Clearing prices under margin calls and the short squeeze
aa r X i v : . [ q -f i n . M F ] F e b Clearing prices under margin calls and the short squeeze
Zachary Feinstein ∗ February 4, 2021
Abstract
In this paper, we propose a clearing model for prices in a financial markets due to margin callson short sold assets. In doing so, we construct an explicit formulation for the prices that wouldresult immediately following asset purchases and a margin call. The key result of this work is thedetermination of a threshold short interest ratio which, if exceeded, results in the discontinuity of theclearing prices due to a feedback loop. This model and threshold short interest ratio is then comparedwith data from early 2021 to consider the observed price movements of GameStop, AMC, and NakedBrand which have been targeted for a short squeeze by retail investors and, prominently, by the onlinecommunity r/WallStreetBets.
Key words: short squeeze, margin call, market clearing
In early 2021, the online community r/WallStreetBets began a targeted campaign of retail investorsto purchase, or otherwise increase the price, of a few specific stocks. Most notably, this investmentcampaign led to large price swings in GameStop Corp. (GME). This, likewise, caused the distress ofcertain hedge funds such as Melvin Capital Management LP due to their large short positions in thisstock. The purpose of this work is to provide a model of prices for assets with large short positions;in particular, those positions are subject to margin calls and can face a short squeeze.Margin calls and the short squeeze are, in some sense, the mirror image of a fire sale and traditionalprice-mediated contagion. That, more traditional, setting considers the situation in which investorssell assets to satisfy regulatory requirements in a stress scenario; those asset liquidations cause thevalue of assets to decrease and, as such, increase the initial stress further. This feedback effect can leadto significant price drops. Fire sales and price-mediated contagion has been studied in, e.g., [3, 1, 6, 2].In contrast, in this work we are focused on a setting in which price increases have feedback effectsresulting in greater asset purchasing and even greater price increases.There are two primary innovations provided by this work. First, we provide an analytical expres-sion for the clearing prices for an asset with non-negligible number of short sales subject to margin ∗ Stevens Institute of Technology, School of Business, Hoboken, NJ 07030, USA, [email protected] alls. This formulation allows for counterfactual testing of different short selling situations as wellas sensitivity of the clearing solution to the various parameters of the system. Second, as a directconsequence of this analytical formulation, we find a threshold short interest ratio which determineswhether margin calls lead to a short squeeze and a resultant discontinuity in the clearing prices. Assuch, this threshold short interest ratio can be used to determine unstable market configurations. Thisthreshold can be utilized by investors who wish to target heavily short sold assets; similarly regulatorscan utilize such a result to determine short selling constraints that promote the tradeoffs betweenmarket efficiency and price stability.The organization of this work is as follows. In Section 2, we provide background information onshort selling and a simple model for margin calls on such obligations. This is extended in Section 3 inwhich we determine the prices which clear a financial market with external investors and the possibilityfor margin calls on short sold assets. Section 4 considers the implication of these clearing prices. Inparticular, a threshold short interest ratio is determined below which the prices are continuous withrespect to the actions of external investors but above which prices can jump. This model is thentested against data from early 2021 in three case studies of stocks targeted by the online communityr/WallStreetBets in Section 5. Notationally, let
S > be the total shares short sold by some financial institution at the initial priceof p > . On these shares, the institution has posted an initial margin of M = (1 + α ) Sp for some α > ; that is, the short selling institution must post all proceeds from the original sale of the assetas well as an additional α proportion in cash. Within the United States, this initial margin α = 50% by Federal Reserve Board Regulation T. This margin is to guarantee that the short selling institutionwill honor its agreement to return the S assets at some future date. For simplicity, we will assumethat there is no interest charged on the short selling firm for borrowing the S shares.As prices change from p to p , the short selling institution must guarantee that they retain at least (1 + µ ) Sp in the margin account. This level µ ∈ (0 , α ) is called the maintenance margin. Within theUnited States, this maintenance margin µ ≥ (see, e.g., Federal Reserve Board Regulation T). Forthe purposes of this work, we ignore the possibility of the firm withdrawing holdings from the marginaccount as the prices drop. Therefore, the margin account is unaffected so long as M ≥ (1 + µ ) Sp .There are two basic ways in which the short selling institution may satisfy this maintenance marginwhen M < (1 + µ ) Sp :(i) the firm can post additional cash in the amount of [(1 + µ ) Sp − M ] + so that the margin accountsatisfies the maintenance margin; or(ii) the firm returns Γ shares of the asset so that M ≥ (1 + µ )( S − Γ) p , i.e., Γ := [ S − M (1+ µ ) p ] + asthe minimal required shares to return.Within this work we assume, except where otherwise explicitly mentioned, that the firm chooses to hysical Units Proportion of ADV Relation Short sold shares
S s S = sV Margin account M = (1 + α ) S m = (1 + α ) s M = mV External capital purchases
C c C = cV Market impacts b β b = β/V Table 1: Summary of notation utilized in this work. solely return shares rather than post additional cash as it has the lower cost as Γ p = [ Sp − M µ ] + ≤ [(1 + µ ) Sp − M ] + . As introduced in the prior section, as prices rise the margin requirements may require a firm that hasshort sold assets to purchase assets back. Within this section, we are interested in the dynamics thatprices take. In particular, we are interested in how prices rise with asset purchases – notably suchconstructions will not depend on which market participant is transacting. We will consider a linearinverse demand function for this purpose, i.e., if x ≥ assets are purchased in the financial marketthen the resulting price is: f ( x ) := 1 + bx for market impact parameter b > . This linear inverse demand function is prominently utilized in thefire sale literature; we refer the interested reader to, e.g., [9, 4, 8]. As highlighted by this linear inversedemand function, we assume without loss of generality that the price at the start of this event is .Furthermore, we assume this initial price of is high enough so that (1 + µ ) S ≤ M , i.e., no margincall occurs without outside intervention; for notation let α = M − SS ∈ [ µ, ∞ ) , i.e., M = (1 + α ) S .Before continuing, we wish to summarize notation utilized in this work. In particular, we considertwo (equivalent) formulations: (i) in physical units of assets and cash or (ii) in proportion to theaverage daily volume [ADV] V > of the asset. These notations are summarized in Table 1. Inparticular, with the proportion of ADV notation, we wish to highlight that s is often called the “shortinterest ratio” or the “days-to-cover ratio.”With the external capital purchases C > , and the possibility of asset purchases to satisfy themargin requirements, the price p must satisfy the clearing equation p = f Cp + (cid:20) S − M (1 + µ ) p (cid:21) + ! . (1)That is, the price must satisfy an equilibrium with the number of assets being purchased externallyto the short selling institution ( C/p ) and those purchased to satisfy the margin requirements ( [ S − M (1+ µ ) p ] + ). The following proposition guarantees there exists some clearing price. In fact, as providedin Lemma 3.2 below, there exists a realized clearing price for which we can give an explicit formulation. Proposition 3.1.
There exists some clearing price p ∈ [1 , f ( C + S )] satisfying (1) . roof. Let Φ( p ) := f ( Cp + [ S − M (1+ µ ) p ] + ) . Then, by monotonicity of the inverse demand function f ,it must follow that Φ(1) ≥ f ( C ) > f (0) = 1 and Φ( f ( C + S )) ≤ f ( C + S ) . As the linear inversedemand function is continuous, existence of a clearing price follows immediately by Brouwer’s fixedpoint theorem.Now, we wish to consider an explicit construction for a clearing price p ∗ of (1) which is realized inthe sense that no assets are purchased that are not forced to occur. As such, this clearing price can beconstructed as the result of the fictitious margin call notion (akin to the fictitious default algorithmof [5, 10]) or a tâtonnement process (as in, e.g., [7, 8]). Lemma 3.2.
Let p ∗ take value: p ∗ = √ bC if C ≤ bS M µ h M − (1+ µ ) S (1+ µ ) S i bS + q (1+ bS ) +4 b [ C − M µ ] if C ≤ bS M µ h M − (1+ µ ) S (1+ µ ) S i (2) = √ βc if c ≤ β α µ h α − µ µ i βs + q (1+ βs ) +4 β [ c − (1+ α ) s µ ] if c > β α µ h α − µ µ i . (3) Then p ∗ is a clearing price to (1) and satisfies the following algorithm:(i) determine the unique positive price assuming no margin call is required, i.e., p ∗ = f ( Cp ∗ ) with p ∗ > ;(ii) if (1 + µ ) Sp ∗ ≤ M then terminate and report p ∗ ;(iii) if (1 + µ ) Sp ∗ > M then determine the unique price resulting in a margin call, i.e., p ∗ = f ( S + h C − M µ i /p ∗ ) with (1 + µ ) Sp ∗ > M .Proof. Following the provided algorithm we wish to show that the resulting clearing price is givenby (2). To do so we will use the specific form of the linear inverse demand function.(i) Assume that the short selling firm is not subject to a margin call. This would occur if (1+ µ ) Sp ∗ ≤ M at the clearing price p ∗ .Rewriting (1) under the no margin call assumption with explicit formulation of the inversedemand function provides the clearing equation: p ∗ = 1 + bCp ∗ . As this is equivalent to a quadratic equation, there trivially exist two possible clearing prices: p ∗ ∈ { ±√ bC } . However, by observation, < √ bC ; therefore, −√ bC < is nota reasonable clearing price. As such, in this no margin call scenario, there must be a uniqueclearing price given by: p ∗ = 1 + √ bC √ βc . (4) e conclude this discussion of the no margin call setting by considering which values of theexternal capital purchase C (or equivalently c ) result in no margin calls. That is, under whichexternal capital purchases is this value an actual clearing solution. In particular, this consistencyholds if and only if Sp ∗ ≤ M µ . By utilizing (4), the external capital purchases must boundedfrom above by, in either physical unit or proportional notation, C ≤ bS M µ (cid:20) M − (1 + µ ) S (1 + µ ) S (cid:21) c ≤ β α µ (cid:20) α − µ µ (cid:21) =: c ∗ . (5)(ii) We now assume that the short selling firm is subject to a margin call. Utilizing the resultsabove, assume that C > bs M µ h M − (1+ µ ) S (1+ µ ) S i – otherwise no margin call would be required andwe recover the form above. In contrast to above, this would occur if (1 + µ ) Sp ∗ > M at theclearing price p ∗ .Rewriting (1) under a margin call with the linear inverse demand function provides the clearingequation: p ∗ = 1 + bS + b (cid:20) C − M µ (cid:21) /p ∗ . As this is equivalent to a quadratic equation, there trivially exist two possible clearing prices p ∗ ∈ bS ± r (1 + bS ) + 4 b h C − M µ i . Before discussing which of these clearing prices is feasible, we wish to show that the clearing pricesare real-valued. That is, we wish to demonstrate that the discriminant (1 + bS ) + 4 b h C − M µ i is nonnegative. Recall C > bS M µ h M − (1+ µ ) S (1+ µ ) S i . By simple substitution and bounding of thecapital purchases: (1 + bS ) + 4 b (cid:20) C − M µ (cid:21) ≥ (1 + bS ) + 4 b M µ (cid:20) bS (cid:18) M − (1 + µ ) S (1 + µ ) S (cid:19) − (cid:21) = (1 + bS ) + 4 M µ (cid:20) M − (1 + bS )(1 + µ ) S (1 + µ ) S (cid:21) = 1 S (cid:18) S (1 + bS ) + 4 M µ (cid:20) M − (1 + bS )(1 + µ ) S µ (cid:21)(cid:19) = 1 S (cid:18) S (1 + bS ) + 4 M µ (cid:20) M µ − (1 + bS ) S (cid:21)(cid:19) = 1 S S (1 + bS ) − S (1 + bS ) M µ + 4 (cid:20) M µ (cid:21) ! = 1 S (cid:18) S (1 + bS ) − M µ (cid:19) ≥ . Now, we wish to consider the possible clearing price bS − q (1+ bS ) +4 b [ C − M µ ] in order to demon-strate that it is not the clearing price. In particular, this price does not require any margin call, .e., S bS − r (1 + bS ) + 4 b h C − M µ i < M µ , as such a relation is equivalent to our condition on whether a margin call will occur C > bS M µ (cid:20) M − (1 + µ ) S (1 + µ ) S (cid:21) . As this price would not require a margin call, it cannot be the clearing price.Therefore, utilizing the existence of a clearing price from Proposition 3.1, the clearing price mustbe p ∗ = 1 + bS + r (1 + bS ) + 4 b h C − M µ i . It can be verified that (1 + µ ) Sp ∗ > M under our condition on C at this clearing price thusverifying our result. For the purposes of this section, we will focus solely on the clearing price given by (3) based on theproportion of ADV formulation as provided in Lemma 3.2. In particular, we will focus on the changein prices at the cutoff threshold for the margin call c ∗ as provided in (5). We wish to highlight thatthe threshold for no margin calls to occur is independent of the size of the short sale as evidencedfrom the form of c ∗ . Definition 4.1.
The size of the short squeeze is the difference between the price with a margin callwith that without a margin call at external capital purchase of c ∗ , i.e., δ = lim c ց c ∗ p ∗ ( c ) − lim c ր c ∗ p ∗ ( c ) with explicit dependence of the clearing price on capital purchases with p ∗ defined as in (3) . We now wish to provide an explicit formulation for the size of the short squeeze and, in particular,necessary and sufficient conditions for the existence of a short squeeze based on the size of the shortinterest ratio.
Theorem 4.2.
The size of the short squeeze is given by δ = h βs − − µ +2 α µ i + .Proof. Consider the form of the clearing price from (3). δ = 12 " βs + s (1 + βs ) + 4 β (cid:20) c ∗ − (1 + α ) s µ (cid:21) − p βc ∗ . o simplify the size of the short squeeze, consider the form of both discriminants at c ∗ . First, forthe case without a margin call, βc ∗ = 1 + 4 1 + α µ (cid:18) α − µ µ (cid:19) = 1 + 2 µ + µ + 4 α − µ − αµ + 4 α µ = (cid:18) − µ + 2 α µ (cid:19) . Second, for the case with a margin call, (1 + βs ) + 4 βc ∗ − βs α µ = 1 + 4 βc ∗ + 2 βs + ( βs ) − βs α µ = (cid:18) − µ + 2 α µ (cid:19) + 2 βs (cid:18) − α µ (cid:19) + ( βs ) = (cid:18) − µ + 2 α µ − βs (cid:19) . Combining these above results leads to the form for the short squeeze: δ = 12 (cid:20) βs + (cid:12)(cid:12)(cid:12)(cid:12) − µ + 2 α µ − βs (cid:12)(cid:12)(cid:12)(cid:12) − − µ + 2 α µ (cid:21) = (cid:20) βs − − µ + 2 α µ (cid:21) + . This result on the size of the short squeeze leads to the important corollary below.
Corollary 4.3.
The size of the short squeeze is strictly positive δ > if and only if s > s ∗ := − µ +2 αβ (1+ µ ) proportion of the ADV has been short sold.Proof. This follows from a trivial application of Theorem 4.2.
In this section we wish to consider three stocks that have been part of a coordinated action from theonline community r/WallStreetBets in early 2021. Specifically we focus on GameStop Corp. (GME),AMC Entertainment Holdings Inc. (AMC), and Naked Brand Group Ltd. (NAKD). All data utilizedin this section was collected from Yahoo Finance with data chosen so as to be timed prior to theactions of the online community r/WallStreetBets in early 2021.To simplify these examples we consider constant parameters α = 0 . and µ = 0 . to conformwith regulatory requirements and provide simple heuristic values. Notably, these values provide aconstant threshold s ∗ ≈ . for the existence of a strictly positive short squeeze. That is, pricesbecome unstable and jump if the shorted quantity s exceeds s ∗ so long as external investors injectmore than c ∗ into the stock. Additionally, to simplify these considerations, we will proceed with an stimated market impact of β = 2 ; we wish to note that, due to the fact that s ∗ is constant withrespect to the market impact, the primary conclusions drawn in these examples are insensitive to thechoice of β . Example 5.1. [GameStop Corp.] In this first case study, we consider GME stock in early 2021. Asof December 15, 2020 – prior to the early 2021 price movements for the stock – there are 69.75 millionshares of the stock outstanding; of those shares, 46.89 million are actually floating and available fortransactions. The ADV for all trading days in 2020 was approximately 6.68 million shares per day.Critically for this work, as of December 15, 2020, a total of 68.13 million shares were shorted. Thatis, the short interest ratio is given by s = 68 . / . ≈ . . Notably this short interest ratio isfar in excess of s ∗ ≈ . . As such, we find that a strictly positive short squeeze occurs with size δ = βs − − µ +2 α µ ≈ . , i.e., the price would jump by more than 1900% due to a short squeeze.With a price per share of approximately $17/share prior to the short squeeze, we would anticipatea sudden jump in prices to over $340/share due to the short squeeze. This is consistent with pricemovements observed in early 2021 leading up to a spike in prices on January 27, 2021. Example 5.2. [AMC Entertainment Holdings Inc.] In this second case study, we consider AMCstock in early 2021. As of January 15, 2021, there are 287.28 million shares of the stock outstanding;of those shares, 114.94 million are actually floating and available for transactions. The ADV for alltrading days in 2020 was approximately 10.70 million shares per day. Critically for this work, as ofJanuary 15, 2021, a total of 44.67 million shares were shorted. That is, the short interest ratio is givenby s = 44 . / . ≈ . . Notably this short interest ratio is in excess of s ∗ ≈ . . As such, we findthat a strictly positive short squeeze occurs with size δ = βs − − µ +2 α µ ≈ . , i.e., the price wouldjump by more than 700% due to a short squeeze. With a price per share of approximately $2.33/shareprior to the short squeeze, we would anticipate a sudden jump in prices to over $18.90/share due tothe short squeeze. This is consistent with price movements observed on in early 2021 with a significantprice spike observed on January 27, 2021. Example 5.3. [Naked Brand Group Ltd.] In this final case study, we consider NAKD stock inearly 2021. As of January 15, 2021, there are 234 million shares of the stock outstanding; of thoseshares, 41.61 million are actually floating and available for transactions. The ADV for all tradingdays in 2020 was approximately 31.23 million shares per day. Critically for this work, as of January15, 2021, a total of 31.26 million shares were shorted. That is, the short interest ratio is given by s = 31 . / . ≈ . . Notably this short interest ratio is below s ∗ ≈ . . As such, we find thatno short squeeze jump in prices would occur. We do, however, wish to note that the margin callsand external purchase of assets will cause prices to rise – possibly very rapidly – even without such arealized jump due to the short squeeze. The relative stability of NAKD stock in early 2021 in contrastto the volatility of GME and AMC may point to a similar conclusion. Conclusion
In this work we developed a formulation that provides the equilibrium price due to feedback effects ofmargin calls on short sales. In doing so, we found two cutoff levels: (i) c ∗ fraction of the ADV thatneeds to be invested into the asset to trigger a margin call and (ii) s ∗ fraction of the ADV that, ifshort sold, will trigger a short squeeze and a rapid jump in prices. We found, numerically, that recentprice movements in several stocks can be attributed to the short squeeze as they have been highlyshort sold – above s ∗ – and, due to increased retail investment, have likely exceeded c ∗ in externalinvestments. The threshold to achieve a short squeeze s ∗ is of wider interest as it provides a thresholdfor which prices can become unstable. This threshold would be of particular interest to regulatorsas the consequences are inherently tied to financial stability as, if s ∗ is exceeded, short squeezes andprice volatility is a natural consequence. References [1] Hamed Amini, Damir Filipović, and Andreea Minca. Uniqueness of equilibrium in a payment systemwith liquidation costs.
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