Disasters Implied by Equity Index Options

David Backus (NYU), Mikhail Chernov (LBS),

and Ian Martin (Stanford)

American Economic Association | Atlanta | January 3, 2010

This version: January 3, 2010

Backus, Chernov, & Martin (NYU)  Disasters in options  1 / 14

Option prices Option prices Outline

How disasters affect asset prices Disasters in macroeconomic data Disasters in options data Extensions

How disasters work

Entropy and cumulants

Entropy: for x > 0

L(x) ≡ log Ex − E log x ≥ 0

Alvarez-Jerman bound

L(m) ≥ E (log r^{j} − log r^{1})

Zin’s “never a dull moment” conjecture

L(m) = \underbrace{κ_{2}(log m)/2!}_{(log)normal term} + \underbrace{κ_{3}(log m)/3! + κ_{4}(log m)/4! + ⋯}_{high-order cumulants (incl disasters)}

Disasters in macro data

Macro disasters: overview

No dynamics: iid consumption growth Tight link between consumption growth and equity returns

g_{t+1} = c_{t+1}/c_{t}
d_{t} = c_{t}^{λ}
log r_{t+1}^{e} = constant + λ log g_{t+1}

Representative agent with power utility

log m_{t+1} = log β − α log g_{t+1}
κ_{j}(log m)/j!

= κ_{j}(log g)(−α)^{j}/j!

“Yaron’s bazooka”

Disasters in macro data

Macro disasters: entropy

Disasters in macro data

Macro disasters: cumulants

Disasters in option prices

Option disasters: overview

Option prices an obvious source of information Put options are bets on low returns Parameters from Broadie, Johannes, and Chernov (JF, 2007)

Disasters in option prices

Option disasters: option prices

Disasters in option prices

Option disasters: option prices

Disasters in option prices

Option disasters: cumulants

Disasters in option prices

Option disasters: components of entropy

High-Order Cumulants

Model  Entropy Variance/2 Odd  Even

Consumption-based model

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

Disasters in option prices

Option disasters: consumption implied by options

Cons Growth Process Based on

Macro Data  Option Prices
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

Summary and extensions

Bottom line

Barro, Longstaff & Piazzesi, Rietz

▶ Disasters contribute to equity premium, entropy ▶ Evident in macro data

We look at options

▶ Smile/smirk suggests something like disasters ▶ More modest than macro data ▶ High entropy suggests it’s not enough to match equity premium

Extensions: reconsider

▶ Tight link between consumption growth and equity returns ▶ Power utility ▶ Consumption growth iid

Summary and extensions

Consumption growth and equity returns

Summary and extensions

Implied “risk aversion” in the option model