I am interested in the theoretical considerations regarding expected values of (Black76/Black-Scholes) option values.If I was clever enough I would start by taking the risk-neutral expectation of the option value (which itself is a risk-neutral expectation due to the FAPF) and see where this brings me. I am expecting a change from risk-neutral to some T-forward measure and on this point I wouldn't be surprised if this is similar to a Swaption (which is an option on a derivative). I give these anecdotes merely to express my best guesses in the hope of geting some advice. Would be very cool to do this from first principles. If someone has done this before I would appreciate some guidance/starting points. If there is some research you can point to even better.thanks g

[$]E[E[x]] = E[x][$]

Of course thank you! My question is actually more around the correct approach to calculate the n-th percentile of the Potential Future Exposure (PFE) of an option during its life.For the sake of completeness I will repost the question.Thanks again

Explain exactly how to calculate your percentile for the underlying stock and I'll tell you how for the option.

Herewith my full approachAssume GBM under real world measure for the (Gold) priceSolve and simulate exact 95-th percentileAt future dates calculate Black76. Few subtleties here. We are simulating spot but we want forward gold, so I just use today's discount factors to forward value the simualted spot until maturityFinallly, for PFE calculate the maximum of these future option valuesI have very little guidance and don't have an exhaustive audience to critique me. Please feel free to do so. thanks

What you are trying to do is still somewhat vague to me. However, given what you have said, a few comments:1. GBM for the underlying, means the underlying S(t) (spot) has a lognormal distribution. You can calculate the percentile points exactly.2. If the forward price F(t) = multiplier S(t), which sound like your model, where the multiplier is just some constant,then F(t) also has a lognormal distribution with exactly known percentile points.3. If your model is that the option price in the future at t, call it C(t), is given bythe Black formula C(t) = C_{Black}(F(t)) with other parameters known, then the distributionof C(t) can be worked out. This is simply a change of variable from one with a known distribution X=F(t)to a new random variable Y = Y(X), where Y = C(t). The transformation is monotone, so one value of Y corresponds to each X. I am going to discuss it now using X and Y. Just remember that X is the forward price, andY is the option price.The density of X is given by [$]p_X(x) dx = p_X(x) |dx/dy| dy[$], so the density of Y is[$] p_Y(y) = p_X(x(y)) |dx/dy|[$]. In your case, [$]p_X[$] is lognormal, [$]x(y)[$] is the inverse of the Black formula, and [$]|dx/dy][$] is the reciprocal of the Delta of the option. The bottom line is that a nice smooth density for the future option value C(t) could be worked out (say in Mathematica),and the 95th percentile of that could be discovered, all without doing simulations. (`Smooth', of course, presumes t < T, the expiration)=============================================================p.s. I will add that inverting the Black formula for the forward, given the option price, is no harder thaninverting it to get the vol. given the option price. So, if you know how to get implied vols. you know how toget the [$]x(y)[$] needed above. Both require a simple root finder.

Thanks very much. You are right in the money for most, however not sure why you add the bit about inverting.Just to reiterate, all I want to do is calculate "PFE for the life of the option". I am curious if what I am doing is grossly incorrect or not too unreasonable, or spot-on would be nice!Unfortunately my Stats is very rusty. Could you explain how you go from the density of X to that of Y?Thanks, very insightful!

You're welcomeRe your first question, you said"Finally, for PFE calculate the maximum of these future option values".This is quite vague. What are the forwards that produce this list of option values? Are they any simulated forward values above a 95th percentile point?If so, then (for a call option) your maximum will tend to [$]+\infty[$] (as the simulation size increases) since arbitrarily large forward values will ultimately be drawn. Since [$]+\infty[$] seems unlikely to be the PFE, your procedure may indeed be grossly incorrect. So, what's missing is that you need to: (i) precisely define the option PFE, and then(ii) precisely explain what you have done to estimate it.Re changing variables, this is standard.For example if X is distributed std normal, what is the density for f(X) = X^3, etc?What if the density of X is just some unspecified p(x)? Think about it, work examples, Wikipedia, student forum, etc.

Thanks.Regarding the forwards that I use:What I have done is assumed GBM for the spot and at future dates I simulate the exact value at a 95th percentile. Then, a bit naively, I use discount factors observed at [$]T=0[$] (today) and I "accrue" my simulated spot to the maturity of the option [$]T[$]. I.e [$]F(t)=S^{95th}(t)\frac{D(0,t)}{D(0,T)}[$]This is the forward that I "plug" into Black76 at each date [$]t[$]. I repeat this for a few discrete steps until [$]T[$], and then simply take the maximum of all these Black76 values as my PFE.Trust this answers (i) and (ii).thanks again

I see. Since what you are doing seems to agree with Duffie, here(http://www.darrellduffie.com/uploads/su ... ro2004.pdf)it seems fine.My only quibble is that I don't see the need to 'simulate' [$]S_{95}(t)[$] as this is easily deduced from the corresponding critical pt of the normal distribution. That is, if [$]X[$] is normally distributed andhas [$]\alpha_1[$] as the [$]95\%[$] percentile point, and [$]Y = e^X[$], then the [$]95\%[$] percentile point for the [$]Y[$]-distribution is [$]\alpha_2 = e^{\alpha_1}[$].For the same reason, now that I understand what you are doing, you can disregard my`change-of-variables' commentary, as this is not needed.

Wonderful.Thanks for the reference.Regarding your quibble; Rest assured, I have applied Itô and solved the GBM such that I have a closed form for my n-th!Thank you very much for the help.