I'm currently reading Mark Joshi's "The concepts and practice of mathematical finance" and I like it very much but there are still some questionsIn the beginning part of the book, differentiation is handled quite strangely. For example, in chapter 4 there is a section on delta of an option.Differentiating the B-S formula w.r.t to spot price the book states thatBut this formula doesn't really show anything. As d1 is dependent on S Well, OK, if we consider C(S,d1,t) and treat them as independent it's OK. But right in the next paragraph a gamma is derivedand here we do use the dependence of d1 on S. From this formula it immediately follows that B-S price is convex. The theorem is right but I just don't get the derivation. Am I missing something here?P.S. And anyway I wanted to say "thank you" for the book to mark

In the derivation of the delta some steps are left to the reader. You have to proceed as you have done and use the relation thatThe above relation is found using the fact that N' is the standard normal density.

here's the general method for deriving greeks from the BS formula. I assume r=0 and t=0 for simplicity; you can put it back in by replacing my K with Ke^-rT and T with T-t. Take the derivative of C with respect to a general quantity x, using chain & product rules...Next bit is the "aha" moment; you notice that (look at the normal density & do a bit of algebra). This means that in your expression below you are correct but you haven't spotted that the second & third terms cancel to zero. Noting also that you can reshuffle the equation and getIf we then specify x=S, the delta expression comes from the first term and the others evaluate to zero. you can set x=sigma, T or r to compute other first-order greeks.As for the gamma formula, it just comes from differentiating the delta again, like this i'm confused by your comment "this formula doesn't show anything".. it absolutely does...

Yeah, my formula for Delta is right. I just didn't note the SN'(d1) = K*exp(-r(T-t))N'(d2) and then second and third term just cancel out. As for gamma the problem wasn't in differentiation of N(d1) I was thinking about those extra terms. But all is clear now, my fault Thanks for clarification... And the point about strangeness was about informal delta hedging arguments when we ignore the derivative of the hedge w.r.t to S but there also is a note that it will be justified in next chapters - and it is so I just confused things a bit.

One good way to confirm if your algebra is correct is to compare it with a simple central difference approximation

OK i was a bit lazy and left the derivation to the reader.there are many ways to do it and we had another thread on the topic already.I am actually putting a chapter on this in the sequel.

Hi Mark, any update on the progress with the sequel? Really looking forward to it, hope it will be published soon!

i havent made a huge amount of progress lately but hope to soon... I have made some progress on some extra chapters for my C++ book, however.

Hi,In the book, for the question 2.43 I would answer that the Vega is negative for K_1<S<K_2 and positive otherwise (the option pays 1 at maturity if the underlying S verifies K_1<S<K_2). Basically you are long volatility when you are OTM and short volatility otherwise...ps: This is approximative since the Vega changes of sign not exactly at K_1, K_2 for a non-normal model.