The dynamic relation between options trading, short selling, and aggregate stock returns

Abstract

We examine the information contained in option trading and short selling using a dynamic VAR model. First, we address whether options and shorts are complements or substitutes. Contrary to existing event studies around option listing introductions, we show short selling and options trading are complements rather than substitutes. Second, we examine which group is relatively more informed. The results indicate that options traders are relatively more informed. Finally, we examine if options are redundant. Our results indicate that options markets are non-redundant.

Appendices

Appendix 1: Variable definitions

Variable name	Definition
SIR	Short interest as a % of shares outstanding
RETURN	Value-weighted return of S&P 500 firms with option and short data
OIRP	Total open interest of puts
OIRC	Total open interest of calls
O/S	Total dollar option volume divided by total dollar share volume

Appendix 2: Derivation of Sims’ test in the context of short interest and returns

We utilize a theorem in time-series econometrics, which states that any time-series process has both invertible and non-invertible representations (see Fuller 1976, pp. 64–66, Theorem 2.6.4). Although stock returns, R_t, may follow a general autoregressive moving-average (i.e., ARMA(p,q)) process, for expositional simplicity, we assume that other uninformed investors, observing current and past stock returns, infer a first-order (invertible) moving average, MA(1) [i.e., ARMA (0,1)], process of the returns:

$${\text{R}}_{\text{t}} = \, (1 \, {-}\uplambda{\text{L}}){\text{u}}_{\text{t}} ,\quad \quad |{\lambda | } < 1.0,$$

(4)

where R_t is the stock return at time t, L is the lag (or backshift) operator (i.e., Lⁿ R_t = R_t−n), u_t is white noise with ${\text{var}}\left( {{\text{u}}_{\text{t}} } \right) = \sigma_{\text{u}}^{2}$, and λ is a parameter that indicates the contribution of u_t−1 to R_t. The autocovariance functions (ACFs) for this MA(1) return process are¹⁴:

$$\begin{aligned} & {\text{var}}\left( {{\text{R}}_{\text{t}} } \right) \, = \, (1 \, +\uplambda^{2} )\upsigma_{\text{u}}^{2} , \\ & {\text{cov}}\left( {{\text{R}}_{\text{t}} ,{\text{R}}_{{{\text{t}} - 1}} } \right) \, = \, -\uplambda \upsigma _{\text{u}}^{2} , \\ & {\text{cov}}\left( {{\text{R}}_{\text{t}} ,{\text{R}}_{{{\text{t}} - {\text{k}}}} } \right) \, = \, 0,\quad {\text{for k}} \ge 2. \\ \end{aligned}$$

(5)

On the other hand, suppose that informed short sellers (or options traders), observing the same current and past stock returns, infer the following (non-invertible) MA(1) process of the returns:

$${\text{R}}_{\text{t}} = \, (1 -\uplambda^{ - 1} {\text{L}}){\text{v}}_{\text{t}} ,\quad \quad |\uplambda|\,\, < 1.0,$$

(6)

where v_t is white noise with ${\text{var}}\left( {{\text{v}}_{\text{t}} } \right) = \sigma_{\text{v}}^{2}$. The ACFs for this MA(1) return process are:

$$\begin{aligned} & {\text{var}}\left( {{\text{R}}_{\text{t}} } \right) \, = \, (1 +\uplambda^{ - 2} )\upsigma_{\text{v}}^{2} , \\ & {\text{cov}}\left( {{\text{R}}_{\text{t}} , {\text{R}}_{{{\text{t}} - 1}} } \right) \, = \, -\uplambda^{ - 1}\upsigma_{\text{v}}^{2} , \\ & {\text{cov}}\left( {{\text{R}}_{\text{t}} ,{\text{R}}_{{{\text{t}} - {\text{k}}}} } \right) \, = \, 0,\quad {\text{ for}}\;\;{\text{k}} \ge 2. \\ \end{aligned}$$

(7)

It is noted that if we set $\upsigma_{\text{v}}^{2} =\uplambda^{2}\upsigma_{\text{u}}^{2}$, then the ACFs in (5) and (7) are identical. Since the return process can be identified in practice by the observed ACFs, the identical ACFs imply that stock return processes in (4) and (6) represent the same return process. That is, for a given return process, uninformed investors and informed short sellers (or options traders) may infer different MA(1) processes.¹⁵

In addition, it is noted that $\upsigma_{\text{v}}^{2}$ is smaller than $\upsigma_{\text{u}}^{2}$:

$$\upsigma_{\text{v}}^{2} <\upsigma_{\text{u}}^{2} .$$

(8)

This is because $\upsigma_{\text{v}}^{2} =\uplambda^{2}\upsigma_{\text{u}}^{2}$ and ∣λ∣ < 1.0. This means that the variance of the one-step-ahead forecast error of the return process in (6) by informed short sellers (i.e., $\upsigma_{\text{v}}^{2}$) would be smaller than the corresponding variance of the return process in (4) by other uninformed investors (i.e., $\upsigma_{\text{u}}^{2}$). However, unlike the u_t process, the v_t process cannot be fully recovered by other uninformed investors from the information about current and past values of stock returns because the process is not invertible. That is, although both short sellers and other uninformed investors observe the same (current and past) returns, under information asymmetry, informed short sellers with a larger information set, $\Omega _{\text{t}}^{*} = \{ {\text{R}}_{{{\text{t}} - {\text{j}}}} ,{\text{ v}}_{{{\text{t}} - {\text{j}}}} ,{\text{ u}}_{{{\text{t}} - {\text{j}}}} ,\,\,{\text{for j}} \ge 0\}$, can forecast future returns better than other uninformed investors with a smaller information set, Ω_t = {R_t−j, u_t−j, for j ≥ 0}.

We obtain an important alternative insight by comparing the corresponding autoregressive representations (ARR) of the moving average representations (MAR) of stock return processes {R_t} in (4) and (6):

$${\text{u}}_{\text{t}} = \, (1 -\uplambda{\text{L}})^{ - 1} {\text{R}}_{\text{t}} = \mathop \sum \limits_{j = 0}^{\infty }\uplambda^{j} {\text{R}}_{t - j} ,\,\quad {\text{and}}$$

(9)

$${\text{v}}_{\text{t}} = (1 -\uplambda^{ - 1} {\text{L}})^{ - 1} {\text{R}}_{\text{t}} = - (\uplambda{\text{L}}^{ - 1} )(1 -\uplambda{\text{L}}^{ - 1} )^{ - 1} {\text{R}}_{\text{t}} = - \sum\limits_{j = 1}^{\infty } {\uplambda^{j} } {\text{R}}_{t + j} .$$

(10)

Note that the innovations {u_t} in the other uninformed investors’ return process are backward-looking, whereas the innovations {v_t} in the informed short sellers’ return process are forward-looking.¹⁶

How is this information asymmetry between informed short sellers and other uninformed investors related to the dynamic relation between short sales and stock returns (i.e., the predictive power of short sales)? Suppose that short sellers have an informational advantage in that they can forecast the firm’s future prospects (or overvaluation) better than other uninformed investors by observing v_t. If informed short sellers use this information in their short-sale decisions, their short sales (or short interest ratio, SIR _t) will be a function of innovation v_t that they observe but other uninformed investors do not:

$$SIR_{t} {\text{ = f}}\left( {{\text{v}}_{\text{t}} } \right) = \sum\limits_{i = 0}^{\infty } {(\theta_{i} L^{i} )} {\text{v}}_{\text{t}} = \sum\limits_{i = 0}^{\infty } {\theta_{i} } {\text{v}}_{{{\text{t}} - {\text{i}}}} ,\quad {\text{with}}\quad \sum\limits_{i = 0}^{\infty } {\theta_{i}^{2} } < \infty$$

(11)

Then, by using v_t in (10), short interest and stock return processes will be related as follows:

$$\begin{aligned} SIR_{{_{\text{t}} }} & = \sum\limits_{i = 0}^{\infty } {(\theta_{i} L^{i} )} {\text{v}}_{\text{t}} = \, \sum\limits_{i = 0}^{\infty } {(\theta_{i} L^{i} )} \left\{ {\left( {1 -\uplambda^{ - 1} {\text{L}}} \right)^{ - 1} {\text{R}}_{\text{t}} } \right\} \\ & = \sum\limits_{i = 0}^{\infty } {(\theta_{i} L^{i} )} \left( { - \, \sum\limits_{j = 1}^{\infty } {\lambda^{j} } {\text{R}}_{t + j} } \right) = \, \sum\limits_{j = - \infty }^{\infty } {\delta_{j} } {\text{R}}_{t - j} \\ \end{aligned}$$

(12)

where δ _j for j = −∞, …, −2, −1, 0, 1, 2, …, ∞ is a function of $\theta_{\text{i}}$ and $\lambda^{\text{j}}$. That is, the informed short sales will be a linear combination of future, current, and past returns; thus, they will be forward-looking.¹⁷

In contrast, suppose that short sellers do not have an informational advantage or they simply do not make short-sale decisions based on their informational advantage. Then, the uninformed short sales will be a function of the innovation that other uninformed investors observe, u_t:

$$SIR_{\text{t}} = {\text{f}}({\text{u}}_{\text{t}} ) = \sum\limits_{i = 0}^{\infty } {(\theta_{i} L^{i} )} {\text{u}}_{\text{t}} = \sum\limits_{i = 0}^{\infty } {\theta_{i} } {\text{u}}_{{{\text{t}} - 1}} ,\quad {\text{with}}\quad \sum\limits_{i = 0}^{\infty } {\theta_{i}^{2} } < \infty$$

(13)

Then, by using u_t in (9), uninformed short sales and stock return processes will be related as follows:

$$\begin{aligned} SIR_{\text{t}} & = \sum\limits_{i = 0}^{\infty } {(\theta_{i} L^{i} )} {\text{u}}_{\text{t}} = \, \sum\limits_{i = 0}^{\infty } {(\theta_{i} L^{i} )} (1 -\uplambda{\text{L}})^{ - 1} {\text{R}}_{\text{t}} \\ & = \sum\limits_{i = 0}^{\infty } {(\theta_{i} L^{i} )} \left( { \, \sum\limits_{j = 0}^{\infty } {\uplambda^{j} } {\text{R}}_{t - j} } \right) = \, \sum\limits_{k = 0}^{\infty } {\delta_{k} } {\text{R}}_{t - k} , \\ \end{aligned}$$

(14)

where δ _k for k = 0, 1, 2, …, ∞ is a function of $\theta_{\text{i}}$ and $\lambda^{\text{j}}$. That is, in this case, the uninformed short sales will only reflect the past and current returns and will not be related to future returns; thus, they will be backward-looking. To summarize, we have shown that under information asymmetry, informed (or informative) short sales (or short interests) are related not only to past and current returns but also to future returns. In contrast, in the absence of information asymmetry, uninformed (or non-informative) short sales (or short interests) are not related to future returns.

Appendix 3: Unit root tests: sample period, 1996:03–2011:12 (Observations 188)

3.1 Using equal-weighted aggregate data

(i) Augmented Dickey–Fuller regression $\Delta x_{t} = a_{0} + ax_{t - 1} + \sum\nolimits_{i = 1}^{m} {\gamma_{i}\Delta x_{t - i} + v_{t} }$ (ii) Phillips–Perron regression $x_{t} = b_{0} + bx_{t - 1} + v_{t} .$
Variables (x t )	Dickey–Fuller test				Phillips–Perron test
	τα				Z(tb)
	1 lag	2 lags	3 lags	4 lags	1 lag	2 lags	3 lags	4 lags
RETURNt	−10.254	−7.469	−6.100	−5.894	−11.885	−11.807	−11.789	−11.808
SIRt	−1.678	−1.713	−1.683	−1.484	−1.742	−1.767	−1.778	−1.769
OIRPt	−1.270	−1.077	−1.026	−0.919	−1.120	−1.121	−1.098	−1.061
OIRCt	−1.493	−1.139	−1.072	−0.922	−1.282	−1.269	−1.221	−1.158
O/St	−5.352	−4.449	−4.305	−3.699	−7.052	−7.068	−7.223	−7.295
RESPt	−3.226	−3.071	−2.732	−2.393	−2.850	−2.952	−2.940	−2.870
RESCt	−3.958	−3.593	−2.936	−2.495	−3.216	−3.358	−3.311	−3.175
RES3t	−3.719	−3.439	−2.866	−2.450	−3.098	−3.229	−3.193	−3.076
RESPCt	−3.741	−2.887	−3.181	−2.655	−3.278	−3.230	−3.201	−3.149

1. Critical values of t-statistics for both τ_α and Z(t_b) are: 1 % = −3.470, 5 % = −2.879, 10 % = −2.576 (Fuller 1976, Tables 8.5.1 and 8.5.2, pp. 371–373). The details of the adjusted t-statistics Z(t_b) can be found in the work of Phillips and Perron (1988)
2. RETURN _t value weighted return of S&P 500 firms WITH option and short data (in other words, not all 500 firms make it into this return calculation), SIR _t short interest as a %, OIRP _t total open interest of puts, OIRC _t total open interest of calls, O/S _t total dollar option volume/total dollar share volume, RESP _t residual in the regression of SIR_t on OIRP_t, RESC _t residual in the regression of SIR_t on OIRC_t, RES3 _t residual in the regression of SIR_t on OIRP_t and OIRC_t, RESPC _t residual in the regression of OIRP_t on OIRC_t

Appendix 4: Complements versus substitutes: equal-weighted aggregate data

H0 (null hypothesis)	– Sum of coeff.	Chi square statistic t-statistic	Significance level
$\Delta SIR_{t} = \alpha + \sum_{j = 1}^{3} \alpha_{j}\Delta SIR_{t - j} + \sum_{j = 0}^{3} \beta_{j}\Delta OIRP_{t - j} + b * RESP_{t - 1}$				(1)
H0: $\sum_{j = 0}^{3} \beta_{j} = 0$	−0.3323	t = −1.3141	0.1888
$\Delta SIR_{t} = \alpha + \sum_{j = 1}^{3} \alpha_{j}\Delta SIR_{t - j} + \sum_{j = 0}^{3} \beta_{j}\Delta OIRC_{t - j} + b * RESC_{t - 1}$				(1)
H0: $\sum_{j = 0}^{3} \beta_{j} = 0$	−0.2931	t = −1.1814	0.2375
$\Delta OIRP_{t} = \alpha + \sum_{j = 1}^{3} \alpha_{j}\Delta OIRP_{t - j} + \sum_{j = 0}^{3} \beta_{j}\Delta SIR_{t - j} + b * RESP_{t - 1}$				(2)
H0: $\sum_{j = 0}^{3} \beta_{j} = 0$	0.2456	t = 2.0759	0.0379
$\Delta OIRC_{t} = \alpha + \sum_{j = 1}^{3} \alpha_{j}\Delta OIRC_{t - j} + \sum_{j = 0}^{3} \beta_{j}\Delta SIR_{t - j} + b * RESC_{t - 1}$				(2)
H0: $\sum_{j = 0}^{3} \beta_{j} = 0$	0.2507	t = 1.7668	0.0772
$\Delta OIRP_{t} = \alpha + \sum_{j = 1}^{3} \alpha_{j}\Delta OIRP_{t - j} + \sum_{j = 0}^{3} \beta_{j}\Delta OIRC_{t - j} + b * RESPC_{t - 1}$				(2)
H0: $\sum_{j = 0}^{3} \beta_{j} = 0$	0.5170	t = 3.5076	0.0005
$\Delta OIRC_{t} = \alpha + \sum_{j = 1}^{3} \alpha_{j}\Delta OIRC_{t - j} + \sum_{j = 0}^{3} \beta_{j}\Delta OIRP_{t - j} + b * RESPC_{t - 1}$				(2)
H0: $\sum_{j = 0}^{3} \beta_{j} = 0$	0.8978	t = 5.0203	0.0000
$SIR_{t} = \alpha + \sum_{j = 1}^{3} \alpha_{j} SIR_{t - j} + \sum_{j = 0}^{3} \beta_{j} OIRP_{t - j}$				(1)
H0: $\sum_{j = 0}^{3} \beta_{j} = 0$	0.0708	t = 1.7560	0.0791
$SIR_{t} = \alpha + \sum_{j = 1}^{3} \alpha_{j} SIR_{t - j} + \sum_{j = 0}^{3} \beta_{j} OIRC_{t - j}$				(1)
H0: $\sum_{j = 0}^{3} \beta_{j} = 0$	0.0890	t = 2.3807	0.0173
$OIRP_{t} = \alpha + \sum_{j = 1}^{3} \alpha_{j} OIRP_{t - j} + \sum_{j = 0}^{3} \beta_{j} SIR_{t - j}$				(2)
H0: $\sum_{j = 0}^{3} \beta_{j} = 0$	0.0198	t = 1.3483	0.1776
$OIRC_{t} = \alpha + \sum_{j = 1}^{3} \alpha_{j} OIRC_{t - j} + \sum_{j = 0}^{3} \beta_{j} SIR_{t - j}$				(2)
H0: $\sum_{j = 0}^{3} \beta_{j} = 0$	0.0318	t = 1.8032	0.0714
$OIRP_{t} = \alpha + \sum_{j = 1}^{3} \alpha_{j} OIRP_{t - j} + \sum_{j = 0}^{3} \beta_{j} OIRC_{t - j}$				(2)
H0: $\sum_{j = 0}^{3} \beta_{j} = 0$	0.0450	t = 1.2636	0.2064
$OIRC_{t} = \alpha + \sum_{j = 1}^{3} \alpha_{j} OIRC_{t - j} + \sum_{j = 0}^{3} \beta_{j} OIRP_{t - j}$				(2)
H0: $\sum_{j = 0}^{3} \beta_{j} = 0$	0.0808	t = 2.0181	0.0436
$\Delta SIR_{t} = \alpha + \sum_{j = 1}^{3} \alpha_{j}\Delta SIR_{t - j} + \sum_{j = 0}^{3} \beta_{j} O/S_{t - j}$				(1)
H0: $\sum_{j = 0}^{3} \beta_{j} = 0$	0.0175	t = 0.3692	0.7120
$O/S_{t} = \alpha + \sum_{j = 1}^{3} \alpha_{j} O/S_{t - j} + \sum_{j = 0}^{3} \beta_{j}\Delta SIR_{t - j}$				(2)
H0: $\sum_{j = 0}^{3} \beta_{j} = 0$	−1.5906	t = −2.1717	0.0299