Topics in Pricing

Lecture 2: Limits to

Johan Hombert (HEC Paris)

Master in Economics – Spring 2020 Explaining mispricing

Need to explain two things:

A. Why does mispricing appear?

I Institutional reasons

I Behavioral reasons

B. Why does mispricing persist? → “Limits to arbitrage”

1. Imperfect diversification

– Fundamental risk

– Noise trading risk

2. Funding constraints

– Capital flows

– Leverage constraints Fundamental risk

I Idea: arbitrageurs are imperfectly diversified ⇒ limits their willingness to exploit mispricing, even if no systematic risk I Risky asset in zero net supply I t = 1: demand shock D ≶ 0, (endogenous) price P I t = 2: payoff Ve ∼ N (V , σ2)

I Risk-free asset, normalize rf = 0 I Mass A of competitive arbitrageurs

I Endowment W1 I Buy X shares of risky asset

I Consume W2 = W1 + X (Ve − P) −γW I CARA utility: u(W2) = −e 2

I Equilibrium γ P = V + σ2D A Index effect (Wurgler and Zhuravskaya, 2002) Index effect (Wurgler and Zhuravskaya, 2002) Noise trading risk (De , Shleifer, Summers, Waldmann, 1990) I OLG model at t: endowment w invested −γc at t + 1: consume, u(ct+1) = −e t+1 I Risk-free asset r every period perfectly elastic supply at price = 1 I “Risky” asset dividend r every period supply = 1, (endogenous) price pt n ∗ 2 I µ noise traders believe Et [pt+1] = Et [pt+1] + θt , θt ∼ N (θ , σθ ) I 1 − µ arbitrageurs hold correct beliefs I Equilibrium

µ(θ − θ∗) µθ∗ γµ2σ2 p = 1 + t + − θ t 1 + r r r(1 + r)2 Noise trading risk (DSSW 1990)

When arbs have horizons:

– “Noise traders create their own space”

– Excess volatility

– Return predictability (reversal)

– Noise traders don’t necessarily die out Noise trading risk: Evidence (Foucault, Sraer, Thesmar, 2011) Noise trading risk: Evidence (Foucault, Sraer, Thesmar, 2011) Noise trading risk: Evidence (Foucault, Sraer, Thesmar, 2011) Arbitrageurs horizon

I Endogenous short horizon because of flow-performance relation

I Can even amplify non-fundamental fluctuations (Shleifer and Vishny, 1997) Slow moving capital (Mitchell, Pedersen and Pulvino, 2007)

I Models: Vayanos and Woolley (2013), Moreira (2019) Arb (Hombert and Thesmar, 2014)

I Can arbs increase the duration of their liabilities? Leverage constraints I Idea: arbitrageurs face funding constraints I Model: issuance impossible, issuance constrained I Key papers: Gromb and Vayanos (2002), Brunnermeier and Pedersen (2009), Gˆarleanuand Pedersen (2011) I Today: simple version of these models I Asset in zero net supply t = 1: trade at price P , supply shock for S+P1−V shares (0

I Risk-free asset, normalize rf = 0 I Mass 1 of competitive arbitrageurs

I Endowment W1; position X1 in asset; max W2 = W1 + X1(V − P1) I Leverage constraint: mP1|X1| ≤ W1 ( or m ∈ [0, 1])

W1 I Equilibrium: P1 = V − S + if W1 < mS, else P1 = V m I What if S < 0? Amplification

I Arbs start at t = 1 with X0 > 0 shares of asset:

W1 = W0 + X0(P1 − P0)

I ⇒ Amplification mechanism (“fire sales”) LoOP violation & Contagion

I Extend to several i = 1, ..., I

t = 1: selling pressure Si +Pi −V , price P I Pi i

I t = 2: payoff V

I Equilibrium V Pi = for all i 1 + λmi   where λ > 0 such that m S − λmi V = W ∑i i i 1+λmi 1

I Violation of Law of One Price

I Contagion Evidence: Gˆarleanuand Pedersen (2011) Chen, Chen, He, Liu, Xie (2020) Chen, Chen, He, Liu, Xie (2020) What determines margins?

I Gromb and Vayanos (2002): assume risk-free debt

Vmin I Ve ∈ [Vmin, Vmax] ⇒ m = 1 − P

I Fostel and Geanakoplos (2015) derive risk-free margin as optimal contract in binomial model Ve ∈ {Vmin, Vmax}

I Brunnermeier and Pedersen (2009): assume Var-at-Risk rule

V˜ −P I α-VaR ⇔ m such that Pr( P < −m) = α

I Adrian and Shin (2014) derive VaR rule as optimal debt contract in presence of risk-shifting What determines margins?

I m depends on distribution of V˜ → how is it estimated?

I Case 1: constant volatility, distribution of V˜ known

I e.g., Gromb-Vayanos 2002

I ⇒ stabilizing margins because P ↓⇒ m ↓

I Case 2: time-varying volatility

I Brunnermeier-Pedersen 2009: ARCH volatility σt+1 = σ¯ + θ|∆Vt | estimated using |∆Pt |

I ⇒ destabilizing margins because P ↓↓⇒ m ↑ (“liquidity spiral”) Margins

Margin on S&P 500 futures Haircut on repo transactions (average across all collateral excluding US treasuries) (Brunnermeier and Pedersen, 2009) (Gorton and Metrick, 2011) Financial intermediaries leverage I Adrian and Shin (2010) Leverage constraints: Risk premium

I Arbs choose Xi0 at t = 0 → endogenize Pi0 I To simplify: no leverage constraint at t = 0

I Si , mi random, realized at t = 1 ⇒ Pi1, λ, W1 also random I Date 0 equilibrium

Cov0(λ, Pi1) Pi0 = E0[Pi1] + < E0[Pi1] E0[1 + λ]

⇒ Expected returns

1 E [Ri1] = − Cov0(λ, Ri1) E0[1 + λ]

I Empirical evidence? I Challenge: proxy for λ I “Intermediary asset pricing”: financial intermediaries balance sheet I Adrian, Etula and Muir (2014) vs. He, Kelly and Manela (2017) Adrian, Etula and Muir (2014)

−1 I λt ∼ (broker-dealer leverage)

I Intuition: ⇔ bad times Adrian, Etula and Muir (2014) He, Kelly and Manela (2017)

−1 I λt ∼ (broker-dealer capital ratio) = leverage

I Intuition: low capital ratio ⇔ bad times

I Opposite to Adrian-Etula-Muir He, Kelly and Manela (2017) Leverage constraints: Welfare

I Need to model noise traders and specify their utility function

I First Welfare Theorem doesn’t apply

I Pecuniary externality: arbs do not internalize that their actions affect the leverage constraints of other arbs through prices

I Gromb and Vayanos (2002), Lorenzoni (2008) References I Adrian, Etula and Muir 2014. Financial intermediaries and the cross-section of asset returns. Journal of Finance I Adrian and Shin 2010. Liquidity and leverage. Journal of Financial Intermediation I Adrian and Shin 2014. Procyclical leverage and value-at-risk. Review of Financial Studies I Brunnermeier and Pedersen 2009. Market liquidity and funding liquidity. Review of Financial Studies I Chen, Chen, He, Liu and Xie. 2020. Pledgeability and asset prices: Evidence from the Chinese corporate markets. I De Long, Shleifer, Summers and Waldmann 1990. risk in financial markets. Journal of Political Economy I Fostel and Geanakoplos 2015. Leverage and default in binomial economies: a complete characterization. Econometrica I Foucault, Sraer and Thesmar 2011. Individual investors and volatility. Journal of Finance I Gˆarleanuand Pedersen 2011. Margin-based asset pricing and deviations from the law of one price. Review of Financial Studies I Gorton and Metrick 2011. Securitized banking and the run on repo. Journal of I Gromb and Vayanos 2002. Equilibrium and welfare in markets with financially constrained arbitrageurs. Journal of Financial Economics I He, Kelly and Manela 2017. Intermediary asset pricing: new evidence from many asset classes. Journal of Financial Economics I Hombert and Thesmar 2014. Overcoming limits of arbitrage: theory and evidence. Journal of Financial Economics I Lorenzoni 2008. Inefficient credit booms. Review of Economic Studies I Mitchell, Pedersen and Pulvino 2007. Slow moving capital. American Economic Review I Moreira 2019. Capital immobility and the reach for yield. Journal of Economic Theory I Shleifer and Vishny 1997. The limits of arbitrage. Journal of Finance I Vayanos and Woolley 2013. An institutional theory of momentum and reversal. Review of Financial Studies I Wurgler and Zhuravskaya 2002. Does arbitrage flatten demand curves for ? Journal of Business