Illiquidity and Higher Cumulants

Abstract

Illiquidity, or the market’s inability to accommodate large trades without a price change, affects the trading and pricing of financial assets even for a market as developed as U.S. equities.¹ The effects of illiquidity are particularly severe for derivative contracts, where even short-term at-the-money (ATM) options written on the largest stocks can have bid-ask spreads on the order of 2|$\%$|⁠.² Large traders and institutional investors, such as mutual and pension funds, respond to illiquidity by trading strategically, that is, accounting for their price impact. Some investors (e.g., J. P. Morgan and Citigroup) have in-house “optimal execution” desks that devise trading strategies to minimize trading costs. Other investors use software and services provided by more specialized trading firms. Such strategic trading stands in contrast to the price-taking behavior commonly assumed in classical asset pricing models.³ Importantly, modern markets are largely dominated by institutions managing billions of dollars and, hence, strategic trading might be a key driver of a typical investors’ behavior.

How are illiquidity and asset prices determined in equilibrium when investors internalize their price impact? The literature on strategic trading addresses this question adopting a CARA-normal framework for tractability. Traders are either risk neutral or have constant absolute risk aversion (CARA) utility functions, and asset payoffs have Gaussian distribution. These assumptions make such models inapplicable to derivative markets, where payoffs are nonlinear functions of the underlying asset prices and, hence, cannot be Gaussian. Notably, multiple derivatives written on the same asset must be studied jointly. Thus, to study illiquidity in derivative markets, we need a model of strategic trading for multiple assets with non-Gaussian payoffs. Our paper develops such a model.

We allow for multiple assets and a general distribution of asset payoffs. Despite significant technical challenges, we characterize equilibrium explicitly and are able to derive its properties analytically. In an application of our theory, we derive several surprising implications regarding option bid-ask spreads and provide supporting evidence. In particular, option bid-ask spreads may decrease in risk aversion, physical variance, and open interest, but they may increase after earnings announcements.

We assume that CARA traders, whom we refer to as liquidity providers (LPs), exchange multiple risky assets for a riskless asset over one period while internalizing their price impact. LPs all have the same risk aversion and are symmetrically informed. The absence of information asymmetry implies that, in our setting, the unique source of price impact is inventory risk.⁴ Trading is organized as a uniform-price double auction: traders simultaneously submit demand functions specifying the number of units of the assets they want to buy as a function of the prices of all assets. All trades are executed at prices that clear the market. Our main innovation (as compared with previous research) is to allow for an arbitrary distribution of the risky asset payoffs under the sole restriction of bounded support.⁵ In addition to LPs, uninformed liquidity demanders (LDs) submit market orders. We express all equilibrium quantities as functions of the aggregate liquidity demand.

In equilibrium, traders need to determine their optimal demand functions (the map from the vector of prices to the vector of positions), knowing the demand functions of all other traders. However, we show that an LP’s problem is, in fact, equivalent to that of a trader just knowing, for each order size, his price impact matrix (i.e., how his trades move prices of all assets at the margin). This is an intuitive representation of the problem. Real-world traders typically have a market impact model that serves as an input to their optimal execution algorithm.⁶ The equilibrium price impact matrix is pinned down by the requirement that it is consistent with the demand functions of all other traders and must be equal to the inverse of the “slope” of the total residual supply of all other traders for any level of liquidity demand. We show that optimality and consistency conditions imply a partial differential equations (PDEs) system for the equilibrium demand. Remarkably, solving this complex system of PDEs can be reduced to solving a single-asset ordinary differential equation (ODE). Such an ODE is linear and, thus, can be solved in closed form.⁷ This ODE characterizes the price function in an economy whose single asset is an index defined by a vector of asset holdings. We establish equilibrium uniqueness in the class of symmetric equilibria with strictly decreasing, continuously differentiable demands, and arbitrage-free equilibrium prices.

We then look at the implications for bid-ask spreads and derive several surprising results.⁸ For example, we show that bid-ask spreads may decrease in LPs’ risk aversion and the size of their inventory when LPs’ risk aversion is high. The key to our surprising result is the interaction between LPs’ inventories and bid-ask spreads. Consider what happens when LPs’ initial inventory increases. The LP will decrease the price (both bid and ask) to reflect the higher inventory cost. Such an effect, specifically a decrease in price when inventory increases, is present in both CARA-normal models and our general model. The effect unique to our model is that the ask price will decrease more than the bid price. A more profound decrease in the ask price makes buying from LPs relatively more attractive, which LPs like: they are more eager to decrease inventory when initial inventory level is higher. Such interaction is absent in CARA-normal models, where the size of inventory affects the price level, but not the bid-ask spread.⁹ The same interaction is behind our other surprising results: bid-ask spreads may decrease in LPs’ risk aversion and physical variance and may increase after earnings announcements. Again, this movement occurs in contrast to CARA-normal models. See, for example, Vayanos and Wang 2012 for a review.

We also derive closed-form expressions for bid-ask spreads and show that they are proportional to LPs’ risk aversion and risk-neutral variance of asset payoffs. These expressions in the general case are remarkably similar to those in the Gaussian benchmark; one needs to make only the minor adjustment of substituting the risk-neutral variance for the physical variance. However, this minor adjustment changes comparative statics dramatically, as the risk-neutral variance depends on model parameters, such as risk aversion. It is this dependence that is responsible for our surprising results mentioned above. The empirical implication of our closed-form expressions is that the cross-section of bid-ask spreads of options is explained by the risk-neutral (not physical) variance of these options’ payoffs.

In the empirical part of the paper, we confront this and other predictions using U.S. options data. In stark contrast to the conventional wisdom, and in line with the surprising implications of our theory, we find (1) a negative relationship between bid-ask spreads and the VIX, commonly interpreted as a proxy for marketwide risk aversion;¹⁰ (2) a nonmonotonic relationship between bid-ask spreads and physical variance; (3) an increase in the bid-ask spread following earnings announcements; and (4) a negative relationship between option bid-ask spreads and the size of LPs’ inventory, proxied for by the options’ open interest.¹¹ In line with our explicit solutions showing that bid-ask spreads are proportional to LPs’ risk aversion and risk-neutral variance of options payoffs we find that (1) bid-ask spreads are positively related to an exact measure of options risk-neutral variance that we construct from option prices using the Carr and Madan 1998 payoff decomposition formula and (2) the betas in the regression of spreads on risk-neutral variances are positively related to proxies of risk aversion.

Our findings stand in stark contrast with existing evidence on stock market liquidity. First, equity market liquidity and the VIX are negatively related (Nagel 2012). Second, a release of public information (disclosure) is associated with improved liquidity for the underlying stock (Healy and Palepu 2001). Third, the association between stock market liquidity and the size of market makers’ inventories is negative (Comerton-Forde et al. 2010). All three relationships hold in the opposite direction for stock options.

Although this contrasting evidence might be initially surprising, both sets of findings are consistent with our theory if we allow for some form of market segmentation. Indeed, we show that unconventional results only hold when LPs’ risk aversion is sufficiently high. In contrast, we recover conventional results when this risk aversion is small. We argue that LPs in options markets might have a much smaller risk-bearing capacity than LPs in equity markets. First, options’ embedded leverage means this capacity becomes exhausted much more quickly. Second, LPs have many options contracts to intermediate, even for a single underlying asset. Much higher average percentage bid-ask spreads in the options markets also point toward a significantly lower risk-bearing capacity for options’ LPs.

1. The Model

There are two time periods |$t\in\{0,1\}$|⁠.¹² A number |$L>2$| of strategic liquidity providers (LPs) trade assets with liquidity demanders (LDs) at |$t=0$| and consume at |$t=1$|⁠.¹³ There are |$N$| risky assets and a risk-free asset (a bond). The bond is a numeraire and, thus, earns a net return of zero. A risky asset |$k$| is a claim to a terminal dividend |$\delta_k$|⁠.

We put forward the following technical restrictions on the model parameters.

Assumption 1 simply requires that there be no redundant securities.¹⁵ Assumption 2 is a natural one. Real-world investors are protected by limited liability, which implies that dividends |$\delta_i$| are nonnegative; hence, there must be a lower bound. An upper bound is also natural when one considers that the resources of any firm are limited, which means that no asset can have an infinite payoff.

As is common in the literature, we assume that LDs’ aggregate trade is characterized by the aggregate supply shock |$s\in\mathbb{R}^N$|⁠, which has full support¹⁶ and is independent of |$\delta.$| As in Klemperer and Meyer, our assumptions imply that equilibrium quantities will depend on the realization of |$s$|⁠, but not its distribution.

Before detailing our equilibrium concept, we define the set of arbitrage-free prices.

As is common in the literature, we focus on arbitrage-free, symmetric Nash equilibria with strictly decreasing demands (hereafter, simply “an equilibrium”).

Definition 2 (i) is simply a Nash equilibrium requirement. Parts (ii) and (iii) are technical; they ensure that the inverse demand, for which we solve when deriving the equilibrium, is well-defined. Part (iv) is required to ensure that the equilibrium is unique. Solving for the equilibrium amounts to solving an ODE, and this requirement places a transversality condition that yields a unique solution. The economic meaning of condition (iv) is as follows. Suppose that, in addition to strategic LPs, there is an arbitrarily small mass of competitive (price-taking) LPs. Then, for prices that are not arbitrage-free, the price-taking LPs would submit infinite demands; thus, the market would not be clear. Hence, there can be no equilibria when prices are not within |${\mathcal A}$|⁠. Thus, requirement (iv) selects, among many potential equilibria, the one that is robust to the presence of a vanishingly small number of competitive LPs.

We will often suppress the second argument, and simply write |$f(q)$|⁠, provided that no confusion could arise. The following remarks are in order.

2. Equilibrium

This section provides a characterization of equilibrium and derives the closed-form solutions.

2.1 Characterization of equilibrium

The next theorem summarizes our equilibrium characterization.

2.2 Closed-form solution

Such an ODE can be solved in a closed form using standard methods.²⁰ In contrast, solving (systems of) PDEs usually presents significant technical challenges. Surprisingly, we now show that solving the (seemingly complex) system of PDEs (5) boils down to solving a linear ODE that is similar to (9). Hence, our approach is tractable even in the case of multiple assets.

Now consider the unrestricted economy. In the symmetric equilibrium, for supply shock realizations |$s=tq$| (⁠|$t\in{\mathbb R}^+$|⁠), it should be optimal for LPs to absorb |$1/L$| fraction of supply shock |$s$|⁠, that is, to trade |$t/L$| units of portfolio |$q$|⁠. Hence, the price LPs bid for |$t/L$| units of portfolio |$q$| in the unrestricted economy, or |$q^\top I(t/L q)$|⁠, should be an optimal bid in the restricted economy. Therefore, |$q^\top I(t/L q)=\iota(t/L)$| should satisfy ODE (10), which completes the first step.

In the second step, we differentiate this equation with respect to |$q$| and then apply (11) to obtain (14) for |$I(q)$|⁠. It then remains to establish the global optimality (1) of |$D(p)=I^{-1}(q)$|⁠. This is highly nontrivial due to the complex, nonconvex nature of (1). In Appendix B.1, we develop novel mathematical techniques to tackle it. The following is true.

One of the key results of the above Proposition is the equilibrium uniqueness. Two features of the model help to achieve it: (a) the bounded support of |$\delta$| and (b) the no-arbitrage restriction (iv) of Definition 2. Essentially, (a) and (b) help to select the unique solution of the ODE (12) among the continuum of solutions. Indeed, one can show that the general solution to (12) is given by |$\hat{\iota}(t;q)=\iota^*(t;q)+C\cdot t^{L-1}$|⁠, where |$C$| is a constant and |$\iota^*(t;q)$| is given by (13).²¹ If |$C<0$|⁠, then |$\hat{\iota}(t;q)$| becomes arbitrary small as |$t$| increases, so |$\hat{\iota}(t;q)$| becomes smaller than the minimum payoff of the portfolio |$q$| (which is finite, because of the boundedness of |$\delta$|⁠). A price less than a minimal payoff clearly violates no-arbitrage restriction. A similar argument applies to the case |$C>0$|⁠, where the price will be above maximal payoff. Thus, the only solution that could satisfy the no-arbitrage restriction is |$\iota^*(t;q)$|⁠. Here, the boundedness of |$\delta$| is important. Without it, the solutions of (12) with |$C\neq0$| will not violate the no-arbitrage restrictions, because the maximum or minimum payoffs on the portfolio |$q$| are not finite.

One can derive an expression for the equivalent martingale measure (EMM) in our economy. Doing so allows us to rewrite the equilibrium objects just described in a more compact way as well as gain additional insights. In what follows we use an asterisk to indicate those moments of |$\delta$| evaluated under the EMM.

Note that |${\zeta(t;q)}$| is an stochastic discount factor (SDF) in a competitive economy, where LPs absorb an order of size |$t\times q$|⁠. Equation (19) shows that |$Z^*$| (i.e., the SDF in the economy with market power) is a weighted average of the SDFs in the competitive economies, where LPs absorb orders of size |$t\times q$| with |$t>1$|⁠. This outcome is intuitive: LPs exercise their market power by charging the price that competitive LPs would charge for absorbing a larger order. Thus, (19) manifests the demand reduction common to auctions of divisible goods (see Ausubel et al. 2014).

The function |$Z^*(q)$| is a Radon-Nikodym derivative for the risk-neutral measure in the economy with the per capita supply shock |$q$|⁠. Therefore, |$Z^*(0)={\exp({-\gamma}x_{0}^\top\delta)}/{E[\exp({-\gamma}x_{0}^\top\delta)]}$| is associated with the risk-neutral measure in the economy without a supply shock. Let the density of |$\delta$| be |$\eta(\delta)$|⁠. In the sequel, we refer to the probability measure with the density |$\eta(\delta) Z^*(0)/E[Z^*(0)]$| as the risk-neutral measure. An asterisk indicates moments under this measure. Below, we will show that bid-ask spreads are related to the second moments of the distribution under this measure.

3. Bid-Ask Spreads

In this section we derive implications of our theory for the joint behavior of bid-ask spreads when payoffs are non-Gaussian. We then test our key predictions using a large panel of exchange-traded options on U.S. stocks.

3.1 Bid-ask spreads: Theory

We define the bid-ask spread for an asset |$k$| as the difference in equilibrium prices when LPs absorb (buy or sell) a small amount of |$n_k$| units of asset |$i$|⁠, normalized by |$n_k$|⁠. We interpret |$n_k$| as a minimal lot size.

The last equality is a direct corollary of Proposition 2. The advantage of such a measure, compared to the one without normalization by |$n_k$|⁠, is that it is independent of the minimal lot size.²²

We start by characterizing bid-ask spreads for the Gaussian case.²³ We are interested in how bid-ask spreads change when (a) risk aversion increases; (b) public information is released; and (c) the riskiness of assets systematically increases,²⁴ modeled as a rescaling of the distribution of |$\delta$|⁠, preserving its mean. To have a shift parameter that affects the riskiness of the assets, in this section we maintain the following assumption.

The comparative statics in Corollary 2 are standard for CARA-normal models. As the risk faced by LPs decreases due to the release of information or a decrease in the asset riskiness |$\sigma$|⁠, they demand a lower compensation for providing liquidity and the bid-ask spread decreases. Similarly, bid-ask spread widens with an increase in risk aversion. In stark contrast to this conventional wisdom, evidence from stock options data (see Section 3.2) documents (1) a negative relationship between bid-ask spreads proxies for marketwide risk aversion; (2) a nonmonotonic relationship between bid-ask spreads and physical variance; and (3) an increase in bid-ask spread following the earnings announcements. We also find a negative relationship between option bid-ask spreads and the size of LPs’ inventory |$|x_{0,k}|$|⁠, proxied for by the option open interest. This result clearly contradicts Equation (21): in a CARA-normal setting, inventory does not affect bid-ask spreads.

Below, we will theoretically demonstrate that the comparative statics of Corollary 2 might change signs when we abandon the Gaussian assumption, in line with our empirical results. We first consider the comparative statics for risk aversion |$\gamma$| and the asset riskiness |$\sigma$| and then derive necessary and sufficient conditions for |$\partial{\operatorname{BA}}_k/\partial\gamma<0$| and |$\partial{\operatorname{BA}}_k/\partial\sigma<0$| in Proposition 3. Next, we will consider the case when LPs’ risk aversion is high and show in Proposition 4 that all results highlighted in Corollary 2 are overturned in that case.

Two main takeaways emerge from Proposition 3. First, the expression for bid-ask spread remains remarkably similar to that in the Gaussian case, with physical variance substituted for the risk-neutral one (compare (21) and (22)). Importantly, the risk-neutral variance depends on model parameters, such as risk aversion, |$\gamma$|⁠. The coskewness terms in (23) arise precisely because of that: we show in the proof that they are proportional to the sensitivity of |${\operatorname{var}}^*(\delta_k)$| to |$\gamma$| (see Lemma 8).

Second, the proposition gives necessary and sufficient conditions for the Gaussian comparative statics with respect to the |$\gamma$| and |$\sigma$| to be overturned. This happens when the term |$\gamma \cdot {\operatorname{coskew}}^*(\delta_k,\delta_k,\delta^\top x_{0})$| is large compared to |${\operatorname{var}}^*(\delta_k)$|⁠. Thus, if the distribution of |$\delta$| has zero higher cumulants or if risk aversion |$\gamma$| is small, we have |${\operatorname{sign}}\left(\frac{\partial}{\partial \gamma} BA_k\right)={\operatorname{sign}}\left(\frac{\partial}{\partial \gamma} BA_k\right)={\operatorname{sign}}\left({\operatorname{var}}^*(\delta_k)\right)>0$|⁠, explaining why in the Gaussian case the bid-ask spreads are always increasing in |$\gamma$| and also why the same is true when |$\gamma$| is small (even if the distribution is non-Gaussian).²⁵

Next, we will show that, when |$\gamma$| increases, the Gaussian comparative statics are necessarily reversed.²⁶

The implications of Proposition 4 are intriguing. Contrary to conventional wisdom based on Corollary 2, (a) physical distribution has only second-order effects on bid-ask spreads and (b) these spreads decrease in |$\gamma$| and depend on LPs’ inventories, |$x_0.$| We now discuss these counterintuitive comparative statics through the lens of formula (23). When |$\gamma$| is small, so is the second term in (23), and the bid-ask spread increases in |$\gamma$|⁠. As |$\gamma$| increases, the risk-neutral measure puts progressively higher weight on the “bad” states of the world, that is, the states where the payoff to LPs’ inventory of asset |$k$|⁠, |$x_{0,k} \delta_k$| is small. Such greater weight put on bad states of the world implies that the risk-neutral distribution of |$x_{0,k} \delta_k $| will be concentrated around small realizations of |$x_{0,k} \delta_k$| and thus skewed to the right, with a relatively small variance. See Figure 1 for an illustration. This makes the right-hand side of (23) negative, implying that the bid-ask spread decreases in |$\gamma$|⁠.

Figure 1

Risk-neutral distribution of |$\boldsymbol{x_{0,k} \delta_k}$| for |$\boldsymbol{\gamma=1,5,15}$|⁠. We assume that |$\boldsymbol{\delta\sim\textbf{Beta}[a,b]}$|⁠, |$\boldsymbol{a=b=2}$|⁠, |$\boldsymbol{N=1}$|⁠, |$\boldsymbol{x_0=1}$|⁠, and |$\boldsymbol{L=5}$|⁠. 

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The intuition underlying Propositions 3 and 4 complements the one presented in the Introduction. We now make a tighter link between our results and the intuition already introduced. First, analogously to Proposition 4, one can show that with positive initial inventory |$x_{0,k}$|⁠, the mid-price decreases in |$\gamma$| for high enough |$\gamma$|⁠. This is intuitive: when LPs are more risk averse the prices are lower to reflect higher inventory risk. It is only possible that bid-ask spread and mid-price both decrease when the ask decreases more than the bid. In the Introduction we argued that the key to ask decreasing more than the bid is the asymmetry in the effect of an increase in inventory on LPs’ marginal utility relative to a decrease in inventory. Technically, such asymmetry is captured by the coskewness terms in Proposition 3. The asymmetry is not due to payoff bounds, but rather due to nonnormality. Indeed, with normal distribution, the marginal utility is linear, and the effect is symmetric.

3.2 Bid-ask spreads: Empirics and discussion

This section documents several surprising empirical patterns in the prices of options on U.S. stocks that are consistent with our model’s predictions.

3.2.1 Data description

We obtain daily option prices and Black-Scholes implied volatilities from Option Research and Technology Services (ORATS), a data provider for historical options quotes and implied volatilities.²⁸ The data set covers the period of January 3, 2007, to May 17, 2021, and contains data for options bids, asks, volume, open interest, implied volatilities, and the price of the underlying asset for all U.S. stocks with traded options.²⁹

Many prices of options on small stocks are stale, and the underlying option contracts have extremely low liquidity. To highlight that small stocks do not drive our findings, we focus on the 145 stocks composing the NASDAQ index as of February 22, 2022. In addition, we prefilter data to exclude options with a volume of fewer than 20 contracts,³⁰ so we consider only those options with more than one and less than 60 days to expiry.

We define the bid-ask spreads as |$(Ask-Bid)/(0.5 (Ask+Bid))$|⁠. Each option contract is uniquely determined by its underlying asset, expiration date, option type (call or put), and strike. To avoid dealing with extremely illiquid contracts, we keep only those option contracts in our panel for which (a) bid prices are positive for both call and put for that strike, (b) bid-ask spreads are positive for both put and call at that strike, and (c) either call or put at that strike have |$Ask<1.5\cdot Bid.$|

3.2.2 Results and discussion

Before we proceed, we make a note about the notation. Throughout the section, we use subscript |$s$| to index stocks, subscript |$k$| to index option contracts on this stock and subscript |$t$| to index dates. We use two proxies for risk aversion: (a) the VIX, the CBOE S&P 500 implied volatility index, and (b) RA, the proxy for risk aversion kindly provided on the website of Bekaert et al. 2021. For each option contract, |$\Sigma^*_{s,k,t}$| denotes the proxy for the risk-neutral variance of its payoff (see below for a description of the construction of the proxy), and |$\text{IV2}_{s,k,t}$| denotes the squared implied volatility for a particular option contract. We also use option volume, open interest (⁠|${\it oi}$|⁠), implied volatility (⁠|${\it IV}$|⁠) (provided by ORATS), and its square (⁠|${\it IV}2={\it IV}^2$|⁠), as well as physical variance of the underlying stocks |$\Sigma_{k,k}$| in our regressions.

Cross-section of bid-ask spreads and risk-neutral variances:

That is, consistent with our theory, we assume that each date the markets for all options on a given stock are integrated, implying that the theoretical relationship of Proposition 3 should hold simultaneously for all options. For each stock and date combination |$(s,t)$|⁠, regression (28) is a linear panel regression across the whole panel of option contracts available on that date for that stock. For robustness, we also run this regression separately only for put and only for call options. Thus, we end up with the following three panels of |$\beta$| estimates: |$\beta^{Call}_{s,t},\ \beta^{Put}_{s,t},$| and |$\beta_{s,t}.$|

Proposition 3 implies the following empirical predictions for the behavior of the |$\beta$| estimates in (28):a

• (1)

  |$\beta^{Call}_{s,t},\ \beta^{Put}_{s,t},$| and |$\beta_{s,t}$| are all positive.

• (2)

  |$\beta^{Call}_{s,t},\ \beta^{Put}_{s,t},$| and |$\beta_{s,t}$| are positively related to risk aversion, as proxied for by either the VIX or RA.

We first look at prediction (1). We find that all three |$\beta$| estimates are positive for more than 99.95|$\%$| of observations in our panel. They have a mean of about |$1$| and a median of about |$0.6$|⁠. These findings are in perfect agreement with our prediction (1). Moving on to prediction (2), we can see from Table 1 that both the VIX and RA correlate positively with all three beta measures, with the RA correlation being significantly higher. We then take a deeper look at our prediction. We compute the correlation between |$\beta$| and proxies of risk aversion for every single stock and find that the correlation between |$\beta$| and RA is positive for 117 of 145 stocks, with a mean of 0.1 and a standard deviation of 0.2, which, with 145 observations, gives a t-statistic of 6. All these findings stand in perfect agreement with our predictions.

As panel C in Tables 2 and 3 shows, in both regressions the coefficient |$\beta_3$| is positive and highly significant, consistent with the theory predictions. One interesting additional test we run is whether |$\Sigma^*_{s,k,t}$| can be replaced by implied volatility. Despite no direct relationship between implied volatility and |$\Sigma^*_{s,k,t}$| theoretically, one might intuitively expect that these quantities are related because both are related to volatility. Therefore, we run a version of (29), with |$\Sigma^*_{s,k,t}$| replaced by squared implied volatility (IV2). As panel B in Tables 2 and 3 shows, |$\beta_3$| is indeed positive for puts, but not for calls, suggesting that for calls, |$\Sigma^*_{s,k,t}$| is capturing a dimension of risk-neutral variance that is not spanned by implied volatility.

Bid-ask spreads and risk aversion, open interest, and physical variance:

We now proceed to our other predictions. If one does not control for the risk-neutral variance, then conventional wisdom, largely based on the CARA-normal setting (see Corollary 2) suggests that the bid-ask spread is increasing in physical variance and risk aversion and that a release of public information should lead to a reduction in spreads. In contrast, Proposition 4 implies that the opposite can be true when risk aversion |$\gamma$| is sufficiently high. We take RA as our main proxy for risk aversion here. The results when RA is replaced with the VIX are very similar and are reported in Tables 8 and 9 in the appendix.

For markets in which liquidity providers have a sufficiently high risk aversion, we predict that bid-ask spreads:

• (1)

  may be decreasing in risk aversion,

• (2)

  may be decreasing in open interest,

• (3)

  may be decreasing in physical variance, and

• (4)

  may increase after the release of public information.

We believe that options markets and the markets of the respective underlying assets are highly fragmented; as a result, risk aversion (the risk-bearing capacity of liquidity providers) differs across them. For example, a trading desk can be intermediating a particular subset of options; capital requirements are extremely high for illiquid option contracts, implying that the effective risk aversion (the ability of dealers to take on balance sheet risk) will be much higher for options than for stocks. Yet, aggregate risk aversion is known to move in cycles, driven by the VIX (see Nagel 2012). Hence, we think about the “true” |$\gamma$| as being proportional to RA, |$\gamma=a RA,$| but with the proportionality constant |$a$| being much higher for options than for stocks. For markets with a small |$a$|⁠, we are in the “normal” regime, and Proposition 3 conforms with the findings for equities: bid-ask spreads are increasing in risk aversion (as proxied for by the VIX, see Nagel 2012).³⁵ For markets with a large |$a$|⁠, we are in the regime of Proposition 4, and the sign of the relationship may reverse.

We provide evidence for the surprising predictions of Proposition 4. Namely, we find a negative relationship between put bid-ask spreads and RA (panel A in Table 3).³⁶ The evidence for calls is similar, but we find that it might not be statistically significant for some empirical specifications (as is the case for panel A of Table 2).³⁷ We speculate that our results for put options are stronger because LPs’ effective risk aversion is higher for puts: LPs tend to have a short position in puts and so are exposed to downside risk, because LDs like to buy puts to get such protection (see Chen et al. 2019). We leave an empirical identification of whether our conjecture is true to future research. To the best of our knowledge, our paper is the first to document (and provide a theoretical foundation for) this surprising and counterintuitive relationship between risk aversion and option bid-ask spreads.

We now proceed to our second novel prediction: a negative relationship between option bid-ask spreads and open interest.³⁸ Although this link has not received much attention in the academic literature, it is an important part of option traders’ folklore. For example, according to Investopedia:³⁹“Open interest also gives you key information regarding the liquidity of an option.... All other things being equal, the bigger the open interest, the easier it will be to trade that option at a reasonable spread between the bid and ask.” Panel A in Tables 2 and 3 confirms this intuition, which is also fully consistent with the result of Proposition 4.

We will now discuss the dependence on physical variance. Panel A of Table 3 shows that for put options, this dependence is quite subtle: the relationship is strongly negative for variances below the panel median and then becomes positive for the part of the sample above the median.⁴⁰ We interpret this result as strong evidence for the importance of nonlinear models in explaining the illiquidity of options. Finally, for completeness, we also include option volume in the list of explanatory variables. Cao and Wei 2010 argue that the relationship between bid-ask spreads and volume is positive, potentially driven by asymmetric information. In contrast, in panel A of both tables, we find a strong, negative relationship between spreads and volume.⁴¹

Bid-ask spreads around earnings announcements:

Finally, we discuss our last empirical prediction, namely, the fact that, contrary to conventional wisdom, bid-ask spreads may increase after public information release. To test this prediction, we compare option bid-ask spreads on the day before an earnings announcement with the spread on the same option on the day after the earnings announcement. We proceed as follows:

• In our panel data set (prefiltered for volume and liquidity, as previously explained), we select all options that exist both before and after the earnings announcement date. For these options, we compute the change in the bid-ask spread,

  $$ \begin{equation} \Delta {\it BA} = {\it BA}_{{\it after\ earnings}} - {\it BA}_{\it before\ earnings}. \end{equation}$$

  (30)

• We then run a contemporaneous panel regression of spread changes on various controls. We include controls to make sure the change in the bid-ask spread is not mechanical.

Table 4 reports the results of the regressions separately for calls and for puts. We introduce three controls in our regression: moneyness change computed over the same time period as |$\Delta {\it BA}$|⁠, number of days to expiry,⁴² and change in the implied volatility. Consistent with our key prediction (and completely inconsistent with the CARA-normal model), we see that the constant term is positive and highly significant. Although all three controls in our regressions are significant, their explanatory power is quite low. The change in bid-ask spreads is dominated by the positive constant term; that is, bid-ask spreads significantly increase after earnings announcements.

4. Relation to the Literature

Our paper is related to two broad strands of the literature: strategic trading and models of asset trading without normality. In our model, information is symmetric, and price effects arise from traders’ limited risk-bearing capacity. We model trade using the classic uniform-price double-auction protocol in which traders submit price-contingent demand schedules. For the single-asset case, see Klemperer and Meyer 1989, Kyle 1989, Vayanos 1999, Wang and Zender 2002, Vives 2011, Rostek and Weretka 2012, Ausubel et al. 2014, Bergemann et al. 2015, Rostek and Weretka 2015b, Du and Zhu 2017, Kyle et al. 2017, and Lee and Kyle 2018; for the multiasset case, see Rostek and Weretka 2015a and Malamud and Rostek 2017.⁴³Antill and Duffie 2017 and Duffie and Zhu 2017 consider models in which the uniform-price auction market is augmented by price discovery sessions.

All of these papers feature traders with marginal utilities that are linear in trade size (which is either assumed directly or follows from the combination of CARA utility and normality of asset payoffs).⁴⁴ With the exception of Du and Zhu 2017, these papers derive linear equilibria with price impact that is constant.⁴⁵ As we previously noted, such models cannot speak to evidence for U.S. options that we present. Du and Zhu 2017 derive nonlinear equilibria when there are two agents, in which case no linear equilibria exist. Du and Zhu also show that nonlinear equilibria often exist. This nonlinearity is not linked to higher moments, which is a fundamental aspect of our paper; instead, in Du and Zhu, it is linked to strategic behavior by traders.⁴⁶ As far as we know, our paper is the first to derive closed-form solutions in a multiasset double auction with nonlinear marginal utility and to link nonlinearities in equilibrium properties with higher moments of asset payoffs.⁴⁷

A large body of literature exists on competitive trading with nonstrategic LPs in setups that deviate from CARA-normal. For example, several papers relax the assumption of normal payoff distributions but either maintain the CARA assumption or assume risk neutrality (see Gennotte and Leland 1990; Ausubel 1990a; Ausubel 1990b; Bhattacharya and Spiegel 1991; DeMarzo and Skiadas 1998, 1999, Yuan 2005; Albagli et al. 2015; Breon-Drish 2015; Pálvölgyi and Venter 2015; Chabakauri et al. 2017). Peress 2003 and Malamud 2015 examine noisy rational expectations equilibria with non-CARA preferences. In all of these papers, liquidity provision is competitive. In contrast, we assume that LPs are strategic and demonstrate that this assumption has notable implications for the (cross-)reversals of option returns.

Our paper is also related to the literature on transaction costs and asset prices; see Heaton and Lucas 1996, Vayanos 1998, Vayanos and Vila 1999, Lo et al. 2004, Acharya and Pedersen 2005, and Buss and Dumas 2019. Our study differs from these studies in that we assume transaction costs to be endogenous. In addition, we demonstrate that commonality in transaction costs (illiquidity) emerges endogenously in our model. This paper speaks to the literature on optimal dynamic execution algorithms for price effects that are exogenous and nonconstant (see Bertsimas and Lo 1998; Almgren and Chriss 2001; Almgren et al. 2005; Huberman and Stanzl 2005; Obizhaeva and Wang 2013). Our paper complements this literature by providing equilibrium foundations for nonlinear price functions.

The paper of Liu and Wang 2016 derives implications of earnings announcements for bid-ask spreads, similar to our paper. Assuming less information asymmetry after earnings announcements, the results in Liu and Wang 2016 imply that expected bid-ask spreads may decrease with information asymmetry. This happens because strategic market makers may optimally shift some trades with some investors to other investors by adjusting bid or ask. We believe their findings are complementary to ours. Our effect operates through changes in uncertainty, while their effect operates through changes in information asymmetry.

Finally, a related strand of the literature considers strategic liquidity provision and uses discriminatory price mechanisms to model trade. Notable examples are the studies of Biais et al. 2000 and Back and Baruch 2004, who also allow for non-Gaussian payoffs. An important difference between these papers and ours is that both Biais et al. and Back and Baruch assume that LPs are risk neutral and assume no inventory risk, whereas our model focuses on inventory risk.

5. Conclusion

We present a tractable model of strategic trading in an economy populated by a finite number of large and strategic CARA investors who trade a finite number of assets with an arbitrary distribution of asset payoffs. We show that departing from the common (but unrealistic) assumption of normal payoffs has far-reaching economic implications for asset illiquidity. More specifically, (a) illiquidity may decrease risk aversion, physical variance, and LPs’ inventory size; and (b) it may increase after earnings announcements. These results are consistent with the evidence for U.S. stock options that we present.

We develop a novel constructive approach to solve for the equilibrium in a multiasset strategic trading model in a closed form. We establish that solving for the equilibrium is equivalent to solving a linear ODE, which can be done using standard methods. It would be instructive to extend, along several relevant dimensions, our departure from the common CARA-normal assumption in strategic trading models. We are currently examining the equilibrium implications of wealth effects (i.e., removing the CARA assumption) and heterogeneity of investors’ wealth. Other extensions worth exploring include the cases of heterogeneity in investors’ risk aversion (as a means to study risk sharing among strategic traders), strategic informed trading, and dynamic strategic trading.

A. A Summary of Notation

B. Proofs

B.1 Proof of Theorem 1

B.2 Proof of Proposition 1

B.3 Proof of Proposition 2

B.4 Proof of Corollary 1

B.5 Proof of Corollary 2

B.6 Proof of Proposition 3

We first formulate a more general version of the proposition.

Hence, for small enough |$\gamma$|⁠, we have |${\operatorname{sign}}\left(\frac{\partial}{\partial \gamma} BA_k\right)={\operatorname{sign}}\left(\frac{\partial}{\partial \sigma} BA_k\right)>0$|⁠.

B.7 Proof of Proposition 4

We first state the more general statement of the proposition, which does not require |$\delta$| to be bounded on both sides. We prove the proposition in the more general form.

Proposition. Suppose that the density of |$\delta$|⁠, |$\eta(\delta)$|⁠, is strictly positive everywhere on its support. Suppose also that for all |$i$|⁠, the distribution of |$\delta_i x_{0,i}$| is bounded from below. Suppose that the dividend vector is given by |$\hat\delta=E[\delta]+\sigma (\delta-E[\delta])$|⁠, where |$\sigma$| is a scalar. Suppose that |$x_{0,k}\neq 0$|⁠. Suppose that the equilibrium characterized in Proposition 2 exists. The claims that follow correspond to that equilibrium. Then, the derivatives |$\frac{\partial}{\partial \gamma} BA_k$| and |$\frac{\partial}{\partial \sigma} BA_k$| are positive for small enough |$\gamma$| and change sign (at least once) as |$\gamma$| increases. Moreover, we have |$BA_k= 4\frac{L-1}{L-2}\gamma^{-1} x_{0,k}^{-2} + O(\gamma^{-1}).$| Correspondingly, for large enough |$\gamma$|⁠: the bid-ask spread decreases in risk aversion |$\gamma$| and initial inventory |$|x_{0,k}|$|⁠. Suppose further that there is a release of public information |$s_p$| of the form |$s_p=\delta+u$|⁠, where |$u\sim N(0,\Sigma)$|⁠. |${\operatorname{BA}}({\mathcal F})$| denotes the bid ask-spreads, given the information |${\mathcal F}$|⁠. Then, |$E[{\operatorname{BA}}(s_p)]-{\operatorname{BA}}(\emptyset)$| is positive for small enough |$\gamma$| and changes sign (at least once) as |$\gamma$| increases.

We start with the following useful lemmas.

Acknowledgement

We are grateful to Itay Goldstein (the editor) and two anonymous referees for their tremendous help throughout the review process. For their valuable feedback, we thank Bruno Biais, Svetlana Bryzgalova, Georgy Chabakauri, Jean-Edouard Colliard, Bernard Dumas, Daniel Ferreira, Thierry Foucault, Mike Gallmeyer (discussant), Craig Holden, Yunzhi Hu (discussant), Ravi Jaganathan, Mina Lee, Dong Lou, Peter Kondor, John Kuong, Albert (Pete) Kyle, Jiasun Li, Igor Makarov, Ian Martin, Milan Martinovic, Konstantin Milbrandt, Anna Obizhaeva, Olga Obizhaeva, Martin Oehmke, Joel Peress, Cameron Pfiffer, Rohit Rahi, Dimitri Vayanos, Gyuri Venter, S.“Vish” Viswanathan (discussant), Kathy Yuan, and Kostas Zachariadis. We are especially grateful to Marzena Rostek for her comments and suggestions. Semyon Malamud gratefully acknowledges financial support from the Swiss Finance Institute and the Swiss National Science Foundation [project no. 100018_192692]. This paper was previously circulated under the title “Asset Prices and Liquidity with Market Power and Non-Gaussian Payoffs.” Supplementary data can be found on The Review of Financial Studies web site.

Footnotes

References