# Options-Pricing Formula with Disaster Risk.

We derive a new options-pricing formula that applies when disaster risk is the dominant force, when the size distribution of disasters is characterized by a power law, and when the economy has a representative agent with Epstein-Zin utility with a constant coefficient of relative risk aversion. Specifically, we consider far-out-of-the-money put options on the overall stock market, corresponding empirically to the S&P 500 in the United States and analogous indices for other countries. The pricing formula applies when the option is sufficiently far out of the money (operationally, a relative exercise price or moneyness of 0.9 or less) and when the maturity length is not too long (operationally, up to 6 months).

In the prescribed region, the elasticity of the put-options price with respect to maturity is close to one. The elasticity with respect to the exercise price is greater than one, roughly constant, and depends on the difference between the power-law tail parameter, denoted [alpha], and the coefficient of relative risk aversion, [gamma]. (This difference has to be positive for various rates of return not to blow up.)

The options-pricing formula involves a term that is proportional to the disaster probability, p. This term depends also on three other parameters: [gamma], [alpha], and the threshold disaster size, [z.sub.0]. If these three parameters are fixed, we can use estimated time fixed effects to gauge the time variations in p. The options-pricing formula depends also on potential changes in p. Specifically, sharp increases in p can get out-of-the-money put options into the money without the realization of a disaster. We find empirically that the probability, q, of a large upward movement in p can be treated as roughly constant.

We show that the theoretical formula conforms with data from 1994 to 2017 on far-out-of-the-money put options on the U.S. stock market and analogous indices over shorter periods for other countries. Our analysis relies on two types of data--indicative prices on over-the-counter (OTC) contracts offered to clients by a large financial firm and U.S. market data provided by Berkeley Options Data Base and OptionMetrics. The Berkeley data allow us to extend the U.S. analysis back to 1983, thereby bringing out the key role of the stock-market crash of October 1987. The OptionMetrics information allows us to check whether the results using U.S. OTC data differ from those using market data. We find that the main results are similar with the two types of data.

Section I lays out the basic rare-disasters framework. Section II works out a formula for pricing of put options within the disaster setting. The analysis starts with a constant probability, p, of disasters and then introduces possibilities for changing pt. Section III sets up the empirical framework and describes the results. The main data comprise OTC information on options prices for far-out-of-the money put options on seven major stock-market indices. These data are supplemented for the United States with longer-term information from Berkeley Options Data Base and with market-based data from OptionMetrics. Section IV concludes.

I. Baseline Disaster Model and Previous Results

We use a familiar setup based on rare-macroeconomic disasters, as developed in Rietz (1988) and Barro (2006, 2009). The model is set up for convenience in discrete time. Real GDP, Y, is generated from

(1) [log([Y.sub.t+1]) = log([Y.sub.t]) + g + [u.sub.t+1] + [v.sub.t+1]],

where, g[greater than or equal to]0 is the deterministic part of growth, [u.sub.t+1] (the diffusion term) is an i.i.d. normal shock with mean 0 and variance [[sigma].sup.2], and vt+1 (the jump term) is a disaster shock. Disasters arise from a Poisson process with probability of occurrence p per period. For now p is taken as constant; later, variations in p over time play a central role. When a disaster occurs, GDP falls by the fraction b, where 0<b[less than or equal to]1. The distribution of disaster sizes is time invariant. (The baseline model includes disasters but not bonanzas.) This jump-diffusion process for GDP is analogous to the one posited for stock prices in Merton (1976, equations [1]-[3]). (1)

In the underlying Lucas (1978)-tree model, which assumes a closed economy, no investment, and no government purchases, consumption, [C.sub.t], equals GDP, [Y.sub.t]. The implied expected growth rate of C and Y is given, if the period length is short, by

(2) [g* = g + (1/2) x [[sigma].sup.2] -p x Eb],

where Eb is the mean of b. In this and subsequent formulas, we use an equal sign, rather than approximately equal, when the equality holds as the period length shrinks to zero.

The representative agent has Epstein-Zin/Weil utility, (2) as in Barro (2009):

(3) [mathematical expression not reproducible]

where [gamma]>0 is the coefficient of relative risk aversion, [theta]>0 is the reciprocal of the intertemporal-elasticity-of-substitution (IES) for consumption, and [rho]>0 is the rate of time preference. As shown in Barro (2009) (based on Giovannini and Weil [1989] and Obstfeld [1994]), with i.i.d. shocks and a representative agent, the attained utility ends up satisfying the form:

(4) [mathematical expression not reproducible]

where the constant [PHI]>0 depends on the parameters of the model. Using equations (3) and (4), the first-order condition for optimal consumption over time follows from a perturbation argument as

(5) [mathematical expression not reproducible]

where [R.sub.t+1] is the gross rate of return on any available asset from time t to time t+1. When [gamma]=[theta]--the familiar setting with time-separable power utility--the term on the left-hand side of equation (5) equals one.

The process for C and Y in equation (1) implies, if the period length is negligible:

(6) [mathematical expression not reproducible]

This condition can be used along with equation (5) to price various assets, including a risk-free bond and an equity claim on a perpetual flow of consumption (that is, the Lucas tree).

Equations (5) and (6) imply that the constant risk-free interest rate is given by

(7) [mathematical expression not reproducible]

Let Pt be the price at the start of period t of an unlevered equity claim on the Lucas tree. Let [V.sub.t] be the dividend-price ratio; that is, the ratio of [P.sub.t] to [C.sub.t]. In the present model with i.i.d. shocks, [V.sub.t] equals a constant, V, so that the growth rate of [P.sub.t] equals the growth rate of [C.sub.t]. The reciprocal of V equals the dividend-price ratio and can be determined from equations (5) and (6) to be

(8) [mathematical expression not reproducible]

The constant expected rate of return on equity, [r.sup.e], is the sum of the dividend yield, 1/V, and the expected rate of capital gain on equity, which equals g*, the expected growth rate of the dividend (consumption). Therefore, [r.sup.e] is the same as equation (8) except for the elimination of the term -g*. (3) The constant equity premium is given from equations (7) and (8) by:

(9) [mathematical expression not reproducible]

The disaster or jump term in equation (9) is proportional to the disaster probability, p. The expression in brackets that multiplies p depends on the size distribution of disasters, b, and the coefficient of relative risk aversion, [gamma]. These effects were calibrated in Barro (2006) and Barro and Ursua (2012) by using the long-term history of macroeconomic disasters for 40 countries to pin down p and the distribution of b. The results accord with an observed average unlevered equity premium of 0.04-0.05 per year if [gamma] is around 3-4.

The diffusion term, [gamma][[sigma].sup.2], in equation (9) is analogous to the expression for the equity premium in Mehra and Prescott (1985) and is negligible compared to the observed average equity premium if [gamma] and [[sigma].sup.2] take on empirically reasonable values. For many purposes--including the pricing of far-out-of-the-money stock options--this term can be ignored.

II. Pricing Stock Options

A. Setup for pricing options

We now discuss the pricing of stock options within our model, which fits into the class of jump-diffusion models. Options pricing within this general class goes back to Merton (1976) and Cox and Ross (1976). The use of prices of far-out-of-the-money put options to infer disaster probabilities was pioneered by Bates (1991). This idea has been applied recently by, among others, Bollerslev and Todorov (2011a, 2011b); Backus, Chernov, and Martin (2011); Seo and Wachter (2016); and Siriwardane (2015).

We derive a pricing solution for far-out-of-the-money put options under the assumption that disaster events (jumps) are the dominant force. Key underlying conditions for the validity of the solution are that the option be sufficiently far out of the money and that the maturity not be too long. Under these conditions, we derive a simple pricing formula that reflects the underlying Poisson nature of disaster events, combined with an assumed power-law distribution for the sizes of disasters. This formula generates testable hypotheses--which we subsequently test--on the relation of the put-options price to maturity and exercise price. The formula also allows for a time-fixed-effects procedure to back out a time series for disaster probability.

Consider a put option on equity in the Lucas tree. To begin, suppose that the option has a maturity of one period and can be exercised only at the end of the period (a European option).

The exercise price or strike on the put option is

(10) [exercise price = [epsilon] x [P.sub.t]],

where we assume 0<[epsilon][less than or equal to]1. We refer to [epsilon], the ratio of the exercise price to the stock price, as the relative exercise price (often described as "moneyness").

The payoff on the put option at the start of period t+1 is zero if [P.sub.t+1][greater than or equal to] [epsilon] x [P.sub.t]. If [P.sub.t+1]<[epsilon] x [P.sub.t], the payoff is [epsilon][P.sub.t]-[P.sub.t+1]. If [epsilon]<1, the put option is initially out of the money. We focus empirically on options that are sufficiently far out of the money ([epsilon] sufficiently below one) so that the diffusion term, u, in equation (1) has a negligible effect on the chance of getting into the money over one period. The value of the put option then hinges on the disaster term, v. Specifically, the value of the put option depends on the probability, p, of experiencing a disaster and the distribution of disaster sizes, b. Further, what will mostly matter is the likelihood of experiencing one disaster. As long as the period (the maturity of the option) is not too long, the chance of two or more disasters has a second-order pricing impact that can be ignored as a good approximation. (4)

Let the price of the put option at the start of period t be [OMEGA] x [P.sub.t]. We refer to [OMEGA], the ratio of the options price to the stock price, as the relative options price. The gross rate of return, [??], on the put option is given by

(11) [mathematical expression not reproducible]

If there is one disaster of size b, the put option is in the money at the start of period t+1 if

[[p.sub.t+1]/[P.sub.t] = (1 + g) x (1 - b) < [epsilon]].

We work with the transformed variable z[equivalent to]1/(1-b), which corresponds to the ratio of normal to disaster consumption. The condition 0<b[less than or equal to]1 translates into z>1, with z tending to infinity as b tends to 1. When expressed in terms of z, the gross rate of return on the put option is modified from equation (11) to:

(12) [mathematical expression not reproducible]

To determine [OMEGA], we use the first-order condition from equation (5), with [R.sub.t+1] given by [??] from equation (12). The results depend on the form of the distribution for z, to which we now turn.

B. Power-law distribution of disaster sizes

Based on the findings for the distribution of observed macroeconomic disaster sizes in Barro and Jin (2011), we assume that the density function for z conforms to a power law: (5)

(13) [f(z) = A[z.sup.-(1+[alpha])], where A > 0, [alpha] > 0, and z [greater than or equal to] [z.sub.0] > 1].

The general notion of this type of power law was applied by Pareto (1897) to the distribution of high incomes. The power-law distribution

has since been applied widely in physics, economics, computer science, and other fields. For surveys, see Mitzenmacher (2003) and Gabaix (2009), who discusses underlying growth forces that can generate power laws. Examples of applications include sizes of cities (Gabaix and Ioannides [2004]), stock-market activity (Gabaix, et al. [2003] and Plerou, et al. [2004]), CEO compensation (Gabaix and Landier [2008]), and firm size (Luttmer [2007]). The power-law distribution has been given many names, including heavy-tail distribution, Pareto distribution, Zipfian distribution, and fractal distribution.

The parameter [z.sub.0]>1 in equation (13) is the threshold beyond which the power-law density applies. For example, in Barro and Ursua (2012), the floor disaster size of [b.sub.0]=0.095 corresponds to [z.sub.0]=1.105. We treat [z.sub.0] as a constant. The condition that f(z) integrate to one from [z.sub.0] to infinity implies [??]. Therefore, the power-law density function in equation (13) becomes

(14) [mathematical expression not reproducible]

The key parameter in the power-law distribution is the Pareto tail exponent, [alpha], which governs the thickness of the right tail. A smaller [alpha] implies a thicker tail.

The probability of drawing a transformed disaster size above z is given by

(15) [mathematical expression not reproducible]

Thus, the probability of seeing an extremely large transformed disaster size, z (expressed as a ratio to the threshold, [z.sub.0]), declines with z in accordance with the tail exponent [alpha]>0.

One issue about the power-law density is that some moments related to the transformed disaster size, z, might be unbounded. For example, in equation (7), the risk-free rate depends inversely on the term E[(1 - b).sup.-Y]. Heuristically (or exactly with time-separable power utility), we can think of this term as representing the expected marginal utility of consumption in a disaster state relative to that in a normal state. When z[equivalent to]1/(1-b) is distributed according to f(z) from equation (14), we can compute

(16) [mathematical expression not reproducible]

The term on the right side of equation (15) is larger when [gamma] is larger (more risk aversion) or [alpha] is smaller (fatter tail for disasters). But, if [alpha][less than or equal to][gamma], the tail is fat enough, relative to the degree of risk aversion, so that the term blows up. In this case, [r.sup.f] equals minus infinity in equation (7), and the equity premium is infinity in equation (9). Of course, in the data, the risk-free rate is not minus infinity and the equity premium is not infinity. Therefore, the empirical application of the power-law density in Barro and Jin (2011) confined [gamma] to a range that avoided unbounded outcomes, given the value of [alpha] that was estimated from the observed distribution of disaster sizes. That is, the unknown [gamma] had to satisfy [gamma]<[alpha] in order for the model to have any chance to accord with observed average rates of return. (6) This condition, which we assume holds, enters into our analysis of far-out-of-the-money put-options prices.

Barro and Jin (2011, Table 1) estimated the power-law tail parameter, [alpha], in single power-law specifications (and also considered double power laws). The estimation was based on macroeconomic disaster events of size 10% or more computed from the long history for many countries of per capita personal consumer expenditure (the available proxy for consumption, C) and per capita GDP, Y. The estimated values of [alpha] in the single power laws were 6.3, with a 95% confidence interval of (5.0, 8.1), for C and 6.9, with a 95% confidence interval of (5.6, 8.5), for Y. (7) Thus, the observed macroeconomic disaster sizes suggest a range for [alpha] of roughly 5-8.

C. Options-pricing formula

To get the formula for [OMEGA], the relative options price, we use the first-order condition from equations (5) and (6), with the gross rate of return, [R.sub.t+1], corresponding to the return [??] on put options in equation (12). We can rewrite this first-order condition as

(17) [mathematical expression not reproducible]

where z[equivalent to]1/(1-b) is the transformed disaster size and 1 + [rho] is an overall discount term, given from equations (5) and (6) (when the diffusion term is negligible) by

(18) [mathematical expression not reproducible]

We can evaluate the right-hand side of equation (17) using the density f(z) from equation (14) along with the expression for [??] from equation (12). The result involves integration over the interval z[greater than or equal to](1+g)/[epsilon] where, conditional on having one disaster, the disaster size is large enough to get the put option into the money. The formula depends also on the probability, p, of having a disaster. Specifically, we have:

(19) [mathematical expression not reproducible]

Evaluating the integral (assuming [gamma]<[alpha] and [epsilon]<[1+g]/[z.sub.0]) leads to a closed-form formula for the relative options price:

(20) [mathematical expression not reproducible]

D. Maturity of the option

Equation (20) applies when the maturity of the put option is one "period." We now take account of the maturity of the option. In continuous time, the parameter p, measured per year, is the Poisson hazard rate for the occurrence of a disaster. Let T, in years, be the maturity of the (European) put option. The density, h, for the number of hits (disasters) over T is given by (8)

(21) [mathematical expression not reproducible]

If pT is much less than 1, the contribution to the options price from two or more disasters will be second-order, relative to that from one disaster. For given p, this condition requires a consideration of maturities, T, that are not "too long." In this range, we can proceed as in our previous analysis to consider just the probability and size of one disaster. Then, in equation (20), p will be replaced as a good approximation by pT.

The discount rate, [rho], and growth rate, g, in equation (20) will be replaced (approximately) by [rho] T and gT. For given [rho] and g, if T is not too long, we can neglect these discounting and growth terms. The impacts of these terms are of the same order as the effect from two or more disasters, which we have already neglected.

When T is short enough to neglect multiple disasters and the discounting and growth terms, the formula for the relative options price simplifies from equation (20) to:

(22) [mathematical expression not reproducible]

Here are some properties of the options-pricing formula:

* The formula for [OMEGA], the ratio of the options price to the stock price, is well-defined if [alpha]>[gamma], the condition noted before that ensures the finiteness of various rates of return.

* The exponent on maturity, T, equals 1.

* The exponent on the relative exercise price, [epsilon], equals 1+[alpha]-[gamma], which is constant and greater than 1 because [alpha]>[gamma]. We noted before that [alpha] ranged empirically between 5 and 8. The corresponding range for [gamma] (needed to replicate an average unlevered equity premium of 0.04-0.05 per year) is between 2.5 and 5.5, with lower [gamma] associating with lower [alpha]. The implied range for [alpha]-[gamma] (taking account of the association between [gamma] and [alpha]) is between 2.5 and 4.5, implying a range for the exponent on [epsilon] between 3.5 and 5.5.

* For given T and [epsilon], [OMEGA] depends on the disaster probability, p; the shape of the power-law density, as defined by the tail coefficient, [alpha], and the threshold, [z.sub.0]; and the coefficient of relative risk aversion, [gamma]. The expression for [OMEGA] is proportional to p.

* For given p and [gamma], [OMEGA] rises with a once-and-for-all shift toward larger disaster sizes; that is, with a reduction in the tail coefficient, [alpha], or an increase in the threshold, [z.sub.0].

* For given p, [alpha], and [z.sub.0], [OMEGA] rises for sure with a once-and-for-all shift in [gamma] if [epsilon][less than or equal to]1, which is the range that we are considering for put options. Note that, in contrast, the Black-Scholes options-pricing formula implies that [OMEGA] is independent of [gamma]. (9)

We can look at the results in terms of the "risk-neutral probability," [p.sup.n], defined as the value of p that would generate a specified relative options price, [OMEGA], when [gamma]=0. The formula for the ratio of the risk-neutral to the objective probability, [p.sup.n]/p, implied by equation (22) is:

(23) [mathematical expression not reproducible]

Note that [p.sup.n]/p depends on the relative exercise price, [epsilon], but not on the maturity, T. If we assume parameter values consistent with the previous discussion--for example, [alpha]=7 and [gamma]=3.5--the implied [p.sup.n]/p is 5.1 when [epsilon]=0.9, 7.8 when [epsilon]=0.8, 12.4 when [epsilon]=0.7, 21.3 when [epsilon]=0.6, and 40.3 when [epsilon]=0.5. Hence, the relative risk-neutral probability associated with far-out-of-the-money put options is sharply above one.

To view it another way, the relative options price, [OMEGA], may seem far too high at low [epsilon], when assessed in terms of the (risk-neutral) probability needed to justify this price. Thus, people who are paying these prices to insure against the risk of an enormous disaster may appear to be irrational. In contrast, the people writing these far-out-of-the-money puts may seem to be getting free money by insuring against something that is virtually impossible. Yet the pricing is reasonable if people have roughly constant relative risk aversion with [gamma] of 3-4 (assuming a tail parameter, [alpha], for disaster size around 7). The writers of these options will have a comfortable income almost all the time, but will suffer tremendously during the largest rare disasters (when the marginal utility of consumption is extremely high).

E. Diffusion term

The formula for [OMEGA], the relative options price, in equation (22) neglects the diffusion term, u, in the process for GDP (and consumption and the stock price) in equation (1). This omission is satisfactory if the put option is sufficiently far out of the money so that, given a reasonable variance [[sigma].sup.2] of the diffusion term, the chance of getting into the money over the maturity T is negligible. In other words, the tail for the normal process is not fat enough to account by itself for, say, 10% or greater declines in stock prices over periods of a few months. Operationally, our main empirical analysis applies to options that are at least 10% out of the money ([epsilon][less than or equal to]0.9) and to maturities, T, that range up to 6 months.

If we consider put options at or close to the money, the diffusion term would have a first-order impact on the value of the option. If we neglect the disaster (jump) term--which will be satisfactory here--we would be in the standard Black-Scholes world. In this setting (with i.i.d. shocks), a key property of the normal distribution is that the variance of the stock price over interval T is proportional to T, so that the standard deviation is proportional to the square root of T. This property led to the result in Brenner and Subrahmanyam (1988) that the value of an at-the-money put option would be proportional to the square root of the maturity.

We, therefore, have two theoretical results concerning the impact of maturity, T, on the relative options price, [OMEGA]. For put options far out of the money (operationally for [epsilon][less than or equal to]0.9), the exponent on T is close to 1. For put options close to the money (operationally for [epsilon]=1), the exponent on T is close to one-half

F. Stochastic Volatility

The asset-pricing formula in equation (22) was derived under the assumption that the disaster probability, p, and the size distribution of disasters (determined by [alpha] and [z.sub.0]) were fixed. (10) We focus here on shifting p, but the results are isomorphic to shifting disaster intensity (reflecting changes in [alpha] and [z.sub.0]).

We can rewrite equation (22) as

(24) [mathematical expression not reproducible]

where [??] is a constant. We can estimate equation (24) with data on [OMEGA] for far-out-of-the-money put options on, say, the S&P 500. Given ranges of maturities, T, and relative exercise prices, [epsilon], we can estimate elasticities of [OMEGA] with respect to T and [epsilon]. We can also test the hypothesis that [[eta].sub.1] p is constant. Using month-end data on put options for several stock-market indices, we estimated monthly fixed effects and tested the hypothesis that these fixed effects were all equal for each stock-market index. The results, detailed in a later section, strongly reject the hypothesis that [[eta].sub.1]p is constant. Instead, the estimated monthly fixed effects fluctuate dramatically, including occasional sharp upward movements followed by gradual reversion over several months toward a baseline value that is close to zero. From the perspective of the model, if we assume that [alpha], [gamma], and [z.sub.0] are fixed, so that [[eta].sub.1] is constant, these shifts reflect variations in the disaster probability, p.

If [gamma]>1, as we assume, equation (8) implies that a once-and-for-all rise in disaster probability, p, lowers the price-dividend ratio, V, if [theta]<1 (meaning that the intertemporal elasticity of substitution, 1/[theta], exceeds 1). (11) Bansal and Yaron (2004) focus on IES>1 because it corresponds to the "normal case" where an increase in the expected growth rate, g*, raises V. Barro (2009) argues that IES>1 is reasonable empirically and, therefore, also focuses on this case.

Generally, the effects on options pricing depend on [theta] and other parameters and also on the stochastic process that generates variations in p, including the persistence of these changes. However, for purposes of pricing stock options, we need only consider the volatility of the overall term, [[eta].sub.1]p, which appears on the right side of equation (24). Our first-round look at the data--that is, the estimated monthly fixed effects--suggested that this term looks like a disaster process. On rare occasions, this term shifts sharply and temporarily upward and leads, thereby, to a jump in the corresponding term in equation (24). We think of this shock as generated by another Poisson probability, q, with a size distribution (for changes in stock prices) involving another power-law distribution, in this case with tail parameter [alpha]*>[gamma]. If this process for changing p is independent of the disaster realizations (which depend on the level of p), then equation (22) is modified to

(25) [mathematical expression not reproducible]

The first term on the right side of equation (25) reflects put-option value associated with the potential for realized disasters, and the second term gauges value associated with changing pt and the effects of these changes on stock prices. (12) The inclusion of [p.sub.t] in the first term is an approximation that neglects the tendency for [p.sub.t] to revert over time toward a baseline value that is close to zero. This approximation for options with relatively short maturity is similar to others already made, such as the neglect of multiple disasters and the ignoring of discounting and expected growth.

We can rewrite the formula in equation (25) as

(26) [mathematical expression not reproducible]

where

(27) [mathematical expression not reproducible]

are constants. (13) The new term involving [[eta].sub.2]>0 turns out to be important for fitting the data on put-options prices. Notably, this term implies [OMEGA]>0 if [p.sub.t]=0 because of the possibility that [p.sub.t] will rise during the life of the option. The preclusion of changing [p.sub.t] (corresponding to [[eta].sub.2]=0) leads, as emphasized by Seo and Wachter (2016), to overestimation of the average level of [p.sub.t] in the sample. (14) In addition, our hypotheses about elasticities of [OMEGA] with respect to T and [epsilon] in equation (26) turn out to accord better with the data when [[eta].sub.2]>0 is admitted.

III. Empirical Analysis

The model summarized by equation (26) delivers testable predictions. First, the elasticity of the put-options price with respect to maturity, T--denoted [[beta].sub.T]--is close to one. Second, for a given value of [??], the elasticity of the put-options price with respect to the relative exercise price, [epsilon]--denoted [[beta].sub.[epsilon]]--is greater than one and corresponds to 1+[alpha]-[gamma]. Given a value of [gamma] and the estimated value of [alpha]*-[alpha] from equation (26), the results can be used to back out estimates of the tail parameters [alpha] and [alpha]*. Finally, the monthly fixed effects provide estimates of each period's disaster probability, [p.sub.t] (or, more precisely, of [p.sub.t] multiplied by the positive constant [[eta].sub.1]). We assess these results empirically by analyzing prices of far-out-of-the-money put options on the U.S. S&P 500 and analogous broad indices for other countries.

A. Data and methodology

Our primary data source is a broker-dealer with a sizable market-making operation in global equities. We utilize over-the-counter (OTC) options prices for seven equity-market indices--S&P 500 (U.S.), FTSE (U.K.), DAX (Germany), ESTX50 (Euro zone), Nikkei (Japan), OMX (Sweden), and SMI (Switzerland). We subsequently check the U.S. results with OTC data against those with market-based information from OptionMetrics. This check is useful because the OTC data do not necessarily correspond to actual trades.

Our data derive from implied-volatility surfaces generated by the broker-dealer for the purpose of analysis, pricing, and marking-to-market. (15) These surfaces are constructed from transactions prices of options and OTC derivative contracts. (16) The dealer interpolates these observed values to obtain implied volatilities for strikes ranging from 50% to 150% of spot and for a range of maturities from 15 days to 2 years and more. Even at very low strikes, for which the associated options seldom trade, the estimated implied volatilities need to be accurate for the correct pricing of OTC derivatives such as variance swaps and structured retail products. Institutional-specific factors are unlikely to influence pricing in a significant way because other market participants can profitably pick off pricing discrepancies among dealers. Therefore, sell-side dealers have strong incentives to maintain the accuracy of their implied-volatility surfaces.

As mentioned, the OTC data source is superior to market-based alternatives in the breadth of coverage for exercise prices and maturities. Notably, the market data tend to be less available for options that are far out of the money and for long maturities. The broad range of strikes in the broker-dealer data is important for our analysis because it is the prices of far-out-of-the-money put options that will mainly reflect disaster risk. In practice, we use put options with exercise prices of 50%, 60%, 70%, 80%, and 90% of spot; that is, we exclude options within 10% of spot.

For maturities, we focus on the range of 30 days, 60 days, 90 days, and 180 days. (17) Our main analysis excludes options with maturities greater than six months because the prices in this range may be influenced significantly by the possibility of multiple disaster realizations and also by discounting and expected growth. However, in practice, the results for one-year maturity accord reasonably well with those for shorter maturities.

Using the data on implied volatilities, we re-construct options prices from the standard Black-Scholes formula, assuming a zero discount rate and no dividend payouts. We should emphasize that this use of the Black-Scholes formula to translate implied volatilities into options prices does not bind us to the Black-Scholes model of options prices. The formula is used only to convert the available data expressed as implied volatilities into options prices. Our calculated options prices are comparable to directly quoted prices (subject to approximations related to discounting and dividend payouts).

B. Estimation of the Model

We estimate the model based on equation (26) with non-linear least-squares regression. In this form, we think of the error term as additive with a constant variance (although we calculate standard errors of estimated coefficients by allowing for serial correlation in the error terms). Log-linearization with a constant-variance error term (that is, a shock proportional to price) is problematic because it understates the typical error in extremely far-out-of-the-money put prices, which are close to zero. That is, this specification gives undue weight to puts with extremely low exercise prices.

In the non-linear regression, we allow for monthly fixed effects to capture the unobserved time-varying probability of disaster, pt, in equation (26). We allow the estimated pt to differ across the seven stock-market indices; that is, we estimate index-time fixed effects. Note that, for a given stock-market index and date, these effects are the same for each maturity, T, and relative exercise price, [epsilon]. We carry out the estimation under the constraint that all of the index-time fixed effects are non-negative--corresponding to the constraint that all pt are non-negative. On average for the seven stock-market indices, the constraint of non-negative monthly fixed effects is binding for 8% of the observations. Only a negligible number of the unconstrained estimates of the fixed effects are significantly negative.

We sample the data at monthly frequency, selecting only month-end dates, to allow for ease of computation with a non-linear solver. The selection of mid-month dates yields similar results. The sample period for the United States in our main analysis is August 1994-February 2017. Because of lesser data availability, the samples for the other stock-market indices are shorter. Subsequently, we expand the U.S. sample back to 1983, particularly to assess pricing behavior before and after the global stock-market crash of 1987. However, we do not use this longer sample in our main analysis because the data quality before 1994 is substantially poorer.

Table 1 shows the estimated equations. The regressions apply to each of the seven stock-market indices individually and also to joint estimation with pooling of all of the data. In the last case, we constrain the estimated coefficients (including the monthly fixed effects) to be the same for each stock-market index.

1. Maturity Elasticities. The estimated elasticities with respect to maturity, [[beta].sub.T], are close to one, as hypothesized. For example, the estimated coefficient for the U.S. S&P 500 is 0.988 (s.e.=0.036) and that for all seven indices jointly is 0.958 (0.027). The only cases in which the estimated coefficient differs significantly from 1 at the 5% level are Japan (NKY), where the estimated coefficient is 0.878 (s.e.=0.032) and Sweden (OMX), estimated coefficient of 0.918 (0.040). The p-values for these estimated coefficients to be statistically different from 1 are 0.0001 and 0.040, respectively. In the main, with the exception of Japan, the results indicate that prices of far-out-of-the-money put options on broad market indices are roughly proportional to maturity, in accordance with the rare-disasters model. This nearly proportional relationship between options price and maturity for far-out-of-the-money put options is a newly documented fact that cannot be explained under the Black-Scholes model.

The unit elasticity of options price with respect to maturity for far-out-of-the-money put options contrasts with the previously mentioned result from Brenner and Subrahmanyam (1988) that prices of at-the-money put options in the Black-Scholes model are proportional to the square root of maturity. This result arises because, with a diffusion process driven by i.i.d. normal shocks, the variance of the log of the stock price is proportional to time and, therefore, the standard deviation is proportional to the square root of time. In contrast, as discussed earlier, the roughly proportional relationship between far-out-of-the-money put prices and maturity arises because, in a Poisson context, the probability of a disaster is proportional to maturity. The resulting pricing formula is only approximate because it neglects, for example, the potential for multiple disasters within the time frame of an option's maturity, omits a diffusion term, ignores discounting and expected growth, and also neglects the tendency of the disaster probability to revert over the life of an option toward a baseline value close to zero. However, for options that are not "too long," these approximations will be reasonably accurate, consistent with the findings on maturity elasticity shown in Table 1.

We verified empirically that the maturity elasticity is close to one-half for at-the-money put options. For the pooled sample with data from all seven stock-market indices, the estimated [[beta].sub.T] is 0.497 (s.e.=0.030). Similar point estimates apply for each of the seven stock-market indices considered individually.

2. Elasticities with respect to exercise price. Table 1 shows estimates of the elasticity, [[beta].sub.[epsilon]], of the put-options price with respect to the relative exercise price (holding fixed the term that involves [[eta].sub.2]q in equation [26]). The coefficient [[beta].sub.[epsilon]] corresponds in the model to 1 + [alpha] - [gamma], where [alpha] is the tail coefficient for disaster sizes and [gamma] is the coefficient of relative risk aversion. The estimated [[beta].sub.[member of]] for the various stock-market indices are all positive and greater than one, as predicted by the model. The estimated coefficients are similar across indices, falling into a range from 4.0 to 4.8. The joint estimate across the seven indices is 4.55 (s.e.=0.17).

Rare-disasters research with macroeconomic data, such as Barro and Ursua (2008) and Barro and Jin (2011), suggested that a [gamma] of 3-4 would accord with observed average (unlevered) equity premia. With this range for [gamma], the estimated values of [[beta].sub.[epsilon]] in Table 1 suggest that [alpha] would be between 6 and 8. This finding compares with an estimate for [alpha] based on macroeconomic data on consumption in Barro and Jin (2011, Table 1) of 6.3 (s.e.=0.8). Hence, the estimates of [alpha] implied by Table 1 accord roughly with those found from direct observation of the size distribution of macroeconomic disasters (based on consumption or GDP).

3. Estimated disaster probabilities. We use the estimated monthly fixed effects for each stock-market index from the regressions in Table 1 to construct time series of estimated (objective) disaster probabilities, [p.sub.t]. Note from equations (26) and (27) that the estimation identifies pt multiplied by the parameter [??], which will be constant i the size distribution of disasters (determined by [alpha] and [z.sub.0]) and the coefficient of relative risk aversion, [gamma], are fixed. When [[eta].sub.1] is constant, the estimated [p.sub.t] for each stock-market index will be proportional to the estimated monthly fixed effect.

To measure the level of pt, we need a value for [eta]1, which depends in equation (27) on [alpha], [z.sub.0], and [gamma]. We assume for a rough calibration that the threshold for disaster sizes is fixed at [z.sub.0] =1.1 (as in Barro and Jin [2011]) and that the coefficient of relative risk aversion is [gamma]=3. We allow the tail coefficient, [alpha], to differ for each stock-market index; that is, we allow places to differ with respect to the size distribution of potential disasters. We use the estimated coefficients from Table 1 for [[beta].sub.[epsilon]] (which equals 1+[alpha]-[gamma] in the model) to back out the implied [alpha], also shown in Table 1. These values range from 6.0 to 6.8, implying a range for [eta]1 from 0.71 to 0.89. Dividing the estimated monthly fixed effects for each stock-market index by the associated [eta]1 generates an estimated time series of pt for each country or region.

These values of pt are shown for the seven stock-market indices in Figure 1, Panel A. Panel B presents the results just for the United States (SPX), with the standard volatility indicator (VIX) included as a comparison. (18) Panel C shows the results for all indices estimated jointly (last column of Table 1). Note that our assumed parameter values, embedded in the computation of [eta]1, influence the levels of the pt series in Figure 1, but not the time patterns.

Table 2 provides summary statistics for the estimated disaster probabilities. These probabilities, shown in Figure 1, Panel A, have high correlations among the countries, with an average pair-wise correlation for the monthly data of 0.89. The average pair-wise correlation with Japan (NKY), 0.80, is notably lower than the others. The high correlations across stock-market indices indicate that the main variations in inferred disaster probabilities reflect the changing likelihood of a common, global disaster.

The mean estimated disaster probability from Table 2 is 6.4% per year for the S&P 500 and 6.3% for all countries jointly. For the other indices, the means range from 4.6% for Japan (NKY) to 9.0% for Sweden (OMX). These estimates can be compared with average disaster probabilities of 3-4% per year estimated from macroeconomic data on rare disasters--see, for example, Barro and Ursua [2008]). However, this earlier analysis assumed that disaster probabilities were constant across countries and over time.

The estimated disaster probabilities in Figure 1, Panel A, are volatile and right-skewed, with spikes during crisis periods and lower bounds close to zero. The U.S. disaster probability hit a peak of 42% per year in October-November 2008, just after the Lehman crisis. Similarly, the other six stock-market indices show their highest disaster probabilities around 40% in October-November 2008. Additional peaks in disaster probability occurred around the time of the Russian and Long-Term Capital Management (LTCM) crises in August-September 1998. In this case, the estimated U.S. disaster probability reached 29% in August 1998.

The patterns found for the U.S. disaster probability mirror results for options-derived equity premia in Martin (2015) and for disaster probabilities in Siriwardane (2015). The U.S. disaster probability is also highly correlated with the Chicago Board Options Exchange's well-known volatility index (VIX), as indicated in Figure 1, Panel B. The VIX, based on the S&P 500 index, is an average of Black-Scholes based implied volatility for puts and calls with maturity between 23 and 37 days and for an array of exercise prices. The correlation between the VIX and our U.S. disaster probability (using month-end data from August 1994 to February 2017) is 0.96 in levels and 0.90 in monthly changes. However, the levels of the two series are very different, with the p series having the interpretation as an objective disaster probability (per year) and the VIX representing an average implied volatility (with units of annual standard deviation of stock-price changes). (19)

The estimated U.S. disaster probability, p, is positively but not that strongly correlated with the indexes of economic policy uncertainty constructed by Baker, Bloom, and Davis (2016). (20) From August 1994 to February 2017, the correlation of our p series with their news-based uncertainty measure was 0.45 and that with their broader uncertainty measure was 0.47. Similar correlations show up between their indicators and the VIX. As an example of deviation, in January-February 2017, our p series was around 1% (compared to a mean of 6.4%), the VIX was at 12-13 (mean of 20.2), the news-based policy uncertainty indicator was close to 200 (mean of 112), and the broader policy uncertainty indicator was about 140 (mean of 106). In other words, disaster probability and the VIX were low (according to the financial markets), while policy uncertainty was high (according to Baker, Bloom, and Davis [2016] and, presumably, most political commentators).

The estimated first-order AR(1) coefficient for the estimated U.S. disaster probability shown in Figure 1, Panel B, is 0.88 (s.e.=0.03), applying at a monthly frequency. This coefficient implies that shocks to disaster probability have a half-life around eight months. The persistence of disaster probabilities for the other stock-market indices (Figure 1, Panel A) is similar to that for the United States, with estimated AR(1) coefficients ranging from 0.85 to 0.89, except for Japan at 0.80. An important inference is that the movements in disaster probability shown in Figure 1 are temporary. The series is associated with occasional sharp upward spikes (involving the probability q), followed by reasonably quick reversion toward a baseline value close to zero.

Although we attributed the time pattern in Figure 1 to variable disaster probability, [p.sub.t], the variations in the monthly fixed effects may also reflect changes in the other parameters contained in the term that multiplies pt in equation (26) and is shown in equation (27) as [??] (21) For example, outward shifts in the size distribution of disasters, generated by reductions in the tail parameter, [alpha], or increases in the threshold disaster size, [z.sub.0], would work like increases in p. (22) Similarly, increases in the coefficient of relative risk aversion, [gamma], would raise [[eta].sub.1]. This kind of change in risk preference, possibly due to habit formation, has been stressed by Campbell and Cochrane (1999). Separation of changes in the parameters of the disaster distribution from those in risk aversion require simultaneous consideration of asset-pricing effects (reflected in Figure 1) with information on the incidence and size of disasters (based, for example, on movements of macroeconomic variables).

4. Coefficients associated with changing disaster probability. The options-pricing formula in equation (26) involves the probability, q, of an upward jump in disaster probability, pt. The probability q enters multiplicatively with [[eta].sub.2], given in equation (27). In other words, [eta]2q is identified in the data. (23) The results in Table 1 show that the estimates of [[eta].sub.2]q range from 0.085 to 0.109, except for Japan at 0.138. Note that the underlying values of [[eta].sub.2]q are assumed to be constant over time for each stock-market index; we consider later whether this restriction is satisfactory. The effect of [[eta].sub.2]q on the options price interacts in equation (26) with the exercise price, [epsilon], to the power [alpha]*-[alpha]. Because the sample for each stock-market index has variation each month in [epsilon], the non-linear estimation identifies [alpha]*-[alpha]. These estimates range, as shown in Table 1, from 8.1 to 11.7.

Note in equation (26) that the put-options price, [OMEGA], depends on the sum of [[eta].sub.1][p.sub.t] and [[eta].sub.2]q, with the second term multiplied by [[epsilon].sup.[alpha]*-[alpha]]. Given that [alpha]*-[alpha] is estimated to be around 9.5, this last term ranges from 0.001 when [epsilon]=0.5 to 0.37 when [epsilon]=0.9. Options pricing depends, accordingly, on an effective probability that weighs the current disaster probability, pt, along with the probability, q, of a sharp upward future rise in pt. From this perspective, it is clear that omitting the chance of future rises in disaster probability--that is, assuming q=0--will result in estimates of [p.sub.t] that are too high on average compared with objective probabilities of disasters. Moreover, this effect will be much more significant at high exercise prices, such as [epsilon]=0.9, than at low ones, such as [epsilon]=0.5. For very low exercise prices, such as [epsilon]=0.5, almost all of the option value reflects the chance of a realization of a disaster during the life of the option. In contrast, for high exercise prices, such as [epsilon]=0.9, the option value depends partly on the possibility of a disaster occurrence and partly on the possibility of pt rising sharply.

As noted before, the term involving q>0 in equation (26) implies [OMEGA]>0 even when [p.sub.t]=0. For example, using the estimated value [[eta].sub.2]q=0.10 (from the pooled sample in Table 1) and taking [p.sub.t]=0, the term [??] in equation (26) is 0.037 when [epsilon]=0.9. That is, the effective probability that determines [OMEGA] can be as high as 4% per year even though [p.sub.t]=0 applies.

5. Long-term results for the United States. A lot of analysis of options pricing, starting with Bates (1991), suggests that the nature of pricing changed in character following the October 1987 stock-market crash. In particular, a "smile" in graphs of implied volatility against exercise price is thought to apply only post-1987. To examine this idea, we expanded our analysis to the period June 1983 to July 1994 by using market-based quotes on S&P 100 index options from the Berkeley Options Data Base. (24) These data derive from CBOE's Market Data Retrieval tapes. Because of the limited number of quotes on out-of-the-money options in this data base, we form our monthly panel by aggregating quotes from the last five trading days of each month. The available Berkeley data allows us to consider relative exercise prices, [epsilon], around 0.9, with maturities, T, close to 30, 60, and 90 days. We also have a small amount of data with [epsilon] around 0.8 and maturity, T, of about 30 days.

Table 3 extends the analysis of put-options pricing from Table 1 to consider U.S. regression estimates over the longer period 1983-2017. In this estimation, the data from Berkeley Options Data Base (for June 1983 to July 1994) relate to the S&P 100 but are treated as comparable to the OTC data (for August 1994-February 2017) associated with the S&P 500. The estimates of the various coefficients are close to those shown in Table 1, which were based on data from August 1994 to February 2017.

As before, we back out a time series for estimated disaster probability, pt, based on the monthly fixed effects, assuming that the parameters in the term [eta]1 in equations (26) and (27) that involves pt are fixed. We use levels for these other parameters similar to those used before (including [[eta].sub.1]=0.72). Figure 2 graphs the resulting time series of estimated U.S. disaster probability. Readily apparent is the dramatic jump in pt at the time of the October 1987 stock-market crash, in which the S&P 500 declined by 20.5% in a single day. The estimated [p.sub.t] reached 156% per year but fell rapidly thereafter. (25) The Persian Gulf War of 1990-1991 caused another rise in disaster probability, to 18-19%.

The bottom part of Table 3 shows statistics associated with the time series in Figure 2. A comparison pre-crash (June 1983-Sept 1987) and post-crash (Oct 1988-July 1994), based on the data from the Berkeley Options Data Base, shows an increase in the typical size and volatility of the estimated disaster probability, [p.sub.t]. The change in mean is from 0.005 to 0.027, and the change in standard deviation is from 0.009 to 0.045. The period August 1994-February 2017, based on OTC data related to the S&P 500, shows further rises in mean and standard deviation--to 0.067 and 0.068, respectively. Thus, the overall suggestion is that the mean and standard deviation of the disaster probability shifted permanently upward because of the October 1987 stock-market crash.

We also checked whether there was a shift around the time of the 1987 crash in the term involving the probability, q, of upward shifts in [p.sub.t]. The result is that the estimated coefficient [[eta].sub.2]q is 0.056 (s.e.=0.019) for the pre-crash period June 1983-Sept 1987 and 0.090 (0.031) for the post-crash period Oct 1988-Feb 2017. The p-value for the statistical significance of the difference between these two estimated coefficients is 0.022. Thus, there is evidence that the crash permanently raised the perceived probability q of future sharp increases in disaster probability.

6. Results with market data on put-options prices. One possible shortcoming of the results in Table 1 is that they are based on underlying OTC data that represent menus of options prices offered to clients by a large financial firm. Although these menus are informed by market transactions, they do not necessarily correspond to actual trades.

To check whether the reliance on OTC data is a problem, we redid the U.S. analysis shown in Table 1 using market-based information from OptionMetrics on far-out-of-the-money put options based on the S&P 500 index. As in Table 1, these data cover options with exercise prices, [epsilon], of 0.5, 0.6, 0.7, 0.8, and 0.9, and maturities of 30, 60, 90, and 180 days. The sample is January 1996 to April 2016. Unfortunately, we lack comparable data for other countries. (26) The regression results with the OptionMetrics data are in Table 4.

The number of observations for the OptionMetrics sample in Table 4 is 3529, compared to 5420 for the U.S. SPX in Table 1. The main reason for the decline in sample size is missing data from OptionMetrics, not the truncation of the sampling interval. Despite the reduction in sample size, it is clear that the OptionMetrics data provide a great deal of coverage over a long period on far-out-of-the-money put options on the S&P 500.

The main inference from Table 4 is that the estimated coefficients and fit using OptionMetrics data are very close to those based on the U.S. OTC data in Table 1. One likely reason for this correspondence is that the producers of the OTC information take account of market data, including those that appear in OptionMetrics. In any event, the closeness in results for OTC and market data for the United States makes us more comfortable with the OTC results for the other six stock-market indices, for which we lack long-term market-based information on put-options prices.

C. Model robustness

In the underlying theory, the asset-pricing formula in equation (26) applies as an approximation--based, for example, on neglecting possibilities of multiple disasters, ignoring terms associated with discounting and growth, neglecting pricing implications of a diffusion term, and ignoring effects from the tendency of pt to revert over time toward a baseline value close to zero. More generally, properties such as [[beta].sub.T]=1 (and constant) and [[beta].sub.[epsilon]]=1+[alpha]-[gamma] (and constant) would not hold precisely. In this section, we explore the empirical robustness of the model estimated in Table 1 under various scenarios.

1. Constancy of the maturity elasticity, [[beta].sub.T]. We re-estimated the regressions in Table 1 while allowing for different values of [[beta].sub.T] over ranges of maturity, T. As an example, we estimated one value of [[beta].sub.T] for T equal to 30 or 60 days and another for T equal to 90 or 180 days. For the United States (SPX), the estimated [[beta].sub.T] is 0.985 (s.e.=0.036) in the low range of T and 0.942 (0.056) in the high range, with a p-value of 0.31 for equality of these two coefficients. Similarly, for all stock-market indices estimated jointly, the estimated [[beta].sub.T] is 0.954 (s.e.=0.028) in the low range of T and 0.911 (0.044) in the high range, with a p-value of 0.18 for equality of these two coefficients.

We also redid the regressions in Table 1 while expanding the sample to include put options with one-year maturity. For the United States (SPX), the estimated [[beta].sub.T] becomes 0.941 (s.e.=0.032), compared to 0.988 (0.036) in Table 1, which allows for maturities only up to six months. For all stock-market indices estimated jointly, the estimated [[beta].sub.T] with the inclusion of one-year maturity becomes 0.905 (s.e.=0.020), compared to 0.958 (0.027) in Table 1.

The general pattern is that the estimated [[beta].sub.T] declines with the inclusion of longer maturities. However, the effects are moderate even for a range of T up to one year. These findings support the underlying approximations in the model but also suggest that the sample should be restricted to options that are not overly long; for example, up to six months.

We also considered whether [[beta].sub.T] is the same over different ranges of exercise price, [epsilon] (knowing that, for [epsilon]=1--at-the-money options--[[beta].sub.T] would be close to 0.5). We re-estimated the regressions in Table 1 while allowing for different values of [[beta].sub.T] over various ranges of [epsilon]. As an example, we estimated one value of [[beta].sub.T] for [epsilon] equal to 0.5, 0.6, or 0.7 and another for [epsilon] equal to 0.8 or 0.9. For the United States (SPX), the estimated [[beta].sub.T] is 1.201 (s.e.=0.125) in the low range of [epsilon] and 0.978 (0.032) in the high range, with a p-value of 0.07 for equality of these two coefficients. Similarly, for all stock-market indices estimated jointly, the estimated [[beta].sub.T] is 1.189 (s.e.=0.085) in the low range of [epsilon] and 0.946 (0.027) in the high range, with a p-value of 0.004 for equality of these two coefficients. Thus, there is some indication that [[beta].sub.T] is lower at high [epsilon] than at low [epsilon]. However, even for [epsilon] as high as 0.9, the estimated [[beta].sub.T] remains close to 1.

2. Stability of coefficients associated with exercise price. We also checked whether the coefficient [[beta].sub.[epsilon]] in Table 1 is stable over various ranges of [epsilon]. As an example, we estimated one [[beta].sub.[epsilon]] for [epsilon] equal to 0.5, 0.6, or 0.7 and another for [epsilon] equal to 0.8 or 0.9. For the United States (SPX), the estimated [[beta].sub.[epsilon]] is 4.66 (s.e.=0.33) in the low range of [epsilon] and 4.34 (0.67) in the high range, with a p-value of 0.49 for a test of the equality of these coefficients. Similarly, for all stock-market indices estimated jointly, the estimated [[beta].sub.[epsilon]] is 4.48 (s.e.=0.17) in the low range of [epsilon] and 4.14 (0.32) in the high range, with a p-value of 0.16 for a test of the equality of these coefficients. Thus, these results are consistent with the stability of the coefficient [[beta].sub.[epsilon]] over ranges of [epsilon].

3. Different sample periods. We checked for stability of the regression coefficients over time by re-estimating the regressions in Table 1 with separate coefficients for the four sub-periods shown in Table 5. These periods are Aug 1994-Sept 2002, Oct 2002-June 2007, July 2007-April 2012, and May 2012-Feb 2017. These intervals were chosen to be of equal length, starting from January 1998, at which point five of the seven stock-market indices have data. The results in Table 5 are for the United States (SPX) and for the pooled sample with data for the seven indices.

The general pattern in Table 5 is that the estimated coefficients are reasonably stable across the sub-periods, although hypotheses of equality of coefficients over time tend to be rejected at usual critical levels. For example, for the maturity elasticity, [[beta].sub.T], the range of estimated values over the four sub-periods is fairly narrow for the U.S. data--1.004 (s.e.=0.035), 1.135 (0.035), 0.912 (0.039), and 1.296 (0.046). Similar results obtain for the pooled sample of seven indices. Despite the narrow range of estimates, the hypothesis of equality is rejected in each case with a p-value of 0.000 because the estimated coefficients have high precision.

Similarly, for the coefficient [[beta].sub.[epsilon]] related to exercise-price elasticity, the range of estimates for the United States is from 3.82 to 6.15, and the hypothesis of equal coefficients is rejected with a p-value of 0.000. Analogous findings apply for the sample of seven stock-market indices.

For the estimated value of [alpha]*-[alpha], the range is wider--from 3.6 to 10.7 for the United States--but the relative lack of precision implies that the p-value for the hypothesis of equal coefficients has a higher p-value, 0.025. A similar pattern applies for the sample of seven stock-market indices.

Finally, the hypothesis of equal values of the coefficient [[eta].sub.2]q fares better in the sense that the range of estimated values is narrow, and the hypothesis of equality of coefficients is accepted at usual critical levels. For the United States, the range is from 0.078 to 0.107, and the p-value for equal coefficients is 0.084. For the seven-index sample, the range is from 0.089 to 0.113, and the p-value for equal coefficients is 0.34. These results support the assumption that, at least since August 1994, the probability, q, of a sharp upward movement in disaster probability, [p.sub.t], is constant. That is, unlike the dramatic variations in [p.sub.t] itself, it seems reasonable to assume time invariance with regard to the volatility associated with potential variations in [p.sub.t].

IV. Conclusions

Options prices contain rich information on market perceptions of rare disaster risks. We develop a new options-pricing formula that applies when disaster risk is the dominant force, the size distribution of disasters follows a power law, and the economy has a representative agent with Epstein-Zin utility. The formula is simple but its main implications about maturity and exercise price accord with U.S. and other data from 1983 to 2017 on far-out-of-the-money put options on broad stock-market indices.

If the coefficient of relative risk aversion and the size distribution of disasters are fixed, the regression estimates of monthly fixed effects provide information on the evolution of disaster probability, [p.sub.t]. The estimated [p.sub.t] is highly correlated across seven major stock-market indices, applicable to the United States, United Kingdom, Euro area, Germany, Japan, Sweden, and Switzerland. All of these series show a sharp peak during the financial crisis of 2008-09. More generally, we think that the estimated time series of [p.sub.t] reflect primarily global disaster risks. Using U.S. data, the peak in the estimated disaster probability is much more dramatic in the stock-market crash of October 1987.

Extensions of the empirical analysis would allow for second-order terms. These terms involve the possibility of multiple disasters, the presence of a diffusion term, allowances for discounting and expected growth, and the tendency for disaster probability to revert over time toward a baseline value close to zero.

This market-based assessment of objective disaster probability should be a valuable indicator of aggregate economic conditions for practitioners, macroeconomists, and policymakers. As an example, in January-February 2017, the estimated U.S. disaster probability was around 1% per year, despite the uncertainty suggested by the volatile political climate. Disaster probability as registered by the financial markets is not the same as political uncertainty.

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Panel A graphs the estimated disaster probabilities for the seven stock-market indices associated with the regressions in Table 1. The annualized disaster probability, [p.sub.jt] for index j, is calculated from the estimated monthly fixed-effect coefficients in the form of equation (26), assuming in the formula for [eta]1 in equation (27) that [z.sub.0]=1.1, [gamma]=3, and [[beta].sub.[epsilon]]=1+[[alpha].sub.j]-[gamma], where [[beta].sub.[epsilon]] is given in Table 1. Panel B is for the United States only (SPX). The VIX measure of volatility is discussed in Chicago Board Options Exchange (2014). Panel C is for the seven indices estimated jointly (last column of Table 1).

This figure presents the estimated U.S. disaster probabilities, [p.sub.t], associated with the regression in Table 3. The underlying data from August 1994 to February 2017 are the OTC data based on the S&P 500 and are the same as those used for the blue graph in Figure 1, Panel B. The data from June 1983 to July 1994 associate with the S&P 100 and are from the Berkeley Options Data Base. The methodology for inferring disaster probabilities from the estimated monthly fixed effects corresponds to that used in Figure 1. Because of missing data, many months before August 1994 do not appear in the figure.

Robert J. Barro and Gordon Liao

Harvard University

December 2017

(*) We appreciate helpful comments and assistance with data from Josh Coval, Ben Friedman, Xavier Gabaix, Tina Liu, Matteo Maggiori, Greg Mankiw, Robert Merton, Richard Roll, Steve Ross, Emil Siriwardane, Jessica Wachter, and Glen Weyl, and participants in the macroeconomics seminar at Harvard University.

(1) Related jump-diffusion models appear in Cox and Ross (1976).

(2) Epstein and Zin (1989) and Weil (1990).

(3) The transversality condition, which ensures that the value of tree equity is positive and finite, is [r.sup.e]>g*.

(4) Similarly, if we allowed for possible bonanzas, we could neglect the chance of a disaster and a bonanza both occurring over the period.

(5) In Kou (2002, p. 1090), a power-law distribution is ruled out because the expectation of next period's asset price is infinite. This property applies because Kou allows for favorable jumps (bonanzas) and, more importantly, he assumes that the power-law shock enters directly into the log of the stock price. This problem does not arise in our context because we consider disasters and not bonanzas, and, more basically, because our power-law shock multiplies the level of GDP (and consumption and the stock price), rather than adding to the log of GDP.

(6) With constant absolute risk aversion and a power-law distribution of disaster sizes, the relevant term has to blow up. The natural complement to constant absolute risk aversion is an exponential distribution of disaster sizes. In this case, the relevant term is bounded if the parameter in the exponential distribution is larger than the coefficient of absolute risk aversion. With an exponential size distribution and constant relative risk aversion, the relevant term is always finite.

(7) Barro and Jin (2011, Table 1) found that the data could be fit better with a double power law. In these specifications, with a threshold of [z.sub.0] = 1.105, the tail parameter, [alpha], was smaller in the part of the distribution with the largest disasters than in the part with the smaller disasters. The cutoff value for the two parts was at a value of z around 1.4.

(8) See Hogg and Craig (1965, p. 88).

(9) See, for example, Hull (2000, pp. 248 ff.). However, this standard result depends on holding fixed the risk-free rate, [r.sup.f]. Equation (7) shows that [r.sup.f] depends negatively on [gamma].

(10) We also assumed that preference parameters, including the coefficient of relative risk aversion, [gamma], are fixed.

(11) Under the same conditions, a fall in [alpha] or a rise in [[sigma].sup.2] reduces V.

(12) The formulation would also encompass effects on stock prices from changing [alpha] or [gamma].

(13) These values are constant if [alpha], [alpha]*, [z.sub.0], ([z.sub.0])*, [gamma], and q are all constant. [alpha]*-[alpha] is identified in our estimation because we have sample variation in relative exercise prices, [epsilon].

(14) As Seo and Wachter (2016) note, these problems appear, for example, in Backus, Chernov, and Martin (2011).

(15) A common practice in OTC trading is for executable quotes to be given in terms of implied volatility instead of the actual price of an option. Once the implied volatility is agreed on, the options price is determined from the Black-Scholes formula based on the readily observable price of the underlying security. Since the Black-Scholes formula provides a one-to-one link between price and implied volatility, quotes can be given equivalently in terms of implied volatility or price.

(16) Dealers observe prices through own trades and from indications by inter-dealer brokers. It is also a common practice for dealers to ask clients how their prices compare to other market makers in OTC transactions.

(17) We omit 15-day options because we think measurement error is particularly serious in this region in pinning down the precise maturity. Even the VIX index, which measures short-dated implied volatility, does not track options with maturity less than 23 days.

(18) A discussion of the VIX is contained in Chicago Board Options Exchange (2014).

(19) Implied volatility is a puzzling object, because it corresponds to the patch needed to get an observed options price to equal the Black-Scholes price. Implied volatility varies not only over time but also systematically across exercise prices, [epsilon], because the underlying assumption of normality deviates from the data. This cross-[epsilon] variation is described as a "smile." The original form of the VIX was based only on at-the-money options and was, therefore, easier to interpret than the current version, which is an average across exercise prices of implied volatilities. There is also systematic variation of implied

volatility with maturity, but this property does not matter much for the standard VIX, which considers options only in a narrow range around 30 days.

(20) Their data are available at policyuncertainty.com.

(21) In the model with i.i.d. shocks, this term does not depend on the intertemporal elasticity of substitution for consumption, 1/[theta], or the rate of time preference, [rho].

(22) Kelly and Jiang (2014, p. 2842) assume a power-law density for returns on individual securities. Their power law depends on a cross-sectional parameter and also on aggregate parameters that shift over time. In contrast to our analysis, they assume time variation in the economy-wide values of the tail parameter, analogous to our [alpha], and the threshold, analogous to our [z.sub.0]. (Their threshold corresponds to the fifth percentile of observed monthly returns.)

(23) We can therefore identify q if we know the value of [??]. If we continue to assume [gamma]=3 and use the estimates of [alpha]* implied by the results in Table 1, the missing element is the threshold, [z*.sub.0]. However, reasonable variations in [z*.sub.0] imply large variations in [eta]2 and, hence, in the estimated q.

(24) Direct access to this database has been discontinued. We thank Josh Coval for sharing his version of the data. We have the data from Berkeley Options Data Base through December 1995.

(25) Note that a disaster probability above 100% per year is well defined in the context of a continuous-time Poisson specification.

(26) We have data from Bloomberg but only since late 2010.
```Table 1 Regressions for Put-Options Prices
OTC Data, 1994-2017

Index                         SPX      FTSE     ESTX     DAX
Country/Region                US       UK       EURO     GER

Aug94-   Jan98-   Jun98-   Jan00-Feb17
Feb17    Feb17    Feb17

[[beat].sub.T]                 0.988    0.994    0.941    0.935
(0.036)  (0.040)  (0.040)  (0.038)
[[beat].sub.[member of]]       4.71     4.68     4.49     4.32
(0.31)   (0.31)   (0.33)   (0.34)
[alpha]*-[alpha]               9.48     9.86     8.88     8.11
(3.01)   (3.63)   (2.87)   (2.27)
[[eta].sub.2]q                 0.090    0.085    0.100    0.101
(0.029)  (0.032)  (0.030)  (0.025)
Implied est of [alpha]         6.71     6.68     6.49     6.32
Implied est of [[eta].sub.1]   0.728    0.733    0.769    0.805
R-squared                      0.974    0.975    0.974    0.973
[sigma]                        0.0013   0.0013   0.0016   0.0016
N                                5420     4600     4500     4120
Mean dep var.                  0.0039   0.0042   0.0055   0.0052
[sigma] dep var.               0.0077   0.0082   0.0098   0.0096

Index                         NKY      OMX      SMI      --
Country/Region                JAP      SWE      SWZ      ALL

Jan 98-  Jan98-   Jan98-   Aug94-Feb17
Feb17    Feb17    Feb17

[[beat].sub.T]                 0.878    0.918    1.003    0.958
(0.032)  (0.040)  (0.040)  (0.027)
[[beat].sub.[member of]]       3.98     4.73     4.80     4.55
(0.29)   (0.32)   (0.31)   (0.17)
[alpha]*-[alpha]              11.69    10.23    11.62     9.52
(3.26)   (3.91)   (3.79)   (2.01)
[[eta].sub.2]q                 0.138    0.109    0.102    0.100
(0.047)  (0.044)  (0.040)  (0.020)
Implied est of [alpha]         5.98     6.73     6.80     6.55
Implied est of [[eta].sub.1]   0.892    0.724    0.713    0.757
R-squared                      0.959    0.967    0.974    0.913
[sigma]                        0.0019   0.0019   0.0012   0.0026
N                                4600     4600     4600     32440
Mean dep var.                  0.0049   0.0056   0.0036   0.0047
[sigma] dep var.               0.0093   0.0103   0.0075   0.0090

This table presents non-linear least-squares regression estimates of
the model for pricing far-out-of-the money put options with variable
disaster probability. We use OTC data on relative put-option prices,
[OMEGA], for seven stock-market indices with maturity, T, of 30, 60,
90, and 180 days and relative exercise price, [epsilon], of 0.5, 0.6,
0.7, 0.8, and 0.9. The estimation corresponds to equation (26): [OMEGA]
= T[[epsilon].sup.1+[alpha]-[gamma]. [[eta].sub.1][p.sub.t] +
[[eta].sub.2][q.sub.[[epsilon]].sup.([alpha]*-[alpha]), where [p.sub.t]
is the disaster probability, a is the tail parameter for disaster
sizes, [alpha]* is the tail parameter for stock-price changes induced
by upward jumps in [p.sub.t], q is the probability of an upward jump in
[p.sub.t], [gamma] is the coefficient of relative risk aversion, and
[[eta].sub.1] and [[eta].sub.2] are constants shown in equation (27).
We use the estimated monthly fixed effects for each stock-market index
to gauge the variations in [[eta].sub.1][p.sub.t] and then use a
calibrated value of [[eta].sub.1] to infer levels of [p.sub.t]. The
results are in Figure 1. The estimation constrains [p.sub.t][greater
than or equal to]0 for each observation. This constraint turns out to
be binding on average for 8% of the observations for the seven
stock-market indices. The column labeled "all" pools the data on the
seven stock-market indices and uses the same coefficients and set of
monthly fixed effects for all indices. The estimated exponent on T,
[[beta].sub.T], should equal 1. The estimated exponent on the first
[epsilon] term, [[beta].sub.[member of]], should equal
1+[alpha]-[gamma]. Implied estimates of a are shown, based on
[gamma]=3. Implied estimates of [[eta].sub.1] are shown, assuming in
equation (27) that the threshold value for disaster size is
[z.sub.0]=1. 1. Cross-section-clustered standard errors (which allow
for serial correlation of the error terms) are in parentheses.

Table 2
Statistics for Estimated Disaster Probabilities

Index              sample            mean    std. dev.   maximum

SPX (US)           1994.08-2017.02   0.064   0.067       0.42
FTSE (UK)          1998.01-2017.02   0.073   0.071       0.40
ESTX (Euro area)   1998.06-2017.02   0.077   0.069       0.38
DAX (Germany)      2000.01-2017.02   0.062   0.065       0.35
NKY (Japan)        1998.01-2017.02   0.046   0.052       0.44
OMX (Sweden)       1998.01-2017.02   0.090   0.081       0.45
SMI (Switz.)       1998.01-2017.02   0.064   0.073       0.39
ALL                1994.08-2017.02   0.063   0.064       0.41

This table his table presents statistics on estimated disaster
probabilities from the regressions in Table 1.
The disaster probabilities are calculated as described in the notes to
Figure 1.

Table 3 Regression for U.S. Put-Options Prices
Berkeley and OTC Data, 1983-2017

[[beta].sub.T]               0.986 (0.035)
[[beta].sub.[member of]]     4.76 (0.30)
[alpha]*-[alpha]            10.39 (3.56)
[[eta].sub.2]q               0.091 (0.034)
R-squared                    0.975
[sigma]                      0.0013
N                              5662

Statistics for Estimated Disaster Probabilities (shown in Figure 2)

Period start   Period end   mean    std. dev.   maximum

June 1983      Feb 2017     0.062   0.104       1.56
June 1983      Sept 1987    0.005   0.009       0.035
Oct 1987       Sept 1988    0.309   0.477       1.56
Oct 1988       Jul 1994     0.027   0.045       0.191
Aug 1994       Feb 2017     0.067   0.068       0.430

The form of the regression corresponds to that for the U.S. SPX in
Table 1. The data from June 1983 to July 1994 are based on the S&P 100
index and are market-based information from the Berkeley Options Data
Base. The data from August 1994 to February 2017 are OTC values based
on the S&P 500, as in Table 1. For the Berkeley data, we formed monthly
panels of put-options prices by aggregating quotes from the last five
trading days of each month. We applied a bivariate linear interpolation
on the implied volatility surface to obtain put prices with granular
strikes at every 10% moneyness interval and maturities ranging from one
to six months. The methodology for inferring disaster probabilities
from the estimated monthly fixed effects corresponds to that used in
Figure 1, with the results shown in Figure 2. Because of missing
information in the Berkeley data, many months before August 1994 do not
appear in the regression or in Figure 2. Cross-section-clustered
standard errors (which allow for serial correlation of the error terms)
are in parentheses.

Table 4 Regression for U.S. Put-Options Prices
OptionMetrics Data, 1996-2016

[[beta].sub.T]              0.944 (0.032)
[[beta].sub.[member of]]    4.48 (0.28)
[alpha]*-[alpha]            8.96 (2.42)
[[eta].sub.2]q              0.090 (0.023)
R-squared                   0.975
[sigma]                     0.0015
N                             3529

This regression corresponds to that for the U.S. SPX in Table 1, except
for the use of market-based OptionMetrics data over the period January
1996-April 2016.

Table 5
Stability over Different Samples of Estimated Coefficients in
Regressions for Put-Options Prices

Coefficient             1994.08-2002.09  2002.10-2007.06
U.S. (SPX)

[[beta].sub.T]           1.004            1.135
(0.035)          (0.035)
[[beta].sub.[epsilon]]   5.06             6.15
(0.29)           (0.53)
[alpha]*-[alpha]        10.70             6.77
(3.38)           (2.19)
[[eta].sub.2]q           0.099            0.078
(0.035)          (0.016)
All (seven stock-market indices)
[[beta].sub.T]           0.963            1.010
(0.031)          (0.054)
[[beta].sub.[epsilon]]   4.82             5.16
(0.24)           (0.41)
[alpha]*-[alpha]         9.35             9.03
(2.79)           (2.09)
[[eta].sub.2]q           0.111            0.092
(0.029)          (0.020)

Coefficient             2007.07-2012.04  2012.05-2017.02  p-value
U.S. (SPX)

[[beta].sub.T]           0.912            1.296            0.000
(0.039)          (0.046)
[[beta].sub.[epsilon]]   3.82             5.11             0.000
(0.40)           (0.55)
[alpha]*-[alpha]         3.61             4.96             0.025
(2.01)           (2.48)
[[eta].sub.2]q           0.107            0.084            0.084
(0.021)          (0.011)
All (seven stock-market indices)
[[beta].sub.T]           0.901            1.150            0.000
(0.028)          (0.048)
[[beta].sub.[epsilon]]   3.48             5.01             0.000
(0.18)           (0.32)
[alpha]*-[alpha]         4.14             6.62             0.038
(1.21)           (3.51)
[[eta].sub.2]q           0.113            0.089            0.34
(0.008)          (0.014)

Note: The estimated coefficients shown for four sub-periods correspond
to those shown for full samples in Table 1. The sub-periods were chosen
to have roughly equal numbers of observations, starting from January
1998, by which the data are available for five of the seven
stock-market indices considered in Table 1. The results apply to the
U.S. (SPX) stock-market index and for the pooled sample of all seven
stock-market indices. The p-values are for the hypothesis that the
associated coefficient is the same across the four sub-periods. The
p-value for the joint hypothesis that all coefficients are equal across
the sub-periods has a p-value of 0.000 for the U.S. SPX and for the
pooled sample of all seven stock-market indices.
```
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