paper presentation: disasters implied by equity index options · 2012. 2. 8. · model #1:...

Paper Presentation:Disasters Implied by Equity Index Options

Áron TóbiásPh.D. Student

Central European UniversityDepartment of Economics

February 3, 2010

Paper Presentation: Disasters Implied by Equity Index Options 1 / 32

References

References


References

References

Backus, D. – Chernov, M. –Martin, I. 2009. “Disasters implied by eq-uity index options.” Unpublished working paper. Available at http://pages.stern.nyu.edu/∼dbackus/GE_asset_pricing/disasters/BCM_disasters_latest.pdf. Date of access: January 21, 2010.

The Economist. 2009. “In Plato’s cave.” January 22.


http://pages.stern.nyu.edu/~dbackus/GE_asset_pricing/disasters/BCM_disasters_latest.pdf



Main Idea

Main Idea


Main Idea

Normality Assumption vs. Facts

“[I]f [daily movements in] the Dow Jones Industrial Average followed anormal distribution, [. . . ] [i]t should have moved by more than 7% onlyonce in every 300,000 years; in the 20th century it did so 48 times.” (TheEconomist, 2009)

Figure 1: Density of the normal distribution contrasted with a “fat-tail”distribution. Source: The Economist.


Main Idea

Normality Assumption vs. Facts

1 Predictions of macroeconomic models based on the lognormalityof consumption growth (and normality of the equity premium) donot fit empirical data. Specifically, predictions of a representative-agent model with power utility can be reconciled with the data onlyif investors are assumed to be implausibly risk-averse.

2 The normality assumption imposed on equity premiums in option-pricing models is also at odds with the data. Implied volatilities ofoptions should be constant in the normal case. However, what is ob-served is a volatility smile (or, rather, smirk): Volatility is decreasingin a put option’s moneyness.


Main Idea

Disasters

In their paper, BCM (2009) introduce additive disasters to1 log consumption growth in a macroeconomic model;2 the equity premium in an option-pricing model.

The results stemming from each of these two, independent, approachesare reassuring:

1 More plausible (lower) values of the risk-aversion parameter are suf-ficient to account for observed data; and

2 the implied volatility smirk can be accounted for

if disasters are allowed for.


Building Tools

Building Tools


Building Tools

Pricing Equations

The return of any asset j is determined by the pricing equation:

Et(mt+1rjt+1) = 1,

where m is the pricing kernel (PK), which is the same for all assets. Therole of the PK is twofold: (i) discounting; (ii) risk adjustment.

Another characterization is the following:

E∗t (rjt+1) = r f

t+1,

where r f is the risk-free rate. In this case, the expectation is taken withrespect to an artificial distribution of the return on asset j , which treats theinvestor as if s/he were risk-neutral. This is the risk-neutral distribution.As the PK has been eliminated in the second equation, its discountingrole is taken over by the risk-free rate, and investors’ risk attitudes areincorporated in the risk-neutral distribution. (Simply put: probabilities ofbad events are scaled up, those of good ones are scaled down).


Building Tools

Two Approaches of Determining the PK

1 In macro models, the PK is computed from proportional consump-tion growth (denoted by g). If preferences are of the power-utility(CRRA) class, then:

mt+1 = βg−αt+1,

where α is the risk-aversion parameter and β is the rate of subjectivediscounting (time preference).

2 In option-pricing models, the value of the PK in any state x ∈ X (X isthe state space) can be determined by using the risk-neutral and trueprobabilities/densities of the specific state, and the risk-free rate:

m(x) =1r f ·

p∗(x)

p(x).

In practice, risk-neutral probabilities are usually estimated from cross-section data, whereas true ones from time-series data.


Building Tools

Entropy of the PK

Central to the analysis by BCM is the entropy of the PK:

L(m) ≡ ln[E(m)]− E[ln(m)].

Entropy is a measure of the uncertainty associated with a random variable.Similarly to the variance, it is always nonnegative and zero if and only ifa random variable is constant with probability one. However, the entropyencompasses also high-order cumulants.


Building Tools

Moments

For any random variable ξ, the moment-generating function (mgf) isdefined as follows for any s ∈ R:

h(s; ξ) ≡ E[exp(sξ)].

Its name comes from the fact that its nth derivative evaluated at zerogives the nth moment of the random variable:

h(n)(0; ξ) = E(ξn), n ∈ N.


Building Tools

Cumulants

For any random variable ξ, the cumulant-generating function (cgf) isdefined as the logarithm of the mgf for any s ∈ R:

k(s; ξ) ≡ ln[h(s; ξ)] = ln{E[exp(sξ)]}

Its nth derivative evaluated at zero gives the nth cumulant of the randomvariable:

κn ≡ k(n)(0; ξ), n ∈ N.

Moments and cumulants are tightly interrelated. For example, it can beshown that κ1 is identical to the mean, κ2 to the variance, and skewnessand excess kurtosis are given, respectively, as κ3/κ

3/22 and κ4/κ

22.


Building Tools

Contributions of Cumulants to Entropy

By the Taylor series of the cgf, it can be proven that the entropy of thePK is related to the cumulants of the logarithm of the PK from the secondorder and above:

L(m) =

∞∑j=2

κj [ln(m)]/j !.


Building Tools

Contributions of Cumulants to Entropy

Here comes the trick of the whole BCM paper. A benchmark assumptionis that ln(m) is normally distributed. The fact that the cumulants ofthe normal distribution for j ≥ 3 are zero implies that if the normalityassumption is tenable, then the entropy of the PK should depend onlyon the variance of the logarithm of the PK. Decompose the entropy asfollows:

L(m) = σ2/2 + Co + Ce ,

where1 σ2/2 represents the “normal component;”2 Co represents the contributions of odd high-order cumulants (j ≥ 3)

to the entropy (e.g. skewness); and3 Ce represents the contributions of even high-order cumulants (j ≥ 4)

to the entropy (e.g. kurtosis).That is, Co and Ce measure departures from normality of the logarithmof the PK.


Building Tools

Alvarez – Jermann Lower Bound

It can be shown that the entropy of the PK is bounded below by theexpected value of the equity premium or excess return (expressed as logdifferences). This is the Alvarez – Jermann lower bound:

L(m) ≥ E[ln(r j)− ln(r f )].

Equity premiums are observed (their mean estimated from a long time se-ries is 4.4%) but the PK is not; therefore, one wants to compute the latterusing a suitable model. In doing so, one must make sure that parametersdetermining the PK are identified in a way so that the Alvarez – Jermannlower bound is not violated.

Additional structure is imposed on the estimation of PK by the Hansen – Jagan-nathan upper bound, which states that the absolute value of the Sharpe ratio(mean excess return divided by its standard deviation) should be no greater thanthe standard deviation of the PK divided by its mean. This upper bound charac-terizes the set of mean-variance efficient returns.


Model #1: Consumption-Based Model

Model #1: Consumption-BasedModel



Determining the PK

In this consumption-based model, the PK is derived from the investor’spreferences and assets are priced using true probabilities. If preferencesare of the CRRA type, the PK is given as follows:

ln(mt+1) = logβ − α ln(gt+1).

BCM assume that the logarithm of consumption growth is specified asfollows:

ln(gt+1) = wt+1 + zt+1,

where wt and zt are independent of each other and over time. More-over, wt is normal. (This represents the first of the terms determiningthe entropy of the PK.) zt is the disaster term introducing high-ordercumulants.



Role of Disasters

If zt ≡ 0 (no disasters), we are at the benchmark case of lognormalconsumption growth. In order for the Alvarez – Jermann lower bound tohold, a corresponding lower bound must be imposed on the risk-aversionparameter: α ≥ 8.47, which lower bound is implausibly high.

However, if we introduce departures from normality of the PK throughhigh-order cumulants of log consumption growth by means of disasters,the lower bound on the risk-aversion parameter imposed by the Alvarez –Jermann lower bound can be relaxed.



Bernoulli Disasters

Suppose that the disaster is a Bernoulli random variable:

zt =

{0 with probability 1− ω,−θ with probability ω

with θ > 0. A disaster which is highly undesirable (θ = 0.3) but infre-quent (ω = 0.01) is able to generate a negative skewness and a positivekurtosis in log consumption growth. The contributions of these high-ordercumulants to the entropy of the PK are quite dramatic, especially if in-vestors are risk-averse (α is high). This follows because

κj [ln(m)] = (−α)jκj [ln(g)],

so the high-order cumulants of log consumption growth are incorporatedin the entropy of PK amplified by the risk-aversion parameter (if α > 1).



Bernoulli Disasters

As a consequence, the lower bound imposed on the risk-aversion pa-rameter decreases to 6.59. However, the dramatic effects of high-ordercumulants on the entropy of the PK disappear if the disaster is less catas-trophic but more frequent.

The conclusion is reversed for booms: The Alvarez – Jermann lower boundis not violated only if the α is very high, even higher than in the benchmarkcase.



Mixed Poisson –Normal Disasters

Now suppose that the disaster is more complicated. Specifically, morethan one disaster can occur, which are distributed normally and inde-pendent of each other. The disasters add up. The number of disastersfollows a Poisson distribution. To sum up:

j ∼ Poisson (ω),

zt | j ∼ N (jθ, jδ2).

The results are qualitatively the same as in the Bernoulli case, butcontributions of the disasters to the entropy of the PK through the high-order cumulants of log consumption growth is even more enormous if αis high. Moreover, the lower bound on the risk-aversion parameter α isfurther relaxed to 5.38.


Model #2: Derivative-Pricing Approach

Model #2: Derivative-PricingApproach



Derivation of Risk-Neutral Distributions

Remember that with the option-pricing approach, preference-based de-termination of the PK is bypassed and risk-neutral distributions are usedinstead. Starting from the Merton model, log equity premium (based onthe S&P 500 index) is specified as a sum of a normal component and a dis-aster term (cf. log consumption growth having been analogously specifiedin the consumption-based model):

ln(r et+1)− ln(r f

t+1) = wt+1 + zt+1,

where

wt ∼ N (µ, σ2),

wt∗∼ N (µ∗, σ∗2),

zt ∼ Poisson – normal mixture (θ, δ, ω2),

zt∗∼ Poisson – normal mixture (θ∗, δ∗, ω∗2).



Derivation of Risk-Neutral Distributions

The parameters of the true and risk-neutral distributions are (i) estimatedfrom time-series data; or (ii) chosen so that the resulting processes matchthese data; or (iii) taken from the preceding literature.

If one calculates risk-neutral distributions in consumption-based models,one observes that these distributions are shifted to the left relative to therespective true ones. For example, if the true distribution of log consump-tion growth is normal, the risk-neutral distribution will be normal as wellwith a lower mean (depending on the risk aversion parameter) and thesame variance (uniform shift to the left). If the true distribution of logconsumption growth is Bernoulli, simply greater weight is put on the badoutcome for the risk-neutral distribution.



Disasters Cheer Up the Model

By allowing for non-normal disasters and contributions of high-order cumu-lants of the equity premium to the entropy of the PK, the model is ableto reproduce the asymmetric smirk of implied volatility for put op-tions. However, the implied distribution of disasters is less skewed andless leptokurtic than calculated from the consumption-based framework.That is, tails are thinner and extreme events are less frequent.



A Caveat

In the consumption-based framework, relative risk aversion is constant(decipher the abbreviation CRRA). However, no structure is imposed onrisk aversion in the option-pricing framework. Therefore, if investors are—plausibly—more risk-averse when returns are negative than when they arepositive, risk premiums on assets might reflect not only merely disastersbut also pricing of disasters, giving them greater weight than CRRA pref-erences.


Summary and Extensions




Summary

Allowing for disasters in consumption growth or equity premiums im-proves the ability of both a macroeconomic and an option-pricingmodel to account for observed equity premiums.

Disasters implied by the option-pricing model are more modest thanmacroeconomic consumption-based evidence would imply.



Extenstions

Assumptions that may be worth relaxing:1 Power utility. Constant relative risk aversion is not supported by the

option-pricing model.2 Representative-agent framework. There are many research direc-

tions in finance aiming at exploring better alternatives.3 IID consumption growth. This assumption is seemingly very tenu-

ous, but BCM argue that it is in fact probably harmless.4 Tight links between dividends and aggregate consumption.


Any questions?


Thank you for your attention.


paper presentation: disasters implied by equity index options · 2012. 2. 8. · model #1:...

Documents