Disasters Implied by Equity Index Options

Transcription

1 Disasters Implied by Equity Index Options David Backus (NYU), Mikhail Chernov (LBS), and Ian Martin (Stanford) University of Glasgow March 4, 2010 This version: March 3, / 64

2 The idea Disasters are infrequent hard to estimate their distribution Idea: infer from option prices (market prices of bets on disasters) We find: disasters apparent in options data the mechanism generating disasters is more modest than what is assumed based on macro data 1 / 64

3 Entropy 2 / 64

4 Entropy 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. 2 / 64

5 Outline Preliminaries: entropy, Alvarez-Jermann bound, cumulants Disasters in macroeconomic data Risk-neutral probabilities Disasters in options data Compare the implications of the two approaches Extensions and related work 3 / 64

6 Alvarez-Jermann bound Pricing relation ( ) E t m t+1 r j t+1 = 1 Entropy: for any x > 0 L(x) log Ex E log x 0 AJ bound (i.i.d. case) L(m) ( E log r j logr 1) 4 / 64

7 Cumulant generating function Cumulants k(s;x) = log Ee sx = j=1 κ j (x)s j /j! Cumulants are almost moments mean = κ 1 (x) variance = κ 2 (x) skewness = κ 3 (x)/κ 3/2 2 (x) (excess) kurtosis = κ 4 (x)/κ 2 2(x) If x is normal, κ j (x) = 0 for j > 2 5 / 64

8 Entropy and cumulants 6 / 64

9 Entropy and cumulants Entropy of pricing kernel L(m) = log Ee log m E log m = k(1,log m) κ 1 (log m) 6 / 64

10 Entropy and cumulants Entropy of pricing kernel L(m) = log Ee log m E log m = k(1,log m) κ 1 (log m) Zin s never a dull moment conjecture L(m) = κ 2 (log m)/2! +κ }{{} 3 (log m)/3!+κ 4 (log m)/4!+ }{{} (log)normal term high-order cumulants (incl disasters) 6 / 64

11 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 ) 7 / 64

12 Alvarez-Jermann bound vs. Hansen-Jagannathan bound Κ s; log m s / 64

13 Alvarez-Jermann bound vs. Hansen-Jagannathan bound Κ s; log m s L m 8 / 64

14 Alvarez-Jermann bound vs. Hansen-Jagannathan bound Κ s; log m s L m Σ m E m / 64

15 Disasters based on macro fundamentals 9 / 64

16 Macro disasters: Model Consumption growth iid Parameter values 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! Match mean and variance of log consumption growth Average number of disasters (ω = 0.01), mean (θ = 0.3) and variance (δ 2 = ) Similar to Barro, Nakamura, Steinsson, and Ursua (2009) 10 / 64

17 Macro disasters: Deviations from normality Pricing kernel log m t+1 = logβ αlogg t+1 L(m) = log Ee log m E log m = k( α;log g)+ακ 1 (log g) 11 / 64

18 Macro disasters: Deviations from normality Pricing kernel log m t+1 = logβ αlogg t+1 L(m) = log Ee log m E log m = k( α;log g)+ακ 1 (log g) Yaron s bazooka κ j (log m)/j! = κ j (log g)( α) j /j! The contribution of higher-order cumulants peaks at j = α α j j! = α 1 α 2... α j 1 α j 11 / 64

19 Macro disasters: Cumulants 2 x 10 3 Contributions Cumulants Contributions 3 x α= α= / 64

20 Macro disasters: Entropy Entropy of Pricing Kernel L(m) Alvarez Jermann lower bound normal Risk Aversion α 13 / 64

21 Macro disasters: Entropy Entropy of Pricing Kernel L(m) Alvarez Jermann lower bound disasters normal Risk Aversion α 13 / 64

22 Macro disasters: Entropy Entropy of Pricing Kernel L(m) Alvarez Jermann lower bound disasters normal booms Risk Aversion α 13 / 64

23 Macro-finance and risk-neutral pricing 14 / 64

24 Macro-finance and risk-neutral pricing Pricing relation q 1 E t ( ) r j t+1 = 1, where q 1 is a price of a one-period riskless bond Translating between preferences and risk-neutral probabilities p(x)m(x) = q 1 p (x) p (x) = p(x)m(x)/q 1 14 / 64

25 Macro-finance and risk-neutral pricing: Examples Normal log consumption growth If logg N (µ,σ 2 ) (true distribution) Then risk-neutral distribution also lognormal with µ = µ ασ 2,σ = σ Poisson log consumption growth If disasters have probability ω and distribution N (θ,δ 2 ) Then risk-neutral distribution has same form with ω = ωexp( αθ+(αδ) 2 /2),θ = θ αδ 2,δ = δ 15 / 64

26 Macro-finance and risk-neutral pricing Pricing relation q 1 E t ( ) r j t+1 = 1, Translating between preferences and risk-neutral probabilities p(x)m(x) = q 1 p (x) p (x) = p(x)m(x)/q 1 16 / 64

27 Macro-finance and risk-neutral pricing Pricing relation q 1 E t ( ) r j t+1 = 1, Translating between preferences and risk-neutral probabilities p(x)m(x) = q 1 p (x) p (x) = p(x)m(x)/q 1 m(x) = q 1 p (x)/p(x) 16 / 64

28 Macro-finance and risk-neutral pricing Pricing relation q 1 E t ( ) r j t+1 = 1, Translating between preferences and risk-neutral probabilities p(x)m(x) = q 1 p (x) p (x) = p(x)m(x)/q 1 m(x) = q 1 p (x)/p(x) Entropy L(m) = L(p /p) = E log(p /p) 16 / 64

29 Disasters in options 17 / 64

30 Disasters in options Put option (bet on low returns) q p t = q 1 E t (b r e t+1) + Estimate p by varying strike price b (cross section) (Breeden and Litzenberger, 1978) Black-Scholes-Merton benchmark Quote prices as implied volatilities [high price high vol] Horizontal line if (log)normal Skew suggests disasters 17 / 64

31 Disasters in options: Data vs normal benchmark sample mean Implied volatility normal Moneyness (std deviations of strike price from ATM) 18 / 64

32 Disasters in options: Merton model Equity returns iid log r e t+1 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: the same with *s Parameter values Choose risk-neutral parameters to match option prices Average number of disasters: ω = ω = 1.5, mean: θ = 0.03, θ = 0.05, variance: δ 2 = , δ 2 = Calibration is based on Broadie, Chernov, and Johannes (2007) 19 / 64

33 Calibrating option parameters 0.2 estimated Merton model Implied Volatility (annual units) Moneyness: difference of return from zero 20 / 64

34 Calibrating option parameters 0.2 estimated Merton model 0.19 Implied Volatility (annual units) smaller θ* Moneyness: difference of return from zero 20 / 64

35 Calibrating option parameters 0.2 estimated Merton model 0.19 Implied Volatility (annual units) smaller θ* smaller δ* Moneyness: difference of return from zero 20 / 64

36 Calibrating option parameters 0.2 estimated Merton model 0.19 Implied Volatility (annual units) smaller θ* smaller δ* 0.15 smaller δ* and positive θ* Moneyness: difference of return from zero 20 / 64

37 Comparing macro- and option-based models Entropy and cumulants of pricing kernel Result: option-based entropy is large Consumption growth implied by option prices Option-based p + power utility p Result: more modest skewness and kurtosis, tail probabilities Option prices implied by consumption growth Macro-based p + power utility p Compute option prices Result: steeper volatility smile Risk aversion implied by options Result: Risk aversion declines with increase in returns 21 / 64

38 Comparing models: components of entropy High-Order Cumulants Model Entropy Variance/2 Odd Even Macro (α = 5.38) % 39% 22% Options % 15% 24% 22 / 64

39 Comparing models: components of entropy High-Order Cumulants Model Entropy Variance/2 Odd Even Macro (α = 5.38) % 39% 22% Options % 15% 24% 23 / 64

40 Comparing models: one more step Levered equity Claim to c λ Log return logr e = λ logg + constant Calibrate λ to match volatility of returns λ = 0.15 / = 4.3 vol(log r e ) vol(log g) 24 / 64

41 Comparing models: consumption implied by options 25 / 64

42 Comparing models: consumption implied by options Calibration Implied α ω θ δ / 64

43 Comparing models: consumption implied by options Calibration Implied α ω θ δ Skew Excess Kurt / 64

44 Comparing models: consumption implied by options Calibration Implied α ω θ δ Skew Excess Kurt Tail prob ( 3 st dev) Great Depression 25 / 64

45 Comparing models: consumption implied by options Calibration Implied α ω θ δ Skew Excess Kurt Tail prob ( 3 st dev) Great Depression Tail prob ( 5 st dev) / 64

46 Comparing models: options implied by macro model 0.25 Implied Volatility (annual units) consumption model (12 months) consumption model (3 months) Moneyness: difference of return from zero 26 / 64

47 Comparing models: options implied by macro model 0.25 Implied Volatility (annual units) consumption model (12 months) consumption model (3 months) option model (12 months) option model (3 months) Moneyness: difference of return from zero 26 / 64

48 Comparing models: risk aversion In option model, implicit risk aversion accounts for Equity premium Prices of options (high entropy) Form differs from power utility Not constant Parameters imply greater aversion to adverse risks 27 / 64

49 Comparing models: risk aversion computation Math RA = log m log g = log(p /p) log r e log r e log g }{{} λ 28 / 64

50 Comparing models: risk aversion computation Math RA = log m log g = log(p /p) log r e log r e log g }{{} λ Example: log r e N (µ,σ 2 ) and N (µ,σ 2 ) RA = [(σ 2 /σ 2 1)log r e + µ (σ 2 /σ 2 )µ ]/σ 2 λ 28 / 64

51 Comparing models: risk aversion computation Math RA = log m log g = log(p /p) log r e log r e log g }{{} λ Example: log r e N (µ,σ 2 ) and N (µ,σ 2 ) RA = [(σ 2 /σ 2 1)log r e + µ (σ 2 /σ 2 )µ ]/σ 2 λ Interpretation If σ = σ, RA is constant If σ < σ, RA decreases with logr e 28 / 64

52 Comparing models: risk aversion variation RA Returns 29 / 64

53 Bottom line Barro (2006), Longstaff and Piazzesi (2004), and Rietz (1988) Disasters account for equity premium Evident in macro data We look at options Disasters evident in option prices More modest than in macro data Suggest high average risk aversion, greater aversion to bad outcomes Imply higher entropy than equity premium 30 / 64

54 Open questions Consumption and dividends Examples: Bansal and Yaron (2004), Gabaix (2009), Longstaff and Piazzesi (2004) Source of apparent risk aversion Exotic preferences Heterogeneous agents Examples: Alvarez, Atkeson, and Kehoe (2009); Bates (2008); Chan and Kogan (2002); Lustig and Van Nieuwerburgh (2005) Time dependence Short rate, predictable returns, stochastic volatility Examples: Drechsler and Yaron (2008), Wachter (2008) 31 / 64

55 Consumption and Dividends So, far they were perfectly correlated: log g d = λlog g log r e = λlog g + constant Can we relax this and is it important? log r e = λlog g + constant+noise 32 / 64

56 Returns and consumption growth log r log g OLS (using Shiller s data) log r e = 3 log g + constant+noise What is the nature of the noise term? 33 / 64

57 Using information in options to model noise 0.25 Implied Volatility (annual units) consumption model (12 months) consumption model (3 months) option model (12 months) option model (3 months) Moneyness: difference of return from zero 34 / 64

58 Consumption and Dividends Add noise: log g d = λlog g + w d t log r e = λlog g + constant+w d t w d t N (0,σ d2 ) Calibration λ = 3 σ d = ( ) 2 = As a result, α = 5.79 (instead of 5.38) 35 / 64

59 Implied volatilities revisited 0.25 Implied Volatility (annual units) consumption model (12 months) consumption model (3 months) option model (12 months) option model (3 months) Moneyness: difference of return from zero 36 / 64

60 Implied volatilities revisited 0.25 Implied Volatility (annual units) consumption model (12 months) consumption model (3 months) option model (12 months) option model (3 months) Moneyness: difference of return from zero 36 / 64

61 Comparing models: consumption implied by options Calibration Implied Implied (w. noise) α ω θ δ / 64

62 Comparing models: consumption implied by options Calibration Implied Implied (w. noise) α ω θ δ Initial conclusions are robust to imperfect correlation between consumption and dividends Returns and consumption seem to share a common jump component 37 / 64

63 Open questions Consumption and dividends Examples: Bansal and Yaron (2004), Gabaix (2009), Longstaff and Piazzesi (2004) 38 / 64

64 Open questions Consumption and dividends Examples: Bansal and Yaron (2004), Gabaix (2009), Longstaff and Piazzesi (2004) Source of apparent risk aversion Exotic preferences Heterogeneous agents Examples: Alvarez, Atkeson, and Kehoe (2009); Bates (2008); Chan and Kogan (2002); Lustig and Van Nieuwerburgh (2005) Time dependence Short rate, predictable returns, stochastic volatility Examples: Drechsler and Yaron (2008), Wachter (2008) 38 / 64

65 Sources of Entropy in Dynamic Representative Agent Models David Backus (NYU), Mikhail Chernov (LBS), and Stanley Zin (NYU) University of Glasgow March 4, 2010 This version: March 3, / 64

66 Understanding dynamic models Are dynamic features important for the disaster story? More generally, how does one discern critical features of modern dynamic models? The size of equity premium is no longer an overidentifying restriction 39 / 64

67 Market-adjusted excess returns Asset Class Value Momentum US stocks 4.3% 6.1% UK stocks 2.7% 10.8% Euro stocks 4.2% 10.9% Jpn stocks 11.3% 4.2% FX 4.9% 2.7% Bonds 0.3% 0.3% Commodities 6.4% 8.8% Annualized alphas relative to the MSCI world equity index in excess of the US Treasury Bill rate Source: Asness, Moskowitz, and Pedersen (2009) 40 / 64

68 Understanding dynamic models Are dynamic features important for the disaster story? More generally, how does one discern critical features of modern dynamic models? The size of equity premium is no longer an overidentifying restriction 41 / 64

69 Understanding dynamic models Are dynamic features important for the disaster story? More generally, how does one discern critical features of modern dynamic models? The size of equity premium is no longer an overidentifying restriction The models are built up from different state variables Which pieces are most important quantitatively? We start by thinking about how risk is priced in these models What is the source of the evident high entropy in the data? We use ACE to characterize this 41 / 64

70 AJ bound, non-i.i.d. case AJ bound L(m) E ( log r j log r 1) + L(q 1 ) }{{} non-i.i.d. piece 42 / 64

71 AJ bound, non-i.i.d. case AJ bound L(m) E ( log r j log r 1) + L(q 1 ) }{{} non-i.i.d. piece Conditional entropy: L t (m t+1 ) = log E t m t+1 E t log m t+1 Average conditional entropy (ACE) L(m) = EL t (m t+1 )+L(E t (m t+1 )) = EL t (m t+1 )+L(q 1 ) EL t (m t+1 ) E ( log r j log r 1) 42 / 64

72 Advantages of average conditional entropy (ACE) Transparent lower bound: expected excess return (in logs) Alternatively, ACE measures the highest risk premium in the economy Conditional entropy is easy to compute; to compute ACE evaluate conditional entropy at steady-state values ACE is comparable across different models with different state variables, preferences, etc. 43 / 64

73 Key models External habit Recursive preferences Heterogeneous preferences / 64

74 α is replaced by 1 α A change in notation Example: CRRA preferences; RA= 5 Old α = 5 New α = 4 45 / 64

75 External habit Equations (Abel/Campbell-Cochrane/Chan-Kogan/Heaton) U t = j=0 β j u(c t+j,x t+j ), u(c t,x t ) = (f(c t,x t ) α 1)/α. Habit is a function of past consumption: x t = h(c t 1 ), e.g., Abel: x t = c t 1. Dependence on habit Abel: f(c t,x t ) = c t /x t Campbell-Cochrane: f(c t,x t ) = c t x t Pricing kernel: m t+1 = β u c(c t+1,x t+1 ) u c (c t,x t ) ( f(ct+1,x t+1 ) = β f(c t,x t ) ) α 1 ( ) fc (c t+1,x t+1 ) f c (c t,x t ) 46 / 64

76 Example 1: Abel (1990) + Chan and Kogan (2002) Preferences: f(c t,x t ) = c t /x t Chan and Kogan have extended the habit formulation: log x t+1 = (1 φ) i=0 Relative (log) consumption φ i log c t i = φlog x t +(1 φ)log c t log s t log(c t /x t ) = φlog s t 1 + log g t Pricing kernel: log m t+1 = logβ+(α 1)log g t+1 αlog(x t+1 /x t ) = logβ+(α 1)log g t+1 α(1 φ)log s t 47 / 64

77 ACE: Abel-Chan-Kogan Pricing kernel: log m t+1 = logβ+(α 1)log g t+1 α(1 φ)log s t Conditional entropy: L t (m t+1 ) = log E t e log m t+1 E t log m t+1 log E t e log m t+1 = logβ+k(α 1;log g) α(1 φ)log s t (= log r 1 ) E t log m t+1 = logβ+(α 1)κ 1 (log g) α(1 φ)log s t L t (m t+1 ) = k(α 1;log g) (α 1)κ 1 (log g) ACE: EL t (m t+1 ) = k(α 1;log g) (α 1)κ 1 (log g) 48 / 64

78 ACE: Abel-Chan-Kogan Pricing kernel: log m t+1 = logβ+(α 1)log g t+1 α(1 φ)log s t Conditional entropy: L t (m t+1 ) = log E t e log m t+1 E t log m t+1 log E t e log m t+1 = logβ+k(α 1;log g) α(1 φ)log s t (= log r 1 ) E t log m t+1 = logβ+(α 1)κ 1 (log g) α(1 φ)log s t L t (m t+1 ) = k(α 1;log g) (α 1)κ 1 (log g) ACE: EL t (m t+1 ) = k(α 1;log g) (α 1)κ 1 (log g) It is exactly the same as in the CRRA case 48 / 64

79 Example 2: Campbell and Cochrane (1999) Preferences: f(c t,x t ) = c t x t Campbell and Cochrane specify (log) surplus consumption ratio directly: log s t = log[(c t x t )/c t ] log s t = φ(log s t 1 log s)+λ(log s t 1 )(log g t κ 1 (log g)). 49 / 64

80 Example 2: Campbell and Cochrane (1999) Preferences: f(c t,x t ) = c t x t Campbell and Cochrane specify (log) surplus consumption ratio directly: log s t = log[(c t x t )/c t ] log s t = φ(log s t 1 log s)+λ(log s t 1 )(log g t κ 1 (log g)). Compare to relative (log) consumption in Chan and Kogan log s t log(c t /x t ) = φlog s t 1 + log g t 49 / 64

81 Example 2: Campbell and Cochrane (1999) Preferences: f(c t,x t ) = c t x t Campbell and Cochrane specify (log) surplus consumption ratio directly: log s t = log[(c t x t )/c t ] log s t = φ(log s t 1 log s)+λ(log s t 1 )(log g t κ 1 (log g)). Compare to relative (log) consumption in Chan and Kogan log s t log(c t /x t ) = φlog s t 1 + log g t 49 / 64

82 Example 2: Campbell and Cochrane (1999) Preferences: f(c t,x t ) = c t x t Campbell and Cochrane specify (log) surplus consumption ratio directly: log s t = log[(c t x t )/c t ] log s t Pricing kernel: = φ(log s t 1 log s)+λ(log s t 1 )(log g t κ 1 (log g)). log m t+1 = logβ+(α 1)logg t+1 +(α 1)log(s t+1 /s t ) = logβ (α 1)λ(log s t )κ 1 (log g) + (α 1)(1+λ(log s t ))log g t+1 + (α 1)(φ 1)(log s t log s) 50 / 64

83 Example 2: Campbell and Cochrane (1999) Preferences: f(c t,x t ) = c t x t Pricing kernel: log m t+1 = logβ+(α 1)logg t+1 +(α 1)log(s t+1 /s t ) = logβ (α 1)λ(log s t )κ 1 (log g) + (α 1)(1+λ(log s t ))log g t+1 + (α 1)(φ 1)(log s t log s) Conditional entropy: L t (m t+1 ) = k((α 1)(1+λ(log s t ));log g) (α 1)(1+λ(log s t ))κ 1 (log g) 51 / 64

84 Additional assumptions To compute ACE, we have to specify λ and log g Conditional volatility of the consumption surplus ratio λ(log s t ) = 1 1 φ b/(1 α) 1 2(log s t log s) 1 σ 1 α In discrete time, there is an upper bound on logs t to ensure positivity of λ In continuous time, this bound never binds so we will ignore it In Campbell and Cochrane, b = 0 to ensure a constant logr 1 Consumption growth is i.i.d. Case 1. logg t+1 = w t+1, w t+1 N (µ,σ 2 ) Case 2. logg t+1 = w t+1 z t+1, z t+1 j Gamma(j,θ 1 ), j = ω 52 / 64

85 ACE: Campbell and Cochrane, Case 1 Conditional entropy: L t (m t+1 ) = ((α 1)(φ 1) b)/2+b(log s t log s) ACE: EL t (m t+1 ) = ((α 1)(φ 1) b)/2 53 / 64

86 ACE: Campbell and Cochrane, Case 1 Conditional entropy: L t (m t+1 ) = ((α 1)(φ 1) b)/2+b(log s t log s) ACE: EL t (m t+1 ) = ((α 1)(φ 1) b)/2 All authors use α = 1 ACE for different calibrations (quarterly) 53 / 64

87 ACE: Campbell and Cochrane, Case 1 Conditional entropy: L t (m t+1 ) = ((α 1)(φ 1) b)/2+b(log s t log s) ACE: EL t (m t+1 ) = ((α 1)(φ 1) b)/2 All authors use α = 1 ACE for different calibrations (quarterly) Campbell and Cochrane (1999): φ = 0.97, b = 0; EL t (m t+1 ) = (0.120 annual) 53 / 64

88 ACE: Campbell and Cochrane, Case 1 Conditional entropy: L t (m t+1 ) = ((α 1)(φ 1) b)/2+b(log s t log s) ACE: EL t (m t+1 ) = ((α 1)(φ 1) b)/2 All authors use α = 1 ACE for different calibrations (quarterly) Campbell and Cochrane (1999): φ = 0.97, b = 0; EL t (m t+1 ) = (0.120 annual) Wachter (2006): φ = 0.97, b = 0.011; EL t (m t+1 ) = (0.098 annual) 53 / 64

89 ACE: Campbell and Cochrane, Case 1 Conditional entropy: L t (m t+1 ) = ((α 1)(φ 1) b)/2+b(log s t log s) ACE: EL t (m t+1 ) = ((α 1)(φ 1) b)/2 All authors use α = 1 ACE for different calibrations (quarterly) Campbell and Cochrane (1999): φ = 0.97, b = 0; EL t (m t+1 ) = (0.120 annual) Wachter (2006): φ = 0.97, b = 0.011; EL t (m t+1 ) = (0.098 annual) Verdelhan (2009): φ = 0.99, b = 0.011; EL t (m t+1 ) = (0.062 annual) 53 / 64

90 ACE: Campbell and Cochrane, Case 2 Conditional entropy: L t (m t+1 ) = (α 1)(1+λ(log s t ))ωθ + ((1+(α 1)(1+λ(log s t ))θ) 1 1)ω + ((α 1)(φ 1) b)/2+b(log s t log s) ACE: use log-linearization around log s EL t (m t+1 ) = ωd 2 /(1+d)+((α 1)(φ 1) b)/2 d = σ θ (α 1)(φ 1) b 54 / 64

91 ACE: Campbell and Cochrane, Case 2 Conditional entropy: L t (m t+1 ) = (α 1)(1+λ(log s t ))ωθ + ((1+(α 1)(1+λ(log s t ))θ) 1 1)ω + ((α 1)(φ 1) b)/2+b(log s t log s) ACE: use log-linearization around log s EL t (m t+1 ) = ωd 2 /(1+d)+((α 1)(φ 1) b)/2 d = σ θ (α 1)(φ 1) b Calibration as above + vol of log g + jump parameters: σ 2 = (0.035) 2 /4 ωθ 2 BNSU: ω = 0.01/4, θ = 0.15 BCM: ω = /4, θ = / 64

92 ACE: Campbell and Cochrane, Case 2 Calibration ACE ACE (case 1) ACE jumps CC + BNSU W + BNSU V + BNSU CC + BCM W + BCM V + BCM / 64

93 Time dependence via external habit No time-dependence in consumption growth Nevertheless: habit with varying volatility may have a substantial impact on the entropy of the pricing kernel Could be relevant for option prices (Du, 2008) 56 / 64

94 Recursive preferences: traditional version Equations (Kreps-Porteus/Epstein-Zin/Weil) U t = [ (1 β)c ρ t + βµ t (U t+1 ) ρ] 1/ρ µ t (U t+1 ) = ( E t U α ) 1/α t+1 IES = 1/(1 ρ) CRRA = 1 α α = ρ additive preferences 57 / 64

95 Recursive preferences: pricing kernel Scale problem by c t (u t = U t /c t, g t+1 = c t+1 /c t ) u t = [(1 β)+βµ t (g t+1 u t+1 ) ρ ] 1/ρ Pricing kernel (mrs) m t+1 = β ( ct+1 c t ) ρ 1 ( Ut+1 µ t (U t+1 ) ( = β g ρ 1 gt+1 u t+1 t+1 µ t (g t+1 u t+1 ) ) α ρ ) α ρ 58 / 64

96 Loglinear approximation Loglinear approximation log u t = ρ 1 log[(1 β)+βµ t (g t+1 u t+1 ) ρ ] = ρ 1 log [(1 ] β)+βe ρlogµ t(g t+1 u t+1 ) b 0 + b 1 logµ t (g t+1 u t+1 ). Exact if ρ = 0 : b 0 = 0, b 1 = β Solve by guess and verify 59 / 64

97 Example 1: Bansal-Yaron Consumption growth Guess value function log g t = g + γ(l)v 1/2 t 1 w 1t v t = v + ν(l)w 2t (w 1t,w 2t ) NID(0,I) log u t = u + ω g (L)v 1/2 t 1 w 1t + ω v (L)w 2t Solution includes ω g0 + γ 0 = γ(b 1 ) i=0 b i 1γ i ω v0 = b 1 (α/2)γ(b 1 ) 2 ν(b 1 ) 60 / 64

98 ACE: Bansal-Yaron Pricing kernel log m t+1 = log β+(ρ 1)g (α ρ)(α/2)ω 2 v0 Conditional entropy + (ρ 1)[γ(L)/L] + v 1/2 t 1 w 1t (α ρ)(α/2)γ(b 1 ) 2 v t + [(ρ 1)γ 0 +(α ρ)γ(b 1 )]v 1/2 t w 1t+1 + (α ρ)ω 2 v0 w 2t+1 L t (m t+1 ) = [(ρ 1)γ 0 +(α ρ)γ(b 1 )] 2 v t /2+(α ρ) 2 ω 2 v0 /2 ACE (Bansal, Kiku, Yaron, 2007; monthly) = = if ρ = α 61 / 64

99 Example 2: Wachter Consumption growth log g t = g + σw 1t + z t λ t = (1 ϕ)λ+ϕλ t 1 + σ λ w 2t (w 1t,w 2t ) NID(0,I) z t j N (jθ,jδ 2 ) j 0 has jump intensity λ t 1 Guess value function log u t = u + ω λ λ t Solution includes ] ω λ = (1 b 1 ϕ) 1 b 1 [e αθ+(αδ)2 /2 1 /α 62 / 64

100 ACE: Wachter Pricing kernel log m t+1 = log β+(ρ 1)x (α ρ)(α/2)[σ 2 +(ω λ σ λ ) 2 ] λ t (e αθ+(αδ)2 /2 1)/α + (α 1)(σw 1t+1 + z t+1 )+(α ρ)(ω λ σ λ )w 2t+1 Conditional entropy (monthly) L t (m t+1 ) = (α 1) 2 σ 2 /2+(α ρ) 2 (ω λ σ λ ) 2 /2 } + λ t {[e (α 1)θ+(α 1)2 δ 2 /2 1] (α 1)θ ACE (monthly) = = if ρ = α 63 / 64

101 Time dependence via recursive preferences Little time-dependence in pricing kernel Nevertheless: interaction of (modest) dynamics in consumption growth and recursive preferences can have a substantial impact on the entropy of the pricing kernel Not clear it s relevant to option prices, but it s a route to magnify the impact of disasters on excess returns 64 / 64