HiReading and studying derivatives textbooks i often find, not well explained, the concepts of marginal and conditional density for pricing options. Besides formulas, which is the real meaning and importance of that in the general pricing framework in a smile environment?Thank you

I think marginal and conditional densities mean the same thing, they mean that we have to keep some variables constant and integrate the pdf wrt the other variables over the entire domain, something like P(-inf<x1<inf and -inf<x2<inf|x3=k3 and x4=k4), so we have marginal distribution as some function of (k3,k4). Maybe in a smile environment, say you are also specifying a SDE for the variance along with the price, the marginal distributions for price can be used for computing confidence intervals for trading (See i am also a student. maybe I am imagining things)

QuoteOriginally posted by: mrblueHiReading and studying derivatives textbooks i often find, not well explained, the concepts of marginal and conditional density for pricing options. Besides formulas, which is the real meaning and importance of that in the general pricing framework in a smile environment?Thank youTake 3 month SPX options, as an example. Learn about the Breeden-Litzenberger relation, which is a way of deducing p(S | S0),the risk-neutral pricing density for options, from today's option chain. If you know theDirac Delta, you will find it trivial -- see my p.s. for the details.Technically, p(S | S0) is a conditional density, conditional on the current index level S0 = 804.But this kind of conditioning is also trivial and not my point. Let a year pass and suppose S0 = 804 again. You are ready to buy a 3 month call and you takep(S | S0) from today (Apr 1, 2009) and compute (*) c(K) = integral Max[S-K,0] p(S | S0) dSI ignore interest rate discounting for simplicity of the argument. Assume rates = 0 in both cases. You find the resulting prices from (*) are way off compared to the market. What went wrong?What was wrong is that you should have really written p(S | S0, todays-smile)for the density you deduced on Apr. 1, 2009. The smile on Apr 1, 2010 is quitedifferent and that's why (*) failed. p(S | S0, todays-smile) is the transition density conditional on today's price and today's smile. If the Apr 1, 2010 smile is much lower (we all hope!), then the market prices for c(K) then will be much lower than those computed from (*) with the Apr 1, 2009 density.In stochastic process models, we write p(S | F0), where F0 = "state-of-the-world" today.It is the same idea. F0, technically called the filtration, is simply a collection of thevariables you are conditioning on. Failure to note this kind of conditioning is the source of egregious errors, suchas using local volatility to price exotic options. =========================================================p.s. As I said, if you know the Dirac Delta, you will see that takingtwo K derivatives of (*) gets you the density p( ) from the option chain.You don't need the whole chain, just the options expiring at the target maturity, 3 months for the example. You will need to smooth the option prices (and extrapolate) to include all possible strikes.

Alan,Great post. Thanks!Can you recommend some literature (your own?) that would give some more detail and workings on this topic, especially on the application of the Breeden-Litzenberger relation and the connection to the Dirac Delta?Many thanks,