Hi,Could someone explain me how, from options' market prices, we can derive the implied density of the underlying. Assume that I observe call(K,T) for some K and T. If I use BS framework I can invert and find implied volatility. This part is ok, I understand.But I don't see how I could draw the implied distribution for the underlying. I guess I would obtain points and after I have to fit some parametric density smoother. But how can I find these points ?ThankDavid

U want to look at Wilmotts book or Dupires work .. its well known. Basically, u can show that the implied pdf is the second derivatives of C(K,T) with respect to K... or more practically u can price up the butterflies and get the implied density

Could you go further with the butterfly ! please

well a butterfly is basically when u are short 2 calls (say) of strike K, and long one call strike K-h and long another call strike K+hthereforebutterfly = C(K-h) + C(K+h) - 2C(K) = C(K) - C'(K)h + .5 *C''(K)*h^2 + C(K) + C'(K)h + .5*C''(K)*h^2-2C(K) = C''(K)*h^2and if I remember correctly the implied pdf isp(St,T) = exp(rT)*C''(K)

Corrado-Su modelImplied volatility skews and stock return skewness and kurtosis implied by stock option priceshttp://www.bus.miami.edu/~tsu/ejf97.pdfhttp:// ... el.xlshave fun!!

The butterfly result quadrature refers to is by Breeden and Litzenberger. However, you probably don't want to look at the original paper (I have a pdf if you REALLY want it, but it is massive: original paper version photographed for archive, presumably). However, the result we are interested is Appendix 15A of Hull. A lot of B&L is about other stuff (not relevant here). Moreover, the B&L discussion is discrete, in Hull it is continuous, and so, neat.Importantly, the result is assuming a CONTINUUM of option prices for the given expiry. (In B&L, the assumption is that there are option prices for every strike, in $0.01 steps.) In reality, we only have a discrete finite set. Thus, we need to interpolate to 'complete' the continuum. For this interpolation: it seems one can choose, but I quite like cubic spline on premium (makes the twice differentiation do'able in code). Because the market is incomplete (there isn't a continuum) it is best not to mortgage your house based on the results - it's just a model.

One more intuition behind the relationship between butterfly prices, and implied pdf is considering the A-D state prices . Butterfly payoff and state price density are both proporional to the probability density of attaining the state, hence butterfly prices can be used to calculate the pdf.(Incidentally, I was also last week working on implied pdf . I tried lognormal mixture of 2 densities, and it seemed to reproduce the volatility smile pretty well for MSFT.)

tx to all

QuoteOriginally posted by: bagheadCorrado-Su modelImplied volatility skews and stock return skewness and kurtosis implied by stock option priceshttp://www.bus.miami.edu/~tsu/ejf97.pdfhttp:// ... el.xlshave fun!!Does anyone have the paper they wrote in 1994 I think ? Because this one which is quite useful is the empirical testing of their original one. Or put another way: Is it worth reading the first one ?