = g(mu)<br /><br />Then use the first of these to eliminate the mu in the second. Thus the call is a function of the stock price and not mu.Entsophynoreply@blogger.comtag:blogger.com,1999:blog-7905515.post-87605895096608958932013-02-12T12:39:31.082-06:002013-02-12T12:39:31.082-06:00Ed did figure out the basic formula, but he didn&#...Ed did figure out the basic formula, but he didn't have the risk-neutrality point, if I'm not mistaken. More importantly, Ed hasn't suffered too much, he's done quite well.Eric Falkensteinhttps://www.blogger.com/profile/07243687157322033496noreply@blogger.comtag:blogger.com,1999:blog-7905515.post-82650627742434464802013-02-12T12:18:17.563-06:002013-02-12T12:18:17.563-06:00Ed Thorpe was generous enough to teach one of thes...Ed Thorpe was generous enough to teach one of these two eggheads how it was done, they then spit shined it with the requisite academic anti-realism varnish, and became famous for it. Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7905515.post-70991943224969999052013-02-12T08:14:35.752-06:002013-02-12T08:14:35.752-06:00Stephane: It isn't obvious that
E[1(Exercise...Stephane: It isn't obvious that <br /><br />E[1(Exercise)] = Q(exercise) = N(d-)<br /><br />d- = (ln(F/K) - sigma² * T/2)/sigma/sqrt(T)<br /><br />Eric Falkensteinhttps://www.blogger.com/profile/07243687157322033496noreply@blogger.comtag:blogger.com,1999:blog-7905515.post-10497085911516939752013-02-12T05:51:43.351-06:002013-02-12T05:51:43.351-06:00There is also what I like to call "the French...There is also what I like to call "the French way", which is actually a probabilistic approach to solving the equation. It might seem pedantic but is quite short and gives physical and meaningfull insight to the terms of the equation :<br /><br />C = discount * E[(S-K)+] = discount * E[(S-K) * 1(Exercise)]<br />Let's split the S and K terms :<br /><br />C = discount * (E[S*1(exercise)] - K * E[1(Exercise)])<br />Where expectations are taken under the Q probability.<br />The second term is easy<br />E[1(Exercise)] = Q(exercise) = N(d-)<br />where d- = (ln(F/K) - sigma² * T/2)/sigma/sqrt(T)<br />where F is the forward price.<br /><br />I really do prefer ln(F/K) to ln(S(0)/K)+rT since we understand where the r comes from, and it easy to adapt the formula to dividends, repo, funding conditions, ... This 'r' is essentially different from the one in the discount term, which is related to the heger funding. The 'r' in d+/- is the risk neutral drift of the spot price.<br /><br />Let work on the first term, slightly more difficult : E[S*1(exercise)]<br />Using a change of numéraire, we can choose S itself as numeraire, and Q(s) the associated measure. Then :<br />E(Q) [S(T)*1(exercise)] = F. E(Qs)[ 1(exercise)] = F.Qs(exercise)<br /><br />Where F is the forward. Usig Girsanov theorem, under Qs the drift of S is r+sigma²/2, hence :<br />Qs(exercise) = N(d+)<br />Where d+ = (ln(F/K) + sigma² * T/2)/sigma/sqrt(T)<br /><br />So now we can write that :<br /><br />C = discount *(F * N(d+) – K N(d-))<br />d+/- = (ln(F/K) +/- sigma² * T/2)/sigma/sqrt(T)<br /><br /><br /><br /><br /><br />Stephane Mysonahttps://www.blogger.com/profile/13160898160306615349noreply@blogger.comtag:blogger.com,1999:blog-7905515.post-61927623304633266002013-02-12T04:16:11.943-06:002013-02-12T04:16:11.943-06:00This comment has been removed by the author.Stephane Mysonahttps://www.blogger.com/profile/13160898160306615349noreply@blogger.com