Martingale Pricing and Single Index Models: Unified Approach with Esscher and Minimal Relative Entropy Measures

Abstract

In this paper, we explore the connection between a single index model under the real-world probability measure and martingale pricing via minimal relative entropy or Esscher transform, within the context of a one-period market model, possibly incomplete, with multiple risky assets and a single risk-free asset. The minimal relative entropy martingale measure and the Esscher martingale measure coincide in such a market, provided they both exist. From their Radon–Nikodym derivative, we derive a portfolio of risky assets in a natural way, termed portfolio G. Our analysis shows that pricing using the Esscher or minimal relative entropy martingale measure is equivalent to a single index model (SIM) incorporating portfolio G. In the special case of elliptical returns, portfolio G coincides with the classical tangency portfolio. Furthermore, in the case of jointly normal returns, Esscher or minimal relative entropy martingale measure pricing is equivalent to CAPM pricing.

1. Introduction

The study of asset pricing is fundamental to financial economics and centers on understanding the relationship between risk and return. On the one hand, traditional models, such as the Capital Asset Pricing Model (CAPM), try to capture this relationship under the real-world probability measure P (Black 1972; Lintner 1965; Sharpe 1964), while, on the other hand, the study of non-arbitrage and of contingent claims pricing has emphasized the role of martingale or risk-neutral measures Q (Bjork 2019; Cochrane 2005; Duffie 2001; Harrison and Pliska 1981; Magill 2002). These two perspectives, while seemingly distinct, share a fundamental goal: to adequately price risk in both idealized complete markets and the more realistic incomplete markets, where not all risks can be perfectly hedged or diversified away (Cochrane 2005; Delbaen 1996).

Unlike the real-world measure P, which is supposed to represent the actual probabilities of outcomes, the risk-neutral measures, Q, are constructed to ensure that, under Q, the discounted—with respect to the risk-free rate—expected future prices are equal to the current market prices. The understanding of the relation between the real-world measure P and the risk-neutral measure Q is essential for the understanding of asset pricing, derivative valuation, and risk management.

A unifying concept that plays a central role in both traditional asset pricing models and the martingale framework is the notion of state prices. As demonstrated by (Duffie 2001, particularly in Chapter 6, Section D), state prices provide a way to determine the expected return of an asset where excess expected returns are proportional to the covariance between an asset’s return and the state-price deflator. This leads to a state-price beta model that resembles the structure of the CAPM, although, instead of relying on the market portfolio’s return, the state-price beta model works within an arbitrage-free pricing framework using state prices.

Under the assumptions of non-arbitrage and market completeness, traditional models like CAPM provide a framework where the pricing of assets is straightforward, and the existence of a unique equivalent martingale measure leads to a unique price of contingent claims. In this idealized setting, risks are either fully priced or diversified away. However, in realistic markets, incompleteness prevails, and unhedgeable risks persist—factors that manifest themselves in the non-uniqueness of the risk-neutral measure. The fact that a family of such measures must be considered reflects the complexities and uncertainties inherent in real-world financial markets. This non-uniqueness problem has been a focal point of numerous theoretical studies aiming to better model real-world asset pricing in incomplete markets (Follmer 2011; Hansen 1991).

In the context of incomplete markets, a challenge arises in choosing a specific martingale measure from the family of possible candidates. Several approaches have been developed to address this issue, each with its own set of criteria (Bjork 2019, chp. 29). Among these, two prominent approaches are the Esscher martingale measure and the minimal relative entropy martingale measure. The Esscher measure, introduced by Gerber and Shiu (1994) in the context of financial markets, adjusts the original probability measure using an exponential factor, making it particularly tractable for certain types of financial models. The minimal relative entropy measure, proposed by Frittelli (2000), minimizes the Kullback–Leibler divergence from the real-world measure to the risk-neutral measure, providing a theoretically well-grounded proposition for selecting a martingale measure in incomplete markets.

This work is placed within the broader context of asset pricing theory and aims to offer results and insights that integrate the real-world and risk-neutral perspectives. It focuses on a one-period market model to explore the relationship between martingale measures and traditional financial models. By doing so, this paper contributes to bridging the gap between pure theory and models that can be applied in both academic and practical contexts, thus extending the efforts in pricing within incomplete markets. The analysis builds on the foundational principles presented in Duffie (2001), particularly on the use of state prices to derive expected returns, but it focuses directly on the exploration of the relationship between martingale measures and traditional financial models like SIM and CAPM. Specifically, it is demonstrated that pricing using the Esscher and minimal relative entropy martingale measures can be interpreted within a single index model framework, and for this, a novel portfolio, termed “portfolio G”, is introduced. This portfolio serves as a bridge between martingale pricing techniques and traditional models like CAPM, offering a new perspective on asset pricing in potentially incomplete markets. While some foundational aspects of this work have been discussed in a preliminary form in the author’s previous book chapter Xanthopoulos (2016), this paper significantly extends and refines this former work.

The remainder of this paper is structured as follows: In Section 2, a short literature review provides the general context by presenting basic theories, ideas, and problems related to this paper. Section 3 introduces the market model, notation, and assumptions used throughout the paper. Section 4 presents the concepts of the minimal relative entropy martingale measure and the Esscher martingale measure, establishing their equivalence within the specified market context. Section 5 focuses on the derivation and properties of portfolio G, which bridges martingale pricing and traditional single index model pricing frameworks. Section 6 examines portfolio G under the assumption of elliptical return distributions, demonstrating that it coincides with the classical tangency portfolio (the maximum Sharpe ratio portfolio). Section 7 further examines the special case of jointly normal returns, showing that in this case, Esscher or minimal relative entropy martingale pricing is equivalent to classical CAPM pricing, with portfolio G being the market portfolio. Finally, Section 8 summarizes and concludes.

2. Literature Review

The Single Index Model (SIM) and the Capital Asset Pricing Model (CAPM) are both simple models that attempt to explain asset returns in a linear fashion using only one risk factor. In SIM, this risk factor can be any single index, such as a sector or macroeconomic variable or any other custom risk factor, while CAPM specifically identifies the market portfolio as the key risk factor. CAPM, however, differs from SIM in that it is an equilibrium model, providing a theoretical basis for how asset prices are determined in a market where supply and demand for assets are balanced.

The SIM was introduced in Sharpe (1963), suggesting that the returns of individual stocks could be linked to a single index, simplifying both the computation and practical application of portfolio analysis. Later, Elton et al. (1976) extended the SIM framework, focusing on its application to portfolio optimization and risk management. Their work explored how using a single index allows for easier estimation of covariance matrices in portfolio analysis. The CAPM, developed independently in Sharpe (1964), Lintner (1965), Mossin (1966), and Black (1972), involved distinct contributions in shaping the theory. Sharpe formalized the relationship between an asset’s expected return and its risk relative to the market, expressed as beta. He showed that the expected return of an asset is proportional to its systematic risk (beta), which measures the asset’s sensitivity to the overall market and that investors are compensated only for systematic risk but not for idiosyncratic risk because the latter can be diversified away. This led to the well-known CAPM formula, $E\left(R_i\right)=R_f+{\beta }_i(E\left(R_m\right)-R_f)$, where $E\left(R_i\right)$ is the expected return on asset i, $R_f$ is the risk-free rate, ${\beta }_i$ is the asset’s sensitivity to the market, and $E\left(R_m\right)$ is the expected return on the market. Lintner extended Sharpe’s work by developing a more rigorous mathematical foundation for CAPM, particularly with respect to the assumptions of market equilibrium, and introducing dynamic considerations, such as how investors rebalance portfolios over time, providing more detailed insights into how risk premia emerge in the market. Mossin provided another derivation of the model using a more formal mathematical approach, leading to the same conclusions as Sharpe and Lintner, and emphasized the importance of investors’ preferences for maximizing returns while minimizing risk in a mean-variance framework. Finally, Black extended the original model by removing the assumption of a risk-free rate, which was a key part of the original CAPM, and introducing the idea that borrowing and lending at a risk-free rate may not be possible for all investors, leading to a modified model where the market portfolio alone explains the pricing of assets. However, CAPM has been subject to extensive criticism over the years. Key critiques focus on its restrictive assumptions, such as market completeness, homogeneous expectations, and the availability of risk-free borrowing. For example, Roll (1977) addressed a conceptual and methodological critique, questioning the observability of the true market portfolio, while Fama and French (2004) demonstrated CAPM’s failure to fully explain the cross-section of stock returns, leading to the development of multi-factor models with additional factors like size, value, profitability, and investment. Various other critiques challenge CAPM’s validity, for example, (Baker et al. 2011; Bali et al. 2011; Buffa et al. 2022; Campbell et al. 2018; Cohen et al. 2005; Frazzini and Pedersen 2014; Hong and Sraer 2016; Kumar 2009; Merton 1987), referring to, among other things, CAPM’s limitations in addressing investor psychology or market anomalies, like, for example, the low-beta anomaly. On the other hand, there are works like Andrei et al. (2023), trying to explain the empirical failure of CAPM.

While CAPM and SIM are based on real-world probabilities, martingale pricing theory, formalized in Harrison and Pliska (1981), uses a risk-neutral or equivalent martingale measure (EMM), ensuring that discounted asset prices follow a martingale process. The Fundamental Theorem of Aset Pricing establishes that no-arbitrage conditions imply the existence of an EMM, under which the discounted prices of all assets evolve as martingales. In complete markets, this EMM is unique, simplifying the valuation of contingent claims. However, in incomplete markets, where the risk factors are more than the tradable assets, there are risks that remain unhedgeable, and the no arbitrage condition implies the existence of an infinity of EMM.

Although CAPM and martingale pricing originate from different theoretical foundations, they are both asset pricing frameworks. Building on the principles of martingale pricing and exploring the concept of the stochastic discount factor or state price density (Duffie 2001) made foundational contributions toward connecting these two frameworks. In complete markets, CAPM can be viewed as a specific case of martingale pricing. In this case, the stochastic discount factor (SDF) is tied to the market portfolio’s return, and only systematic risk is priced, consistent with CAPM’s relationship between beta and expected return. In incomplete markets, however, Duffie extended the pricing framework by incorporating non-tradable risks, which CAPM overlooks. This generalization allows for a more comprehensive asset pricing model, addressing risks that cannot be fully hedged and providing a deeper understanding of pricing dynamics in both complete and incomplete markets.

Nevertheless, the problem of which EMM is chosen or should be chosen by market participants in incomplete markets has prompted extensive academic research. Several approaches have been proposed and studied, each with its own advantages and limitations: The Minimal Martingale Measure (MMM), introduced in Föllmer and Schweizer (1991), minimizes the variance of the hedging error. While simple to compute, it may introduce bias and is less applicable to models with complex dynamics such as jumps or stochastic volatility. The Esscher transform, from Gerber and Shiu (1994), adjusts the probability measure so that the discounted price process becomes a martingale. It is widely used in models with jumps or discontinuities. However, its reliance on model assumptions can limit its applicability in more complex markets. The Minimal Entropy Martingale Measure (MEMM), by Frittelli (2000), minimizes the relative entropy between the real-world and risk-neutral measures, aligning well with utility maximization approaches. Although it reduces informational loss, it is obviously sensitive to the initial probability measure and can also be computationally intensive. Another significant strand of literature resorts to indifference pricing (Carmona 2008; Henderson and Hobson 2004). The utility indifference price is the price at which an investor is indifferent between having a contingent claim or not, given their risk preferences. The risk-neutral measure chosen under utility indifference pricing is typically not unique but reflects the specific investor’s risk profile. For example, exponential utility often leads to the selection of the minimal entropy martingale measure (MEMM). However, solving the optimization problem involved can be complex and is subjective since it depends on the utility function. Finally, Øksendal and Sulem (2006) uses game theory to select the EMM, viewing the market as a strategic game between an agent and “nature”, each one having differing objectives. The equilibrium of this game determines the EMM. Solving the equilibrium can be complex, and the approach may require strong assumptions about market structure. Apart from martingale pricing, where incompleteness poses challenges in an obvious way, incompleteness also affects the traditional asset pricing models like CAPM; see, for example, (Geanakoplos and Shubik 1990; Hur and Chung 2017). Finally, it should be stated that recent research has been focusing on enhancing asset pricing through machine learning methods, for example, (Bagnara 2024), which focuses on how machine learning methods have recently been applied to empirical asset pricing, Rebentrost et al. (2024), which explores the application of quantum computing to financial portfolio optimization, and Fecamp et al. (2020), which focuses on how to use deep learning techniques to compute hedging strategies in markets with sources of incompleteness, including techniques such as illiquidity, non-tradable risk factors, discrete hedging dates, and transaction costs.

3. Preliminaries and Notation

We consider a one-period financial market as follows: There are two trading days, $t_0$ and T. The uncertainty about the state of the world that will be realized at time T is described by a probability measure P defined on the state space $(\mathsf{\Omega },\mathcal{F})$. We assume the existence of a riskless asset bearing a risk-free interest rate r so that an investment of 1 unit at time $t_0$ is worth $1+r$ at time T. Additionally, we assume the existence of m risky traded assets, denoted as $S^{\left(1\right)},\dots ,S^{\left(m\right)}$. We use the following notation:

• $S_0^{\left(j\right)}$: the price of asset $S^{\left(j\right)}$ at time $t_0$.
• $S_0=(S_0^{\left(1\right)},\dots ,S_0^{\left(m\right)})$: the vector of initial prices of the risky assets.
• $S_T^{\left(j\right)}$: the random variable of the price of asset $S^{\left(j\right)}$ at time T.
• $S_T=(S_T^{\left(1\right)},\dots ,S_T^{\left(m\right)})$: the vector of prices of the risky assets at the end of the period.
• $R_j:={\displaystyle \frac{S_T^{\left(j\right)}}{S_0^{\left(j\right)}}}-1$: the return on asset $S^{\left(j\right)}$.
• $R=(R_1,\dots ,R_m)$: the vector of returns of the risky assets.

A probability measure Q on $(\mathsf{\Omega },\mathcal{F})$ is called a martingale measure for this market if and only if $E_Q\left[S_T\right]=S_0(1+r)$; equivalently, if and only if $E_Q\left[R_j\right]=r$ for all $j=1,\dots ,m$. We denote by $\mathcal{M}$ the set of all martingale measures for this market. A martingale measure Q is called equivalent to P if Q and P attach zero probabilities to exactly the same events. The existence of equivalent martingale measures is equivalent to the non-existence of arbitrage opportunities in the market (Delbaen and Schachermayer 1996; Harrison and Pliska 1981).

4. Esscher and Minimal Relative Entropy Martingale Measures for a One-Period Model

In the context of incomplete markets, the martingale measure is not unique, and the question of which measure the market eventually chooses for pricing has been the subject of extended academic research. Two prominent approaches that have emerged from different theoretical approaches are the Esscher martingale measure and the minimal relative entropy martingale measure. While both measures ensure the martingale property for the discounted price processes, they approach the problem of market incompleteness in distinct ways. The Esscher measure originates from actuarial science and is closely related to the concept of risk-neutral valuation, whereas the minimal relative entropy measure is rooted in information theory, focusing on minimizing the informational “distance” between the real-world measure and the risk-neutral measure. In this section, we provide the basic background for these measures and demonstrate their equivalence within a one-period market model.

4.1. The Radon–Nikodym Derivative

The Radon–Nikodym derivative is a fundamental concept in measure theory and probability theory, especially when changing between two probability measures. Let P be the real-world probability measure, and Q be another measure that is absolutely continuous with respect to P (meaning that any event with zero probability under P also has zero probability under Q and denoted $Q\ll P$). According to the Radon–Nikodym theorem, there exists a function known as the Radon–Nikodym derivative and denoted by ${\displaystyle \frac{dQ}{dP}}$, which satisfies the following:

$${\mathbb{E}}_Q\left[X\right]={\mathbb{E}}_P\left[X·{\displaystyle \frac{dQ}{dP}}\right]$$

4.2. The Esscher Transform and the Esscher Martingale Measure

The Esscher transform was originally introduced in Esscher (1932), within the field of actuarial science and specifically in the context of collective risk theory, as a method for adjusting the original probability distribution of aggregate claims. By tilting the distribution using an exponential factor, Esscher provided a way to derive a new probability distribution that reflects the desired risk characteristics. This adjusted distribution could be used to calculate insurance premiums in a way that accounts for both the frequency and severity of claims. Later, Gerber and Shiu (1994) applied the Esscher transform to option pricing to adjust the real-world probability measure to a new measure under which the discounted prices of underlying assets are martingales, suggesting thus a way to obtain a unique price in an incomplete market setting.

It should be recalled that, under the First Fundamental Theorem of Asset Pricing, a market that operates under real-world measure P is free of arbitrage if and only if there exists an equivalent martingale measure Q (i.e., a measure Q such that $Q\ll P$ and $P\ll Q$), under which the discounted, under the risk-free rate, asset prices are martingales. In other words, under Q, all assets’ expected returns are equal to the risk-free rate.

The Esscher martingale measure Q is defined by the Radon–Nikodym derivative:

$${\displaystyle \frac{dQ}{dP}}={\displaystyle \frac{exp(\gamma ·R)}{{\mathbb{E}}_P[exp(\gamma ·R)]}}$$

(1)

where $R=(R_1,R_2,\cdots ,R_m)$ is the vector of asset returns, and $\gamma =({\gamma }_1,{\gamma }_2,\cdots ,{\gamma }_m)\in {\mathbb{R}}^m$ is the Esscher parameter vector that needs to be determined and which, in general, reflects different risk aversion levels or loading factors in premium calculations. The parameter $\gamma$ is chosen to satisfy the martingale condition, which requires that the expected value of the return under the new measure Q equals the risk-free rate:

$${\mathbb{E}}_Q\left[R_i\right]=r\mathrm{for}\mathrm{all}i=1,\cdots ,m$$

(2)

Substituting the expression for the Radon–Nikodym derivative into the martingale condition gives the following:

$${\mathbb{E}}_Q\left[R_i\right]={\mathbb{E}}_P\left[R_i·{\displaystyle \frac{dQ}{dP}}\right]={\mathbb{E}}_P\left[R_i·{\displaystyle \frac{exp(\gamma ·R)}{{\mathbb{E}}_P[exp(\gamma ·R)]}}\right]=r$$

Multiplying both sides of the last equation by ${\mathbb{E}}_P[exp(\gamma ·R)]$ results in the following:

$${\mathbb{E}}_P\left[R_{i}exp(\gamma ·R)\right]=r{\mathbb{E}}_P\left[exp(\gamma ·R)\right]$$

(3)

This last set of equations must hold for each asset i, leading to a system of equations that determines a unique Esscher parameter vector $\gamma =({\gamma }_1,\cdots ,{\gamma }_m)$, which adjusts the original measure P to the martingale measure Q.

In other words, the Esscher martingale measure Q is obtained by solving the following system of non-linear equations

$${\mathbb{E}}_P\left[R_{i}exp\left(\sum _{j=1}^m{\gamma }_j{R}_j\right)\right]=r·{\mathbb{E}}_P\left[exp\left(\sum _{j=1}^m{\gamma }_j{R}_j\right)\right]\forall i\in \{1,\cdots ,m\}$$

to obtain the Esscher parameter vector $\gamma$, which reflects the degree of risk adjustment needed for the transformation of the real-world probability measure P into the risk-neutral measure Q.

4.3. The Minimal Relative Entropy Martingale Measure

The minimal relative entropy martingale measure, introduced in Frittelli (2000), takes a different approach by minimizing the Kullback–Leibler divergence (or relative entropy) from the real-world measure P to the risk-neutral measure Q. This approach is grounded in information theory and aims to find the measure Q that is “closest” to P while still ensuring that the discounted asset prices are martingales.

The relative entropy, or Kullback–Leibler divergence, between two probability measures P and Q is defined as:

$$D_{\mathrm{KL}}(Q\parallel P)={\mathbb{E}}_P\left[ln\left({\displaystyle \frac{dQ}{dP}}\right)\right]$$

Relative entropy measures the “information loss” or “distance” when moving from measure P to measure Q. It is always non-negative and satisfies $D_{\mathrm{KL}}(Q\parallel P)=0$ if and only if $Q=P$ almost everywhere. While it is not a true metric (since it is not symmetric), it is often used to quantify the deviation between probability distributions.

In our case, we aim to find the probability measure Q that minimizes the relative entropy with respect to the true probability measure P, ensuring that Q is a martingale measure. The problem is formalized as follows:

$$\underset{{\displaystyle \frac{dQ}{dP}}}{min}{\mathbb{E}}_P\left[{\displaystyle \frac{dQ}{dP}}ln\left({\displaystyle \frac{dQ}{dP}}\right)\right]$$

(4)

subject to the following constraints:

• ${\displaystyle \frac{dQ}{dP}}\ge 0$
• ${\mathbb{E}}_P\left[{\displaystyle \frac{dQ}{dP}}\right]=1$ (Normalization constraint)
• ${\mathbb{E}}_Q\left(R_j\right)=r$ for all $j=1,\cdots ,m$ (Martingale constraint)

Proposition 1. 

The solution to the relative entropy minimization problem (4) is given by:

$${\displaystyle \frac{dQ}{dP}}={\displaystyle \frac{exp(\gamma ·R)}{{\mathbb{E}}_P[exp(\gamma ·R)]}}$$

(5)

where $\gamma =({\gamma }_1,\cdots ,{\gamma }_m)$ is the solution of the system

$${\mathbb{E}}_P\left[R_{i}exp(\gamma ·R)\right]=r·{\mathbb{E}}_P\left[exp(\gamma ·R)\right]\forall i\in \{1,\cdots ,m\}$$

(6)

Proof. 

(See also Frittelli 2000)

The Lagrangian of this constrained optimization problem is defined as follows:

$$\mathcal{L}\left({\displaystyle \frac{dQ}{dP}},\lambda ,\gamma \right)={\mathbb{E}}_P\left[{\displaystyle \frac{dQ}{dP}}ln\left({\displaystyle \frac{dQ}{dP}}\right)\right]+\lambda \left(1-{\mathbb{E}}_P\left[{\displaystyle \frac{dQ}{dP}}\right]\right)+\sum _{j=1}^m{\gamma }_j\left(r-{\mathbb{E}}_Q\left(R_j\right)\right)$$

(7)

where $\lambda$ and $\gamma =({\gamma }_1,\cdots ,{\gamma }_m)$ are the relevant Lagrange multipliers corresponding to the normalization and martingale constraints, respectively.

To find the optimal ${\displaystyle \frac{dQ}{dP}}$, we apply variational calculus to differentiate the Lagrangian with respect to ${\displaystyle \frac{dQ}{dP}}$ and set the derivative equal to zero. The derivative of the relative entropy term ${\mathbb{E}}_P\left[{\displaystyle \frac{dQ}{dP}}ln\left({\displaystyle \frac{dQ}{dP}}\right)\right]$ with respect to ${\displaystyle \frac{dQ}{dP}}$ turns out to be:

$${\displaystyle \frac{\delta \mathcal{L}}{\delta \left(\frac{dQ}{dP}\right)}}=ln\left({\displaystyle \frac{dQ}{dP}}\right)+1+\lambda -\sum _{j=1}^m{\gamma }_j{R}_j=ln\left({\displaystyle \frac{dQ}{dP}}\right)+1+\lambda -\gamma ·R$$

(8)

Setting this equal to zero and solving for ${\displaystyle \frac{dQ}{dP}}$ we obtain:

$${\displaystyle \frac{dQ}{dP}}=exp\left(-1-\lambda +\sum _{j=1}^m{\gamma }_j{R}_j\right)=exp(-1-\lambda )exp(\gamma ·R)$$

(9)

This last equation, together with the normalization constraint ${\mathbb{E}}_P\left[{\displaystyle \frac{dQ}{dP}}\right]=1$, implies the following:

$$exp(-1-\lambda )={\displaystyle \frac{1}{{\mathbb{E}}_P[exp(\gamma ·R)]}}$$

(10)

Therefore, from Equations (9) and (10), we have that the optimal solution for ${\displaystyle \frac{dQ}{dP}}$ is given by the following expression:

$${\displaystyle \frac{dQ}{dP}}={\displaystyle \frac{exp\left(\gamma R\right)}{{\mathbb{E}}_P\left[exp(\gamma ·R)\right]}}$$

(11)

Furthermore, by a change of measure, the martingale constraints ${\mathbb{E}}_Q\left(R_j\right)=r$ imply that ${\mathbb{E}}_P\left[R_j{\displaystyle \frac{dQ}{dP}}\right]=r$ and substituting here the expression of Equation (11), we have that ${\mathbb{E}}_P\left[R_j{\displaystyle \frac{exp\left(\gamma ·R\right)}{{\mathbb{E}}_P\left[exp\left(\gamma ·R\right)\right]}}\right]=r$ which is clearly equivalent to the system

$${\mathbb{E}}_P\left[R_{j}exp(\gamma ·R)\right]=r{\mathbb{E}}_P\left[exp(\gamma ·R)\right]$$

(12)

This last system of equations determines the Lagrange multipliers $\gamma =({\gamma }_1,\cdots ,{\gamma }_m)$. □

The form of the solution is identical to the Esscher martingale measure, with the same vector $\gamma$ determined by the martingale conditions. The minimal relative entropy measure is particularly appealing in incomplete markets because it provides a theoretically justified way to select the “least biased” martingale measure relative to the real-world measure (Frittelli 2000).

Proposition 2. 

In a one-period market model, the Esscher martingale measure and the minimal relative entropy martingale measure are identical, provided they both exist.

Proof. 

As shown in the previous subsections, both the Esscher martingale measure and the minimal relative entropy martingale measure involve an exponential adjustment of the original probability measure P. The key observation is that the parameter vector $\gamma$, which determines this adjustment, is derived as the unique solution to the same system of martingale conditions in both approaches. Therefore, the resulting measures must coincide. □

5. The Induced-Equivalent Portfolio G

In this section, we introduce and explore the concept of portfolio G which emerges naturally from the Esscher or minimal relative entropy martingale measure by treating the Esscher parameter vector (equivalently the martingale constraint Lagrange multipliers), discussed earlier, as determinants of the weighting of the assets that compose G. This portfolio plays a central role in linking the martingale pricing framework to more traditional financial models, like Single Index Models or the Capital Asset Pricing Model (CAPM). By analyzing the construction and properties of portfolio G, we see how it can serve as a bridge between these two approaches.

Proposition 3. 

Let Q be a probability measure such that

$${\displaystyle \frac{dQ}{dP}}={\displaystyle \frac{exp(\gamma ·R)}{{\mathbb{E}}_P[exp(\gamma ·R)]}}$$

(13)

with $\gamma =({\gamma }_1,\dots ,{\gamma }_m)\in {\mathbb{R}}^m$. Let $\mathsf{\Gamma }={\sum }_{j=1}^m{\gamma }_j$ and $g_j={\displaystyle \frac{{\gamma }_j}{\mathsf{\Gamma }}}$. Consider the portfolio G consisting of positions on the assets $S^{\left(j\right)}$ with respective weights $g_j$ for all $j=1,\dots ,m$. Then the martingale condition $E_Q\left[R_i\right]=r$ is equivalent to

$${\mathbb{E}}_P\left[R_{i}exp(\gamma ·R)\right]=r{\mathbb{E}}_P\left[exp(\gamma ·R)\right]$$

(14)

or equivalently to

$$r-{\mathbb{E}}_P\left(R_i\right)={\displaystyle \frac{{\mathit{Cov}}_P(R_i,exp\left(\mathsf{\Gamma }{R}_G\right))}{{\mathbb{E}}_P(exp\left(\mathsf{\Gamma }{R}_G\right))}}$$

(15)

where $R_G={\sum }_{j=1}^m{g}_j{R}_j$.

Proof. 

$$E_Q\left[R_i\right]=r\iff {\mathbb{E}}_P\left[R_i{\displaystyle \frac{dQ}{dP}}\right]=r\iff {\displaystyle \frac{{\mathbb{E}}_P\left[R_{i}exp(\gamma ·R)\right]}{{\mathbb{E}}_P\left[exp(\gamma ·R)\right]}}=r\iff$$

$${\mathbb{E}}_P\left[R_{i}exp(\gamma ·R)\right]=r{\mathbb{E}}_P\left[exp(\gamma ·R)\right]$$

Subtracting ${\mathbb{E}}_P\left(R_i\right){\mathbb{E}}_P(exp(\gamma ·R))$ from both sides and dividing by ${\mathbb{E}}_P(exp(\gamma ·R))$, we obtain:

$$r-{\mathbb{E}}_P\left(R_i\right)={\displaystyle \frac{{\mathrm{Cov}}_P(R_i,exp(\gamma ·R))}{{\mathbb{E}}_P(exp(\gamma ·R))}}$$

Thus,

$$r-{\mathbb{E}}_P\left(R_i\right)={\displaystyle \frac{{\mathrm{Cov}}_P(R_i,exp\left(\mathsf{\Gamma }{R}_G\right))}{{\mathbb{E}}_P(exp\left(\mathsf{\Gamma }{R}_G\right))}}$$

 □

Definition 1. 

Let P be a probability measure. The equivalent portfolio G consists of positions on the traded assets $S^{\left(j\right)}$, $j=1,\dots ,m$ with respective weights $g_j={\displaystyle \frac{{\gamma }_j}{\mathsf{\Gamma }}}$, where

• $\mathsf{\Gamma }={\sum }_{j=1}^m{\gamma }_j$
• $\gamma =({\gamma }_1,\dots ,{\gamma }_m)$ is the solution to the system of equations

  $$r-{\mathbb{E}}_P\left(R_h\right)={\displaystyle \frac{\mathit{Cov}(R_h,exp\left(\mathsf{\Gamma }{R}_G\right))}{{\mathbb{E}}_P(exp\left(\mathsf{\Gamma }{R}_G\right))}}$$

  (16)

  with $R_G={\sum }_{j=1}^m{g}_j{R}_j$ denoting the return of portfolio G.

Remark 1. 

It is a straightforward calculation to see that if $R_{\mathsf{\Pi }}={\sum }_{j=1}^m{w}_j{R}_j$ denotes the return of a portfolio consisting of positions on the risky assets $S^{\left(1\right)},\dots ,S^{\left(m\right)}$ with respective weights $w_1,\dots ,w_m$ (i.e., $w_1+\dots +w_m=1$), then:

$$r-{\mathbb{E}}_P\left(R_{\mathsf{\Pi }}\right)={\displaystyle \frac{\mathit{Cov}(R_{\mathsf{\Pi }},exp\left(\mathsf{\Gamma }{R}_G\right))}{{\mathbb{E}}_P(exp\left(\mathsf{\Gamma }{R}_G\right))}}$$

(17)

Definition 2 

(The Generalized Beta). Let Π be some portfolio consisting of positions on the risky traded assets $S^{\left(1\right)},\dots ,S^{\left(m\right)}$ and let F be another asset, portfolio, or even a contingent claim in this market. The generalized beta of F with respect to Π is defined as follows:

$${\beta }^{\ast }(F,\mathsf{\Pi }):={\displaystyle \frac{\mathit{Cov}(R_F,exp\left(\mathsf{\Gamma }{R}_G\right))}{\mathit{Cov}(R_{\mathsf{\Pi }},exp\left(\mathsf{\Gamma }{R}_G\right))}}$$

(18)

We can now show that, independently of the distribution of the various assets’ returns, the Esscher martingale (or equivalently, the minimal relative entropy) pricing criterion is equivalent to a SIM pricing scheme. More precisely, the price obtained when using the martingale criterion is the same as the price obtained when using a SIM pricing scheme, where the classical $\beta$ coefficient has been replaced by the generalized ${\beta }^{\ast }$ coefficient.

Theorem 1. 

Let Π be a portfolio consisting of positions on $S^{\left(1\right)},\dots ,S^{\left(m\right)}$, and let F be an asset or even a contingent claim. Let $R_F={\displaystyle \frac{F_T}{F_0}}-1$ be the return of F. Consider the following equation:

$${\mathbb{E}}_P\left(R_F\right)-r={\beta }^{\ast }(F,\mathsf{\Pi })({\mathbb{E}}_P\left(R_{\mathsf{\Pi }}\right)-r)$$

(19)

where

$${\beta }^{\ast }(F,\mathsf{\Pi })={\displaystyle \frac{\mathit{Cov}(R_F,exp\left(\mathsf{\Gamma }{R}_G\right))}{\mathit{Cov}(R_{\mathsf{\Pi }},exp\left(\mathsf{\Gamma }{R}_G\right))}}$$

(20)

with $\gamma =({\gamma }_1,\dots ,{\gamma }_m)$ the unique solution of the system

$$r-{\mathbb{E}}_P\left(R^{\left(h\right)}\right)={\displaystyle \frac{\mathit{Cov}(R_h,exp\left(\mathsf{\Gamma }{R}_G\right))}{{\mathbb{E}}_P(exp\left(\mathsf{\Gamma }{R}_G\right))}}\mathrm{for}h=1,\dots ,m$$

(21)

Then, $F_0$ satisfies Equation (19) if and only if $(1+r)F_0=E_Q\left(F_T\right)$, where Q is the Esscher (equivalently the minimal relative entropy) martingale measure.

Proof. 

$${\mathbb{E}}_P\left(R_F\right)-r={\displaystyle \frac{\mathrm{Cov}(R_F,exp\left(\mathsf{\Gamma }{R}_G\right))}{\mathrm{Cov}(R_{\mathsf{\Pi }},exp\left(\mathsf{\Gamma }{R}_G\right))}}({\mathbb{E}}_P\left(R_{\mathsf{\Pi }}\right)-r)\iff$$

$${\mathbb{E}}_P\left(R_F\right)-r={\displaystyle \frac{\mathrm{Cov}(R_F,exp\left(\mathsf{\Gamma }{R}_G\right))}{(r-{\mathbb{E}}_P\left(R_{\mathsf{\Pi }}\right)){\mathbb{E}}_P(exp\left(\mathsf{\Gamma }{R}_G\right))}}({\mathbb{E}}_P\left(R_{\mathsf{\Pi }}\right)-r)\iff$$

$${\mathbb{E}}_P\left(R_F\right){\mathbb{E}}_P(exp\left(\mathsf{\Gamma }{R}_G\right))+\mathrm{Cov}(R_F,exp\left(\mathsf{\Gamma }{R}_G\right))=r{\mathbb{E}}_P(exp\left(\mathsf{\Gamma }{R}_G\right))\iff$$

$${\mathbb{E}}_P(R_{F}exp\left(\mathsf{\Gamma }{R}_G\right))=r{\mathbb{E}}_P(exp\left(\mathsf{\Gamma }{R}_G\right))\iff$$

$${\mathbb{E}}_P\left(\left({\displaystyle \frac{F_T}{F_0}}-1\right)exp\left(\mathsf{\Gamma }{R}_G\right)\right)=r{\mathbb{E}}_P(exp\left(\mathsf{\Gamma }{R}_G\right))\iff$$

$${\displaystyle \frac{1}{F_0}}{\mathbb{E}}_P(F_{T}exp\left(\mathsf{\Gamma }{R}_G\right))=(1+r){\mathbb{E}}_P(exp\left(\mathsf{\Gamma }{R}_G\right))\iff$$

$$(1+r)F_0={\mathbb{E}}_P\left(F_T{\displaystyle \frac{dQ}{dP}}\right)$$

 □

6. The Portfolio G When Returns Are Elliptical

In our earlier analysis, we considered a general approach to deriving the portfolio G. Here, we restrict the analysis to the class of elliptical returns (Fang 2018). These distributions generalize multivariate normal distributions and offer freedom to model asset returns with heavier or lighter tails but are still symmetric. Within this framework, we demonstrate that portfolio G coincides with the classical tangency portfolio, also known as the maximum Sharpe ratio portfolio.

Theorem 2. 

When returns R are elliptical, the portfolio G derived from the Esscher (or minimal relative entropy) martingale measures coincides with the classical tangency portfolio (maximum Sharpe ratio portfolio).

Proof. 

Let the returns R of the risky assets be elliptical, meaning that R can be expressed as follows:

$$R=\mu +AZ$$

(22)

where $\mu$ is the vector of expected returns, Z is a standardized elliptical random vector with mean zero and covariance matrix I, and A is a matrix such that the covariance matrix of returns is $\mathsf{\Sigma }=A{A}^T$.

The Esscher (or minimal relative entropy) martingale measure Q is obtained via the Radon–Nikodym derivative of the following form:

$${\displaystyle \frac{dQ}{dP}}={\displaystyle \frac{exp(\gamma ·R)}{{\mathbb{E}}_P[exp(\gamma ·R)]}}$$

(23)

where, as we have seen in Proposition 3, $\gamma$ is the vector that solves the following martingale condition:

$${\mathbb{E}}_P[Rexp(\gamma ·R)]=r{\mathbb{E}}_P[exp(\gamma ·R)]$$

(24)

By Equation (22), the right-hand side of Equation (24) becomes the following:

$$r{\mathbb{E}}_P[exp(\gamma ·R)]=r{\mathbb{E}}_P[exp\left({\gamma }^T(\mu +AZ)\right)]=rexp\left({\gamma }^T\mu \right){\mathbb{E}}_P[exp\left({\gamma }^{T}AZ\right)]$$

(25)

and since, for elliptical distributions,

$${\mathbb{E}}_P[exp\left({\gamma }^{T}AZ\right)]=exp\left({\displaystyle \frac{1}{2}}{\gamma }^T\mathsf{\Sigma }\gamma \right)$$

(26)

it follows that

$$r{\mathbb{E}}_P[exp(\gamma ·R)]=rexp\left({\gamma }^T\mu \right)exp\left({\displaystyle \frac{1}{2}}{\gamma }^T\mathsf{\Sigma }\gamma \right)$$

(27)

On the other hand, by Equation (22), the left-hand side of Equation (24) becomes the following:

$${\mathbb{E}}_P[Rexp(\gamma ·R)]={\mathbb{E}}_P[(\mu +AZ)exp\left({\gamma }^T(\mu +AZ)\right)]$$

(28)

which, after trivial rearrangements, is equivalent to the following:

$${\mathbb{E}}_P[Rexp(\gamma ·R)]=exp\left({\gamma }^T\mu \right)\left[\mu {\mathbb{E}}_P[exp\left({\gamma }^{T}AZ\right)]+A{\mathbb{E}}_P[Zexp\left({\gamma }^{T}AZ\right)]\right]$$

(29)

Differentiating now, with respect to $\gamma$, the Moment Generating Function of ${\gamma }^{T}AZ$, and using Equation (26), we have that

$${\nabla }_{\gamma }{\mathbb{E}}_P[exp\left({\gamma }^{T}AZ\right)]={\nabla }_{\gamma }exp\left({\displaystyle \frac{1}{2}}{\gamma }^T\mathsf{\Sigma }\gamma \right)=exp\left({\displaystyle \frac{1}{2}}{\gamma }^T\mathsf{\Sigma }\gamma \right)\mathsf{\Sigma }\gamma$$

(30)

On the other hand,

$${\nabla }_{\gamma }{\mathbb{E}}_P[exp\left({\gamma }^{T}AZ\right)]={\mathbb{E}}_P[{\nabla }_{\gamma }exp\left({\gamma }^{T}AZ\right)]=A{\mathbb{E}}_P[Zexp\left({\gamma }^{T}AZ\right)]$$

(31)

Then, Equations (30) and (31), together with the assumption that $A^{-1}$ exists, readily imply that

$${\mathbb{E}}_P[Zexp\left({\gamma }^{T}AZ\right)]=A^{-1}exp\left({\displaystyle \frac{1}{2}}{\gamma }^T\mathsf{\Sigma }\gamma \right)\mathsf{\Sigma }\gamma$$

(32)

Now, Equation (29) together with Equations (26) and (32) imply that

$${\mathbb{E}}_P[Rexp(\gamma ·R)]=exp\left({\gamma }^T\mu \right)\left[\mu exp\left({\displaystyle \frac{1}{2}}{\gamma }^T\mathsf{\Sigma }\gamma \right)+A{A}^{-1}exp\left({\displaystyle \frac{1}{2}}{\gamma }^T\mathsf{\Sigma }\gamma \right)\mathsf{\Sigma }\gamma \right]$$

(33)

which is clearly equivalent to

$${\mathbb{E}}_P[Rexp(\gamma ·R)]=exp\left({\gamma }^T\mu \right)\left(\mu +\mathsf{\Sigma }\gamma \right)exp\left({\displaystyle \frac{1}{2}}{\gamma }^T\mathsf{\Sigma }\gamma \right)$$

(34)

Now the martingale condition of Equation (24), together with Equations (27) and (34), imply that

$$exp\left({\gamma }^T\mu \right)\left(\mu +\mathsf{\Sigma }\gamma \right)exp\left({\displaystyle \frac{1}{2}}{\gamma }^T\mathsf{\Sigma }\gamma \right)=rexp\left({\gamma }^T\mu \right)exp\left({\displaystyle \frac{1}{2}}{\gamma }^T\mathsf{\Sigma }\gamma \right)$$

which is equivalent to the relation

$$\mu +\mathsf{\Sigma }\gamma =r\mathbf{1}⟺\gamma ={\mathsf{\Sigma }}^{-1}(r\mathbf{1}-\mu )$$

(35)

The portfolio G is defined by the weights $g_i$, which are derived from the components of $\gamma$:

$$g_i={\displaystyle \frac{{\gamma }_i}{\mathsf{\Gamma }}},\mathrm{where}\mathsf{\Gamma }=\sum _{j=1}^m{\gamma }_j$$

(36)

Substituting in this last Equation (36) the expression (35) for $\gamma$, we obtain the following:

$$g_i={\displaystyle \frac{{\left[{\mathsf{\Sigma }}^{-1}(r\mathbf{1}-\mu )\right]}_i}{{\sum }_{j=1}^m{\left[{\mathsf{\Sigma }}^{-1}(r\mathbf{1}-\mu )\right]}_j}}={\displaystyle \frac{{\left[{\mathsf{\Sigma }}^{-1}(\mu -r\mathbf{1})\right]}_i}{{\sum }_{j=1}^m{\left[{\mathsf{\Sigma }}^{-1}(\mu -r\mathbf{1})\right]}_j}}$$

(37)

Finally, it is a standard result (Bodie et al. 2018) that the tangency portfolio weights are given by the following:

$$w_i^{\mathrm{Tangency}}={\displaystyle \frac{{\left[{\mathsf{\Sigma }}^{-1}(\mu -r\mathbf{1})\right]}_i}{{\sum }_{j=1}^m{\left[{\mathsf{\Sigma }}^{-1}(\mu -r\mathbf{1})\right]}_j}}$$

(38)

It is clear now from Equations (37) and (38) that portfolio G and the tangency portfolio coincide. □

7. The Case of Jointly Normal Returns

In this section, we make the assumption that asset returns are jointly normal. In this case, using Stein’s lemma for covariance leads to interesting simplifications; $-\mathsf{\Gamma }$ obtains a natural interpretation as the Sharpe ratio of portfolio G measured in terms of its volatility, the generalized beta coefficient with respect to portfolio G takes the familiar form of the classical beta coefficient, and more importantly, the Esscher or minimal relative entropy martingale pricing is equivalent to the classical CAPM pricing, with portfolio G playing the role of the market portfolio.

So suppose that $R_{\mathsf{\Pi }}$ and $R_G$ are jointly normal. Then, according to Stein’s lemma:

$$\mathrm{Cov}(R_{\mathsf{\Pi }},exp\left(\mathsf{\Gamma }{R}_G\right))=\mathsf{\Gamma }{\mathbb{E}}_P[exp\left(\mathsf{\Gamma }{R}_G\right)]\mathrm{Cov}(R_{\mathsf{\Pi }},R_G)$$

(39)

and the following remarks follow:

Remark 2. 

Suppose that $R_{\mathsf{\Pi }}$ and $R_G$ are jointly normal. Then,

$$\mathsf{\Gamma }={\displaystyle \frac{r-{\mathbb{E}}_P\left(R_{\mathsf{\Pi }}\right)}{Cov(R_{\mathsf{\Pi }},R_G)}}$$

(40)

In particular, if portfolio Π is taken to be G, then the previous equation implies that

$$-\mathsf{\Gamma }={\displaystyle \frac{{\mathbb{E}}_P\left(R_G\right)-r}{Var\left(R_G\right)}}$$

which is equivalent to

$$-\mathsf{\Gamma }\mathrm{stdv}\left(R_G\right)={\displaystyle \frac{{\mathbb{E}}_P\left(R_G\right)-r}{\mathrm{stdv}\left(R_G\right)}}$$

(41)

This last equation shows that $-\mathsf{\Gamma }$ is a measure of the Sharpe ratio of this particular portfolio G in standard deviation units.

Proof. 

Immediate from Equation (17) and Stein’s Lemma. □

Remark 3. 

If $R_F$ and $R_G$ are jointly normal, and if $R_{\mathsf{\Pi }}$ and $R_G$ are also jointly normal, then:

$${\beta }^{\ast }(F,\mathsf{\Pi }):={\displaystyle \frac{\mathit{Cov}(R_F,R_G)}{\mathit{Cov}(R_{\mathsf{\Pi }},R_G)}}$$

(42)

In particular,

$${\beta }^{\ast }(F,G):={\displaystyle \frac{\mathit{Cov}(R_F,R_G)}{\mathit{Var}\left(R_G\right)}}$$

(43)

This coincides with the familiar formula for beta in a classical SIM.

Proof. 

Immediate from Stein’s Lemma and the definition of generalized beta (18). □

Corollary 1. 

Suppose $R_F$ and $R_G$ are jointly normal. Then the minimal relative entropy (equivalently Esscher) martingale pricing is equivalent to CAPM pricing, where the role of the market portfolio is played by portfolio G.

Proof. 

Theorem 1 and Equation (43) imply:

$${\mathbb{E}}_P\left(R_F\right)-r={\beta }^{\ast }(F,G)({\mathbb{E}}_P\left(R_G\right)-r)$$

(44)

and:

$${\beta }^{\ast }(F,G)={\displaystyle \frac{\mathrm{Cov}(R_F,R_G)}{\mathrm{Var}\left(R_G\right)}}$$

(45)

Thus, the pricing relationship under the minimal relative entropy measure is the same as under CAPM, with portfolio G acting as the market portfolio. □

8. Summary and Conclusions

In this work, we explored the relationship between martingale measures and traditional asset pricing models within the context of a one-period market with multiple risky assets and a risk-free asset. We saw that the minimal relative entropy martingale measure and the Esscher martingale measure coincide in such a market, provided they exist. The Radon–Nikodym derivative of the Esscher or minimal relative entropy martingale measure allowed us to derive a portfolio naturally, termed portfolio G, which serves as a bridge between martingale pricing and single index models like the CAPM.

In particular, our analysis revealed that, when asset returns are elliptical, portfolio G coincides with the classical tangency portfolio, that is, the maximum Sharpe ratio portfolio. Furthermore, we saw that when asset returns are jointly normal, martingale pricing under these measures and CAPM pricing are equivalent. The fact that portfolio G coincides with the tangency portfolio in this setting suggests that martingale pricing can be interpreted within a well-established framework, thus providing a useful tool for both academics and practitioners.

However, given the criticism that CAPM has received, one naturally wonders whether this criticism reflects a valid criticism of the choice of the minimal relative entropy or the Esscher martingale measure as the “appropriate” martingale measure in an incomplete market.

The results of this paper naturally invite the more general question of investigating conditions and mechanisms through which martingale measures can be represented as single (or even multi) index models via specific functions and portfolios. Research in this direction would be significant to our understanding of asset pricing in both complete and, in particular, incomplete markets. For example, one could consider the following general question: Given a martingale measure Q, does there always exist a function f and a corresponding portfolio $G=G\left(Q\right)$ such that pricing under Q is equivalent to a Single Index Model of the form

$$E\left(R\right)-r={\beta }^{\ast }\left(E\left(R_G\right)-r\right),$$

where

$${\beta }^{\ast }={\displaystyle \frac{\mathrm{Cov}(R,f\left(R_G\right))}{\mathrm{Cov}(R_G,f\left(R_G\right))}},$$

with R representing the return of some asset and $R_G$ the return of the portfolio G? In the case of Q being the Esscher or the minimal relative entropy martingale measure, an answer has been given in this paper. However, one could possibly explore different classes of martingale measures and identify necessary and sufficient conditions for the existence of f and G. Then, uniqueness questions would arise, and in the case of non uniqueness, the characterization of all possible f and G that satisfy the equivalence could be investigated.