wilmott.com

We would come across this formula almost in every other bookσloc=(2*∂C/∂T/(K^2*∂2C/∂K2))^.5My question is:1. Is σloc a. σloc(K,T) or b. σloc(S(t),t). And the so called local vol surface that is created by using this formula does it have strike as x axis? Or is it spot?2. If I were to price an option using monte carlo with the help of this local vol surface, how do I go about it ?

QuoteOriginally posted by: kooladWe would come across this formula almost in every other bookσloc=(2*∂C/∂T/(K^2*∂2C/∂K2))^.5My question is:1. Is σloc a. σloc(K,T) or b. σloc(S(t),t). And the so called local vol surface that is created by using this formula does it have strike as x axis? Or is it spot?2. If I were to price an option using monte carlo with the help of this local vol surface, how do I go about it ?1. It is (K,T) in the first formula you posted, but you should think of them as "dummys"; once you have sigloc(K,T) all K> 0 < T < Tmax, you have sigloc(S(t),t) needed for your next question.2. Option values for any fixed maturity T <= Tmax are found by simulating dS(t) = sigloc(S(t),t) S(t) dW(t) by the Euler method:(a) S= S(0), t= 0; then repetition of S += sigloc(S,t) S Z sqrt(dt); t += dt, where Z ~ N(0,1), until you achieve S(T). (That's one trial)(b) averaging w(S(T)), where w( ) is the option payoff function, over N simulation trials. p.s. The same method will work if the option value is the expectation of any path functional of {S(t)}_(0 <= t <= T),which covers just about everything. However, that doesn't mean those values are necessarily any good, even if theplain vanilla option values are matched to a market.

Thanks Alan..What vol do you use at t=0?

I would just extrapolate the source market's at-the-money implied vols to T=0. (one number)

Your scheme has no drift. does the local vol in your specification include all the information about transition probabilities and hence you don't need the drift anymore (the implied distribution is already shifted)?