Mixture Models and the Limits of Static Volatility Scenarios

# Pricing path-dependent with Mixture of Models


# Pricing path-dependent with Mixture of Models












I am reading Piterbarg's paper "Mixture of Models: A Simple Recipe for a ... Hangover?" (SSRN Link) and found myself confused about the example provided. Starting from the idea of using a weighted average of derivative security prices computed using different “simple” models (the so-called “mixture of models”, or “ensemble of models”, approach) as a simple way to add stochastic volatility to virtually any model. Perhaps, the mixture of normal distributions (i.e. a distribution with a density that is a weighted average of two or more Gaussian densities with different volatilities) has heavy tails.

To use a mixture model to price path-dependent products, one must specify the mixture model dynamics, particularly how probabilities (weights) of different volatility scenarios evolve over time. This is done in the lognormal mixture model (Brigo and Mercurio), where the risk-neutral density of the asset is a mixture of lognormal densities, and an analytical diffusion coefficient is specified. The square of the local volatility $\nu(t,y)$ is a weighted average of the squared basic volatilities $\sigma_1^2(t), \dots, \sigma_N^2(t)$, with weights as functions of the marginal lognormal densities:

$$ \nu^2(t, y) = \sum_{i=1}^N \Lambda_i(t, y) \sigma_i^2(t) = \frac{\sum_{i=1}^N \lambda_i \sigma_i^2 p_i(S)}{\sum_{i=1}^N \lambda_i p_i(S)}. $$

In the example provided, the probabilities (weights) of the volatility scenarios remain constant over time. The basic interpretation of defining the mixture model with the above static approach is that assumes the weights are fixed from time $t_0$ all the way to the final maturity $T$. However, this uncertainty is "resolved in the next millisecond", but the non-dynamic stochastic volatility model only specifies what is going to happen to those volatility scenarios at a fixed time in the future, and not what happens in between now ($t_0$) and then ($T$). This static approach, where the value of the derivative in the mixture model is given by:

$$ V_{\text{MM}} = \sum_{i=1}^N p_i V(S_i), $$

is unsuitable for valuing path-dependent derivatives, as it does not account for how volatility uncertainty evolves. While this works for simple contracts like European options (the actual volatility path has no bearing on the option value, as only the average volatility, terminal distribution, between now and the option’s expiry matters), it fails for contracts requiring an understanding of underlying market dynamics.

For instance, the example considers a simple compound option. Using the mixture model, its value computed as a direct valuation approach is:

$$ V_{\text{MM}}(0, S_0) = p_1 E_1 \left[ \max \left( K_1 - S_{T_1}, E_1^{T_1} \left[ (K_2 - S_{T_2})^+ \right] \right) \right] + p_2 E_2 \left[ \max \left( K_1 - S_{T_1}, E_2^{T_1} \left[ (K_2 - S_{T_2})^+ \right] \right) \right]. $$

But the path-dependent options, such as the compound option of the example, do depend on the path between now and maturity. The continuation value approach, $\tilde{V}_{MM}$ attempts look at the value at $T_1$ and then discount it back - but still using the same fixed weights when its computing the expected value at $T_1$ refusing to take account of how the probability of being in a particular volatility regime may have changed by $T_1$, depending on the path taken by the underlier. At $T_1$, the compound option value is:

$$ V_{\text{MM}}(T_1, S) = \mathbb{1}_{\{S > S^*\}} H(S) + \mathbb{1}_{\{S \leq S^*\}} (K_1 - S), $$ $$ H(S) = p_1 E_1^{T_1} \left[ (K_2 - S_{T_2})^+ \mid S_{T_1} = S \right] + p_2 E_2^{T_1} \left[ (K_2 - S_{T_2})^+ \mid S_{T_1} = S \right]. $$

$$ \tilde{V}_{\text{MM}}(0, S_0) = p_1 E_1 \left[ V_{\text{MM}}(T_1, S_{T_1}) \right] + p_2 E_2 \left[ V_{\text{MM}}(T_1, S_{T_1}) \right]. $$

However, these two valuations are inconsistent:

$$ V_{\text{MM}}(0, S_0) \neq \tilde{V}_{\text{MM}}(0, S_0). $$

This inconsistency arises because the mixture model fails to account for the dynamics of volatility uncertainty. Can someone clarify the differences between these valuations and confirm whether my understanding of the paper is correct?

## Answer by Andrea (score 1, accepted)

https://quant.stackexchange.com/a/81537

The key sentence to understand the problem is

this uncertainty is "resolved in the next millisecond"

The mixture of model is self-consistent if used properly: this means that once the uncertainty is resolved, it must stay resolved.

At time 0, scenario 1 or 2 is selected and it must stay selected forever.

This means that: if time 0 were January 1st 2019, then forever afterwards you must use only one value of the volatility (which one???).

What happens in reality is very different, and this time 0 is not fixed: this is the inconsistency. Today, tomorrow, the day after: the "time 0" is constantly rolled forward.

In the formula

$$ \tilde{V}_{\text{MM}}(0, S_0) = p_1 E_1 \left[ V_{\text{MM}}(T_1, S_{T_1}) \right] + p_2 E_2 \left[ V_{\text{MM}}(T_1, S_{T_1}) \right]. $$

you are not allowed to use $V_{\text{MM}}$ inside $E_1$ or $E_2$, and if you do, you get nonsense.

If you replace $V_{\text{MM}}$ with $E_1$ inside $E_1$ (and for 2), then it is obviously correct, but, tomorrow you will likely be using $V_{\text{MM}}$ rather than one of $E_1$ or $E_2$.

The same effect happens for models with a very high time-non-homogeneity (local vol?), if you parameterise in a rolling way and not absolute (think of carry, roll down, different theta, forward value...).

The mixture is much more problematic, because the time-non-homogeneity is huge and concentrated in a single point, as opposed to be allocated to a longer period of time where the parameter term structure is steep.

ADDITTION

One might wonder if this simple recipe could not be turned into a self-consistent model?

Not easily I think: if you flipped the coin every single day and the scenario only applied to the following day, and then you flip again: this is a self consistent model, but it does not allow for a simple closed formula. It is basically a stochastic volatility model with 0 correlation between vol and stock.