Can someone show me how to replicate (S^a -K)+?

losemind · February 20th, 2008, 2:07 am

Hi,The payoff of the option is max(S_T^a - K, 0),where a is a constant. S_T is the stock price at maturity T. K is the constant strike price. My questions are:What is the smartest way (least amount of work) to price the above option?How to hedge for such option? Thanks a lot!

amit7ul · February 20th, 2008, 3:50 am

i guess, in BS framework(the lognormal assumption)if s_T is lognormal then Log(s_T) would be normal and also a.Log(s_T) also would be normal..so you can easily find the drifts and variance of underlying y=s_T^a..then use those to find an approximate price

daveangel · February 20th, 2008, 7:46 am

You can get an "exact" solution and I think Mr Wilmott's book has got it - if I remember correctly you have to adjust the drift rate by0.5 * a * (a-1) * (vol ^2)and the volatility gets scaled by a. I am assuming that a >= 1you plug the above into your BS model and bob's your uncle (in this case MR Wilmott)hth

amit7ul · February 20th, 2008, 8:03 am

ds=r.s.dt+v.s.dz where v*sqrt(T)=standard deviation of log(s_T)now i apply ito's lemma on y=a.log(s) and getdy=a/s ds + 0.5*(-a/s^2)*ds*dsdy=a.(r-0.5*v^2).dt + a.v.dzso my vol gets scaled by a(matching with davenagel) and drfit also gets scaled by a(earlier it was r-0.5*v^2 and now its a times that)drift is not matching with whats given in daveangel's post.

daveangel · February 20th, 2008, 8:33 am

substitute U = S^apayoff becomes max(U(T) - K,0) - standard call optionif dS/S = rdt + vol * dzapply ito to U = S^aand u get dU/U = (a*r + .5*a*(a-1)*vol^2) * dt + a*vol * dzI could be wrong (has happenned before!) and as I said its in Wilmott's book

WillK · February 20th, 2008, 9:13 am

amit7ul : Ur result is the same daveangel had, it's just a problem of notation : we indeed have dy=d(lnU) and you were so comparing the "drift term" of d(lnU) to the "drift term" of dU/U.In analogy with Black-Scholes notation, the drift refers rather to the "drift term" of dU/U.

losemind · February 21st, 2008, 2:06 am

Am I right in saying that for S^a, the drift doesn't matter at all, since in BS formulae the drift doesnot appear. So all I need to have is a new volatility, and plug the new volatility together with the old r (the original risk-free rate) in to the BS formulae and that should be the answer. It's different than what you got below.Why?

daveangel · February 21st, 2008, 7:46 am

QuoteAm I right in saying that for S^a, the drift doesn't matter at all, since in BS formulae the drift doesnot appearNO - you would be very wrong in saying that. The drift does appear in BS but it is the risk-neutral drift. As I have pointed out below, you need to adjust your carry r as follows:rp = ar + 0.5 * a * (a - 1) * vol * volwhere rp is the cost of carry for your power optionand the vol gets scaled by a.volp = a * vol