There is a statement in John Hull's book Options, Futures and Other Derivatives 9th page 633 for the relation between implied volatility function (IVF) and implied distribution of asset in future time.

When it is used in practice the IVF model is recalibrated daily to the prices of plain vanilla options. It is a tool to price exotic options consistently with plain vanilla options. As discussed in Chapter 20 plain vanilla options define the risk-neutral probability distribution of the asset price at all future times. It follows that the IVF model gets the risk-neutral probability distribution of the asset price at all future times correct. This means that options providing payoffs at just one time (e.g., all-or-nothing and asset-or-nothing options) are priced correctly by the IVF model. However, the model does not necessarily get the joint distribution of the asset price at two or more times correct. This means that exotic options such as compound options and barrier options may be priced incorrectly.

I can not understand that, IVF guarantees the model match the market price of vanilla option for all strike $K$ and all maturity $T.$ And the implied distribution of asset in future time is totally determined by the market price: $$ p(S^*,t^*;K,T) = e^{r(T - t^*)}\dfrac{\partial^2 V}{\partial K^2}.$$

Here, market value: $V,$ maturity $T,$ strike $K,$ spot price of asset: $S^*,$ current time: $t^*.$

Can anyone give me a clear explanation?

  • $\begingroup$ Could you please clarify what your question is. From you last few sentences, it seems like you want to know why an arbitrage free continuum of market prices of European options implies unique marginal distributions. However, the text that you cite is mostly about two models with the same marginal distributions having different transition densities and thus leading to different prices for path-dependent options. The latter is what @dm63 's answer addresses. $\endgroup$ – LocalVolatility Oct 17 '17 at 11:24
  • $\begingroup$ @LocalVolatility, Could I understand as we know $p(S^*,t^*;K_1,T_1)$ and $p(S^*,t^*;K_2,T_2),$ at time time $t^*$ but still don't know $p(K_1,T_1;K_2,T_2)?$ $\endgroup$ – A.Oreo Oct 17 '17 at 12:01
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    $\begingroup$ As explained in @dm63's answer: Let $X_1$ and $X_2$ denote two random variables of known distributions. Can you infer the distribution of $X_1/X_2$ from the available info ? No because you would require the joint pdf while you only have the marginals. This is exactly the same here, you know the distribution of the returns $X_1 = \ln(S_{T_1}/S_0)$ and $X_2 = \ln(S_{T_2}/S_0)$, but you cannot infer that of the forward return $X_2/X_1 = \ln(S_{T_2}/S_{T_1})$. $\endgroup$ – Quantuple Oct 17 '17 at 14:54
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    $\begingroup$ For your understanding of obtainign the pdf from the option prices, see this answer. $\endgroup$ – will Oct 17 '17 at 16:59

One example: even if you know the implied distribution at all future times T, you know nothing about the change in price between two future times T1 and T2. An exotic depending on the price change from T1 to T2 thus cannot be priced.

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  • $\begingroup$ That means we can not use IFV to price the path dependent exotics? $\endgroup$ – A.Oreo Oct 17 '17 at 12:07
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    $\begingroup$ Well it's the same with any model fitted on vanilla option prices I guess since these options do not integrate any information on the forward returns distribution (only unconditional returns). This means that you are then forced to trust the forward distributions embedded in your model assumptions (e.g. local vol vs. stoch vol both fitted on the same vanilla options). But I would not call IVF a model. Rather a mere parameterisation of the implied volatility smiles (similar to SVI). $\endgroup$ – Quantuple Oct 17 '17 at 14:58
  • $\begingroup$ @Quantuple so vol curve, local vol, vol surface just force you to trust the forward distributions at different level(single maturity, single strike and both )? $\endgroup$ – A.Oreo Oct 17 '17 at 15:13
  • $\begingroup$ @Quantuple I have a new emergent question of implied stochastic vol, could you give me help? $\endgroup$ – A.Oreo Oct 22 '17 at 12:27

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