A note on heterogeneous beliefs with CRRA utilities

The setup is exactly the same as in Brown & Rogers (2009) and readers are assumed to be familiar with this paper. The only difference is that agents now have CRRA utility, with some integer coefficient of relative risk aversion, assumed to be greater than 1.

As before there is a single risky asset producing dividends continuously in time. There is also a riskless asset in zero net supply. Agents all optimise their expected utility of consumption, but they each work under their own measure. Agent $j$ is assumed to work under measure $\mathbb{P}^{j}$. There is also a reference probability measure, which is denoted by $\mathbb{P}^{0}$.

We also assume that the agents have CRRA utility, with integer coefficient of relative risk aversion. Thus, we take:

$\displaystyle U_{j}(t,x)=e^{-\rho_{j}t}\dfrac{x^{1-R}}{1-R}$

for some integer $R\in\{2,3,...\}$. The case in which agents have a constant of relative risk aversion equal to one is the case of logarithmic agents and was considered in Brown & Rogers (2009).

The paper of Brown & Rogers (2009) tells us that the state price density satisfies:

$\displaystyle\zeta_{t}\nu_{j}=U_{j}^{\prime}(t,c_{t}^{j})\Lambda_{t}^{j}$

where $\zeta_{t}$ is the state price density, $\nu_{j}$ is a constant particular to agent $j$ and $\Lambda_{t}^{j}$ is the change of measure martingale for agent $j$ ¹¹1See Brown & Rogers (2009) for further explanation and derivation of this expression.. Substituting for our expression for $U_{j}$ and rearranging gives:

$\displaystyle c_{t}^{j}=\zeta_{t}^{-1/R}\left(\dfrac{e^{-\rho_{j}t}\Lambda_{t}^{j}}{\nu_{j}}\right)^{1/R}$

Since the total consumption must equal the total output of the economy, we obtain:

$\displaystyle\zeta_{t}=\delta_{t}^{-R}\left(\sum_{i}\left(\dfrac{e^{-\rho_{i}t}\Lambda_{t}^{i}}{\nu_{j}}\right)^{1/R}\right)^{R}$

(2.1)

where $\delta_{t}$ is the dividend process produced by the risky asset. Using this state price density we can then compute the wealth of agents, the price of the risky asset and the riskless rate.

The wealth of agent $j$ is given by:

	$\displaystyle w_{t}^{j}$	$\displaystyle=\mathbb{E}_{t}^{0}\left[\int_{t}^{\infty}\dfrac{c_{u}^{j}\zeta_{u}}{\zeta_{t}}du\right]$
		$\displaystyle=\zeta_{t}^{-1}\mathbb{E}_{t}^{0}\left[\int_{t}^{\infty}\left(\Lambda_{u}^{j}e^{-\rho_{j}u}/\nu_{j}\right)^{1/R}\delta_{u}^{1-R}\left(\sum_{i}\left(e^{-\rho_{i}u}\Lambda_{u}^{i}/\nu_{i}\right)^{1/R}\right)^{R-1}du\right]$		(2.2)

Let $\gamma_{i}=\log\nu_{i}$. We will assume that

$\displaystyle d\Lambda_{t}^{j}=\Lambda_{t}^{j}\alpha_{j}dX_{t}$

(2.3)

where $(X_{t})_{t\geq 0}$ is a standard Brownian motion under the reference measure $\mathbb{P}^{0}$. Then we have that

$\displaystyle\left(e^{-\rho_{i}u}\Lambda_{u}^{i}/\nu_{i}\right)^{1/R}=\exp\{-\dfrac{(\rho_{i}u+\gamma_{i})}{R}+\dfrac{\alpha_{i}}{R}X_{u}-\dfrac{1}{2}\dfrac{\alpha_{i}^{2}}{R}u\}$

Since $R$ is an integer²²2Note that we have not needed the assumption that $R$ is an integer until this point, we may perform a multinomial expansion on the final term in expression (2.2) to give:

$$\left(\sum_{i}\left(e^{-\rho_{i}u}\Lambda_{u}^{i}/\nu_{i}\right)^{1/R}\right)^{R-1}=\sum_{|\beta|=R-1}\binom{R-1}{\beta}\exp\{-\dfrac{(\rho\cdot\beta u+\gamma\cdot\beta)}{R}\\ +\dfrac{\alpha\cdot\beta}{R}X_{u}-\dfrac{1}{2}\dfrac{\alpha^{2}\cdot\beta}{R}u\}$$

Here, $\alpha$ and $\rho$ denote the vectors of all the different $\alpha_{j}^{\prime}s$ and $\rho_{j}$’s respectively. The $\binom{R-1}{\beta}$ term denotes the multinomial coefficient $\binom{R-1}{\beta_{1},\beta_{2},...,\beta_{J}}$. The summation is over all vectors $\beta$ of length $J$, with each entry taking values in $\{0,1,...,R-1\}$ and such that $\sum_{i}\beta_{i}=R-1$.

As in Brown & Rogers (2009), we assume that the dividend at time $u$ is given by:

$\displaystyle\delta_{u}=\delta_{0}\exp\{\sigma X_{u}+\left(\alpha^{*}\sigma-\dfrac{\sigma^{2}}{2}\right)u\}$

Putting this together yields:

$$w_{t}^{j}=\delta_{0}^{1-R}\zeta_{t}^{-1}\int_{t}^{\infty}\sum_{|\beta|=R-1}\binom{R-1}{\beta}\mathbb{E}_{t}^{0}\big{[}\exp\{-\dfrac{\rho_{j}u+\gamma_{j}}{R}+\dfrac{\alpha_{j}}{R}X_{u}-\dfrac{1}{2}\dfrac{\alpha_{j}^{2}}{R}u\\ +(1-R)\sigma X_{u}+(\alpha^{*}\sigma-\dfrac{\sigma^{2}}{2})(1-R)u-\dfrac{\rho\cdot\beta u+\gamma\cdot\beta}{R}+\dfrac{\alpha\cdot\beta}{R}X_{u}-\dfrac{1}{2}\dfrac{\alpha^{2}\cdot\beta}{R}u\}\big{]}du$$

It is then straightforward to compute the conditional expectation. We obtain:

$$w_{t}^{j}=\delta_{0}^{1-R}\zeta_{t}^{-1}\int_{t}^{\infty}\sum_{|\beta|=R-1}\binom{R-1}{\beta}\exp\{\left(\dfrac{\alpha_{j}+\alpha\cdot\beta}{R}+(1-R)\sigma\right)X_{t}\\ +\dfrac{1}{2}\left(\dfrac{\alpha_{j}+\alpha\cdot\beta}{R}+(1-R)\sigma\right)^{2}(u-t)\}\\ \exp\{-\dfrac{\gamma_{j}+\gamma\cdot\beta}{R}-\left[\dfrac{\rho_{j}+\rho\cdot\beta}{R}+\dfrac{\alpha_{j}^{2}+\alpha^{2}\cdot\beta}{2R}+\left(\dfrac{\sigma^{2}}{2}-\alpha^{*}\sigma\right)(1-R)\right]u\}du$$

Finally, we may compute the integral to obtain:

$$w_{t}^{j}=\delta_{0}^{1-R}\zeta_{t}^{-1}\sum_{|\beta|=R-1}\binom{R-1}{\beta}\exp\{\left(\dfrac{\alpha_{j}+\alpha\cdot\beta}{R}+(1-R)\sigma\right)X_{t}-\dfrac{\gamma_{j}+\gamma\cdot\beta}{R}\}\\ \exp\{-\left[\dfrac{\rho_{j}+\rho\cdot\beta}{R}+\dfrac{\alpha_{j}^{2}+\alpha^{2}\cdot\beta}{2R}+\left(\dfrac{\sigma^{2}}{2}-\alpha^{*}\sigma\right)(1-R)\right]t\}\\ \left[\dfrac{\rho_{j}+\rho\cdot\beta}{R}+\dfrac{\alpha_{j}^{2}+\alpha^{2}\cdot\beta}{2R}+\left(\dfrac{\sigma^{2}}{2}-\alpha^{*}\sigma\right)(1-R)-\dfrac{1}{2}\left(\dfrac{\alpha_{j}+\alpha\cdot\beta}{R}+(1-R)\sigma\right)^{2}\right]^{-1}$$

where we have assumed that the final line of this expression is positive for all $\beta$, in order that the integral is finite³³3A sufficient condition for the integral to be finite is that $\displaystyle\min_{i}(\rho_{i}+\dfrac{\alpha_{i}^{2}}{2})+\left(\dfrac{\sigma^{2}}{2}-\alpha^{*}\sigma\right)(1-R)-\dfrac{1}{2}\max_{i}\{((R-1)\sigma-\alpha_{i})^{2}\}\geq 0$ . We may also use our expression for $\delta_{t}$ to obtain:

$$w_{t}^{j}=\delta_{t}^{1-R}\zeta_{t}^{-1}\sum_{|\beta|=R-1}\binom{R-1}{\beta}\exp\{\dfrac{\alpha_{j}+\alpha\cdot\beta}{R}X_{t}-\dfrac{\gamma_{j}+\gamma\cdot\beta}{R}-\left[\dfrac{\rho_{j}+\rho\cdot\beta}{R}+\dfrac{\alpha_{j}^{2}+\alpha^{2}\cdot\beta}{2R}\right]t\}\\ \left[\dfrac{\rho_{j}+\rho\cdot\beta}{R}+\dfrac{\alpha_{j}^{2}+\alpha^{2}\cdot\beta}{2R}+\left(\dfrac{\sigma^{2}}{2}-\alpha^{*}\sigma\right)(1-R)-\dfrac{1}{2}\left(\dfrac{\alpha_{j}+\alpha\cdot\beta}{R}+(1-R)\sigma\right)^{2}\right]^{-1}$$

(2.4)

To compute the stock price, we may simply sum the wealth (or compute the net present value of future dividends directly) to obtain:

$$S_{t}=\delta_{t}^{1-R}\zeta_{t}^{-1}\sum_{|\beta|=R}\binom{R}{\beta}\exp\{\dfrac{\alpha\cdot\beta}{R}X_{t}-\dfrac{\gamma\cdot\beta}{R}-\left[\dfrac{\rho\cdot\beta}{R}+\dfrac{\alpha^{2}\cdot\beta}{2R}\right]t\}\\ \left[\dfrac{\rho\cdot\beta}{R}+\dfrac{\alpha^{2}\cdot\beta}{2R}+\left(\dfrac{\sigma^{2}}{2}-\alpha^{*}\sigma\right)(1-R)-\dfrac{1}{2}\left(\dfrac{\alpha\cdot\beta}{R}+(1-R)\sigma\right)^{2}\right]^{-1}$$

(2.5)

Using the expression for the state price density given in (2.1), we obtain the following expression for the price-dividend ratio:

$$S_{t}/\delta_{t}=\left(\sum_{i}\exp\{-\dfrac{\rho_{i}}{R}t-\dfrac{\gamma_{i}}{R}+\dfrac{\alpha_{i}}{R}X_{t}-\dfrac{\alpha_{i}^{2}}{2R}t\}\right)^{-R}\\ \sum_{|\beta|=R}\binom{R}{\beta}\frac{\exp\{\dfrac{\alpha\cdot\beta}{R}X_{t}-\dfrac{\gamma\cdot\beta}{R}-\left[\dfrac{\rho\cdot\beta}{R}+\dfrac{\alpha^{2}\cdot\beta}{2R}\right]t\}}{\left[\dfrac{\rho\cdot\beta}{R}+\dfrac{\alpha^{2}\cdot\beta}{2R}+\left(\dfrac{\sigma^{2}}{2}-\alpha^{*}\sigma\right)(1-R)-\dfrac{1}{2}\left(\dfrac{\alpha\cdot\beta}{R}+(1-R)\sigma\right)^{2}\right]}$$

We will now derive an expression for the riskless rate. Define

$\displaystyle L_{t}\equiv\delta_{t}^{R}\zeta_{t}=\sum_{|\beta|=R}\binom{R}{\beta}\exp\{\dfrac{\alpha\cdot\beta}{R}X_{t}-\dfrac{\gamma\cdot\beta}{R}-\left[\dfrac{\rho\cdot\beta}{R}+\dfrac{\alpha^{2}\cdot\beta}{2R}\right]t\}$

Performing an Itô expansion on $L$ we obtain:

$\displaystyle dL_{t}=L_{t}(\bar{\alpha}_{t}dX_{t}-\bar{\rho}_{t}dt)$

where

	$\displaystyle\bar{\alpha}_{t}$	$\displaystyle=L_{t}^{-1}\sum_{|\beta|=R}\binom{R}{\beta}\dfrac{\alpha\cdot\beta}{R}\exp\{\dfrac{\alpha\cdot\beta}{R}X_{t}-\dfrac{\gamma\cdot\beta}{R}-\left[\dfrac{\rho\cdot\beta}{R}+\dfrac{\alpha^{2}\cdot\beta}{2R}\right]t\}$
	$\displaystyle\bar{\rho}_{t}$	$\displaystyle=L_{t}^{-1}\sum_{|\beta|=R}\binom{R}{\beta}\left(\dfrac{\rho\cdot\beta}{R}+\dfrac{\alpha^{2}\cdot\beta}{2R}-\dfrac{1}{2}\left(\dfrac{\alpha\cdot\beta}{R}\right)^{2}\right)\exp\{\dfrac{\alpha\cdot\beta}{R}X_{t}-\dfrac{\gamma\cdot\beta}{R}-\left[\dfrac{\rho\cdot\beta}{R}+\dfrac{\alpha^{2}\cdot\beta}{2R}\right]t\}$

An Itô expansion on $\zeta_{t}=L_{t}\delta_{t}^{-R}$ then gives:

$\displaystyle d\zeta_{t}=\zeta_{t}(-r_{t}dt-\kappa_{t}dX_{t})$

where

$\displaystyle r_{t}=\bar{\rho}_{t}+R\sigma(\alpha^{*}+\bar{\alpha}_{t})-\dfrac{\sigma^{2}R(R+1)}{2},\qquad\kappa_{t}=R\sigma-\bar{\alpha}_{t}$

We will now explore the volatility and drift of the stock. First define:

$\displaystyle Z_{t}=\sum_{|\beta|=R}\frac{\binom{R}{\beta}\exp\{\dfrac{\alpha\cdot\beta}{R}X_{t}-\dfrac{\gamma\cdot\beta}{R}-\left[\dfrac{\rho\cdot\beta}{R}+\dfrac{\alpha^{2}\cdot\beta}{2R}\right]t\}}{\dfrac{\rho\cdot\beta}{R}+\dfrac{\alpha^{2}\cdot\beta}{2R}+\left(\dfrac{\sigma^{2}}{2}-\alpha^{*}\sigma\right)(1-R)-\dfrac{1}{2}\left(\dfrac{\alpha\cdot\beta}{R}+(1-R)\sigma\right)^{2}}$

We may then deduce that:

$\displaystyle dZ_{t}=Z_{t}(\tilde{\alpha}_{t}dX_{t}-\tilde{\rho}_{t}dt)$

where

	$\displaystyle\tilde{\alpha}_{t}$	$\displaystyle=Z_{t}^{-1}\sum_{|\beta|=R}\frac{\binom{R}{\beta}\dfrac{\alpha\cdot\beta}{R}\exp\{\dfrac{\alpha\cdot\beta}{R}X_{t}-\dfrac{\gamma\cdot\beta}{R}-\left[\dfrac{\rho\cdot\beta}{R}+\dfrac{\alpha^{2}\cdot\beta}{2R}\right]t\}}{\dfrac{\rho\cdot\beta}{R}+\dfrac{\alpha^{2}\cdot\beta}{2R}+\left(\dfrac{\sigma^{2}}{2}-\alpha^{*}\sigma\right)(1-R)-\dfrac{1}{2}\left(\dfrac{\alpha\cdot\beta}{R}+(1-R)\sigma\right)^{2}}$
	$\displaystyle\tilde{\rho}_{t}$	$\displaystyle=Z_{t}^{-1}\sum_{|\beta|=R}\frac{\binom{R}{\beta}\left(\dfrac{\rho\cdot\beta}{R}+\dfrac{\alpha^{2}\cdot\beta}{2R}-\dfrac{1}{2}\left(\dfrac{\alpha\cdot\beta}{R}\right)^{2}\right)\exp\{\dfrac{\alpha\cdot\beta}{R}X_{t}-\dfrac{\gamma\cdot\beta}{R}-\left[\dfrac{\rho\cdot\beta}{R}+\dfrac{\alpha^{2}\cdot\beta}{2R}\right]t\}}{\dfrac{\rho\cdot\beta}{R}+\dfrac{\alpha^{2}\cdot\beta}{2R}+\left(\dfrac{\sigma^{2}}{2}-\alpha^{*}\sigma\right)(1-R)-\dfrac{1}{2}\left(\dfrac{\alpha\cdot\beta}{R}+(1-R)\sigma\right)^{2}}$

Using expression (2.5), we may then deduce that the stock price process satisfies:

$\displaystyle dS_{t}=S_{t}\{(\sigma+\tilde{\alpha}_{t}-\bar{\alpha}_{t})dX_{t}+(\bar{\rho}_{t}-\tilde{\rho}_{t}+\sigma\alpha^{*}+(\tilde{\alpha}-\bar{\alpha})(\sigma-\bar{\alpha}))dt\}$

In particular, we may note that the volatility is given by:

$\displaystyle\sigma_{t}^{S}=\sigma+\tilde{\alpha}_{t}-\bar{\alpha}_{t}$

In order to understand the volume of trade, we must first understand the portfolio held by the different agents. Note that the wealth of agent $j$ must satisfy:

$\displaystyle dw_{t}^{j}=\pi_{t}^{j}(dS_{t}+\delta_{t})-c_{t}^{j}dt+(w_{t}^{j}-\pi_{t}^{j}S_{t})r_{t}$

(2.6)

where $\pi_{t}^{j}$ denotes the amount of the risky asset held by agent agent $j$ at time $t$. But we also know that the wealth of agent $j$ is given by equation (2.4). If we perform an Itô expansion on equation (2.4), then we may compare with equation (2.6) and read off the portfolio held by agent $j$.

Proceeding in this way, first let

$\displaystyle Z_{t}^{j}=\sum_{|\beta|=R-1}\frac{\binom{R-1}{\beta}\exp\{\dfrac{\alpha_{j}+\alpha\cdot\beta}{R}X_{t}-\dfrac{\gamma_{j}+\gamma\cdot\beta}{R}-\left[\dfrac{\rho_{j}+\rho\cdot\beta}{R}+\dfrac{\alpha_{j}^{2}+\alpha^{2}\cdot\beta}{2R}\right]t\}}{\left[\dfrac{\rho_{j}+\rho\cdot\beta}{R}+\dfrac{\alpha_{j}^{2}+\alpha^{2}\cdot\beta}{2R}+\left(\dfrac{\sigma^{2}}{2}-\alpha^{*}\sigma\right)(1-R)-\dfrac{1}{2}\left(\dfrac{\alpha_{j}+\alpha\cdot\beta}{R}+(1-R)\sigma\right)^{2}\right]}$

Then we may write:

$\displaystyle dZ_{t}^{j}=Z_{t}^{j}(\tilde{\alpha}_{t}^{j}dX_{t}-\tilde{\rho}_{t}^{j}dt)$

where

$\displaystyle\tilde{\alpha}_{t}^{j}Z_{t}^{j}=\sum_{|\beta|=R-1}\frac{\binom{R-1}{\beta}\dfrac{\alpha_{j}+\alpha\cdot\beta}{R}\exp\{\dfrac{\alpha_{j}+\alpha\cdot\beta}{R}X_{t}-\dfrac{\gamma_{j}+\gamma\cdot\beta}{R}-\left[\dfrac{\rho_{j}+\rho\cdot\beta}{R}+\dfrac{\alpha_{j}^{2}+\alpha^{2}\cdot\beta}{2R}\right]t\}}{\left[\dfrac{\rho_{j}+\rho\cdot\beta}{R}+\dfrac{\alpha_{j}^{2}+\alpha^{2}\cdot\beta}{2R}+\left(\dfrac{\sigma^{2}}{2}-\alpha^{*}\sigma\right)(1-R)-\dfrac{1}{2}\left(\dfrac{\alpha_{j}+\alpha\cdot\beta}{R}+(1-R)\sigma\right)^{2}\right]}$

and $\tilde{\rho}_{t}^{j}$ is another process, not currently of interest to us. By Itô’s formula:

$\displaystyle dw_{t}^{j}=w_{t}^{j}\{(\sigma+\tilde{\alpha}_{t}^{j}-\bar{\alpha}_{t})dX_{t}+(\bar{\rho}_{t}-\tilde{\rho}_{t}^{j}+\sigma\alpha^{*}+(\tilde{\alpha}_{t}^{j}-\bar{\alpha}_{t})(\sigma-\bar{\alpha}_{t})\}$

(2.7)

Comparing (2.7) with (2.6), we obtain the proportion of risky asset held by agent $j$ as:

$\displaystyle\pi_{t}^{j}=\frac{w_{t}^{j}(\sigma+\tilde{\alpha}_{t}^{j}-\bar{\alpha}_{t})}{S_{t}(\sigma+\tilde{\alpha}_{t}-\bar{\alpha}_{t})}$

(2.8)

In exactly the same manner as Brown & Rogers (2009), we may then perform an Itô expansion on $\pi_{t}^{j}$. We may then interpret the quadratic variation of $\pi_{t}^{j}$ as the volume of trade of agent $j$. The calculation is omitted here, because it is straightforward and the analysis adds little to our understanding. Future work could explore the behaviour of this volume of trade.