I'm not sure outrun's formula works, but not for that reason.If you follow through my method 1, you can see the value function for delta is[$]delta(S_0) = \frac{e^{-r T}}{S_0} E_0[S_T \, 1_{\{S_T > K\}}][$],which seems different to me than his prescription. His prescription, if I am reading it right, will give e^(-r T) Phi(d2(S0)), which is not delta. You have to differentiate the backwards variable, not the forward variable.

QuoteOriginally posted by: AlanI'm not sure outrun's formula works, but not for that reason.If you follow through my method 1, the value function associated to delta is[$]delta(S_0) = \frac{e^{-r T}}{S_0} E_0[S_T \, 1_{\{S_T > K\}}][$],which seems different to me than his prescription. His prescription, if I am reading it right, will give e^(-r T) Phi(d2(S0)), which is not deltaJust looking in Joerg K/Daniel W's book page 416. They discuss Greeks using Malliavan Calculus. For a plain BS option the delta is the discounted payoff multiplied by a score function within the Expectation function. In this case it is given bys / (S(0) sig T^1/2)where s = Z_N == Gaussian variate used to generate S(T) = S(t_N). It's like an integration by parts, it does look similar. "The interest of this approach was to avoid the differentiation of the payoff functionin the simulation process."

So it seems the method you have mentioned above is the "Liklihood Ratio" method. As per Glasserman and what outrun says, it doesn't matter that the payoff is not differentiable at S=K since the probability is zero. I really find it amazing that nobody has written a simple accessible introductory text about Monte-Carlo methods in finance. That usually makes me believe that not many people properly understand it. In which case that also makes me think, what do people in banks use? Anyway, the likelihood ratio method as above seems to reply on having an analytical solution for the underlying. What if you don't have that?

I suspect all 3 solns I wrote work for any model that is (conditionally) translationally invariant in X=log S.Many equity models will satisfy this, including most stoch. vol. models, exp Levy models, etc.[The formula with Phi(d2) should then be interpretted as Prob(ST > K) in that case].My advice is that you should try to confirm all 3 solns by doing a MC for -BS-Heston '93-any exp Levy model-any other stoch. vol. model with the needed invariance (say the GARCH diffusion)In the first 3 cases, you can compare with an analytic delta. The last one you could compare with,let's say, a pde soln.Then, having done all that, you can write your text ==========================================================================p.s. The discussion at eqn (3.12) and below in my article here has some relevance to this.All the types of models I mention have the same structure as (3.12) in their option value formula.

QuoteI really find it amazing that nobody has written a simple accessible introductory text about Monte-Carlo methods in finance. Nick Webber has IMO. With VBA code. And the numerical analysis of MC is underdeveloped in general.And the MC was first applied to finance in 1977 (Phelim Boyle).

I looked through the contents of the book and what I could see in preview and I couldn't find any discussion of computing sensitivities, do you have a page number?

The book does have 1 exercise, it is still a good intro to MC. K&W have a chapter as mentioned. Or you could try Malliavan calculus if you are feeling ambitious.

I'm a newb. One step at a time. I just found it amazing that you can calculate an option price very easily using MC, as soon as you ask the next question "What about the greeks?" then you reach the forefront of quant finance knowledge so quickly. In physics it took me a year before I realised what the "edge of knowledge" on my PhD topic was

QuoteThe most basic approach is to price an option with MC and then price it again with MC but then do it for S+h and S-h to get the delta/gamma. That's what my 5 year old son would do.Doesn't mean it's correct This approach is probably not even wrong, mathematically speaking.Hand-waving might be used but I can't see how you reconcile the fact that Wiener curve is continuous everywhere and differentiable nowhereSo the S+h and S-h would be an issue but I can be corrected by our resident stochastics experts. It's a can of worms looking for an optimal size h. Numerical Differentiation 101. Kienitz and Wetterau have a chapter on all this stuff. MC probably wasn't built for this kinds of computation?

QuoteOriginally posted by: CuchulainnQuoteThe most basic approach is to price an option with MC and then price it again with MC but then do it for S+h and S-h to get the delta/gamma. That's what my 5 year old son would do.Doesn't mean it's correct This approach is probably not even wrong, mathematically speaking.Hand-waving might be used but I can't see how you reconcile the fact that Wiener curve is continuous everywhere and differentiable like this nowhereSo the S+h and S-h would be an issue but I can be corrected by our resident stochastics experts. It's a can of worms looking for an optimal size h. Numerical Differentiation 101. Kienitz and Wetterau have a chapter on all this stuff. MC probably wasn't built for this kinds of computation?

Peter Jaeckel: greeks for MC In particular, see the fun with path recycling! More scary - but not unsurprising - stuffQuoteFor Gamma, the situation is even worse. In this case,again, we are essentially averaging over a sequence ofzeros and ones,4 only this time they are not resulting fromthe path pair or trio terminating in the money or out of themoney (i.e. as if we sampled a Heavyside function), butinstead the value one is only returned if the terminatingspot levels of the path trio straddle the strike of the option.In other words, the calculation of Gamma by explicit finitedifferencing for options with payoff functions that exhibit akink anywhere is equivalent to carrying out a Monte Carlosampling computation over a Dirac spike, i.e. a non-zerovalue is only ever obtained if the spot value at maturity isright at the strike.

QuoteOriginally posted by: barnyAs per Glasserman and what outrun says, it doesn't matter that the payoff is not differentiable at S=K since the probability is zero. But the context IMO is calculus and not probability and computing greeks at K??? Sets with measure zero are not the issue here.The discontinuity in the payoff affects the computation of delta and gamma.

QuoteOriginally posted by: outrunThat not the forefront really. :DThe most basic approach is to price an option with MC and then price it again with MC but then do it for S+h and S-h to get the delta/gamma. That's what my 5 year old son would do.Doing that you'll see that there is a lot of sample noise, and that it can be reduced by re-using the sample random numbers so that scenarios will be similar. ..That's what I've seen at risk departments at banks. Banks also use the same technique to reduce day-to-day variation in the VAR. With limited number of MC paths the sample noise will be big, and since VAR is just an imprecise measure or risk anyways with little absolute meaning, people are more interest in changes in VAR.But as we've seen the "bump and revalue" is essentially wrong. It depends in a non-trivial way on your delta S, too small causes problems and too big also. And then there are the problems that Cuch mentions written about by Glasserman, Jaeckel etc.