Disasters Implied by Equity Index Options David Backus (NYU) Mikhail Chernov (LBS) Ian Martin (Stanford GSB) November 18, 2009 Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 1 / 31
Summary Problem: disasters infrequent hard to estimate their distribution Solution: infer from option prices (market prices of bets on disasters) What we find disasters apparent in options data more modest than disasters in macro data Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 2 / 31
Outline Preliminaries: cumulants, entropy, AJ bound Three objects of interest in financial economics: true probabilities p(x), risk-neutral probabilities p (x), stochastic discount factor m(x) Macro-finance: m and p Option-pricing: p and p Third possibility: m and p p (x) = R f p(x)m(x) m(x) = 1 R f p (x) p(x) Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 3 / 31
Cumulants Cumulant generating function of a random variable x k(s) = log Ee sx = κ j (x)s j /j! j=1 Cumulants are closely related to moments κ 1 = k (0) = mean κ 2 = k (0) = variance, σ 2 κ 3 = k (0) = σ 3 skewness κ 4 = k (0) = σ 4 excess kurtosis If x is normal, κ j = 0 for j > 2 Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 4 / 31
Entropy Entropy, L(x), of a random variable x > 0 is log Ex E log x A measure of the variability of x Hans-Otto Georgii (quoted by Hansen and Sargent): When Shannon had invented his quantity and consulted von Neumann on what to call it, von Neumann replied: Call it entropy. It is already in use under that name and, besides, it will give you a great edge in debates because nobody knows what entropy is anyway. Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 5 / 31
Alvarez-Jerman bound Entropy: for x > 0 L(x) log Ex E log x 0 AJ bound L(m) E ( log r j log r 1) Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 6 / 31
Alvarez-Jerman bound Entropy: for x > 0 L(x) log Ex E log x 0 AJ bound L(m) E ( log r j log r 1) Relates a measure of variability of the stochastic discount factor to a risk-adjusted measure of expected returns Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 6 / 31
Alvarez-Jerman bound vs. Hansen-Jagannathan bound Entropy: for x > 0 L(x) log Ex E log x 0 AJ bound L(m) E ( log r j log r 1) HJ: for x > 0 HJ bound HJ(x) σ(x) Ex 0 HJ(m) Er j r 1 σ(r j r 1 ) Relates a measure of variability of the stochastic discount factor to a risk-adjusted measure of expected returns Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 6 / 31
Entropy and cumulants Entropy of pricing kernel L(m) = log Ee log m E log m = κ j (log m)/j! j=2 Zin s never a dull moment conjecture L(m) = κ 2 (log m)/2! + κ 3 (log m)/3! + κ 4 (log m)/4! + }{{} high-order cumulants (incl disasters) In a lognormal model, all the higher cumulants are zero Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 7 / 31
Entropy and cumulants Can calculate contribution of odd cumulants and even cumulants separately eg, κ j (x)s j /j! = [k(s) + k( s)]/2 j even Since m(x) = R f p (x)/p(x), we have L(m) = L(p /p), and hence L(m) = log Ep /p E log p /p = log p(x) [p (x)/p(x)] E log p /p x = E log p /p (aka relative entropy or Kullback-Leibler divergence ) Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 8 / 31
Entropy and cumulants 0.05 0.5 1.0 1.5 2.0 s 0.05 0.10 Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 9 / 31
Entropy and cumulants 0.05 0.5 1.0 1.5 2.0 s 0.05 0.10 L m Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 9 / 31
Entropy and cumulants 0.05 0.05 0.5 1.0 1.5 2.0 s L m Σ m E m 0.10 Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 9 / 31
Plan of attack Modelling assumptions i.i.d. Tight link between consumption growth and equity returns Representative agent with power utility [when needed] Parameter choices Match mean and variance of log consumption growth Ditto log equity return Base disasters on macroeconomic evidence (Barro, Barro-Ursua) Or on equity index options Compare macro- and option-based examples Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 10 / 31
Macro disasters: environment Consumption growth and equity return g t+1 = c t+1 /c t d t = ct λ log rt+1 e = constant + λ log g t+1 Power utility Yaron s bazooka log m t+1 = log β α log g t+1 κ j (log m)/j! = κ j (log g)( α) j /j! Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 11 / 31
Macro disasters: Poisson-normal mixture Consumption growth log g t+1 = w t+1 + z t+1 w t+1 N (µ, σ 2 ) z t+1 j N (jθ, jδ 2 ) j 0 has probability e ω ω j /j! Parameter values Match mean and variance of log consumption growth Jump probability (ω = 0.01), mean (θ = 0.3), and variance (δ 2 = 0.015 2 ) [similar to Barro, Nakamura, Steinsson, and Ursua] Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 12 / 31
Macro disasters: entropy Cumulant generating functions ( ) k(s; log g) = sµ + s 2 σ 2 /2 + ω e sθ+s2 δ 2 /2 1 ( ) k(s; log m) = s log β sαµ + s 2 α 2 σ 2 /2 + ω e sαθ+s2 α 2 δ 2 /2 1 Entropy ( ) L(m) = α 2 σ 2 /2 + ω e αθ+α2 δ 2 /2 1 + αωθ, Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 13 / 31
Macro disasters: entropy 0.4 0.35 Entropy of Pricing Kernel L(m) 0.3 0.25 0.2 0.15 0.1 0.05 Alvarez Jermann lower bound normal 0 0 2 4 6 8 10 12 Risk Aversion! Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 14 / 31
Macro disasters: entropy 0.4 0.35 Entropy of Pricing Kernel L(m) 0.3 0.25 0.2 0.15 0.1 0.05 Alvarez Jermann lower bound disasters normal 0 0 2 4 6 8 10 12 Risk Aversion! Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 14 / 31
Macro disasters: entropy 0.4 0.35 Entropy of Pricing Kernel L(m) 0.3 0.25 0.2 0.15 0.1 0.05 Alvarez Jermann lower bound disasters normal booms 0 0 2 4 6 8 10 12 Risk Aversion! Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 14 / 31
Macro disasters: cumulants 2 x 10 3 Contributions Cumulants 1 0 1 2 1 0 0.1 0.05 0 Contributions 3 x 10 3 2 3 4 5 6 7 8 9 10!=2 2 3 4 5 6 7 8 9 10!=10 2 3 4 5 6 7 8 9 10 Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 15 / 31
Macro disasters: cumulants High-Order Cumulants Model (α = 10) Entropy Variance/2 Odd Even Normal 0.0613 0.0613 0 0 Poisson disaster 0.5837 0.0613 0.2786 0.2439 Poisson boom 0.0266 0.0613 0.2786 0.2439 Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 16 / 31
Macro disasters: equity premium 2 1.5 Entropy and Equity Premium 1 0.5 0 sample mean = AJ bound entropy equity premium 0.5 0 2 4 6 8 10 12 Risk Aversion! Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 17 / 31
Option disasters: overview Options an obvious source of information, but... Options on equity, not consumption Determine risk-neutral, not true distribution True distribution has the usual lack of data problems Plan of attack Estimate risk-neutral distribution from options Estimate true distribution two ways Compare options implied by macro-based disaster model Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 18 / 31
Risk-neutral probabilities: examples Normal log consumption growth If log g N (µ, σ 2 ) (true distribution) Then risk-neutral distribution also lognormal with µ = µ ασ 2, σ = σ Poisson log consumption growth Jumps have probability ω and distribution N (θ, δ 2 ) Risk-neutral distribution has same form with ω = ω exp[ αθ + (αδ) 2 /2], θ = θ αδ 2, δ = δ Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 19 / 31
Option disasters: information in option prices Put option (bet on low returns) q p t = 1 R f E t (b r e t+1) + Strategy Estimate p by varying strike price b (cross section) Estimate p and Rf from time series data Black-Scholes-Merton benchmark Quote prices as implied volatilities (high price high vol) Horizontal line if lognormal Skew suggests disasters Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 20 / 31
Option disasters: Merton model Equity returns i.i.d. log rt+1 e = log r 1 + w t+1 + z t+1 w t+1 N (µ, σ 2 ) z t+1 j N (jθ, jδ 2 ) j 0 has probability e ω ω j /j! Risk-neutral distribution: ditto with s Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 21 / 31
Option disasters: parameter values Choose (µ, σ, ω, θ, δ) to match distribution of equity returns Jumps: ω = 1.512, θ = 0.0259, δ = 0.0229 Equity premium: µ + ωθ Variance of equity returns: σ 2 + ω(θ 2 + δ 2 ) Set (ω, θ, δ ) to match option prices Jumps: ω = ω, θ = 0.0482, δ = 0.0981 Set σ = σ Set µ to satisfy pricing relation (1/r f )E r e = 1 All of this from Broadie, Chernov, and Johannes (JF, 2007) Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 22 / 31
Option disasters: implied volatility for 3mo options 0.2 0.19 estimated Merton model Implied Volatility (annual units) 0.18 0.17 0.16 smaller!* smaller "* 0.15 smaller "* and positive!* 0.06 0.04 0.02 0 0.02 0.04 0.06 Moneyness: difference of return from zero Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 23 / 31
Option disasters: components of entropy High-Order Cumulants Model Entropy Variance/2 Odd Even Consumption-based models Normal (α = 10) 0.0613 0.0613 0 0 Poisson (α = 10) 0.5837 0.0613 0.2786 0.2439 Poisson (α = 5.38) 0.0449 0.0177 0.0173 0.0099 Option-based model Option model 0.7647 0.4699 0.1130 0.1819 Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 24 / 31
Comparing macro- and option-based models Direct comparison of entropy and cumulants Consumption growth implied by option prices Scale option-based p to consumption Find p using power utility Result: more modest skewness and kurtosis, tail probabilities Option prices implied by consumption growth Find macro-based p using power utility Scale to equity returns Compute option prices Result: steeper volatility smile Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 25 / 31
Comparing models: consumption implied by options Consumption Process Based on Cons Growth Option Prices α 5.38 10.07 ω 0.0100 1.3864 θ 0.3000 0.0060 δ 0.1500 0.0229 Skewness 11.02 0.31 Excess Kurtosis 145.06 0.87 Tail prob ( 3 st dev) 0.0090 0.0086 Tail prob ( 5 st dev) 0.0079 0.0002 Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 26 / 31
Comparing models: options implied by consumption 0.25 Implied Volatility (annual units) consumption model (12 months) 0.2 0.15 0.1 consumption model (3 months) 0.05 0.06 0.04 0.02 0 0.02 0.04 0.06 Moneyness: difference of return from zero Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 27 / 31
Comparing models: options implied by consumption 0.25 consumption model (12 months) Implied Volatility (annual units) 0.2 0.15 0.1 option model (12 months) option model (3 months) consumption model (3 months) 0.05 0.06 0.04 0.02 0 0.02 0.04 0.06 Moneyness: difference of return from zero Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 27 / 31
Risk aversion in the option model Risk aversion implied by arbitrary pricing kernel RA log m log g = log(p /p) log r e log r e log g Risk aversion not constant ( state dependent ) Parameters imply greater aversion to adverse risks Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 28 / 31
Risk aversion in the option model 35 30 25 20 RA 15 10 5 0 5 0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25 Returns Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 29 / 31
Bottom line Barro et al; Longstaff & Piazzesi; Rietz Back out asset pricing implications (p ) from assumptions on preferences (m) and real-world probability distributions (p) Disasters contribute to equity premium, entropy Evident in macro data We look at options Estimate p from time series of market returns and p from cross-section of option prices Implies very high entropy Smile/smirk suggests something like disasters But more modest than macro data Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 30 / 31
Open questions Sources of apparent risk aversion Exotic preferences Heterogeneous agents Examples: Alvarez, Atkeson, and Kehoe; Bates; Chan and Kogan; Du; Guvenen; Lustig and Van Nieuwerburgh; Longstaff and Wang Consumption and dividends Examples: Bansal and Yaron, Gabaix, Longstaff and Piazzesi Time dependence Short rate, predictable returns, stochastic volatility Examples: Drechsler and Yaron, Wachter, Shaliastovich Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 31 / 31