Disasters Implied by Equity Index Options

Transcription

1 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

2 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

3 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

4 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

5 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

6 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

7 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

8 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

9 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

10 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

11 Entropy and cumulants s Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 9 / 31

12 Entropy and cumulants s L m Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 9 / 31

13 Entropy and cumulants s L m Σ m E m 0.10 Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 9 / 31

14 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

15 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

16 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 = ) [similar to Barro, Nakamura, Steinsson, and Ursua] Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 12 / 31

17 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

18 Macro disasters: entropy Entropy of Pricing Kernel L(m) Alvarez Jermann lower bound normal Risk Aversion! Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 14 / 31

19 Macro disasters: entropy Entropy of Pricing Kernel L(m) Alvarez Jermann lower bound disasters normal Risk Aversion! Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 14 / 31

20 Macro disasters: entropy Entropy of Pricing Kernel L(m) Alvarez Jermann lower bound disasters normal booms Risk Aversion! Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 14 / 31

21 Macro disasters: cumulants 2 x 10 3 Contributions Cumulants Contributions 3 x != != Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 15 / 31

22 Macro disasters: cumulants High-Order Cumulants Model (α = 10) Entropy Variance/2 Odd Even Normal Poisson disaster Poisson boom Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 16 / 31

23 Macro disasters: equity premium Entropy and Equity Premium sample mean = AJ bound entropy equity premium Risk Aversion! Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 17 / 31

24 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

25 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

26 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

27 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

28 Option disasters: parameter values Choose (µ, σ, ω, θ, δ) to match distribution of equity returns Jumps: ω = 1.512, θ = , δ = 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 = , δ = 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

29 Option disasters: implied volatility for 3mo options estimated Merton model Implied Volatility (annual units) smaller!* smaller "* 0.15 smaller "* and positive!* Moneyness: difference of return from zero Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 23 / 31

30 Option disasters: components of entropy High-Order Cumulants Model Entropy Variance/2 Odd Even Consumption-based models Normal (α = 10) Poisson (α = 10) Poisson (α = 5.38) Option-based model Option model Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 24 / 31

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

32 Comparing models: consumption implied by options Consumption Process Based on Cons Growth Option Prices α ω θ δ Skewness Excess Kurtosis Tail prob ( 3 st dev) Tail prob ( 5 st dev) Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 26 / 31

33 Comparing models: options implied by consumption 0.25 Implied Volatility (annual units) consumption model (12 months) consumption model (3 months) Moneyness: difference of return from zero Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 27 / 31

34 Comparing models: options implied by consumption 0.25 consumption model (12 months) Implied Volatility (annual units) option model (12 months) option model (3 months) consumption model (3 months) Moneyness: difference of return from zero Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 27 / 31

35 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

36 Risk aversion in the option model RA Returns Backus, Chernov & Martin (Stanford GSB) Disasters implied by options 29 / 31

37 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

38 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