Are You Committing Model Abuse?

Charles Wurtz, Ph.D., managing director of Xticket Systems, shows how the use of garden-variety models to value exotic instruments can lead to risk-blindness on a massive scale.

In the years immediately after its inception, the Black-Scholes model accomplished something that no other model had ever achieved before: it allowed traders to generate solid, theoretical values for derivatives. As Garman, Kohlhagen, Whaley and others followed, Wall Street players came to have faith that soon there would be a model that could value any instrument, any time and any place.

This mythical generic model, however, has yet to emerge. Instead, countless quants are taking time-honored models, such as the tried-and-true Black-Scholes, and modifying them with "correction factors" so that they will appear to work for more complex instruments than they have been designed to price. Once a model has been "kluged" (that's techie lingo for loaded down with fix-factors), it is often propagated throughout an institution's system. This process-from the creation of fudged models to their widespread distribution-is known as model abuse, and this phenomenon has tremendous implications for global risk management.

I'd like to cite one truly hair-raising example of model abuse that I encountered during my years as a software provider. It involves a very large broker-dealer who had an insurance company client that wanted to hedge its mega-portfolio against a sudden downturn in the market. Ironically, it wasn't a case of the stereotypical derivatives salesman pushing complex structures-but its exact opposite. In this case, the insurance company's manager explained a complex structure the company wanted to purchase.

90-day Wonder

It was a 90-day European put option on the Dow index struck at 15 percent below the level of the Dow at the time the contract was signed. Since the manager was pretty sure he wanted to retain coverage all year, he asked if the dealer could embed a call option into the put that would allow the insurer to "renew" its option for the next quarter, restruck at the Dow on the expiration date minus 15 percent. This renewal feature would apply whether or not the put option had expired worthless. The insurer also wanted to pay a single, lump-sum premium up front to keep this "renewal" option for a year. The manager's rationale for locking in such long-term protection was the low level of volatility at the time of the transaction; he thought that perhaps he could get a reduced premium on an out-of-the-money option.

The dealer was very pleased with the idea. The insurance company had already sold itself on a mildly unusual structure, and he smelled a fat commission. With visions of dollar signs flashing through his head, the dealer hustled his structure to the option desk's quants to quote a price before the insurer could shop the option to another dealer. The quants, however, were not exactly sure how to price this option. Sure, the European put was straightforward, but the embedded call was giving them problems.

Finally, the brightest rocket scientist of them all came up with the following: he used Black-Scholes to value the original option, using the volatility of the S&P 100 index. He then assumed that volatility would remain constant and that the insurer would "renew" its option for all four quarters. So he multiplied his original Black-Scholes valuation by four-effectively charging the client for four out-of-the-money put options-and then doubled the figure to give the salesman some room for negotiation.

When the salesman called the insurance company with the "inflated" quote, he was surprised that they snapped up the option at the stated price without even a token attempt to dicker. Pleased with his success, the dealer decided to play around with his new option a bit in the hopes of selling it to other insurance companies.

Reality Check

It was at this point that our salesman decided to price his option using a stochastic model of volatility. This model predicts future volatilities based on probability statistics. When plugged into the option pricing model, our salesman came to a sickening realization: the 15 percent decline in the Dow was, statistically speaking, much more likely to occur over the coming year than he had anticipated when he sold the option to the insurance company. His quants had failed to take into consideration that, because the entire premium would be set up front, the bank would not be able to adjust the option's price according to revised volatility estimates.

The bottom line was that the dealer sold an option for about one-tenth of its actual value. That was why the insurer had snapped up the option so eagerly: according to their internal analysis, which employed a stochastic volatility model at the outset, the dealer had offered them an incredible, once-in-a-lifetime bargain. The dealer was then forced to spend a bundle hedging the seemingly innocuous option that had once been considered "such a good idea."

Close Enough

Although this is a particularly dramatic example, fudge factors are used in theoretical pricing every day. Consider the widespread practice of using "implied volatility." In a typical case, you would retrieve a series of, say, market prices for options on the S&P 100. Then you would run Black-Scholes calculations for these options using a whole series of volatilities until you arrived at a theoretical option value between the bid and the ask price on the exchange. The "guesstimate" volatility that you used to arrive at the intermediate option price is then known as your "implied" volatility. This volatility figure is later used to price other options, such as individual stock options, options on the S&P 500, etc.

When companies try to measure their risk on a global basis, the effects of fudge factors are amplified exponentially. For example, in order to measure portfolio-wide risk, many firms attempt to break down their portfolios, which typically contain a wide variety of instruments, from the simple to the complex, into simple cash flows. These cash flows are then added together in order to produce a single number that reflects the firm's total global exposure.

Of course, this process is fraught with problems. Models that can, say, break up a simple bond into its component cash flows are not likely to be accurate for swaptions, no matter how they are altered. And other global risk methodologies, some of which involve summing deltas and gammas generated by different models, run into similar apples-and-oranges type conundrums.

The result? Most models work very well within some very narrowly defined parameters and can be tweaked outside these parameters so they spit out reasonable-looking numbers. Whether or not those numbers actually mean anything is an entirely different story.

While I do not have any magical solution to suggest at this time, I do think that it's time for all of us in the risk management profession to stop hiding behind rationalizations such as, "It's a large portfolio, so modeling discrepancies and approximations will largely cancel each other out."

In the long run, close enough is not good enough. Monumental, strategic decisions could-and are-being carried out based upon numbers that have been approximated, fudged, kluged and corrected so many times that they are effectively meaningless.