Why Vanilla Option Prices Do Not Determine Multi-Time Exotic Payoffs
Summary
The document clarifies what an implied volatility function calibrated to vanilla options can and cannot establish. If the model is recalibrated to match vanilla option prices across strikes and maturities, it reproduces the risk-neutral distribution of the underlying at each individual future time. This supports consistent pricing of payoffs that depend on the asset price at just one date, such as digital or asset-or-nothing options.
Matching those separate distributions does not specify how the asset price evolves jointly across multiple dates. In particular, the marginals at two future times do not reveal the distribution of the price change between them or the dependence between those prices. Exotic contracts whose payoff depends on a path or on prices at several times, including barriers and compound options, can therefore be mispriced. The answer gives a conceptual explanation rather than a numerical example, and it does not describe a particular calibration procedure beyond daily recalibration to vanilla prices.
Key ideas
- Vanilla option prices across strikes identify a risk-neutral distribution at a given maturity.
- An implied volatility function calibrated to vanilla prices can price single-date payoffs consistently with them.
- Separate distributions at each future date do not determine the joint distribution across dates.
- Path-dependent and multi-date options may be mispriced even when vanilla prices are matched.
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# IVF and implied distribution of underlying in John Hull's book
# IVF and implied distribution of underlying in John Hull's book
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?
## Answer by dm63 (score 3, accepted)
https://quant.stackexchange.com/a/36488
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.