Incorporating fat tails in financial models using entropic divergence measures

The evolving literature on updating models for portfolio optimization builds upon the pioneering work of Black and Litterman [4] (BL). BL consider variants of Markowitz’s model where the subjective views of portfolio managers are used as constraints to update models of the market using ideas from Bayesian analysis. Their work focuses on Gaussian framework with views restricted to linear combinations of expectations of returns from different securities. Since then a number of variations and improvements have been suggested (see, e.g., [34], [35] and [37]). Recently, Meucci [33] proposed ‘entropy pooling’ where the original model can involve general distributions and views can be on any characteristic of the probability model. Specifically, he focuses on approximating the original distribution of asset returns by a discrete one generated via Monte-Carlo sampling (or from data). Then a convex programming problem is solved that adjusts weights to these samples so that they minimize the $I$-divergence from the original sampled distribution while satisfying the view constraints. These samples with updated weights are then used for portfolio optimization. Earlier, Avellaneda et al. [3] used similar weighted Monte Carlo methodology to calibrate asset pricing models to market data (see also Glasserman and Yu [21]). Buchen and Kelly in [5] and Stutzer in [41] use the entropy approach to calibrate one-period asset pricing models by selecting a pricing measure that correctly prices a set of benchmark instruments while minimizing $I$-divergence from a prior specified model, that may, for instance be estimated from historical data (see also the recent survey article [31]).

Zhou et.al in [43] consider a statistical learning framework for estimating default probabilities of private firms over a fixed horizon of time, given market data on various ‘explanatory variables’ like stock prices, financial ratios and other economic indicators. They use the entropy approach to calibrate the default probabilities (see also Friedman and Sandow [18]).

Another popular application of entropy based ideas to finance is in valuation of non-hedgeble payoffs in incomplete markets. In an incomplete market there may exist many probability measures equivalent to the physical probability measure under which discounted price processes of the underlying assets are martingales. Fritelli ([19]) proposes using a probability measure that minimizes the $I$-divergence from the physical probability measure for pricing purposes (he calls this the minimal entropy martingale measure or MEMM). Here, the underlying financial model corresponds to a continuous time stochastic process. Cont and Tankov ([6]) consider, in addition, the problem of incorporating calibration constraints into MEMM for exponential Levy processes (see also Kallsen [28]). Jeanblanc et al. ([26]) propose using polynomial-divergence (they call it $f^{q}$-divergence distance) instead of $I$-divergence and obtain necessary and sufficient conditions for existence of an EMM with minimal polynomial-divergence from the physical measure for exponential Levy processes. They also characterize the form of density of the optimal measure. Goll and Ruschendorf ([22]) consider general $f$-divergence in a general semi-martingale setting to single out an EMM which also satisfies calibration constraints.

A brief historical perspective on related entropy based literature may be in order: This concept was first introduced by Gibbs in the framework of classical theory of thermodynamics (see [20]) where entropy was defined as a measure of disorder in thermodynamical systems. Later, Shannon [39] (see also [30]) proposed that entropy could be interpreted as a measure of missing information of a random system. Jaynes [24], [25] further developed this idea in the framework of statistical inferences and gave the mathematical formulation of principle of maximum entropy (PME), which states that given partial information/views or constraints about an unknown probability distribution, among all the distributions that satisfy these restrictions, the distribution that maximizes the entropy is the one that is least prejudiced, in the sense of being minimally committal to missing information. When a prior probability distribution, say, $\mu$ is given, one can extend above principle to principle of minimum cross entropy (PMXE), which states that among all probability measures which satisfy a given set of constraints, the one with minimum relative entropy (or the $I$-divergence) with respect to $\mu$, is the one that is maximally committal to the prior $\mu$. See [29] for numerous applications of PME and PMXE in diverse fields of science and engineering. See [12], [1], [14], [23] and [27] for axiomatic justifications for Renyi and Tsallis entropy. For information theoretic import of $f$-divergence and its relation to utility maximization see Friedman et.al. [17], Slomczynski and Zastawniak [40].

In this article, we restrict attention to examples related to portfolio optimization and model calibration and build upon ideas proposed by Avellaneda et al. [3], Buchen and Kelly [5], Meucci [33] and others. We first note the well known result that for views expressed as finite number of moment constraints, the optimal solution to the $I$-divergence minimization can be characterized as a probability measure obtained by suitably exponentially twisting the original measure. This measure is known in literature as the Gibbs measure and our analysis is based on the well known ideas involved in Gibbs conditioning principle (see, for instance, [13]). As mentioned earlier, such a characterization may fail when the underlying distributions are fat-tailed in the sense that their moment generating function does not exist in some neighborhood of origin. We show that one reasonable way to get a good change of measure that incorporates views¹¹1In this article, we often use ‘views’ or ‘constraints’ interchangeably in this setting is by replacing $I$-divergence by a suitable ‘polynomial-divergence’ as an objective in our optimization problem. We characterize the optimal solution in this setting, and prove its uniqueness under technical conditions. Our definition of polynomial-divergence is a special case of a more general concept of f-divergence introduced by Csiszar in [8], [9] and [10]. Importantly, polynomial-divergence is monotonically increasing function of the well known relative Tsallis Entropy [42] and relative Renyi Entropy [38], and moreover, under appropriate limit converges to $I$-divergence.

As indicated earlier, we also consider the case where the expert views may specify marginal probability distribution of functions of random variables involved. We show that such views, in addition to views on moments of functions of underlying random variables are easily incorporated. In particular, under technical conditions, we characterize the optimal solution with these general constraints, when the objective may be $I$-divergence or polynomial-divergence and show the uniqueness of the resulting optimal probability measure in each case.

As an illustration, we apply these results to portfolio modeling in Markowitz framework where the returns from a finite number of assets have a multivariate Gaussian distribution and expert view is that a certain portfolio of returns is fat-tailed. We show that in the resulting probability measure, under mild conditions, all assets are similarly fat-tailed. Thus, this becomes a reasonable way to incorporate realistic tail behavior in a portfolio of assets. Generally speaking, the proposed approach may be useful in better risk management by building conservative tail views in mathematical models.

Note that a key reason to propose polynomial-divergence is that it provides a tractable and elegant way to arrive at a reasonable updated distribution close to the given prior distribution while incorporating constraints and views even when fat-tailed distributions are involved. It is natural to try to understand the influence of the choice of objective function on the resultant optimal probability measure. We address this issue for a simple example where we compare the three reasonable objectives: the total variational distance, $I$-divergence and polynomial-divergence. We discuss the differences in the resulting solutions. To shed further light on this, we also observe that when views are expressed as constraints on probability values of disjoint sets, the optimal solution is the same in all three cases. Furthermore, it has a simple representation. We also conduct numerical experiments on practical examples to validate the proposed methodology.

In Section 2, we outline the mathematical framework and characterize the optimal probability measure that minimizes the $I$-divergence with respect to the original probability measure subject to views expressed as moment constraints of specified functions. In Section 3, we show through an example that $I$-divergence may be inappropriate objective function in the presence of fat-tailed distributions. We then define polynomial-divergence and characterize the optimal probability measure that minimizes this divergence subject to constraints on moments. The uniqueness of the optimal measure, when it exists, is proved under technical assumptions. We also discuss existence of the solution in a simple setting. When under the proposed methodology a solution does not exist, we also propose a weighted least squares based modification that finds a reasonable perturbed solution to the problem. In Section 4, we extend the methodology to incorporate views on marginal distributions of some random variables, along with views on moments of functions of underlying random variables. We characterize the optimal probability measures that minimize $I$-divergence and polynomial-divergence in this setting and prove the uniqueness of the optimal measure when it exists. In Section 5, we apply our results to the portfolio problem in the Markowitz framework and develop explicit expressions for the posterior probability measure. We also show how a view that a portfolio of assets has a ‘regularly varying’ fat-tailed distribution renders a similar fat-tailed marginal distribution to all assets correlated to this portfolio. Section 6 is devoted to comparing qualitative differences on a simple example in the resulting optimal probability measures when the objective function is $I$-divergence, polynomial-divergence and total variational distance. In this section, we also note that when views are on probabilities of disjoint sets, all three objectives give identical results. We numerically test our proposed algorithms on practical examples in Section 7. Finally, we end in Section 8 with a brief conclusion. All but the simplest proofs are relegated to the Appendix.

We thank the anonymous referee for bringing Friedman et al [18] to our notice. This article also considers $f$-divergence and in particular, polynomial-divergence (they refer to an equivalent quantity as $u$-entropy) in the setting of univariate power-law distribution which includes Pareto and Skewed generalized $t$-distribution. It motivates the use of polynomial-divergence through utility maximization considerations and develops practical and robust calibration technique for univariate asset return densities.

To shed further light on Example 2, for $M>0$, consider a probability distribution

Where $I_{A}(\cdot)$ denote the indicator function of the set $A$, that is, $I_{A}(x)=1$ if $x\in A$ and $0$ otherwise. Let $\lambda_{M}$ denote the solution to

Solutions such as ${f}_{\lambda_{M}}$ above are typically not representative of many applications, motivating the need to have alternate methods to arrive at reasonable posterior measures that are close to $\mu$ and satisfy constraints such as (3) and (4) while not requiring that the optimal solution be obtained using exponential twisting.

We now address this issue using polynomial-divergence. We first define $f$-divergence introduced by Csiszar (see [8], [9] and [10]).

Note that $I$-divergence corresponds to the case $f(u)=u\log u$. Other popular examples of $f$ include

In this section we consider $f(u)=u^{\beta+1},\ \beta>0$ and refer to the resulting $f$-divergence as polynomial-divergence. That is, we focus on

It is easy to see using Jensen’s inequality that

is achieved by $\nu=\mu$.

Our optimization problem ${\bf O_{2}}(\beta)$ may be stated as:

subject to (3) and (4). Minimizing polynomial-divergence can also be motivated through utility maximization considerations. See [17] for further details.

In the following assumption we specify the form of solution to ${\bf O_{2}}(\beta)$:

To solve (12) for $\mbox{\boldmath{$\lambda$}}\in\Lambda$ one may resort to numerical root-search techniques. A wide variety of these techniques are available in practice (see, for example, [32] and [36]). In our numerical experiments, we used FindRoot routine from Mathematica which employs a variant of secant method.

Alternatively, one can use the dual approach of minimizing a convex function over the set $\Lambda$. In the proof of Theorem 3 we have shown one convex function whose stationary point satisfies a set of equations which is equivalent and related to (12). Therefore, it is possible recover a strongly feasible solution to (12) if one exists. Note that, contrary to the case with $I$-divergence where the dual convex optimization problem is unconstrained, here we have a minimization problem of a convex function subject to the constraints (9) and (13).

Proposition 2 below shows the existence of a solution to ${\bf O_{2}}(\beta)$ under a single random variable, single constraint settings. We then apply this to a few specific examples.

Note that for any random variable $X$, a non-negative function $g$ and a positive integer $n$, we always have $E[g(X)^{n+1}]\geq E[g(X)]E[g(X)^{n}]$, since random variables $g(X)$ and $g(X)^{n}$ are positively associated and hence have non-negative covariance.

As we note in the proof in the Appendix, the uniqueness of the solution follows from Theorem 3. Standard methods like Newton’s, secant or bisection method can be applied to numerically solve (14) for $\lambda$.

Note that an optimal solution to ${\bf O_{2}}(\beta)$ may not exist when Assumption 2 does not hold. For instance, in Proposition 2, it may not exist for $a>\frac{E[g(X)^{n+1}]}{E[g(X)]E[g(X)^{n}]}$. In that case, selecting $\tilde{f}$ and associated $\lambda$ so that $\frac{\tilde{E}[g(X)]}{E[g(X)]}$ is very close to $\frac{E[g(X)^{n+1}]}{E[g(X)]E[g(X)^{n}]}$ is a reasonable practical strategy.

We now discuss how such solutions may be achieved for general problems using a weighted least squares approach. Let $\lambda$ denote $(\lambda_{1},\lambda_{2},\ldots,\lambda_{k})$ and $\nu_{\mbox{\boldmath{$\lambda$}}}$ denote the measure defined by right hand side of (11). We write ${\bf c}$ for $(c_{1},c_{2},\ldots,c_{k})$ and re-express the optimization problem ${\bf O_{2}}(\beta)$ as ${\bf O_{2}}(\beta,{\bf c})$ to explicitly show its dependence on ${\bf c}$. Define

Consider ${\bf{c}}\notin\mathcal{C}$. Then, ${\bf O_{2}}(\beta,{\bf c})$ has no solution. In that case, from a practical viewpoint, it is reasonable to consider as a solution a measure $\nu_{\mbox{\boldmath{$\lambda$}}}$ corresponding to a ${\bf c}^{\prime}\in\mathcal{C}$ such that this ${\bf c}^{\prime}$ is in some sense closest to ${\bf c}$ amongst all elements in $\mathcal{C}$. To concretize the notion of closeness between two points we define the metric

between any two points ${\bf x}=(x_{1},x_{2},\ldots,x_{k})$ and ${\bf y}=(y_{1},y_{2},\ldots,y_{k})$ in $\mathbb{R}^{k}$, where ${\bf w}=(w_{1},w_{2},\ldots,w_{k})$ is a constant vector of weights expressing relative importance of the constraints : $w_{i}>0$ and $\sum_{i=1}^{k}w_{i}=1$.

Let ${\bf c_{0}}:=\arg\inf_{{\bf c}^{\prime}\in\bar{\mathcal{C}}}d({\bf c},{\bf c}^{\prime})$, where $\bar{\mathcal{C}}$ is the closure of $\mathcal{C}$. Except in the simplest settings, $\mathcal{C}$ is difficult to explicitly evaluate and so determining ${\bf c_{0}}$ is also no-trivial.

The optimization problem below, call it ${\bf\tilde{O}_{2}}(\beta,t,{\bf c})$, gives solutions $({{\bf\lambda}_{t}},{\bf y}_{t})$ such that the vector ${\bf c}_{t}$ defined as

has $d$-distance from ${\bf c}$ arbitrarily close to $d({\bf c_{0}},{\bf c})$ when $t$ is sufficiently close to zero. From implementation viewpoint, the measure $\nu_{\lambda_{t}}$ may serve as a reasonable surrogate for the optimal solution sought for ${\bf O_{2}}(\beta,{\bf c})$. (Avelleneda et al. [3] implement a related least squares strategy to arrive at an updated probability measure in discrete settings).

subject to

This optimization problem penalizes each deviation $y_{j}$ from $c_{j}$ by adding appropriately weighted squared deviation term to the objective function. Let $({{\bf\lambda}_{t}},{\bf y}_{t})$ denote a solution to ${\bf\tilde{O}_{2}}(\beta,t,{\bf c})$. This can be seen to exist under a mild condition that the optimal ${\bf\lambda}$ takes values within a compact set for any $t$. Note that then ${\bf c}+{\bf y}_{t}\in\mathcal{C}$.

Formally, our optimization problem ${\bf O_{3}}$ is:

subject to:

where $g(\mathbb{x})$ is a given marginal density function of $\mathbb{X}$, and

for $i=1,2\ldots,k$. For presentation convenience, in the remaining paper we only consider equality constraints on moments of functions (as in (19)), ignoring the inequality constraints. The latter constraints can be easily handled as in Assumptions (1) and (2) by introducing suitable non-negativity and complementary slackness conditions.

Some notation is needed to proceed further. Let $\mbox{\boldmath{$\lambda$}}=(\lambda_{1},\lambda_{2},\ldots,\lambda_{k})$ and

Further, let $f_{\mbox{\boldmath{$\lambda$}}}(\mathbb{x},\mathbb{y})$ denote the joint density function of $(\mathbb{X},\mathbb{Y})$, $f_{\mbox{\boldmath{$\lambda$}}}(\mathbb{y}|\mathbb{x})\times g(\mathbb{x})$ for all $\mathbb{x},\mathbb{y}$, and $E_{\mbox{\boldmath{$\lambda$}}}$ denote the expectation under $f_{\mbox{\boldmath{$\lambda$}}}$.

For a mathematical claim that depends on x, say $S(x)$, we write $S(x)$ for almost all $x\,\,w.r.t.\,\,g(x)dx$ to mean that $m_{g}(x|\,S(x)\,\,\text{is false})=0$, where $m_{g}$ is the measure induced by the density $g$. That is, $m_{g}(A)=\int_{A}g(x)\,dx$   for all measurable subsets $A$.

In Theorem 5, we develop conditions that ensure uniqueness of a solution to ${\bf O_{3}}$ once it exists.