If you have a basket option where the basket consist of 10 assets. Lets now assume that the payoff on this Basket is defined as:max(K,w*((p_t/p_0 - 1))K = strike price, p_t is the price of each of the asset at the maturity of the option, p_0 if the value of the underlying at time 0 and w is a vector of weights. Lets also assume that we will price this option using MC in the BS-framework.The question is as follows: Is it the Asset price p that are being simulated or is it the return that are being simulated? (I would think it should be the asset as BS assumes that p is lognormal which is not the same as saying that the return is lognormal - also the correlation derived from time-series data will be different in these cases)Thanks beforehand

Yes it is the asset price which is lognormal and the return which is (approx) normal

Ok - and that means that you will simulate the asset price and from there figure out the payoff given the payoff definition, is that right?

Yes, that's right. Don't ask me about hedging it however, or about sensitivity to correlation assumptions etc!!

Thanks - ok I won't do that then. However, an additional question - I guess the result will be "very" wrong if you decided to simulate using returns instead of prices - or will it matter?

I guess you can choose any model that fits reality, but don't pass it off as a BS type model. Obviously only the BS model claims to allow a replication price. If you are modelling returns as lognormal with drift equal to the risk free rate you are probably committing a horrendous crime.

That is also my belief, glad you agree.In all cases no matter how the payoff is defined it seems to me that the only thing that is correct is to simulate the asset price - as it generally is the asset price that a process is assumed for. Only the concept of a risk free rate - would seem a bit strange if it was not an asse we were simulating.Looking forward to your comments..

It is usually best if you use the same model for asset X alone and the same asset X in the basket; otherwise your hedges tend to leak.To get the hedges, note that if you write a program to price the basket option, your price will be a function of the the 10 asset prices you feed the program: V = V(p1, p2, ..., pn)so if you run the same program with slighlty different prices, say, V = V(p1 + epsilon, p2, ..., pn) V = V(p1, p2+epsilon, ..., pn) V = V(p1, p2, ..., pn+epsilon)the differences between these bumped prices and the base case gives you the delta risk with respect to the assets. Similarly, there will be n individual volatilities fed to the pricing program; doing the bumps then gives you the ten vega risks.If you are using MC to price the basket option, you must ensure that the same random numbers (or, rather, same paths) are used in the base and perturbed cases or else noise will kill you.Because MC has inherent noise in computing the risks, it is often better to use some approximate method for pricing the basket option, to get smoother & better hedges

HiMaybe I can add a word in defense of returns.Why can't you model the returns?? Following BS they are (in the limit) normaly distributed, so you run a path where you sample at small time intervals the returns r_i. Then, you take their sum say R. Of course, the final price will be S_t=S_0(1+R). This is approximate using exp(x)-1 almost equal to x. So there you go. Of course in BS there is no point in doing so, and in fact, there is no point in simulating the path of a price since the SDE has a closed form solution with an exponential (without forgetting -sigma^2/2).My point: In spirit the data consisting of returns over a path of the price implies its final price... (definitely true in the limit. after all that is how one sees the BS sde: dS/S=...) Anyway, I tried a quick simulation with BS modelling the returns and I get close to the BS price.Of course, when one calculates R, one needs all the returns, while often (Markov stuff???) one can usually forget the past values of the price as we progress in finite time steps until maturity.I can't claim I know what I am talking about... there are probably some issues about the approx above and the large sum (when the partition is small) maybe the simulation is not robust (in whatever sense). zq

To be clear, I have nothing against modelling returns. Just that if you use lognormal returns you are not in BS world.

Sorry....my bad...I guess one way to see that they can't be log-normal (in any reasonable model not just BS) is that returns may (unfortunately) be negative.zq

I'm a bit confused by this discussion. The returns are just the asset prices normalized to an initial price of $1, with 1 subtracted. Modeling returns or modeling prices is the same thing, there is a trivial correspondence.For estimation purposes, you almost always want to use historical returns to set parameters. Return distributions are usually much more independent and stable than price distributions. Also, future returns are generally the things to simulate. Prices in finance are generally irrelevant, you care what happens to $1,000 invested in stock A, not how much one share of stock A goes up or down in price. Moreover, prices have complex distributions, returns are often close to independent and identically distributed Normal, or can be transformed to that.Your basket return is just the return on a portfolio holding w_i/p0_i of asset i. The straightforward approach is to compute the historical values of this portfolio (if possible, or proxy them if not), measure the historical distribution of log return of that portfolio, and plug those parameters into Black-Scholes or whatever model you use.