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A blog on financial markets and their regulation
Kelly, Lustig and Nieuwerburgh have written an NBER Working Paper (Bryan T. Kelly, Hanno Lustig, Stijn Van Nieuwerburgh, “Too-Systemic-To-Fail: What Option Markets Imply About Sector-wide Government Guarantees”, NBER Working Paper No. 17149, June 2011) explaining banking index option spreads during the global financial crisis in terms of the effect of sector-wide government guarantees:
Investors in option markets price in a collective government bailout guarantee in the financial sector, which puts a floor on the equity value of the financial sector as a whole, but not on the value of the individual firms. The guarantee makes put options on the financial sector index cheap relative to put options on its member banks. The basket-index put spread rises fourfold from 0.8 cents per dollar insured before the financial crisis to 3.8 cents during the crisis for deep out-of-the-money options. The spread peaks at 12.5 cents per dollar, or 70% of the value of the index put. The rise in the put spread cannot be attributed to an increase in idiosyncratic risk because the correlation of stock returns increased during the crisis.
I am not convinced about this because the “No more Lehmans” policy implied a guarantee on individual firms and not merely on the sector as a whole. I propose an alternative explanation for the counter intuitive movement of the index spread based on the idea that the market knew the approximate scale of subprime losses but did not know which banks would take those losses. What securitization had done was to spread the risk across the whole world and nobody knew where the risk had ultimately come to rest. However, the total amount of the toxic securities could be estimated and the ABX index provided a market price for what the average losses would be on these securities. In the macabre language that was popular then, the market knew how many murders had taken place, but did not know where the bodies were buried. The interesting implication of this model is that when a “body” (large loss) turns up in one place (bank X), that immediately reduces the chance that a “body” would turn up elsewhere (bank Y) because there were only a fixed number of “bodies” to discover. The fact that bank X has a huge loss reduces the losses that other banks are likely to suffer because the total scale of losses is known.
A simple numerical example using the Black Scholes model would illustrate the application of this idea to the basket-index put spread. I consider a banking sector with only two stocks A and B each of which is trading at 100. Assuming equal number of shares outstanding, the index is also 100. Consider a put option with a strike of 85 with a volatility of 20% and for simplicity an interest rate of 0 (we are in a ZIRP world!). The put option on each of the two stocks is priced at 2.16 by the Black Scholes formula. Since the two stocks are identical the price of a basket of options (half an option each on each of the two stocks) is also 2.16. To value the index put at the same strike, assume that the correlation between the two stocks is 0.50. The standard formula for the variance of a sum implies an index volatility of 17.32% and using a lognormal approximation and the Black Scholes model, the index option is priced at 1.49. The basket-index put spread is 2.16 – 1.49 = 0.67.
Consider now the crisis situation and assume that the correlation rises to 0.60 but nothing else changes. The stock option prices are unchanged, but the higher correlation raises the index volatility to 17.89% and the index put is now worth 1.63. The basket-index put spread declines to 0.53. During the crisis the actual data shows that the spread rose instead of declining as correlations rose. This is the puzzle that Kelly et al are trying to solve.
I now solve the same puzzle using the “where are the bodies buried” model. In this framework, the simple Black Scholes diffusion is supplemented by a jump risk representing the risk that a “body” would be discovered in one of the banks. Assume for simplicity that there is only “body” to be discovered and that the discovery of that “body” would reduce the value of the affected stock by 25%. As far as the index is concerned, there is no uncertainty at all. One of the stocks goes to 75 and the other remains at 100 (though we do not know which stock would be at which price) and so the index drops to 0.50 x 75 + 0.50 x 100 = 87.50. Assuming the same correlation (0.6) and volatility as before the index put option price rises from 1.63 to 4.98 because the put option is now much closer to the money.
As far as either of the two stocks is concerned, the position is more complicated. There is a 50% chance that a “body” turns up at that bank in which case the stock would trade at 75; there is also a 50% chance that there is no “body” in that bank in which case its stock should trade at 100. Let us make the reasonable assumption that the 50% objective probability is also the risk neutral probability. Before we know where the “body” is buried, the stock price of either bank would be 0.50 x 75 + 0.50 x 100 = 87.50. Note the interesting negative dependence in the tail, if a “body” is discovered in one bank, its price would fall from 87.50 to 75, but the price of the other bank would rise from 87.50 to 100.00 because it is now clear that there is no “body” there.
Option valuation in this situation can no longer use Black Scholes because of the jump risk. Adapting the basic idea of the Merton jump model, we can value this put as follows. If the stock price jumps to 100, the Black Scholes put option price would be 2.16 as computed earlier. But if the price jumps to 75, the Black Scholes put price rises dramatically to 12.58 (the put is now actually in the money). Since the risk neutral probabilities of these two events are 50%, the value of the stock option (before we know where the “body” is buried) is 0.50 x 2.61 + 0.50 x 12.58 = 7.37. The basket-index put spread is now 7.37 – 4.98 = 2.39.
The “where are the bodies buried” model produces a rise in the basket-index put spread from 0.67 to 2.39 without any government guarantees at all. At the same time, the basket-index call option spread shows very little change – this is what Kelly et al found in the actual data as well.
We can elaborate and complicate this basic model in many ways. Of course, there can be more than two banks, they may be of different sizes, there may be more than one “body” to be discovered, the number of “bodies” may be uncertain, the effect of a “body” on the stock price may also be uncertain (random jump size). None of this would change the essential feature of the model – a negative tail dependence between the various bank stock prices.
The key purpose of the model is to demonstrate the pitfalls of using correlation to measure dependence relationships when it comes to tail risk. The dependence in the middle of the distribution (the diffusion process) can be large, positive and rising while the dependence in the left tail is becoming sharply negative. This is the phenomenon that Kelly et al seem to be ignoring completely.