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The Return of Jump Modeling

Is Stephen Kou's model more accurate than Black Scholes?

By Barclay Leib

How would it feel to be flying in an airplane knowing your altimeter is systematically flawed? Trapped in a cloud bank, you might think you're 1,000 feet above the ground, but you would never really know for sure until it's too late.This is not unlike the predicament of options theory with regard to the famous Black-Scholes options-pricing model.

For over a quarter-century, the Black-Scholes option-pricing formula has been the mainstay behind the entire derivatives industry. Every day, thousands of prices are calculated with some variation of the model, and large sums of money put at risk. Yet academics and practitioners alike have long known that one key assumption of the model—that market volatility is constant—is empirically incorrect. Because of this, Black-Scholes systematically underprices out-of-the-money options, forcing market-makers to build complicated adjustments such as skew and kurtosis into their pricing.

While many academics have argued that only commodities such as oil and gold were candidates for jump moves, there's no denying that more and more stocks have been witnessing large and sudden percentage-price jumps. The headline-grabbing jumps have recently been to the downside, but takeover-driven vaults higher are, of course, a more common staple of American markets. Even the major equity indices hit sudden air pockets of complete illiquidity last month.

Is it finally time to chuck Black-Scholes and move on to some more sophisticated pricing platform? One 28-year old Columbia assistant professor, Steven Kou, thinks so.

Old flaw

One quiet Thursday night in early April, when most derivatives traders were either ordering a second beer in the local bar or snoozing on the train ride home, Kou addressed a small group of curious quantitative types huddled in a ballroom of New York's Downtown Club. Surrounded by dark-paneled walls recalling a more gentlemanly era in finance, they heard a potentially revolutionary new presentation on a rather old topic called jump diffusion.

When the market rock ‘n' rolls, it really rocks; but when it doesn't, price behavior can be exceedingly quiet.

After Fischer Black and Myron Scholes successfully introduced their model to the world in the early 1970s, academics immediately set out to tinker with it. They knew that the Black-Scholes assumption of constant volatility was wrong, but they did not know just how much that mattered. By 1976, Robert Merton of Harvard introduced a paper called "Option Pricing When Underlying Stock Returns Are Discontinuous.” In it, he developed an alternative model to Black-Scholes that incorporated not just a volatility parameter, but a parameter for how much annual volatility was accounted for by jump moves, and another parameter for the number of jumps one might expect in a given year. The assumption was that changes in an asset price should be divided into two distinct distributions: small continuous changes that are attributed to normal supply-and-demand imbalances (something technically called the Geometric Brownian motion with constant volatility), and occasional jump moves resulting from the release of new and vital information.

Using his model, Merton was able to show that the fair value of out-of-the-money options was typically higher than that when derived from Black-Scholes, while at-the-money options were slightly cheaper. The more jumps per year and the greater the magnitude of each jump, the more pronounced this effect would be. In effect, Merton could explain at least some of the empirically observed options pricing smile so difficult for Black-Scholes.

But barring a few oil traders and quantitative sticklers, Wall Street largely yawned. People did so because jump diffusion was more difficult to calculate with three inputs instead of one, and the model also abrogated various zero-sum hedging rules so loved by academics in their search for closed-form solutions. There was also no compelling case that such a segmented distribution applied to any but a handful of markets. Now, a quarter-century later, Kou's work is once again reviving this discussion.

Keep it simple

Kou's model is elegant in its simplicity and seeks to comply with two empirical observations in the real world. First, markets seem to assign a greater probability to outlying return outcomes than either a normal or lognormal distribution would imply. We intuitively see this when out-of-the-money options trade at a higher implied volatility—so-called fat tails—in a volatility smile. Conversely, the returns associated with markets also appear to be leptokurtic—a fancy term indicating that despite the chance markets may move dramatically, prices and investment returns often stay more tightly bunched than a normal or lognormal distribution would imply. In other words, when the market rock ‘n' rolls, it really rocks; but when it doesn't, price behavior can be exceedingly quiet. The combined effect on a return distribution compared to a normal distribution is pictured above using returns data from the Dow Jones Industrial Average since 1993.

Kou starts with a basic assumption of continuous markets and uses the Brownian motion to account for times when a market is moving normally. Then he adds to this assumption, however, the possibility of jump moves. If jumps are assumed to be zero, the equation simply reverts back to a more traditional valuation model. But when jumps are greater than zero, he weights the value of these jumps not by the normal density function found in Black-Scholes, nor as Poisson events as Merton did in 1976, but by LaPlace's double-exponential distribution of returns. Without getting too technical, Kou's use of the double exponential distribution as opposed to a normal or lognormal distribution helps his model create a high peak of returns and also a wider dispersion of overall returns on the extremes. Kou claims that much of the stochastic volatility work of the 1980s helped explain the occurrence of fat tails, but that his approach is the first also to achieve the high-peak leptokurtic-type return behavior as well in a closed-form and simple manner. If he is right, and if his equation works, Black-Scholes models may be headed for the wastebasket.

Kou has tested his model on only a limited amount of empirical data to date—Japanese Libor caplets as priced by the market in May 1998—but his model fit the real-world premium levels quite closely, despite the existence of a traditional volatility smile in that data. He is also awaiting comments from the father of jump diffusion, Robert Merton, to whom he recently sent a copy of his paper.

Good-enough models

But hold on a second. Are these changes really that important? Haven't we been making do with Black-Scholes for almost 30 years, and are any of the changes suggested by Kou or other academics really that significant?

Sometimes it takes a market event to goad trading houses and academics alike into action. Work on stochastic volatility option models truly took off after the equity crash of 1987. More recently, the massive dose of equity volatility in the U.S. equity markets since last fall has rekindled interest in this field.

These days, at least a few option market-making practitioners value the jump-diffusion model highly—even before Kou's recent elaboration of it. Trent Cutler of the San Francisco-based equity options market-making firm Cutler Group LP, regularly goes to great lengths to train new employees to look at options in jump-diffusion terms. "We probably spend as much time on this as anybody,” says Cutler, "and we actually get some pretty interesting outcomes, perhaps even a pricing advantage, although it's hard to quantify how much this has been worth to us.”

Cutler uses his jump-diffusion models as a way to explain initially aberrant-looking pricing. "We'll look at something that appears out of line, put in an extra jump, and see how much that effects the pricing. Then we'll look out into the real world and see what the market is focusing on. Often we can explain away an odd-looking pricing phenomenon. Other times, we spot an opportunity. We think the model works well in terms of noncontinuous movements and systematic changes in volatility over time and stock prices.”

Looking back over the past few years, Cutler notes that the whole feel of the equity options world and his firm's procedures for dealing with it started to change dramatically in late November 1998. "Before Thanksgiving of that year, we would think of our risk on a given stock as maybe 80 percent of its value to the downside and maybe twice its value on the upside,” he says. "We were a market-maker in Onsale, short just a little bit of gamma at the time, and that stock went from 20 to 100 in three or four trading days. That teeny bit of negative gamma lost us a whole bunch of money. It also changed the whole paradigm. Lots of weird things that we had never seen before started happening. It took us about six weeks to figure it all out, but it suddenly became clear that we had to assume that stocks could go up five times their current value. Any sort of jump was possible and liquidity started to decline.”

Cutler is quick to point out that the jump component in pricing was probably more important back then than it is now. "As overall volatility rises as it has recently,” he explains, "the relative importance of jump moves declines. Distributions are so wide, adding or subtracting a single jump to our model hardly matters any more, whereas it used to matter a great deal.”

Other practitioners are well aware of jump-diffusion modeling, but use it sparingly. Emanuel Derman, head of risk management (CK) at Goldman Sachs, has seen the model used at his firm in specific merger-arbitrage situations, "for which it was quite useful,” he says, but seldom in other situations. Mike Kelly, a quantitatively oriented trader at Onyx Capital Management LLC, doesn't use jump diffusion in his pricing and hedging decisions, but finds it theoretically quite appealing. "It describes the world much better than any Gaussian model,” he says, "and when I previously worked at JP Morgan we had one analyst there who swore by it. He used to say that all the effort spent modeling smiles was a waste of time, and that we should focus just on this. Jump diffusion does help explain away the smile phenomenon.”

Short dates

Of note, jump models are particularly important for short-dated options when the gamma value of the option is high. Assume a discrete jump, and anything could happen to a short-dated option. Assume that volatility is variable, and longer-dated options respond even more. Stochastic volatility models thus have far more impact on longer-dated options than the jump-diffusion model typically does. Jump diffusion matters more when the option expires next week or next month as opposed to next year.

The leptokurtic tendency of the DJIA since 1973 can be seen by comparing the Dow Jones daily return (from 10/29/1993–4/19/2000) and the normal density with the same mean and st. dev.

Building some sort of jump-diffusion assumption into exotic option pricing could also be extremely important as models develop further. In a 1997 Federal Reserve working paper, Chunsheng Zhou showed through numerical examples that ignoring jump risk might lead to "serious biases” in path-dependent option pricing with both long and short maturities. Specifically, he argues that mis-specifying the distribution of prices as a diffusion process, as opposed to a jump-diffusion process, "will understate the knock-out probability for short maturity options, but significantly overstate the knock-out probability for long-maturity options.”

Using 25 percent volatility and a knock-out level 20 percent under his at-the-money strike, Zhou showed that a two-week at-the-money down-and-out call priced with a jump-diffusion model would show a fair value approximately 39 percent less than a similar-termed option priced with a lognormal diffusion process. This is fairly intuitive, since the risk of knocking-out increases a great deal once a jump possibility is allowed. The knock-out feature decreases the chance of the option taking on a greater value.

On the other hand, the fair-value premium of a two-year at-the-money down-and-out call priced with jump diffusion would be 10.55 percent greater than a similarly termed option priced with a lognormal distribution. In this latter instance, the jump assumption allows a greater probability of moving away from a barrier and then remaining relatively still at the new level. The jump assumption actually increases the value of longer-dated barrier options.

Do any exotic option traders spend much time thinking about these issues? Ron Dirusso, managing director in charge of Lehman Brothers currency options, doesn't think so. "Anytime you start messing with the distribution of prices with a path-dependent option, the impact of jump-option pricing is going to be magnified,” he says. "The impact on short-dated pricing is particularly severe, but the average dealer out there doesn't know or care about this. He trades off of the number Fenics provides him as a benchmark—whether that number is right or wrong.” Fenics is a popular software tool used by more than 90 percent of major banks in their currency option pricing.

Not everyone thinks jump-diffusion is the greatest thing since sliced bread.

Nassim Taleb, options savant, derivatives trader and president of Empirica Capital Management, for one, is not a fan of jump-diffusion models. He claims that uttering the mere term "jump diffusion” betrays a lack of understanding of the stochastic process. "Of course we know that the market moves by jumps. That's not a big discovery,” he states. "It is far more tractable and far more effective to use models that create fat tails in a more effective way. This work was pioneered by Hull and White in 1987 with their stochastic volatility models, and was followed in turn by new regime switching models of which jump diffusion is just a special case,” states Taleb. He has yet to read Kou's new paper, however.

Others are hesitant to embrace jump diffusion even after seeing Kou's recent work. Alan Lewis, chairman of Envision Financial Systems and the author of the recent book, Option Valuation under Stochastic Volatility, thought Kou did a reasonably good job modeling the volatility smile phenomenon. He warns, however, that the paper did not adequately broach the topic of skew. He thus suggests that this new model may not solve all the modeling problems for a market like the Standard & Poor's 500, where out-of-the-money puts quite regularly command a lofty volatility premium compared with out-of-the-money calls. Lewis's own book tries to tackle such topics.

Meanwhile, Mamdouh Barakat of MBRM-MB Risk Management, continues to suggest that it is not so important which model one uses—Black-Scholes with a volatility smile adjustment, a finite-difference model with a local volatility surface, jump diffusion or GARCH—but how one applies the model used. According to this viewpoint, if the inputs used are not correct and well-calibrated, then none of the models will improve pricing and hedging performance. But if the inputs are calibrated correctly, "then the choice of which model to use would depend on the speed of calculation of that model, and which model provides the most accurate sensitivities. Indeed, if the requirement of the model is to price both American and European style options based on a consistent volatility term structure, then that factor would dictate that you would need to use the finite difference model.”

Toothpaste theory

Two final points bear mentioning regarding jump-diffusion theory, particularly as we empirically reexamine the current equity markets.

"The impact on short-dated pricing is particularly severe, but the average dealer doesn't know or care. He trades off of the number Fenics provides him—whether that number is right or wrong.”
—John Dirusso

First, in an equity world where the risk of jumps seems ever-present, the record concentration of holdings by many mutual funds (as well as the public) in large-capitalization high-technology companies could someday exacerbate a jump decline. Vincente Zaragoza, president of the Pentium Fund, says that "of all times in history, we have the greatest consensus of the masses right now. If this consensus ever changes and the funds start selling, you could see something of a black hole. It worries me to death.” Zaragoza appropriately was reducing his positions before April's spate of violent trading.

Second, within the three-pronged world of equities, foreign exchange and fixed income, we now have a tri-block currency world (U.S. dollar, yen and the euro), and global real yields that are closer to convergence than they used to be in past decades. Given current central-banker tendencies in macro policy-setting, and assuming that there is always some constant amount of volatility seeking an escape valve within the global capital markets (perhaps a bold assumption, but perhaps not), then the foreign exchange and fixed-income markets may be providing less of an escape valve than they have in the past. Kiran Kumaran, principal for the options-oriented Kiran Kumaran Investment, refers to this as the "toothpaste tube theory of volatility: if you squeeze the tube in the middle, the volatility has to go somewhere.” At present, equities may be winning by default, and thus the chance of jump moves in this arena becomes greater than elsewhere.

If all of the tech talk on options theory above has left you dizzy or perhaps worried that your bank is exposed to risks it doesn't even know about, that's all to the good. If you knew about jump diffusion, but never considered such a model for equity trading, the recent behavior of prices might suggest you at least consider otherwise. The next time you hear some stock opening down 25 percent in value on the opening, considering jump diffusion today might help save you a bit of money then. Sometimes complexity is worth it.

For a copy of Kou's paper see http://www.ieor.columbia .edu/~kou/expo.pdf.