Equilibrium-based volatility models of the market portfolio rate of return (peacock tails or stotting gazelles)

Abstract

We introduce a theoretical and empirical method of studying equilibrium-consistent volatility models. We implement it with the market portfolio’s return, which is central to financial risk management. Within an equilibrium framework, we study two families of such models. One is deterministic volatility, represented by current popular models. The other is in the “constant elasticity of variance” family, in which we propose new models. Theoretically, we show that, together with constant expected returns, the latter family tends to have better ability to forecast. Empirically, our proposed models, while as easy to implement as the popular ones, outperform them in three out-of-sample forecast evaluations of different time periods, by standard predictability criteria. This is true particularly during high-volatility periods, whether the market rises or falls.

Appendix: MLE for Model 1 and Model 3

For Model 3, we obtain MLEs for $(\mu ,S)$ as follows. We first write down the process for the rate of return, $y_{t+1}$:

$$\begin{aligned} y_{t+1}= & {} \mu +\sigma _{t+1} \varepsilon _{t+1} ;\quad \varepsilon _{t+1} \sim N(0,1);\quad E(\varepsilon _i \varepsilon _j )=0,\quad \forall \, i\ne j \nonumber \\ \sigma _{t+1}^2= & {} \frac{S}{M_t }. \end{aligned}$$

(25)

Ignoring the constant terms, the log-likelihood functions for the parameters given one observation and for the parameters given the N-day sample, respectively, are

$$\begin{aligned} l^{s}= & {} -\frac{1}{2}\log \sigma _s^2 -\frac{(y_s -\mu )^{2}}{2\sigma _s^2 } \nonumber \\= & {} \frac{1}{2}\log M_{s-1} -\frac{1}{2}\log S-\frac{(y_s -\mu )^{2}M_{s-1} }{2S}, \nonumber \\ L= & {} -\frac{N}{2}\log S-\frac{\sum \limits _{s=1}^N [ (y_s -\mu )^{2}M_{s-1} ]}{2S}. \end{aligned}$$

(26)

From the first-order condition of maximizing L, we have

$$\begin{aligned} \hat{{\mu }}=\frac{\sum \limits _{s=t-N+1}^t {y_s M_{s-1} } }{\sum \limits _{s=t-N+1}^t {M_{s-1} } },\quad \quad \hat{{S}}=\frac{\sum \limits _{s=t-N+1}^t {(y_s -\hat{{\mu }})^{2}M_{s-1} } }{N}, \end{aligned}$$

(27)

and, consequently, obtain Model 3, described in Eq. (19). We could have fine-tuned N (for instance, using cross-validation) to produce better forecasts, but in this study we set $N=10$ for simplicity. We note that Model 3 can be thought of as a “weighted moving average” model, weighted by the past prices, and thus is very similar to the MLE of its reference model, the “moving average” model (i.e., constant $\sigma ^2$ and $\mu$).

We obtain the standard error of the estimators, from their asymptotic covariance matrix, $(N\hat{{\mathcal {I}}})^{-1}$, where $\hat{{\mathcal {I}}}$ is the estimator of the information matrix. We obtain $\hat{{\mathcal {I}}}$ from the sample version of the expected Hessian matrix,

$$\begin{aligned} \hat{{\mathcal {I}}}\mathop {=}\limits ^{a}\left[ {{\begin{array}{ll} {\frac{\sum \limits _{s=t-N+1}^t {M_{s-1} } }{N\hat{{S}}}}&{} 0 \\ 0&{} {\frac{1}{2\hat{{S}}^{2}}} \\ \end{array} }} \right] , \end{aligned}$$

(28)

where $\mathop {=}\limits ^{a}$ denotes asymptotic equivalence. We note the similarity of the above asymptotic covariance matrix, Eq. (28), to that of the constant volatility model. From the standard error of $\hat{{S}}$, we can find the standard error of $\hat{{\sigma }}_{{t+1}^2}^\mathrm{MLE}$, which is $\left( {\sqrt{\frac{2}{N}}} \right) \hat{{\sigma }}_{{t+1}^2}^\mathrm{MLE}$.

For Model 1, we basically follow the estimation procedures developed in Engle (1982) and Bollerslev (1986). In a general form, denoting $\Theta$ as the unknown parameters to be estimated, we have the process for the rate of return, $y_{t+1}$:⁴³

$$\begin{aligned}&y_{t+1} =\mu _{t+1} +\sigma _{t+1} \varepsilon _{t+1} ;\quad \varepsilon _{t+1} \sim N(0,1);\quad E(\varepsilon _i \varepsilon _j )=0,\quad \forall \, i\ne j, \nonumber \\&\mu _{t+1} =m(\Theta ,\mu _t ,\sigma _t^2 ), \nonumber \\&\sigma _{t+1}^2 =s(\Theta ,\mu _t ,\sigma _t^2 ), \end{aligned}$$

(29)

where $m(\cdot )$ and $s(\cdot )$ are differentiable functions. Using Eq. (29), we can take the first derivative of $\mu _{t+1}$ and $\sigma _{t+1}^2$ w.r.t. $\Theta$:

$$\begin{aligned} \frac{\partial \mu _{t+1} }{\partial \Theta }= & {} \frac{\partial m}{\partial \Theta }+\frac{\partial m}{\partial \mu _t }\frac{\partial \mu _t }{\partial \Theta }+\frac{\partial m}{\partial \sigma _t^2 }\frac{\partial \sigma _t^2 }{\partial \Theta }, \nonumber \\ \frac{\partial \sigma _{t+1}^2 }{\partial \Theta }= & {} \frac{\partial s}{\partial \Theta }+\frac{\partial s}{\partial \mu _t }\frac{\partial \mu _t }{\partial \Theta }+\frac{\partial s}{\partial \sigma _t^2 }\frac{\partial \sigma _t^2 }{\partial \Theta }. \end{aligned}$$

(30)

Ignoring the constant terms, the conditional log-likelihood functions for $\Theta$ given one observation and for $\Theta$ given the whole sample, respectively, are

$$\begin{aligned} l^{t}=-\frac{1}{2}[\log \sigma _t^2 +(y_t -\mu _t )^{2}(\sigma _t^2 )^{-1}],\quad \quad \quad L=\sum _{t=1}^D {l^{t}} , \end{aligned}$$

(31)

where D is the sample size. Taking the partial derivative of $l^{t}$ w.r.t. $\Theta$ gives the score (vector) for one observation:

$$\begin{aligned} \frac{\partial l^{t}}{\partial \Theta }=-\frac{1}{2}\left( \frac{\partial \sigma _t^2 }{\partial \Theta }\right) \left[ (\sigma _t^2 )^{-1}-(\sigma _t^2 )^{-2}(y_t -\mu _t )^{2}\right] +\left( \frac{\partial \mu _t }{\partial \Theta }\right) \left( \sigma _t^2 )^{-1}(y_t -\mu _t \right) . \end{aligned}$$

(32)

Equations (30) and (32) together define an iterative procedure to find the score (vector) for the whole sample.

We use a numerical procedure to maximize the log-likelihood function L in Eq. (31). As in Engle (1982) and Bollerslev (1986), we apply the Beetrndt, Hall, Hall and Hausman (1974, BHHH) algorithm to find the MLEs.⁴⁴ By the law of iterated expectations and after simplifying, the expected Hessian matrix is

$$\begin{aligned} \hbox {E}H=-D\hbox {E}\left[ \frac{(\sigma _t^2 )^{-2}}{2}\left( \frac{\partial \sigma _t^2 }{\partial \Theta }\right) \left( \frac{\partial \sigma _t^2 }{\partial \Theta }\right) '({\partial \sigma _t^2})^{-1}\left( \frac{\partial \mu _t }{\partial \Theta }\right) \left( \frac{\partial \mu _t }{\partial \Theta }\right) '\right] \end{aligned}$$

(33)

where “${\prime }$” is the “transpose” operator. The sample version of (33) is $\hat{{H}}=-\sum \nolimits _{t=1}^{D}\left[ \frac{(\sigma _t^2 )^{-2}}{2}\left( \frac{\partial \sigma _t^2 }{\partial \Theta }\right) \left( \frac{\partial \sigma _t^2 }{\partial \Theta }\right) '({\partial \sigma _t^2})^{-1}\left( \frac{\partial \mu _t }{\partial \Theta }\right) \left( \frac{\partial \mu _t }{\partial \Theta }\right) '\right]$. In the implementation, we use the robust asymptotic covariance matrix (i.e., the “sandwich” form). Denoting $\hat{{V}}=\sum \nolimits _{t=1}^{D}\left( \frac{\partial l^{t}}{\partial \Theta })(\frac{\partial l^{t}}{\partial \Theta }\right) '$ as the “Outer Product of Gradient” estimator of the Hessian matrix, the robust asymptotic covariance matrix is $\hat{{H}}^{-1}\hat{{V}}\hat{{H}}^{-1}$.

We set the initial value, $\sigma _1^2$, to be the long-run mean of $\sigma _t^2$, $\hbox {E}\sigma _t^2$.⁴⁵ By the assumptions of i.i.d. $\varepsilon _t ,\;\;\forall t$ and continuous compounding, we take the unconditional expectation on both sides of Eq. (13) and obtain

$$\begin{aligned} \hbox {E}\sigma _t^2 =\frac{\alpha _0 }{M_0 }\exp \left[ (-t+1)\mu +\frac{1}{2}\sum _{s=1}^{t-1} \hbox {E} \sigma _s^2\right] +(\alpha _1 +\alpha _2 +\mu ^{2}-\mu )\hbox {E}\sigma _{t-1}^2 +\hbox {E}(\sigma _{t-1}^2 )^{2}. \end{aligned}$$

(34)

Further assuming that $M_0$ is finite, that the unconditional variance of $\sigma _t^2$ exists, and that $\hbox {E}\sigma _t^2 <2\mu$, asymptotically (i.e., $t\rightarrow \infty )$ we have

$$\begin{aligned} \hbox {E}\sigma _t^2 \mathop {=}\limits ^{a} \;\frac{\hbox {E}[(\sigma _t^2 )^{2}]}{1+\mu -\mu ^{2}-\alpha _1 -\alpha _2 }. \end{aligned}$$

(35)

In the implementation, we use Eq. (35) as $\sigma _1^2$, where $\hbox {E}[(\sigma _t^2 )^{2}]$ is approximated by the forth moment of the sample, $\frac{1}{D}\sum \nolimits _{t=1}^D {\left( y_t -\frac{1}{D}\sum \nolimits _{s=1}^D {y_s } \right) ^{4}}$.⁴⁶