Sentiment and speculation in a market with heterogeneous beliefs - - PowerPoint PPT Presentation

Sentiment and speculation in a market with heterogeneous beliefs

Ian Martin Dimitris Papadimitriou September, 2020

Martin & Papadimitriou Sentiment and speculation September, 2020 1 / 42

Introduction

Agents disagree about the probabilities of good/bad news Optimists go long; pessimists go short If the market rallies, optimists get rich; if the market sells off, pessimists get rich So prices embed ex post winners’ beliefs This sentiment effect boosts volatility, and hence risk premia Sentiment induces speculation: agents trade at prices that they think are not warranted by fundamentals, in anticipation of adjusting their positions in future

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Related literature

An incomplete and somewhat arbitrary list

Heterogeneous beliefs

◮ Keynes (1936, Chapter 12); Harrison and Kreps (1978);

Scheinkman and Xiong (2003); Geanakoplos (2010); Simsek (2013); Basak (2005); Banerjee and Kremer (2010); Atmaz and Basak (2018); Zapatero (1998); Jouini and Napp (2007); Bhamra and Uppal (2014); Kogan, Ross, Wang, and Westerfield (2006); Buraschi and Jiltsov (2006); Sandroni (2000); Boroviˇ cka (2020); Blume and Easley (2006); Cvitani´ c, Jouini, Malamud, and Napp (2011); Chen, Joslin, and Tran (2012); . . . Heterogeneous risk aversion

◮ Dumas (1989); Chan and Kogan (2002); Longstaff and Wang

(2012); . . .

Martin & Papadimitriou Sentiment and speculation September, 2020 3 / 42

Setup

Agents indexed by h ∈ (0, 1) are endowed with one unit of a risky asset The asset evolves from t = 0, . . . , T on a binomial tree with exogenous terminal payoffs The interest rate is normalized to zero Agent h thinks the probability of an up-move is h Agents have log utility over terminal wealth No learning (today; see the paper for results with learning)

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0.0 0.2 0.4 0.6 0.8 1.0 h 0.5 1.0 1.5 2.0 2.5 3.0 α = β = 5 α = β = 1 α = 2, β = 4 α = 9, β = 5

The mass of agents with belief h follows a beta distribution, pdf f(h) ∝ hα−1(1 − h)β−1 where α, β > 0

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Equilibrium (1): individual optimization

Solve backwards: the price of the risky asset is pd or pu next period Agent h has wealth wh at the current node If he chooses to hold xh units of the asset, then wealth next period is wh − xhp + xhpu in up state and wh − xhp + xhpd in the down state So portfolio problem is max

xh h log [wh − xhp + xhpu] + (1 − h) log [wh − xhp + xhpd]

First order condition: xh = wh

• h

p − pd − 1 − h pu − p

• Martin & Papadimitriou

Sentiment and speculation September, 2020 6 / 42

Helpful to rewrite the FOC in terms of the risk-neutral probability

• f an up-move, p∗, which is defined via p = p∗pu + (1 − p∗)pd:

p∗ = p − pd pu − pd We then have xh = wh

• h

p − pd − 1 − h pu − p

• =

wh pu − pd h − p∗ p∗(1 − p∗)

Martin & Papadimitriou Sentiment and speculation September, 2020 7 / 42

If agent h behaves optimally, wealth next period is wh + xh(pu − p) = wh h p∗ in the up-state, and wh − xh(p − pd) = wh 1 − h 1 − p∗ in the down-state In either case, all agents’ returns are linear in their beliefs, h This is true at every node

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The wealth distribution

Everyone starts with equal wealth at time 0 After m up and n down steps, person h’s wealth is λpathhm(1 − h)n Aggregate wealth equals p, so 1 λpathhm(1 − h)nf(h) dh = p and hence λpath = B(α, β) B(α + m, β + n)p

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0.2 0.4 0.6 0.8 1.0 h 0.5 1.0 1.5 2.0 wealth share

· d du duu

After m up and n down steps, agent h’s share of aggregate wealth is wh p = B(α, β) B(α + m, β + n)hm(1 − h)n The richest agent is h = m/(m + n), who looks right in hindsight

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Equilibrium (2): market clearing

From the FOC, xh = B(α, β) B(α + m, β + n)hm(1 − h)np

• wh
• h

p − pd − 1 − h pu − p

• In aggregate the agents hold one unit of the asset:

1 xh f(h) dh = 1 The equilibrium price is therefore p = (m + α)pdpu + (n + β)pupd (m + α)pd + (n + β)pu

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In terms of risk-neutral probability, at time t = m + n, p∗ = Hm,tpd Hm,tpd + (1 − Hm,t)pu where Hm,t = m + α t + α + β = 1 hwh f(h) p dh is wealth-weighted average belief For comparison, in a homogeneous economy with up-prob H, p∗ = Hpd Hpd + (1 − H)pu

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h=1 h=0 h=Hm,t h=p* shorts balanced levered optimists representative agent all cash

Share of wealth agent h invests in the risky asset is h − p∗ Hm,t − p∗ Representative agent—“Mr. Market”—with h = Hm,t invests fully in the risky asset The agent with h = p∗ invests fully in the bond

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Agents disagree on the risk premium agent h’s perceived risk premium = (h − p∗)(Hm,t − p∗) p∗(1 − p∗) But they agree on objectively measurable quantities, such as risk-neutral variance = (Hm,t − p∗)2 p∗(1 − p∗) Notice that risky share of agent h = h − p∗ Hm,t − p∗ = agent h’s risk premium risk-neutral variance In particular, the risk premium perceived by the representative agent equals risk-neutral variance

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A pricing formula

Result

If the risky asset has terminal payoffs pm,T then its initial price is p0 = 1

T

• m=0

cm pm,T where cm = T m B(α + m, β + T − m) B(α, β)

Result (Signing the effect of heterogeneity on prices)

If 1/pm,T is convex (concave) in m, the price p0 falls (rises) as heterogeneity increases

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Example 1: geometric payoffs, uniform belief distn.

p = 0.96 p = 1.00 H0,0 = 0.50 p* = 0.29 p = 0.68 p = 0.75 H0,1 = 0.33 p* = 0.20 p = 1.69 p = 1.50 H1,1 = 0.67 p* = 0.50 p = 0.56 p = 1.13 p = 2.25

p: price. p: price in homogeneous economy. Hm,t: identity of rep agent. p∗: risk-neutral prob (cutoff between longs and shorts).

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0.2 0.4 0.6 0.8 1.0 h

• 2
• 1

1 2 3 Sharpe ratio

· d u

• Mr. Market perceives a higher Sharpe ratio in “up” than “down”

This is the opposite of what any individual thinks

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Example 2: A risky bond

T = 50 periods to go. Uniform beliefs Bond defaults (recover 30) in the bottom state. Else pays 100 In order of increasing pessimism:

◮ h = 0.50 thinks default prob is less than 10−15 ◮ h = 0.25 thinks default prob is less than 10−6 ◮ h = 0.10 thinks default prob is less than 0.006% ◮ h = 0.05 thinks default prob is less than 8% ◮ h = 0.01 thinks default prob is more than 60%

Initially, h = 0.50 is the representative agent What price does the bond trade at?

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Example 2: A risky bond

T = 50 periods to go. Uniform beliefs Bond defaults (recover 30) in the bottom state. Else pays 100 In order of increasing pessimism:

◮ h = 0.50 thinks default prob is less than 10−15 ◮ h = 0.25 thinks default prob is less than 10−6 ◮ h = 0.10 thinks default prob is less than 0.006% ◮ h = 0.05 thinks default prob is less than 8% ◮ h = 0.01 thinks default prob is more than 60%

Initially, h = 0.50 is the representative agent What price does the bond trade at? at $95.63

Martin & Papadimitriou Sentiment and speculation September, 2020 18 / 42

Example 2: A risky bond

T = 50 periods to go. Uniform beliefs Bond defaults (recover 30) in the bottom state. Else pays 100 In order of increasing pessimism:

◮ h = 0.50 thinks default prob is less than 10−15 ◮ h = 0.25 thinks default prob is less than 10−6 ◮ h = 0.10 thinks default prob is less than 0.006% ◮ h = 0.05 thinks default prob is less than 8% ◮ h = 0.01 thinks default prob is more than 60%

Initially, h = 0.50 is the representative agent What price does the bond trade at? at $95.63 Who would go short, at this price?

Martin & Papadimitriou Sentiment and speculation September, 2020 18 / 42

Example 2: A risky bond

T = 50 periods to go. Uniform beliefs Bond defaults (recover 30) in the bottom state. Else pays 100 In order of increasing pessimism:

◮ h = 0.50 thinks default prob is less than 10−15 ◮ h = 0.25 thinks default prob is less than 10−6 ◮ h = 0.10 thinks default prob is less than 0.006% ◮ h = 0.05 thinks default prob is less than 8% ◮ h = 0.01 thinks default prob is more than 60%

Initially, h = 0.50 is the representative agent What price does the bond trade at? at $95.63 Who would go short, at this price? everyone below h = 0.48!

Martin & Papadimitriou Sentiment and speculation September, 2020 18 / 42

Example 2: A risky bond

T = 50 periods to go. Uniform beliefs Bond defaults (recover 30) in the bottom state. Else pays 100 In order of increasing pessimism:

◮ h = 0.50 thinks default prob is less than 10−15 ◮ h = 0.25 thinks default prob is less than 10−6 ◮ h = 0.10 thinks default prob is less than 0.006% ◮ h = 0.05 thinks default prob is less than 8% ◮ h = 0.01 thinks default prob is more than 60%

Initially, h = 0.50 is the representative agent What price does the bond trade at? at $95.63 Who would go short, at this price? everyone below h = 0.48! Who will stay short?

Martin & Papadimitriou Sentiment and speculation September, 2020 18 / 42

Example 2: A risky bond

T = 50 periods to go. Uniform beliefs Bond defaults (recover 30) in the bottom state. Else pays 100 In order of increasing pessimism:

◮ h = 0.50 thinks default prob is less than 10−15 ◮ h = 0.25 thinks default prob is less than 10−6 ◮ h = 0.10 thinks default prob is less than 0.006% ◮ h = 0.05 thinks default prob is less than 8% ◮ h = 0.01 thinks default prob is more than 60%

Initially, h = 0.50 is the representative agent What price does the bond trade at? at $95.63 Who would go short, at this price? everyone below h = 0.48! Who will stay short? marginal agent p∗ in period 0, 1, 2, . . . is h = 0.48, 0.31, 0.22, . . .

Martin & Papadimitriou Sentiment and speculation September, 2020 18 / 42

Example 2: A risky bond

T = 50 periods to go. Uniform beliefs Bond defaults (recover 30) in the bottom state. Else pays 100 In order of increasing pessimism:

◮ h = 0.50 thinks default prob is less than 10−15 ◮ h = 0.25 thinks default prob is less than 10−6 ◮ h = 0.10 thinks default prob is less than 0.006% ◮ h = 0.05 thinks default prob is less than 8% ◮ h = 0.01 thinks default prob is more than 60%

Initially, h = 0.50 is the representative agent What price does the bond trade at? at $95.63 Who would go short, at this price? everyone below h = 0.48! Who will stay short? marginal agent p∗ in period 0, 1, 2, . . . is h = 0.48, 0.31, 0.22, . . .; only h < 0.006 stay short to the bitter end

Martin & Papadimitriou Sentiment and speculation September, 2020 18 / 42

10 20 30 40 50 t 40 50 60 70 80 90 100 price heterogeneous homogeneous

Figure: The risky bond’s price over time following consistently bad news.

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10 20 30 40 50 t

• 20
• 15
• 10
• 5

5 10 xh,t h = 0.001 h = 0.01 h = 0.1 h = 0.25 h = 0.5 h = 0.75

Figure: The number of units of the risky bond held by different agents, xh,t, plotted against time.

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10 20 30 40 50t 10 20 30 40 leverage static dynamic

Figure: The evolution of the median investor’s leverage over time, assuming bad news arrives each period.

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Price is low at time zero because all investors—even “reasonable”

• nes—worry about the short-term effect of bad news on sentiment

The risk-neutral probability of default, δ∗, is 6.25% δ∗ = 1 1 + εT = O(1/T) In the homogeneous economy, it is less than 10−14 δ∗ = 1 1 + ε (2T − 1) = O

• 2−T

Polynomial / exponential dichotomy holds for any finite α, β; and if “recovery value” is greater than 100 (lottery ticket) Sentiment makes long-dated extreme securities far more valuable

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Example 3: A diffusion limit

Slice the period from 0 to T into 2N short periods Cox–Ross–Rubinstein terminal payoffs, pm,T = e2σ

• T

2N (m−N)

Tune down per-period disagreement by parametrizing α = β = θN Low θ: lots of disagreement. θ → ∞: homogeneous economy Convenient to index agents by their z-score, the number of s.d. by which they are more/less optimistic than the mean As N → ∞, everyone perceives returns as lognormal with volatility annualized return vol0→t = θ + 1 θ + t

T

• σ

But they disagree on risk premia. . .

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Result (Subjective expectations)

The (annualized) expected return of the asset from 0 to t from the perspective of a trader z is: 1 t log E(z) R0→t = θ + 1 θ + t

T

zσ √ θT + θ + 1 θ θ +

t 2T

θ + t

T

σ2

• In particular, the cross-sectional average expected return is
• E1

t log E(z) R0→t = (θ + 1)2 θ +

t 2T

• θ
• θ + t

T

2 σ2 Disagreement is the cross-sectional standard deviation of expected returns: disagreement = θ + 1 θ + t

T

σ √ θT

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Result (Option pricing and the volatility term structure)

The time 0 price of a call option with maturity t and strike price K is C(t, K) = p0Φ(d1) − KΦ(d1 − σt √ t) where d1 = log (p0/K) + 1

2

σ2

t t

• σt

√t and

• σt =

θ + 1

• θ(θ + t

T)

σ In particular, short-dated options have σ0 = θ+1

θ σ and long-dated options

have σT =

• θ+1

θ σ.

Martin & Papadimitriou Sentiment and speculation September, 2020 25 / 42

T t σ

θ+1 θ

σ

θ+1 θ

σ implied physical Figure: The term structures of implied and physical volatility.

Variance risk premium 1

T (var∗ log R0→T − var log R0→T) = σ2 θ

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An illustrative calibration

Data Model 1mo implied vol 18.6% 18.6% 1yr implied vol 18.1% 18.2% 2yr implied vol 17.9% 17.7% 1yr cross-sectional mean risk premium 3.8% 3.2% 1yr disagreement 4.8% 4.2% 10yr cross-sectional mean risk premium 3.6% 1.8% 10yr disagreement 2.9% 2.8% T = 10, σ = 12%, θ = 1.8 Despite being highly stylized, the model generates predictions of broadly the right order of magnitude across multiple dimensions

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2 4 6 8 10 t 5 10 15 20 % avg RP disagreement physical vol implied vol

Figure: Volatility term structures in the baseline calibration with θ = 1.8.

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2 4 6 8 10 t 10 20 30 40 50 60 70 % avg RP d i s a g r e e m e n t physical vol implied vol

Figure: Volatility term structures in a “crisis” calibration with θ = 0.2.

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The variance risk premium

We introduce an identity var∗ X − var X = Rf cov

• M, (X − κ)2

where κ = (E X + E∗ X)/2 is a constant (This is a general result, relying only on absence of arbitrage) From perspective of the representative agent, this specializes to var∗ log R0→T − var log R0→T = cov(z) M(z)

0→T, (log R0→T)2

VRP reflects the fact that bad times correlate with extreme values

• f log R0→T . . . but why is this true in our model?

Martin & Papadimitriou Sentiment and speculation September, 2020 30 / 42

p0 strike

• 100

risk premium (%) calls puts

Figure: Expected excess returns on options of different strikes, as perceived by the rep agent. Solid: heterogeneous beliefs. Dashed: homogeneous.

Homogeneous case: deep OTM calls have very high risk premia Here, rep agent perceives that they are so overvalued due to the presence of optimists that they earn negative risk premia

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Investors’ attitudes to speculation

As people have different beliefs but agree on market prices, they have different stochastic discount factors The properties of these heterogeneous SDFs reflect different views

• n Sharpe ratios and on the value of speculation

Result

The maximum Sharpe ratio (as perceived by investor z) is finite if θ > 1 and equals MSR(z)

0→T =

• θ

√ θ2 − 1 exp     

• z

√ θ + (θ + 1) σ √ T 2 θ (θ − 1)      − 1

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The gloomy investor

The gloomy investor who perceives the smallest MSR (at all horizons t) has z = zg, zg = −θ + 1 √ θ σ √ T This investor perceives zero instantaneous MSR, but a positive MSR to time T associated with a contrarian strategy: buy if the market sells off, sell if the market rallies MSR(zg)

0→T =

• θ

√ θ2 − 1 − 1

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• 3
• 2
• 1

1 2 3 z 0.5 1.0 1.5 2.0 2.5 3.0 Sharpe ratio dynamic static Figure: Max Sharpe ratio (annualized) as perceived by investor z. Baseline calibration.

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Investors have target prices

Result

Terminal wealth of agent z is

W(z)(pT) = p0

• θ + 1

θ exp 1 2 (z − zg)2 − 1 2(1 + θ)σ2T

• log
• pT/K(z)2

Target price K(z) for investor z—i.e., investor z’s ideal outcome—is log K(z) = E(z) log pT + (z − zg)σ √ θT Gloomy investor z = zg wants to be proved right Extremists are happiest if the market moves more than they expect

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K(zg) 1 K(0) R0→T 1 R0→T

(z)

gloomy median Return on wealth against return on the market

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1 K(1) K(0) R0→T 1 +1 s.d. median

Dashed: return on wealth against return on the market Solid: MSR return Log scale on x-axis

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MSR strategies are very short OTM options They are mean-variance-efficient; can (as always) use them for beta pricing with zero alphas Other side of the coin: if betas are calculated wrt, say, the return

• n the market, MSR strategies will earn large alphas

But they are economically unnatural: our investors would prefer to invest fully in cash than to invest any money at all in an MSR strategy

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Example 4: A Poisson limit

A stylized model of disagreement in credit or insurance markets Asset subject to jumps that arrive according to a Poisson process Agents disagree over the jump arrival rate Mean agent perceives arrival rate ω Agent z perceives arrival rate ω(1 − zσ) If q jumps occur in total, terminal payoff is e−qJ (for some fixed J) Again, we can solve for everything in closed form

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Everyone thinks arbitrarily high Sharpe ratios are attainable Can index agents by the arrival rate they perceive For example, if q jumps have occurred by time t, ωrep,t = ω + σ2ωt 1 + σ2ωt q t − ω

• “is” the representative agent, and

ω∗

t =

1 + qσ2 1 − σ2ωT (eJ − 1) + σ2ωteJ eJω is the agent who is out of the market ω∗

t is the risk-neutral arrival rate: it equals the CDS rate (price of a

very short-dated CDS contract that pays $1 if there is a jump)

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2 4 6 8 10 t 0.05 0.10 0.15 0.20 0.25 0.30 arrival rate CDS (het) rep agent (het) CDS (hom) rep agent (hom) 2 4 6 8 10 t 0.5 1.0 1.5 R0→t

(z)

3 s . d . p e s s i m i s t 0.9 s.d. optimist 0.9 s.d. optimist mean investor 2 s.d. pessimist 3 s.d. pessimist

Figure: Left: ωrep,t and ω∗

t on a sample path with jumps at times t = 4 and 5.

Right: The wealth of four agents (z = −3, −2, 0, 0.9) on the same path.

ω = 0.05, σ = 1, T = 10, e−J = 1/2 Even though individuals have stable beliefs, the CDS rate and rep agent’s perceived arrival rate spike after a jump Similar to patterns documented by Froot and O’Connell (1999) and Born and Viscusi (2006)

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Summary

Sentiment generates volatility, speculation, and volume Extreme scenarios become much more important for pricing In a diffusion limit, a variance risk premium emerges, with implied volatility higher than true volatility Moderate investors are contrarian, “short vol”, liquidity suppliers Mean-variance-efficient returns are very short deep-OTM options; they do not interest our investors despite their high Sharpe ratios In a Poisson limit, CDS rates spike after jumps occur, even though all investors perceive constant arrival rates

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