Can any assumptions be made about the pay-off of an exotic option? For example, might we say the distribution of the pay-off a vanilla option would be Normal?I have built a valuation tool that estimates the price of a replicating delta-hedging strategy through Monte Carlo methods by trading the structure of a forward curve. It seems that a histogram of the pay-offs have two relative maximums. Can anyone explain this?Are options (/ real options) prices logNormally distributed, or does the standard assumption not hold given convexity?Thanks

You cannot make any useful general statements about the pay-off distribution of an exotic option. Locally, and away from critical points in time and space, the price path of any derivative that can be replicated via a dynamic trading strategy will inherit some behavioral characteristics from the underlying asset(s), such as continuity and normality in the case of an underlying asset with a price that undergoes a diffusion process.For alternative perspectives that may be more in line with the way your questions were formulated, please stay tuned for list.

Looks like a topic for Student?

Thanks guys. I'll stay tuned. Happy to share any other information

Notation: random variables in caps. Dummy args/realizations in small-caseThis is std. change-of-variable for a density. Given a density (say lognormal) [$]p_{S_T}(s)[$], what is the density [$]p_X(x)[$] of [$]X \equiv f(S_T)[$]? Work it out.Suppose [$]p_{S_T}(s)[$] is lognormal with a peak at 110, and [$]f(s) = \max(s-100,0)[$]. Then, you are going to find that [$]p_X(x)[$] looksroughly lognormal to the right of [$]x=0[$] with a peak at 10, plus it has a Dirac mass at 0. Perhaps these are your two peaks. For simple barriers,similar ideas apply; you will get smooth densities that end in Dirac masses where the barriers kick in. =================================================================================================p.s. Definitely student forum. If this all seems confusing, practice with this. Suppose [$]p_X(x)[$] is a Gaussian. Then what is the density of [$]Y = f(X) = X^2[$]? What is the transformation rule for general [$]y = f(x)[$]?

Hi Alan,Thanks for the advice - and thanks for the feed back regarding Students being more appropriate and the positive response despite this.Regarding the two questions (1. what is the p.d.f. of R.V. X=f(S)? 2. what is the pdf of Y=X^2?)... the best I can seem to do is literally substitute the new random variable into the standard pdf for that distribution. Is there supposed to be some additional step?

Hint: [$] p(x) \, dx = p(x(y)) \left| \frac{dx}{dy} \right| dy [$].A little googling about the key words and Jacobians should answer it all.

A long time ago (early 1980's?) Robert Merton did some published work on the return distribution of vanilla options, maybe you can find some ideas there. It would obviously matter a great deal if you are considering a single position or a portfolio.