# Why is the LMM with mixture dynamics (Brigo & Mercurio) inconsistent for the pricing of exotics?

I am reading about the LMM with lognormal-mixture dynamics. Consider the following dynamics for the forward rate $$F_{i}(t)$$ fixing at $$T_{i-1}$$ and paying at $$T_i$$: \begin{align} dF_{i}(t) = (F_i (t) +\gamma) \sigma_i^I dt \ , \end{align} where $$\gamma$$ is a constant shift and $$\sigma_i^I$$ is a random variable for the volatility drawn at time $$0^+$$ from a discrete or continuous distribution (the superscript $$I$$ denotes the state). These dynamics have the convenience that the price of a European call/put simply is the expectation of Black-Scholes prices over all possible scenarios for the volatility, and one could quite easily calibrate the states on the market smile of the caplet.

In an article by Piterbarg, "Mixture of Models: A Simple Recipe for a... Hangover?", it is argued that such a dynamics cannot be used for the pricing of exotics (one should for example first transform the dynamics into a local vol function).

However, I fail to understand his argument, especially on page 4. The author takes as an example a simple compounded put option with two states {1, 2} for the volatility. He then argues that when time passes by and if the market doesn't change, the continuation value $$H(S)$$ at time $$T_1$$ (first exercise date) will be calculated under the same mixture as follows: \begin{align} H(S) = p_1 E^1_{T_1}\{(K_2 - S_{T_2})^+ |S_{T_1} = S \} + p_2 E^2_{T_1}\{(K_2 - S_{T_2})^+ |S_{T_1} = S \} \ . \end{align} First of all, what I don't understand is that when a book of exotics is marked-to-market on a daily basis, even if the market doesn't change significantly, a new calibration procedure could lead to different calibrated states simply because the time to maturity changes (i.e. we get closer to $$T_1$$ and $$T_2$$ in the example above).

Does anybody has a clear explanation (either intuitively or mathematically) on why such a models can or can't be used in practice? Intuitively I would think that such a model would perform at least better than a simple LMM (only one state with probability 1) when it comes to hedging the smile. I understand that theoretically the model isn't very realistic, but I am looking at the problem from a practitioner's point of view. Any insights on the problematic would be useful.

• In that short note Piterbarg is discussing a method by which a price is a mixture of prices from different models and comes to the conclusion that such approaches are "self-inconsistent and cannot be used for valuation" (see his abstract). Imho this note is a clear enough explanation, both intuitively and mathematically, that such models cant't be used in practice. As far as I can tell the problem of re-calibration when time passes that you mention does not even arise. ... Jun 7, 2022 at 9:17
• In Piterbarg's note I nowhere see the LMM mixture dynamics you are starting with. What has one to do with the other? Please provide more context. Jun 7, 2022 at 9:17
• @KurtG. In his note the dynamics he uses are the same as the LMM mixture dynamics but with only a single LIBOR, with two discrete states {1,2} and with $\gamma=0$. I understand that this model is inconsistent because it only gives a stochastic view on integrated variance and not on the path taken by the volatility process (hence the problem discussed in his paper). From a theoretical pov I understand the problem with this. Jun 7, 2022 at 10:30
• However, in practice, if you see the pricing of exotics as an extrapolation exercise of your hedging instruments with a frequent recalibration on those instruments, I do see some arguments (e.g. quick calibration and simulation) that makes this model attractive if the exotic to price isn't too strongly dependent on the volatility path. Jun 7, 2022 at 10:30

My impression is that Piterbarg criticizes the mixture approach because it is not consistent with the stochastic volatility (SV) world. In the SV world, instanteneous volatility (or variance) continuously evolves over time, although it's not observable. In the mixture model, volatility discountinuously evolves immediately after $$t=0$$, which is not an ideal picture in the SV world. In short, if you're religious to the SV world, then, the mixture model is nonsense.
However, there's an approach to view volatility an uncertain quantity (it's unobservable) and to assume it as a distribution. There's a technique called randomization. For example, see Jacquier & Shi (2019) where the initial variance of the Heston model is assumed as a random variable. In this respect, the mixture approach makes some sense. IMHO, I think the mixture model may not be perfect but is better than the simple (fixed volatility) model.