Disasters Implied by Equity Index Options

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

and Ian Martin (Stanford) Boston University | February 11, 2011

This version: February 14, 2011

Backus, Chernov, & Martin (NYU)  Disasters in options  1 / 45

Summary

The idea

▶ Problem: disasters infrequent ⇒ hard to estimate their distribution ▶ Solution: infer from option prices

What we find

▶ Disasters apparent in options data ▶ More modest than disasters in macro data

Why this is harder than we thought

▶ Barro data gives us “true” distribution of consumption growth ▶ Option prices give us “risk-neutral” distribution of returns

Outline

Preliminaries: entropy, cumulants, plan Disasters in macroeconomic models Digression: risk-neutral probabilities Disasters in option models Comparing models

Preliminaries

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.”

Preliminaries

Entropy bound

Entropy is a measure of dispersion: for x > 0

L(x) ≡ log Ex − E log x ≥ 0

Pricing relation: there exists m > 0 such that

E_{t} (m_{t+1}r_{t+1}) = 1

Entropy bound

L(m) ≥ E (log r − log r^{1})

Preliminaries

Cumulants

Cumulant generating function

k(s; x) = log Ee^{sx} = ∑_{j=1}^{∞}κ_{j}(x)s^{j}/j!

Cumulants are almost moments

mean = κ_{1}
variance = κ_{2}
skewness = κ_{3}/κ_{2}^{3/2}
(excess) kurtosis = κ_{4}/κ_{2}^{2}

Preliminaries

Entropy and cumulants

Entropy of pricing kernel

L(m) = log Ee^{log m} − E log m
= k(1; log m) − E log m = ∑_{j=2}^{∞}κ_{j}(log m)/j!

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)}

Plan of attack

Modeling assumptions

▶ iid ▶ Tight link between consumption growth and equity returns ▶ Representative agent with power utility [if needed]

Parameter choices

▶ Match mean and variance of log consumption growth ▶ Ditto log equity return ▶ Base “disasters” on Barro’s macroeconomic evidence ▶ Or on equity index options

Compare macro- and option-based examples

Macro disasters

Macro disasters: environment

Consumption growth and “equity” return are iid

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

Power utility

log m_{t+1} = log β − α log g_{t+1}

Macro disasters

Macro disasters: the bazooka

Cumulant generating functions

k(s; log m) = k(−αs; log g)

Yaron’s “bazooka”

κ_{j}(log m)/j! = κ_{j}(log g)(−α)^{j}/j!

Macro disasters

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} = 0.15^{2}) [similar to Barro, Nakamura, Steinsson, and Ursua]

Macro disasters

Macro disasters: entropy

Cumulant generating functions

k(s; log g) ≡ log Ee^{s log g} = k(s; w) + k(s; z)
k(s; w) ≡ log Ee^{sw} = sμ + (sσ)^{2}/2
k(s; z) ≡ log Ee^{sz} = ω (e^{sθ+(sδ)^{2}/2} − 1)

Entropy

L(m) = (−ασ)^{2}/2 + ω (e^{−αθ+(αδ)^{2}/2} − 1) + αωθ,

Macro disasters

Macro disasters: cumulants

Macro disasters

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

Macro disasters

Macro disasters: equity premium

Risk-neutral probabilities

Digression: risk-neutral probabilities

Notation: states x have (true) probabilities p(x) Risk-neutral probabilities p^{∗}

p^{∗}(x) = p(x)m(x)/q^{1}
m(x) = q^{1}p^{∗}(x)/p(x)
q^{1} = Em (1-period bond price)

Entropy (aka “relative entropy” or “Kullback-Leibler divergence”)

L(m) = L(p^{∗}/p) = E log(p/p^{∗})

Risk-neutral probabilities

Risk-neutral probabilities: power utility

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

Option disasters

Option disasters: overview

Options an obvious source of information ...

▶ ... about risk-neutral distribution of equity returns

Critical ingredients

▶ Option prices ▶ Merton model ▶ Estimated parameters ▶ Implied volatilities

Option disasters

Option disasters: information in option prices

Put option (bet on low returns)

q_{t}^{p} = q_{t}^{1}E_{t}^{∗}(b − r_{t+1}^{e})^{+}

Strategy

▶ Estimate p^{∗} by varying strike price b (cross section)

Black-Scholes-Merton benchmark

▶ Quote prices as implied volatilities (high price ⇔ high vol) ▶ Horizontal line if lognormal ▶ “Skew” suggests disasters

Option disasters

Option disasters: Merton model

Equity returns iid

log r_{t+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

Option disasters: parameter values

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 (q^{1}E^{∗}r^{e} = 1)

Later: 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})

All from Broadie, Chernov, and Johannes (JF, 2007)

Option disasters

Option disasters: implied volatility

Comparing models

Comparing macro and option models

Approach 1: compare pricing kernels

▶ Required: estimated p from daily data on equity returns

Approach 2: compare consumption growth distributions

▶ Required: connections between g and r^{e}, p and p^{∗}

Approach 3: compare option prices

▶ Required: connections between g and r^{e}, p and p^{∗}

Comparing models

Comparing pricing kernels: 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

Comparing models

Comparing pricing kernels: cumulants

Comparing models

Comparing consumption growth

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

Comparing models

Comparing option prices

Comparing models

Comparing option prices

Comparing models

Comparing models

All of these comparisons point in the same direction

▶ Macro disasters more pronounced than option disasters

Reconsidering

Reconsidering ...

Our models are based on

▶ iid ▶ Tight link between consumption growth and equity returns ▶ Representative agent with power utility

Let’s take a closer look at the last two

Reconsidering

Reconsidering power utility

“Risk aversion” implied by arbitrary pricing kernel

RA ≡ −\frac{∂ log m}{∂ log g} = −\frac{∂ log(p^{∗}/p)}{∂ log r^{e}} · \frac{∂ log r^{e}}{∂ log g}

Reconsidering

Reconsidering power utility

Reconsidering

Reconsidering the link between g and r^{e}

Reconsidering

Reconsidering the link between g and r^{e} Recapitulation

Barro, Longstaff & Piazzesi, Rietz

▶ Disasters contribute to equity premium [entropy] ▶ Evident in macro data ▶ Range of opinion on magnitude

We look at options

▶ Smile/smirk suggests something like disasters ▶ Prices available even for outcomes that don’t occur in sample ▶ Implied disasters less severe than macro data ▶ High entropy from options suggests it’s not enough to match equity

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