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- Mixture Models
Mixture Models
Having examined the traditional stochastic volatility models, we will now briefly look at a more radical approach.
Complex models have large numbers of parameters that must be calibrated with heavy numerical algorithms. Then when one comes to calculate the Greeks, the chances are that the heavy numerics will lead to unstable results.
The Heston and SABR style models are regarded as providing an acceptable balance between realism and simplicity. However, often they are not as fast nor as numerically stable as one would like in an ideal world.
The idea of a mixture model is to write down the simplest possible 'stochastic' volatility model. We imagine that we have a number of volatility states.
In the simplest case, we might have just 2 states. At time zero, that is at the moment we perform the valuation, we toss a coin to decide which of the wo volatility states will be used for the remainder of the contract.
In the simplest version 
and 
might be constant. Alternatively, one can make them time dependent.
The seductiveness of a mixture model comes from its simplicity. After the initial toss, the spot follows a BS process with volatility that is the outcome of the toss. Then, if we know the BS formula 
for any particular contract, we can evaluate the value in the mixture model via:
One can use this pricing formula to value vanilla options at various strikes and then back out the corresponding IVs.
You will find that the smile generated is pretty convincing in spite of the simplicity of the model.
The convexity of the smile is controlled by the spacing between the volatility states. If the volatility states are allowed to be time dependent, then one can control the convexity at various expiry dates.
An immediate drawback is that there is no spot-volatility correlation that could be used to add skew to the smile. If, however, one is modelling a smile that is not skewed then there is nothing wrong with using the mixture model for European contracts. A European contract depends only on the smile at expiry, and therefore as long as the smile is good, the dynamic causing the smile is irrelevant.
However, we would like to use our stochastic volatility model to value path dependent contracts, and here there is a major problem. All the uncertainty in the volatility is concentrated into a single instant at the time of valuation. We do not truly believe that the smile will disappear after that instant. When we next come to value our contract a few seconds later, we will make the same assumption but this time concentrating the volatility uncertainty at the new valuation time.
piterbarg's example in his paper in 2003 renders those mixture models that are expressed merely as distributions without actual arbitrage free dynamics useless and dangerous for general path dependent derivative contracts.
However, later we will find them useful in extending our understanding of stochastic volatility pricing.