Topics in Asset Pricing
Lecture 2: Limits to Arbitrage
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 Long, 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 dividend 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 short 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 capital structure (Hombert and Thesmar, 2014)
I Can arbs increase the duration of their liabilities? Leverage constraints I Idea: arbitrageurs face funding constraints I Model: equity issuance impossible, debt 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 (margin or haircut 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 assets 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 loan 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: deleveraging ⇔ 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 bond markets. I De Long, Shleifer, Summers and Waldmann 1990. Noise trader 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 Financial Economics 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 stocks? Journal of Business