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04/04/2012

Too Systemic to Fail: What Option Markets Imply about Sector-Wide Government Guarantees


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ABSTRACT

A conspicuous amount of aggregate tail risk is missing from the price of financial sector crash insurance during the 2007-2009 crisis. The difference in costs of out-of-the-money put options for individual banks, and puts on the financial sector index, increases four-fold from its pre-crisis level. At the same time, correlations among bank stocks surge, suggesting the high put spread cannot be attributed to a relative increase in idiosyncratic risk. We show that this phenomenon is unique to the financial sector, that it cannot be explained by observed risk dynamics (volatilities and correlations), and that illiquidity and no-arbitrage violations are unlikely culprits. Instead, we provide evidence that a collective government guarantee for the financial sector lowers index put prices far more than those of individual banks, explaining the divergence in the basket-index spread. By embedding a bailout in the standard one-factor option pricing model, we can closely replicate observed put spread dynamics. During the crisis, the spread responds acutely to government intervention announcements.

I. INTRODUCTION

During the 2007-2009 crisis, an episode of elevated systemic risk, the price of crash insurance for the US financial sector was surprisingly low. Out-of-the-money (OTM) put options on the financial sector stock index were cheap relative to OTM put options on the individual banks that comprise the index. The difference between the cost of a basket of individual bank put options and the cost of a strike-matched financial sector index put option reached 15.9 cents per dollar insured in March 2009, or 60% of the cost of the index put. Before the onset of the crisis, this basket-index put spread never exceeded 3.8 cents on the dollar.

The size of the basket-index spread in the financial sector is puzzling. The basket of put options provides insurance against both sector-wide and idiosyncratic bank stock crashes, while the index put option only insures against a sector-wide crash. Standard option pricing logic therefore implies that a disproportionate increase in idiosyncratic risk (relative to aggregate risk) is needed to explain the dramatic increase in the basket-index put spread. This creates a puzzle because, as is common in times of market turbulence, the correlation of financial stocks also surged throughout the crisis. The drastic rise in idiosyncratic risk necessary to explain the put spread counter-factually implies a sharp decrease in stock return correlations.

These two facts, the simultaneous increase in financial sector correlations and the financial sector basket-index put spread, are at odds with the implications of standard asset pricing models. If anything, the standard model suggests that the rapid increase in return correlations should have raised the price of OTM index options relative to the option basket, causing the put spread to shrink.

We found that the increase in the basket-index put spread is much larger in the financial sector than in any other sector. In addition, our findings only pertain to put option prices. Consistent with the logic of the standard model, the prices of the basket of call options and the index call options converge in all sectors during the crisis. A standard single-factor Black-Scholes model cannot reconcile the financial put spread dynamics with rising correlations. We show that option- implied correlations from financial sector puts fall, while realized correlations rise. Simultaneously, the implied correlations backed out from call options rise substantially.

Typically, index options are labeled as expensive because their prices consistently exceed values implied by standard models (Bondarenko, 2003). Driessen, Maenhout, and Vilkov (2009) argue that this is because index options provide a valuable hedge against increases in correlation, while individual name options do not possess this feature. We find that index put options in the financial sector are different. They are always less expensive than those for other sectors, and they become especially cheap during the crisis.

We provide direct and indirect evidence that a sector-wide bailout guarantee was largely responsible for the divergence of individual and index put prices during the recent crisis. The anticipation of future government intervention during a financial sector collapse lowers the market price of crash insurance. In effect, implicit bailout guarantees are crash insurance subsidies for anyone holding stock in the banking sector, and this subsidy drives down the market prices that investors were willing to pay for the traded, private version of insurance. Since any individual bank may still fail amid a collective guarantee, or the failure of a single firm may not be sufficient to trigger government intervention, the downward pressure on individual bank puts is much weaker than the effect on index puts. The government’s guarantee flattens the well-documented volatility skew for put options on the financial sector index, but has little effect on the individual bank put skew. We find no evidence of skew flattening for non-financial indices.

After embedding a government guarantee in the standard option pricing model, the realized volatility and correlation dynamics in the financial sector produce a model-implied put spread that is strikingly similar to that in the data. This analysis provides indirect evidence that a government guarantee can account for dynamics of the basket-index put spread over our sample. The most accurate match of the spread requires a government bailout that truncates the sector-wide equity return at a 55% loss. By comparing the bailout-adjusted and the standard bailout-free cost of an option-based hedge against a financial sector crash, we obtain a dollar estimate of the value of the government guarantee for the financial sector. According to this estimate, government support to banks’ equity was $0.63 billion before mid-2007 and rose to $42.38 billion between mid-2007 and mid-2009. It peaked at well over $150 billion. Our findings imply a substantial reduction in the cost of equity for systemically risky financial firms, consistent with the findings of Gandhi and Lustig (2010), who show that large banks have risk-adjusted equity returns 5% per annum lower than those of the smallest banks, a difference that they attribute to an implicit guarantee that absorbs the tail risk of large banks.1

Furthermore, an event study of the financial sector put spread evolution provides direct evidence of option price sensitivity to government announcements. The spread increases on average by 1.64 cents (64%) in the first five days after government announcements that ex ante increase the probability of a bailout, while it decreases on average 1.92 cents (23%) after announcements that have the opposite effect (and after adjusting for contemporaneous changes in financial sector risk). The largest absolute increase in the spread was registered in the first five days after the US Congress approved the TARP bailout.

We can rule out a number of alternative explanations. Transactions costs can be ruled out because the basket-index put spread constructed with the most costly combination of bid and ask quotes is still quite large. Liquidity differences across various types of options (index versus individual, puts versus calls, or financial firms versus non-financials) are inconsistent with the put spread arising because of illiquidity. Mispricing due to capital constraints, counter-party risk, and short sale restrictions are unlikely culprits. A trade that takes advantage of the basket-index spread ties up relatively little capital (due to implicit leverage in options) and occurs through exchanges with a clearing house in the middle. These option positions are marked-to-market daily and ultimately guaranteed by the AAA-rated Options Clearing Corporation. The short sale ban was in place only for a brief portion of our sample, applied equally to individual and index options, and market makers were exempted from it. Nor do short sale lending fees for financial stocks line up with the put spread dynamics that we document.

Our work connects to various strands of the literature. First, it is linked to the problem of measuring systemic risk in the financial sector.2 Our findings highlight a fundamental complication in inferring systemic risk from market prices. All else equal, the basket-index spread for OTM put options would be a natural measure of systemic risk: the smaller the basket-index spread in a sector, the larger the amount of systemic risk in that sector. However, in sectors that benefit from an implicit or explicit collective guarantee, an increase in the basket-index spread may occur when systemic risk peaks and the collective bailout guarantee is more likely to kick in. Hence, the anticipation of future government intervention is embedded in market prices today and makes them less informative about the true nature of tail risk. This feedback from anticipated corrective action to market prices echoes the problem of a board of directors looking at share prices to fire a CEO in the presence of rational investors anticipating this behavior (Bond, Goldstein, and Prescott, 2010).

The effects of too-systemic-to-fail government guarantees remain highly uncertain and intensely researched. A number of papers measure the impact of these guarantees on the total value of the firm. Lucas and McDonald (2006, 2010) take an option-based approach to valuing guarantees extended to Fannie Mae and Freddie Mac. Veronesi and Zingales (2010) use CDS data to measure the value of government bailouts to bondholders and stockholders of the largest financial firms from the Paulson plan. They estimate that this plan increased the total value of banks’ balance sheets by $131 billion. Our paper focuses exclusively on equity. Ex ante, the anticipation of future bailouts of bondholders and other creditors invariably benefits shareholders (see Kareken and Wallace [1978]). Furthermore, during the crisis, there may be massive uncertainty about the resolution regime, especially for large financial institutions. The government is aware that bankruptcy costs start well before the value of bank equity actually hits zero. As a result, collective guarantees will inevitably tend to benefit shareholders ex post as well. A contribution of this paper is to demonstrate that financial sector guarantees can massively prop up bank equity value.

Other recent studies have also examined the relative pricing of derivative securities. Coval, Jurek, and Stafford (2009) compare the prices of CDX tranches to those of index options prior to and during the financial crisis, and they conclude that CDX tranches are overpriced relative to index options.3 Driessen, Maenhout, and Vilkov (2009), Carr and Wu (2009) and Schurhoff and Ziegler (2011) study prices of index versus individual options. Finally, the option pricing model we develop borrows from the work of Backus, Chernov, and Martin (2011) and Martin (2011).

The rest of the paper is organized as follows. After defining index and basket put and call spreads and their relationship in Section 2, we document their empirical behavior in the financial sector and in all other non-financial sectors in Section 3. The basket-index put spread is the focus of our analysis since this quantity directly captures the relative cost of crash insurance for the sector index versus individual stocks. Section 4 looks at two alternative measures of the cost of crash insurance, implied correlations and the volatility skew. Section 5 adjusts the basket- index spread for changes in volatility with the help of a simple Black-Scholes model. Section 6 introduces a bailout guarantee into Black-Scholes, and shows that this model helps account for the observed basket-index spread dynamics. Section 7 finds direct evidence for our collective government guarantee hypothesis in the events of the 2007-2009 crisis. Section 8 studies and rules out potential alternative explanations, including counterparty risk, mispricing, short sales constraints, hedging costs and liquidity. The last section concludes. Technical details are relegated to a separate appendix.

II. MEASURING THE COST OF SECTOR CRASH INSURANCE: THE BASKET-INDEX SPREAD

Equity options markets are especially well-suited to gauge the market’s perception of too-systemic- to-fail guarantees. Since guarantees only kick in during a financial crisis, their effect should be most visible in the prices of assets that mostly reflect tail risk, like put options. One may insure against a common financial sector crash by buying puts on each individual financial institution, or by buying a put on the financial sector index. In this section we propose a comparison of these insurance schemes that is useful for identifying investor perceptions of government guarantees.

We focus on a traded sector index i comprised of different stocks j. Si,j and si,j are the price per share and number of shares outstanding, respectively, for stock j in index i. The dollar cost of the index, i.e., the total market cap of all the firms in the index, is given by 1  while the price level of the index, Si, is a constant fraction 1/scalei of the total index market cap (thus scalei  2 We use 3  to denote the price of a basket of put options on all stocks: 4 We use 5  to denote the price of a put option on the sector index (similarly for calls).

The basket of put options provides insurance against both common and idiosyncratic stock price crashes, while the index put option only insures states of the world that prompt a common crash. The difference between the costs of these insurance schemes is informative about the relative importance of aggregate and idiosyncratic risks, and is also informative about sector-wide government guarantees.

Strike-Matched Basket. To align our comparison between insurance costs, we impose that the total strike price of the two schemes are equal, an approach that we refer to as “strike-matching.” To do so, we first choose index strike price Ki to match a given Δ.4 Second, we search for options on individual stocks in the index (all of which must share the same Δ, though this may be different from the index Δ) such that their strike prices Ki,j (j = 1, 2, . . . ,Ni) satisfy 

6

The strike price of the index (in dollars) equals the share-weighted sum of the individual strike prices as in Equation (4).

With strike-matching, the cost of the basket has to exceed the cost of the index option by no arbitrage, which bounds the basket-index spread below from zero. The payoffs at maturity satisfy the following inequality:

7
To see why, first note that, for each j, si,j max 7

This implies that 9 This also means that 10 11because the left hand side is non-negative. Since the payoff from the option basket exceeds that of the index option, its cost must be weakly higher as well.

We also note that the Δ of the index option can differ slightly from the moneyness of the option basket. In Appendix B, we consider an alternative method for constructing the basket that uses index and individual options that all have the same moneyness, hence the Δ is equalized across the two insurance schemes. As results in the appendix show, the conclusions from spreads based on either matching scheme are identical.5

Cost Per Dollar Insured. To compare prices across time, sectors, and between puts and calls, we define the cost per dollar insured as the price of an option position divided by the dollar amount that it insures. We then define the basket-index put spread as the difference in the per dollar costs of basket and index insurance:

12

Call spreads are defined analogously.

III. THE BASKET-INDEX SPREAD IN THE DATA

This section describes the behavior of basket-index option spreads observed in the data. We find that OTM put options on the index were cheap during the financial crisis relative to the individual stock options, while OTM index calls were relatively expensive. This pattern is much more pronounced for the financial sector than for non-financial sectors.


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A. Data

We use daily option data from January 1, 2003 until June 30, 2009. This includes option prices on the nine S&P 500 sector index exchange-traded funds (ETFs) traded on the CBOE.6 As ETFs trade like stocks, options on these products are similar to options on an individual stock. The nine sector ETFs conveniently have no overlap and collectively span the entire S&P 500. Appendix A contains more details and lists the top 40 holdings in the financial sector ETF.7 We also use individual option data for all members of the S&P 500. The OptionMetrics Volatility Surface provides daily European put and call option prices that have been interpolated over a grid of time-to-maturity (TTM) and option Δ, and that are adjusted to account for the American option feature of the raw option data.8 These constant maturity and constant moneyness options are available at various intervals between 30 and 730 days to maturity and at values of (absolute) Δ ranging from 20 to 80. We focus primarily on options with 365 days to maturity and Δ of 20. Implied volatility data are from the interpolated implied volatility surface of OptionMetrics. We use CRSP for returns, market cap, and number of shares outstanding for sector ETFs and individual stocks. We calculate realized volatility of index and individual stock returns, as well as realized correlations between individual stocks. All of our calculations track the varying composition of the S&P 500 index (as well as the sector indices) to maintain consistency between the composition of the option basket and the index option each day.9

B. Main Facts

Panel I in Table I provides summary statistics for the basket-index spread, in cents per dollar insured, using the strike-matched approach with Δ = 20 and TTM = 365. The first two columns report results for the financial sector. Columns three and four report results for a value-weighted average of the eight non-financial sectors. The last two columns report the differences in the spread between the financial and non-financial sectors. An increase in the spread between the basket and the index means index options become cheaper relative to the individual options. We report statistics for three samples: The entire January 2003 to June 2009 sample, the January 2003 to July 2007 pre-crisis sample, and the August 2007 to June 2009 crisis sample.

Table I. Basket-Index Spreads on Out-of-the-Money Options

 

Table-I

This table reports summary statistics for the basket-index spread in the cost of insurance per dollar insured for financials, non-financials and their difference (financials minus non-financials) . Numbers reported are in cents per dollar of strike price. The full sample covers 1/2003-6/2009. The pre-crisis sample covers 1/2003-7/2007. The crisis sample covers 8/2007-6/2009. Δ is 20. In the top half of the table, time-to-maturity is 365 days, in the bottom half it is 30 days. Spreads are constructed using strike-matching as described in Section 3.

Over the pre-crisis sample, the mean spread for OTM puts is 1.7 cents per dollar in the financial sector, and 2.3 cents in the non-financial sectors. During the crisis, the mean put spread is 5.9 cents per dollar for financials and 3.7 cents for non-financials. While there is an across-the-board increase in the put spread from pre-crisis to crisis, the increase is much more pronounced for financials (3.4 times versus 1.6 times). The largest basket-index put spread for financials is 15.9 cents per dollar, recorded on March 6, 2009. It represents 60% of the cost of the index option on that day. On that same day, the difference between the spread for financials and non-financials peaks at 10.2 cents per dollar insured. Prior to the crisis, the put spread for financials never exceeds 3.8 cents on the dollar, and it never exceed the non-financial put spread by more that 0.4 cents.

Across the entire sample and all sectors, the average basket-index spread for OTM calls is smaller than for puts: 1.0 cents for financials and 2.0 cents for non-financials. OTM call spreads rise slightly in the crisis, reaching 1.1 cents on average for financials and 2.3 cents for non-financials.

Appendix Table B reports results for our second approach to constructing the basket-index spread in which the Δs of the two insurance schemes are equalized. We see the same pattern as with the strike-matching approach. The time series correlation between these two measures is over 99%. Basket-index spreads are somewhat larger when we match the share-weighted strike price. This is because strike-matching uses individual options that have slightly higher Δ than index options used, which increases spreads. The average Δ-matched put spread during the crisis is 3.8 cents per dollar for financials (compared to 5.9 cents in Table I). The maximum spread is 12.5 cents per dollar insured (compared to 15.9). This number represents 70% of the cost of the index put on March 6, 2009 (compared to 60%). On that same day, the difference between the put spread for financials and non-financials peaks at 9.1 cents per dollar.

The top panel of Figure 1 plots financial sector put prices for the entire sample. The solid line shows the cost of the basket of put options per dollar insured and the dashed line plots the cost of the financial sector put index. Before the crisis, the put spread (dotted line) is small and essentially constant at less than two cents per dollar. During the crisis, it increases as the index option gradually becomes cheaper relative to the basket of puts. The basket cost occasionally exceeds 30 cents per dollar while the cost of the index put rarely rises above 20 cents.

Figure 1. Cost Per Dollar Insured - Financial Sector

Figure-1

Figure-1-legend

The bottom panel of Figure 1 plots call option prices and the call spread. During the crisis, the difference between index calls and the basket of individual calls remains unchanged from its pre-crisis level. We find essentially the same results for call spreads in all other sectors.

Figure 2 compares the put spread of financials and non-financials over time (the dotted lines from the previous figure). For non-financials (solid line), the basket-index spread remains very low until October 2008. For financials (dashed line) on the other hand, the put spread starts to widen in August 2007 (the asset-backed commercial paper crisis), spikes in March 2008 (the collapse of Bear Stearns), and then spikes further after the bailouts of Freddie Mac and Fannie Mae and the Lehman Brothers bankruptcy in September 2008. After a decline in November and December of 2008, the basket-index spread peaks a second time with the rescue of AIG in March 2009. The dotted line plots the difference in put spread between the financial sector and non-financial sectors. This difference is positive throughout the crisis, except for a few days in November of 2008. It increases from the summer of 2007 to October 2008, falls until the end of 2008, and increases dramatically from January to March 2009. None of the eight non-financial sectors experiences anywhere near the large run up in put spreads seen in the financial sector.

Figure 2. Basket-Index Spread in Cost Per Dollar Insured Inferred from Puts

Figure-2

Figure-2-legend

C. The Effect of Time-to-Maturity

Panel II of Table I studies the cost of insurance when TTM is 30 days instead of 365 days. As we show later, these shorter maturity option contracts are more liquid. Naturally, all basket-index spreads are smaller for shorter-dated options since option prices increase with TTM. However the spread patterns are the same as in Panel I. The average put spread for financials is 1.4 cents per dollar in the crisis, up from 0.4 cents pre-crisis. This represents an increase by a factor of 3.4. Per unit of time (that is, relative to the ratio of the square root of maturities), the put spread increase during the crisis is larger for TTM = 30 options than for TTM = 365 options. The 30-day spread reaches a maximum of 4.0 cents on the dollar, or 52% of the cost of the index option on that day.

For non-financials, the put spread increases by a factor of 1.7 (from 0.5 before the crisis to 0.8 cents during the crisis). Call spreads for both financials and non-financials increase during the crisis, by a factor of 1.8 for financials and 1.5 for non-financials.

D. The Effect of Moneyness

Table II reports the cost of insurance for the basket versus the index as a function of moneyness (Δ). It follows the format of Table I. Option prices are naturally higher when options are further in-the-money (ITM), and results show that basket-index spreads also increase in moneyness. However, the proportional increase in the basket-index spread from pre-crisis to crisis is larger for OTM put options than for at-the-money (ATM) puts. The put spread increases by a factor of 3.4 for Δ = 20, 3.7 for Δ = 30, 3.0 for Δ = 40, and 2.6 for Δ = 50. For non-financials, the put spread increase during the crisis is far smaller than for financials across moneyness. The difference between financials and non-financials (reported in the last column) during the crisis is much larger for OTM puts (2.2 cents per dollar at Δ = 20 and 1.0 at Δ = 50). As a fraction of the average crisis cost per dollar insured for financial sector index puts, the financials minus non-financials put spreads are larger for deep OTM options (22% for Δ = 20 versus 5% for Δ = 50). Similarly, the difference in maximum put spread (as a fraction of the financials index crisis maximum) falls from 38% to 22% as moneyness increases from Δ = 20 to Δ = 50.

Table II. Summary Statistics for Spreads on Options Sorted by Moneyness

Table-II
This table reports summary statistics for the basket-index spread in the cost of insurance per dollar insured for financials, non-financials and their difference (financials minus non-financials) using strike-matching. Numbers reported are in cents per dollar of strike price. The full sample covers 1/2003-6/2009. The pre-crisis sample covers 1/2003-7/2007. The crisis sample covers 8/2007-6/2009. Spreads are constructed using strike-matching as described in Section 3.

E. Bid-Ask Spreads of Options

To ensure that the increase in the basket-index put spread is not solely due to wider bid-ask spreads during the crisis, we reconstruct an alternative basket-index spread series using raw option price quotes rather than the interpolated volatility surface provided by OptionMetrics. This also serves as a check that OptionMetrics interpolated prices do not suffer from inaccurate extrapolation or reliance on illiquid contracts. To summarize, results from raw options data combined with accounting for bid-ask spreads and contract liquidity generates put spreads that are qualitatively identical, and quantitatively very similar, to the results we reported above.

For this analysis, we construct synthetic options with constant maturity (365 days) and constant Δ (30) by interpolating raw option prices in a similar vein as OptionMetrics. There are two key differences with the OptionMetrics methodology that makes our approach particularly robust. First, we restrict the universe of raw options to those with positive open interest in order to ensure a minimum degree of liquidity.10 Second, when constructing synthetic options with constant maturity and constant Δ, we strictly interpolate and never extrapolate. In particular, we require at least one option with Δ above 30 and one with Δ below 30, and similarly require one option with maturity greater than 365 and one with maturity less than 365. Often a stock has only one option near Δ = 20, which is why we construct synthetic options with Δ = 30. Finally, to account for bid-ask spreads, all individual option prices are set equal to the bid price, and all index option prices are set equal to their ask price. This results in the most conservative spread in prices of index puts versus the basket of individual puts, so that the bid-ask-adjusted put spread is always narrower than the spread calculated from midquotes.

The resulting “net of transaction costs” basket-index put spread has very similar behavior to the Δ = 20 and TTM = 365 spread series documented above. Their correlation is 96% (0.93%) over the entire (crisis) sample. The “net of transaction costs” put spread for financials is 1.1 cents per dollar before the crisis, rising to 3.5 cents during the crisis. For non-financial, the spread goes from 1.7 to 2.1 cents. The result is an additional increase of two cents per dollar for financials relative to non-financials, quantitatively consistent with the 2.7 cents estimate presented earlier.

F. Sector Analysis

Table III compares the basket-index spread for all nine sectors of the S&P 500. The only other industries which experience significant increases in the basket-index spreads during the crisis are the consumer discretionary sector and the materials sector. Major components of this sector are car manufacturers (Ford and GM) and parts suppliers (e.g., Goodyear and Johnson Controls). This sector also includes retail, home construction (e.g., D. R. Horton and KB Home), hotels (e.g., Marriott and Harrah’s) and other businesses with substantial direct and indirect real estate exposure.11 The basket-index spread peaks at 12.4 cents per dollar insured for this industry, increasing from an a pre-crisis average of 2.9 cents per dollar insured to 5.1 (rising by a factor of 1.8 over the pre-crisis level, versus a factor of 3.4 for financials). It is conceivable that this sector benefits more than other non-financial sectors when the collective guarantee for the financial sector kicks in. The auto industry also benefited directly from a federal government bailout in fourth quarter of 2008. The materials sector ETF has similarly large exposure to businesses benefitting from government guarantees. Examples include US Steel, whose large customers include the automotive and construction industries, and Weyerhaeuser, which produces building materials and operates a large real estate development segment.

Table III. Basket-Index Spreads on Out-of-the-Money Options in Other Sectors

Table-III

This table reports the average basket-index put spread in the cost of insurance per dollar insured for the nine S&P 500 sector ETFs. Numbers reported are in cents per dollar of the strike price. The full sample covers 1/2003-6/2009. The pre-crisis sample covers 1/2003-7/2007. The crisis sample covers 8/2007-6/2009. Δ is 20, time to maturity is 365 days. Sectors are listed in descending order by mean crisis spread. The last column reports the increase in sector put spread from pre-crisis to crisis in cents per dollar and percentage increase over pre-crisis spread, respectively. Spreads are constructed using strike-matching as described in Section 3.

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NOTES

1 In a seminal paper on this topic, O’Hara and Shaw (1990) document large positive wealth effects for shareholders of banks who were declared too-big-to-fail by the Comptroller of the Currency in 1984, and negative wealth effects for those banks that were not included.

2See Acharya, Pedersen, Philippon, and Richardson (2010); Adrian and Brunnermeier (2010); Brownless and Engle (2010); Huang, Zhou, and Zhu (2011) for recent advances in systemic risk measurement.

3Note that a comparison of single-name CDS and the CDX index (modulo changes in the index composition through defaults) is different because the cost of a basket of credit default swaps has to be equal to the CDX index to rule out arbitrage opportunities.

4 The Δ of an option is the derivative of the option price with respect to the underlying asset price. While put options have negative Δ, we use the convention of taking the absolute value, so that all Δs are positive. Δ measures the moneyness of an option, with low values such as 20 indicating out-of-the-money options and high values such as 80 indicating in-the-money options. Short-dated at-the-money forward options have a Δ of approximately 50.

5 We also compare index and basket put prices using options positions that share the same sensitivity to changes in stock return volatility (the so called option “vega”). With the vega-matched approach, spreads between index reported below. Detailed estimates from our vega-matched put price comparison are available upon request.

6 We use SPDR ETFs. SPDRs are a large ETF family traded in the United States, Europe, and Asia-Pacific and managed by State Street Global Advisors. Options on SPDR sector ETFs are physically settled and have an American-style exercise feature.

7 Our sample length is constrained by the availability of ETF option data. For the financial sector (but not for all non-financial sectors), we are able to go back to January 1999. The properties of our main object of interest, the basket-index put spread for financials, do not materially change if we start in 1999.

8 The option price adjustment performed by OptionMetrics converts prices of American options into equivalent European option prices. This allows us to compare them to the European option price formula we later compute in our model.

9 Our results remain unchanged when we focus on the subset of firms that remain in the financial sector index throughout our sample.

10Results are similar if we instead require that contracts have positive volume.

11Discretionary spending of U.S. consumers experienced the largest post-war decrease during the last quarter of 2008.

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Comments

Where is the author note and attribution? It is probably somehwere linked to this article... but really should appear right here on same page with the article itself.

Hi Jon,

They are on the last page of the article. The article is actually 3 pages long.

The comments to this entry are closed.

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