Why the Minimal Entropy Martingale Measure Is Not Intrinsically Bayesian

# Minimal entropy martingale measure and Bayes estimated under Kullback-Laibller divergence loss function


# Minimal entropy martingale measure and Bayes estimated under Kullback-Laibller divergence loss function












We know that no unique equivalent measure exists in an incomplete market. Therefore, we need to choose a pricing measure equivalent to the physical measure based on a criterion. One typical approach in such a situation is to use the concept of minimal entropy martingale measure which minimizes the distance between the pricing measure $\mathbb{Q}$ and the physical measure $\mathbb{P}$ in the entropy sense.

In the Bayes framework, the main objective is to find the posterior distribution of unknown parameters contained in the model by minimizing a loss function that measures the distance between the estimated parameters and their respective true parameters (the distance can be taken to be defined in terms of density function rather than parameters). There are different choices for the loss function. For example, the squared error loss function, absolute value error loss function, and weighted squared error loss function can be used for minimization purposes. Another possible choice is a loss function that is defined in terms of the Kullback-Laibller divergence. Let $f(x)$ be a density function for a continuous random variable $X$, characterized by the parameter $\Theta$. Then, the Kullback error loss function (KEL) is given by \begin{equation}\label{ref37} \text{KL}(\Theta \parallel \hat{\Theta}) = \text{KL}\big(f(x;\Theta) \parallel f(x; \hat{\Theta})\big) = \int_{\mathcal{A}}\log\frac{f(x; \Theta)}{f(x; \hat{\Theta})}f(x; \Theta) dx, \end{equation}

The Bayes estimator is the one that minimizes the expectation of the Kullback error loss function.

I wonder if the Bayse estimate parameters resulting from the above minimization problem can be interpreted as risk-neutral parameters. I see some connection between the Bayese estimated under the KLD loss function and the Minimal entropy martingale measure.

## Answer by Draman (score 1)

https://quant.stackexchange.com/a/75467

It is an interesting observation and a bit of a stretch. However note that the KL loss function is merely a divergence function used in many applications as modelers need it. In the case of the MEMM the probabilty measure found is the closest one to the historical measure that is risk neutral. The KLD has nothing to do with risk neutrality intrinsically.