# Quantifying the Nightmare Scenarios

Dartmouth’s Eric Zitzewitz is one of my favorite co-authors, and a whiz at tracking financial markets. And when he mentioned to me last week that a close look at the options markets told an interesting tale of fear, I asked him to share his observations. Here goes.
Quantifying the Nightmare Scenarios
By Eric Zitzewitz
A Guest Post

There’s no shortage of fear about the economy. But just how fearful should we be? Perhaps financial markets can provide some guidance.
There’s a neat mathematical trick, by which we can use option prices to quantify the probability of the stock market falling by various amounts. Breeden and Litzenberger (1978) show that by comparing the prices of options at adjacent strike prices, you can calculate the approximate value of securities that would pay \$1 if the underlying stock traded in a certain range on expiry day.

“Option prices suggest that there is a very real chance of, dare I write it, another Great Depression.”

(Economists know these as Arrow-Debreu securities; they approximate what option traders call “butterfly spreads” when strike prices are close together.)
For example, using last Friday’s options prices, we can calculate that it would cost 10 cents to buy a portfolio of options that pays \$1 if the S&P 500 falls below 250 on December 18, 2010. If markets were risk-neutral (I’ll come back to this), we could infer that the market thought there was a 10 percent probability that the value of U.S. stocks could fall to one-third their current value by the end of next year. Such a drop would leave the index down to one-sixth of its peak level in late 2007. By way of comparison, in the Great Depression the value of stocks fell to between one-sixth and one-seventh of their earlier values.
In other words, option prices suggest that there is a very real chance of, dare I write it, another Great Depression.
Here is a graph of the probability distribution for the value of the S&P 500 in December 2010 — as implied by the options markets. The red line uses option prices from the end of September 2008 (after Lehman’s collapse); the blue line uses prices from Friday. As you can see, the probability of a substantial drop in the value of stocks has moved from relatively remote to quite substantial. Not only has the distribution shifted to the left, but the left tail has gotten thicker, suggesting there’s much more concern about extremely bad outcomes. (The dashed line indicates where I have used the pricing of the last two options to infer prices of options that are further out of the money.)

This graph yields a pretty interesting story. The most likely scenario is the S&P 500 ends in 2010 between 800 and 1,000, which is up 5 percent to 30 percent from today. But today’s market is being held down by the prospect of a Depression-like decline in stocks.
What do these bad-news scenarios look like? Surely the value of financial firms would be zero, but as these are only around 10 percent of today’s index, the dismal outcomes must be more widespread. Profits would have to fall significantly, with little hope of medium-term recovery. Such a decline in profits would go hand in hand with a sustained decline in incomes across all sectors.
There’s an important caveat to all this. Even when the market price of a bundle of options paying \$1 if “the S&P 500 is below 250 in December 2010” costs 10 cents, we cannot infer that there’s a 10 percent chance of this happening. Since this security would help hedge against extreme wealth losses, investors should be willing to pay an insurance premium. Furthermore, the investors buying these securities could be panicking and overpaying for them, and the more sanguine may be unable to offset this fear if their money is tied up elsewhere.
Regardless of whether it reflects risk aversion, panic, or a true probability, the 10 cent price being paid for a dollar of Depression insurance highlights the fears that are holding stocks down. Policymakers have been trying to reassure investors that they understand the risks of depression and will do what is needed to avoid them. These graphs provide a measure of how far they have to go in convincing us.