Affine Disagreement and Asset Pri cingBy Hui Chen, Scott Joslin, and Ngoc-Khanh Tran∗Models of heterogeneous beliefs can generate richimplications for trading and asset pricing (see Su-leyman Basak (2005) for a recent survey). Whenstudying such models, aggregation often leads todifficulty in computing equilibrium outcomes. Inthis paper, we introduce a flexible framework tomodel heterogeneous beliefs in t he economy, whichwe refer to as “affine disagreement” about funda-mentals. Affine processes (see Darrel Duffie, JunPan, and Kenneth Singleton (2000)) are appealingas they provide a large degree of flexibility in mod-elling the conditional means, volatilities, and jumpsfor various quantities of interest while remains ana-lytically tractable. Our affine heterogeneous beliefsframework allows further for stochastic disagree-ment among agents about growth rates, volatilitydynamics, as well as t he likelihood of jumps andthe distribution of jump sizes.Disagreement about rare disasters provides an in-teresting case study for our framework. Researchby Thomas A. Rietz (1988), Francis Longstaff andMonika Piazzesi (2004), R obert J. Barro (2006) andothers show that the possible occurrence of raredisasters that result in severe losses in consump-tion can have large impact on asset prices. How-ever, the relatively short sample period and lackof historical precedents of disaster events (at leastin the US) make it difficult to precisely measurethe frequencies of disasters or the size of their im-pact. Together, these suggest that there is likely tobe large disagreements among market participantsabout disasters, and such disagreements can signif-icantly affect asset prices.A number of interesting implications arise fromheterogeneous beliefs about disasters. The model∗All three authors are affiliated with MIT SloanSchool of Management, 50 Memorial Drive, Cam-bridge, MA 02142. Chen: (e-mail: [email protected]).Joslin: (e-mail: [email protected]). Tran: (e-mail:[email protected]). We thank Jakub Jurek (the discus-sant), Leonid Kogan, Monika Piazzesi, and Jiang Wangfor helpful comments.endogenously generates variation in the risk-freerates, asset prices, and th e equity risk premiumthrough variation in the distribution of wealth. Innormal times, optimistic agents (who believe d is-asters are less frequent and likely to be less severe)accumulate wealth, which leads to a gradual declinein the equity premium. When disasters strike, thepessimistic agents become relatively more wealthy,resulting in jumps in the equity premium.I. An Affine Heteroge neous BeliefsFrameworkWe consider an endowment economy. Thestochastic environment is summarized by theMarkov state variable Xt, which reflects informa-tion about both the aggregate endowment andagents’ beliefs. We now show how one can chooseXtto model a broad class of disagreement over thedynamics of the economy.A. BeliefsThere are two agents (A, B), each being the rep-resent ative of her own class, who possess heteroge-neous beliefs about the dynamics of Xt. Agent Abelieves that Xtfollows an affine jump diffusion:dXt= µAtdt + σAtdWAt+ dJAt,where WAtis a standard Brownian motion, µAt=KA0+ KA1Xt, and σAt(σAt)⊤= HA0+ HA1· Xt.The term JAtis a pure jump process with inten-sity λAt= λA0+ λA1· Xt, and its jump size has dis-tribution νA, with moment generating function φ.We summarize agent A’s beliefs with the probabil-ity measure PA. For simplicity, we suppose that A’sbeliefs are correct. The method is easily m odifiedto the case where neither agent has correct beliefs.Agent B has an equ ivalent probability measurePB. The differences in beliefs are characterized bythe Radon-Nikodym derivative ηt≡ Et[dPB/dPA].12 PAPERS AND PROCEEDINGS MAY 2010We assume thatηt= ea·Xt−It,(1)where Itis locally deterministic satisfyingdIdt= a · µAt+12kσAtak2+ λAt(φ(a) − 1) ,(2)which ensures that ηtis a PA-martingale.It follows from t he specification of ηtthat, underB’s beliefs, Xtfollows an affine jump diffusiondXt= µBtdt + σBtdWBt+ dJBt,where WBtis a standard Brownian motion un derPB, andi. µBt= µAt+ σAt(σAt)⊤aii. σBt= σAtiii. λBt= φ(a)λAtiv. dνB/dνA(z) = eaz/φ(a)Intuitively, the Radon-Nikodym derivative ex-presses the differences in beliefs by having agent Bassign a higher (lower) probability th an A to thosestates where ηtis high (low). For example, if ηtisincreasing with a component of Xt, then B thinksthat higher values of this component are more likelythan A. In other words, B believes the drift for thiscomponent is larger. Similarly, if ηtjumps at thesame time when Xtjumps, then A and B will dis-agree about the likelihood of jump s. In particular,if the jump in ηtis positive, then B believes that t helikelihood of such a jump is higher than A. More-over, if the jump size in ηtvaries with the jump sizein Xt, then A and B will disagree about the jumpsize distribution as well. Thus, this setup can ac-commodate both disagreement about the frequencyof jumps as well as the conditional distribution ofjump sizes.Finally, while we specify the differences in beliefsexogenously (agents “agree to disagree”), this doesnot preclude agents’ beliefs from arising throughBayesian updating based on different informationsets. For example, when the state variables andsignals follow a Gaussian process, Bayesian updat-ing can lead to heterogeneous beliefs in the form of(1–2).B. Equilibrium Asset PricesWe assume that the agents have constant relativerisk aversion (CRRA) preferences:Ui(C) = EPi0Z∞0e−ρt(Cit)1−γ/(1 − γ)dt,for i = A, B. In addition, we assume that (i) mar-kets are complete, (ii) log aggregat e consump tion,ct= log(Ct), is linear in Xt(ct= c · Xt), and(iii) agents are endowed with some fi xed fraction(θA, θB= 1 − θA) of aggregate consumption.We first solve for the equilibrium consumption al-locations through t he planner’s problem, and thenuse the individual consumption for agent A to deter-mine the stochastic discount factor with respect toher beliefs. The equilibrium consumption of agentA isCAt=11 +˜ζ1γtCt,where˜ζt= ζηtis the stochastic weight that theplanner p laces on agent B, which is linked to theinitial allocation of wealth and the differences inbeliefs. The stochastic discount factor (SDF) underA’s beliefs isMt= e−ρt(CAt)−γ= e−ρt1 +˜ζ1/γtγC−γt.(3)With this stochastic discount factor, we can pricea large class of assets (e.g. riskless bonds andthe aggregate
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