Shorting in Speculative Markets

• DOI: http://doi.org/10.1111/jofi.12871
• Published date: 01 April 2020
• Date: 01 April 2020
• Author: JOSÉ A. SCHEINKMAN,MARCEL NUTZ

THE JOURNAL OF FINANCE •VOL. LXXV, NO. 2 •APRIL 2020

Shorting in Speculative Markets

MARCEL NUTZ and JOS ´

E A. SCHEINKMAN∗

ABSTRACT

In models of trading with heterogeneous beliefs following Harrison-Kreps, short sell-

ing is prohibited and agents face constant marginal costs-of-carry. The resale option

guarantees that prices exceed buy-and-hold prices and the difference is identified as

a bubble. We propose a model where risk-neutral agents face asymmetric increas-

ing marginal costs on long and short positions. Here, agents also value an option to

delay, and a Hamilton-Jacobi-Bellman equation quantifies the influence of costs on

prices. An unexpected decrease in shorting costs may deflate a bubble, linking finan-

cial innovations that facilitated shorting of mortgage-backed securities to the collapse

of prices.

ASKINDLEBERGER AND ALIBER (2005)OBSERVE, many classical economists ar-

gued that the purchase of securities for resale rather than for investment

income is what drives asset price bubbles. To explain such speculation in a dy-

namic equilibrium model, Harrison and Kreps (1978) study risk-neutral agents

with fluctuating heterogeneous beliefs. In their model, long positions can be fi-

nanced at a constant interest rate and short selling is ruled out. The buyer

of an asset thus acquires both a stream of future dividends and an option

to resell, which together with fluctuating beliefs guarantees that speculators

are willing to pay more for an asset than they would pay if they were forced

to hold the asset to maturity, that is, what risk-neutral investors are will-

ing to pay to be able to speculate. Scheinkman and Xiong (2003) consider

a model in which heterogeneous beliefs result from agents’ overconfidence

on different public signals and added trading costs. They show that these

models generate a correlation between trading volume and overpricing,1a

∗Marcel Nutz is with Columbia University. Jose Scheinkman is with Columbia University,

Princeton University, and NBER. We are indebted to Pete Kyle and Xunyu Zhou for fruitful

discussions and to two referees, an Associate Editor, and Stefan Nagel, the Editor, for comments

and suggestions that greatly improved the paper. An earlier version was entitled “Supply and

Shorting in Speculative Markets.” Nutz’s research is supported by an Alfred P. Sloan Fellowship

and NSF Grants DMS-1512900 and DMS-1812661. Wehave read The Journal of Finance disclosure

policy and have no conflicts of interest to disclose.

Correspondence: Jose Scheinkman, Department of Economics, Columbia University, 420 West

118th Street, New York, NY 10027; e-mail: js3317@columbia.edu.

1See also Berestycki et al. (2019).

DOI: 10.1111/jofi.12871

C2020 the American Finance Association

995

996 The Journal of Finance R

characteristic associated with several major bubble episodes over the last three

centuries.2

Another stylized fact is that bubble implosions often follow increases in sup-

ply.For instance, the implosion of the dotcom bubble was preceded by a massive

increase in the float of Internet shares.3Similarly, while the South Sea bubble

lasted less than one year, the amount of outstanding shares of the South Sea

Company (SSC) more than doubled during that period and many other joint-

stock companies were established.4However, the assumption of risk-neutral in-

vestors facing constant marginal costs in the earlier literature on disagreement

and bubbles implies that supply is irrelevant for pricing.5Hong, Scheinkman,

and Xiong (2006) analyze a two-period model with risk-averse investors in

which unexpected increases in supply can deflate bubbles. The economics are

straightforward—when agents are risk averse, their marginal valuation for a

risky asset decreases with the amount they hold.

Short selling an asset can be seen as a source of additional supply. The col-

lapse of prices for mortgage-backed securities (MBSs) in 2007 was preceded by

a series of financial innovations that facilitated shorting: the creation of stan-

dardized credit default swaps (CDS) for MBS in 2005, the introduction of traded

indexes for subprime mortgage-backed credit derivatives in 2006, and the use of

CDS to construct synthetic collateralized debt obligations (CDOs) that allowed

Wall Street to satisfy the global demand for U.S. AAA mortgage bonds with-

out going through the relatively slow process of originating new mortgages.6

The amounts shorted were substantial. Cordell, Huang, and Williams (2011)

estimate that synthetic CDOs, issued mostly after 2005H2, more than doubled

the amount of BBB Home Equity Bonds placed in CDOs between 1998 and

2007.7It is unlikely that this supply would have been absorbed without any

price impact. In any case, starting in the second half of 2007, prices appear to

exhibit substantial discounts relative to fundamentals.8

2See, for example, Carlos, Neal, and Wandschneider (2006) on the South Sea bubble, Hong and

Stein (2007) on the Roaring Twenties, Ofek and Richardson (2003) and Cochrane (2002)onthe

Internet bubble, and Xiong and Yu (2011) on the Chinese warrant bubble.

3See Ofek and Richardson (2003).

4The directors of the SSC understood that bubble companies competed with the SSC’sconversion

scheme and could deflate its own bubble. Harris (1994) documents that the Bubble Act of 1720,

which banned joint-stock companies unless if authorized by Royal Charter,was issued at the behest

of the company to limit the competition for capital.

5Except for the assumption of positive net supply, questions concerning the supply of the asset

subject to bubbles are also ignored in the rational bubbles literature (Santos and Woodford (1997)).

6See Scheinkman (2014) for a summary or Lewis (2015) for an excellent detailed account.

7BBB tranches of Home Equity Bonds were an important part of the CDO machine that trans-

formed subprime mortgages into AAA-rated bonds.

8Beltran, Cordell, and Thomas (2017) provide a methodology to calculate the intrinsic value of

a CDO and apply it to market data (see their Appendix A). They attribute the low prices to the

increase in information asymmetry between buyers and sellers who followed the downgrades of

MBS securities by rating agencies in summer 2007. Analyzing the pricing of index CDS postcrisis,

Stanton and Wallace (2011) suggest that the pricing reflected a limited supply of insurance of

asset-backed securities, presumably relative to demand.

Shorting in Speculative Markets 997

In this paper, we propose a finite-horizon continuous-time model with ntypes

of investors who trade a single asset and aim to maximize expected cumula-

tive net gains from trade. These investors are risk neutral, face a constant

interest rate, and have fluctuating heterogeneous beliefs about the evolution

of a Markov state that determines the asset’s payoff. In contrast to previous

literature, shorting is allowed. Investors pay costs that are proportional to the

square of their positions but the constant of proportionality that defines the cost

of going short may be larger than the corresponding constant for going long.

This asymmetry between the costs of going short and long is a well-known

feature of financial markets (see, e.g., D’Avolio (2002)). The assumptions in the

earlier literature correspond to infinite costs for short positions and constant

marginal costs for long positions. The costs in our model, by contrast, can be

thought of as capturing the monetary costs of holding a position (in particular,

increasing costs of capital), as well as risks that we do not explicitly model,

such as market-wide liquidity shocks that would force agents to liquidate their

positions at unfavorable prices or the recall risk faced by short sellers.

Since costs are quadratic, an agent’s marginal valuation of an asset will de-

crease as their position increases, as would be the case for risk-averse agents.

We therefore view our setup as an alternative to a much less tractable model

with risk aversion, with many of the same forces present.9In particular, we

show that an increase in the aggregate supply of the asset decreases equilib-

rium prices. Importantly,using the two cost coefficients as separate parameters

allows us to impose asymmetric costs and study the impact of changes in rel-

ative costs on prices. By contrast, traditional models with risk aversion treat

longs and shorts symmetrically or rule out shorts by imposing portfolio con-

straints.

We model the asset’s equilibrium price as a function of time and the current

state. Types that expect prices to increase on average over the next instant

choose to go long, with the size of their position depending on the difference

between their expected price changes and the marginal cost of carrying long

positions. The other types choose to go short, by amounts that depend on their

expected price changes and the cost of carrying short positions. Equilibrium

requires that the longs absorb the shorts plus an exogenous supply. Theorem 1

below shows that there exists a unique equilibrium price function and that it

can be characterized by a partial differential equation (PDE). This equation is

of Hamilton-Jacobi-Bellman type with a novel form. In particular, the optimiza-

tion runs over the ways to divide agents into two groups at any time and state;

at the optimum, these are optimists (holding long positions in equilibrium) and

pessimists (holding shorts). A noteworthy feature is that supply enters math-

ematically as a running cost (i.e., like intermediate consumption in Merton’s

problem). Theorem 1also quantifies how these costs influence the effect of

9One difference from models of disagreement that use risk aversion to avoid no-shorting con-

straints is that the presence of holding costs allows an equilibrium to exist even when agents

disagree about perceived arbitrage opportunities.