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The Sensitivity of Vega

In the foreign exchange market, second-order derivatives like vomma and vanna could make you pay dearly for your ignorance.

By Andrew Webb

Vomma and vanna. They may sound like a couple of Greek game show tarts, but for the big interbank foreign exchange options players, they can represent millions of dollars. Who or what the hell are they?

Let’s start with the definition of vega, which is the change in option value to a 1 percent move in volatility. Understanding the vega of an option is an important input to the design and maintenance of an effective hedge. As vega rises, so the hedging requirements increase, so ignoring it can leave a trader painfully underhedged and ultimately nursing substantial losses.

But understanding vega will only get you so far when it comes to running a book of foreign exchange exotics. Because of the vagaries of such instruments as reverse knockouts, you need to know more. For instance, how sensitive is the option’s vega to changes in implied volatility? What, in other words, is its vomma?

And, for that matter, while you’re at it, how sensitive is the option’s vega to changes in the spot rate? What is its vanna?

Dangerous Greeks You Can’t Ignore
Vega: The change in option value for a 1 percent move
Vomma: The sensitivity of vega to changes in implied volatility
Vanna: The sensitivity of vega to changes in the spot rate

Vomma and vanna are represented as DVega/dVol and DVega/dSpot respectively—an intimidatiang pair of terms—and inquiries about them are likely to receive little more than a quizzical look outside the exotic foreign exchange business. But there, they are critically important—and the consequences of ignorance are brutally clear. “If you don’t incorporate them into your pricing functions, you’ll enjoy extreme—but brief—popularity, sell lots of cheap exotic options, and then go spectacularly bust,” says Rashid Hoosenally, managing director of the global risk strategy group at Deutsche Bank in London.

Vomma and vanna have been fairly common knowledge in vanilla markets for years, but they haven’t been taken too seriously. In the case of vomma, the effect on vanilla options has been compensated for since the mid-1980s as a matter of course by hiking the volatility on out-of-the-money options compared with at-the-money (ATM) options (the smile curve). When exotics started booming, however, it quickly became clear that one couldn’t fudge exotic volatility as one could in vanillas to allow for the effect of vomma. In a Black-Scholes-Merton vanilla framework, the market will simply mark the volatility on a 20-delta call at a higher volatility than its ATM equivalent, because that adds into the price the fact that it is convex to changes in volatility.

“You can’t do that with exotics,” says Howard Savery, senior vice president for business development at DerivaTech Consulting LLC. “For most exotics, there is no single volatility point on the smile curve, which, when put into a Black-Scholes-Merton model, will return a proper price that reflects the adjustment that should be made for DVega/dVol. In some cases, there is actually no single volatility point on or off the smile curve that results in the proper price.” Under those circumstances, a “change the volatility by half a volatility point and—hey, we have the right price!” approach clearly doesn’t work. Instead, it’s a case of looking at the convexity curve itself and inferring some price parameters from it, or taking the route of a stochastic volatility model or some kind of Monte Carlo simulation.

“If you don’t incorporate them into your pricing, you’ll sell lots of cheap exotic options.”
—Rashid Hoosenally
Deutsche Bank

As exotics took off, it also became clear that the effect of vomma and vanna in certain parts of the exotic world was far more dramatic than in vanilla products. In particular, it required a close examination of the real cost of hedging products such as barriers. “Exotic trades have a different and sometimes higher exposure to these second-order Greeks than they have to traditional delta and gamma,” says Tim Owens, managing director in charge of risk advisory at Chase London.

More generally, although traders were previously aware that the vega of options fluctuated over volatility and time, they began to recognize the need to pay attention to the cost of hedging those effects at the outset of a trade when making a price. “On the sell-side of the market, this is an essential component of the business that anybody managing a portfolio has to know about,” says Craig Puffenberger, global head of foreign exchange options at Credit Suisse First Boston in New York. “In addition to estimating the cost of hedging the vega exposure over the life of an option, in the case of a barrier option you are also trying to figure out what the expected life of the option will be. That’s an essential part of trying to estimate what the actual add-on effect might be to a simple model’s output.”

The knockout challenge

If there’s a particular exotic option that causes market-makers sleepless night in terms of vomma and vanna (not to mention other Greeks), it’s probably the reverse knockout. In the case of a call, the standard structure starts in the money, with the knockout barrier set above spot. If the market rallies and hits the barrier, the option ceases to exist—it’s the combination of this discontinuous payoff function and an in-the-money termination that makes for such exciting times.

The worst aggravation occurs when spot approaches the barrier. Take the case of a 90 call with a reverse knockout at 110, sold when spot is at 100. Say prices rally and, with a week to expiration, the spot is at 109. According to the model, the knockout isn’t worth much, because, although it’s 19 units in the money, it has a good chance of knocking out. On the other hand, if it doesn’t knock out, the payout is likely to be quite near maximum. The bottom line is that as spot rallies toward the barrier, the stakes increase.

A reverse knockout may have a delta of opposite sign to a vanilla option (more chance of it knocking out the nearer the barrier it gets), hence the decline in its value as the barrier approaches. The sensitivity of its vega to changes in volatility (vomma) is also far higher than for a comparable vanilla in-the-money call. This is because a 1 point rise in volatility increases the probability of the option being knocked out—resulting in the ultimate change in vega, a factor that obviously doesn’t apply in a vanilla situation. For the same underlying reason, the sensitivity of vega to spot (vanna) is also substantially higher as spot nears the barrier.

This makes vega hedging an excruciating process. The instinctive reaction is to sell an option with a strike as close as possible to the spot level at which the reverse knockout’s vega peaks. Unfortunately, while that may provide vega neutrality at that level, the protection is likely to prove temporary since any change in volatility or spot will immediately dislodge it. As a result, there are people who prefer to leave some of these trades without a vega hedge at all. “In some circumstances, I would prefer to leave a dangerous trade roughly unhedged, but in a controlled way, rather than accurately mishedged,” says Nassim Taleb, president of Empirica Capital LLC, a Greenwich, Conn.-based derivatives hedge fund.

“The foreign exchange market had to learn the intricacies of vomma in order to survive.”
—Howard Savery
Derivatech Consulting

In the case of a reverse knockout, vomma only displays these inconvenient tendencies as the spot price heads north. In the opposite direction—at, say, a spot level of 90—the option displays characteristics similar to an ATM vanilla option. A double no-touch (or range trade), by contrast, has discontinuity on both sides of the spot lattice, so it has a higher vomma but a more symmetrical vanna. (As a result, when traders are asked for a double no-touch, they are more concerned with vomma; when asked for a reverse knockout, they’re still concerned with vomma but also rather more concerned—than with a no-touch—about vanna.)

More headaches

In the case of barriers, traders attempting to hedge the risks of vomma and vanna have the additional entertainment of coping with “barrier effects.” Inevitably, certain spot levels are seen as especially significant for any of a number of reasons—breaking into new ground, old resistance, Fibonacci levels and so forth. Equally inevitably, barriers have a tendency to be clustered at those levels, and so various participants have a vested interest in attacking or defending them.

A good example of the sort of odd spot effects this causes was recently seen at 1.4650 in U.S. dollar/Canadian dollar. A trader who’d clearly written an option with a barrier at that level attempted to defend it. The market, observing an extremely large bid in front of the level, proceeded to deluge him in sales until the unfortunate trader threw in the towel. The market then bought back from him at a lower price as he unwound the position. The sudden and choppy spot moves (and attendant shifts in volatility) that result from this sort of activity make hedging for vomma and vanna a decidedly dynamic process.

This is obviously particularly tricky for large trades, which can also (rather surprisingly for foreign exchange) pose liquidity problems. Although the vanilla options market in many crosses is extremely deep, the effect of vomma and vanna can still pose something of a challenge. “If you sell $1 billion of a reverse knockout, you may well have to sell $3 billion of a particular strike to hedge some of your upside vega on it,” says an exotics trader at a major European bank. “That’s OK in something like euro/U.S. dollar, but not so clever in something like cable.”

Different fish

Given the obvious importance of vomma and vanna in foreign exchange exotics, it appears surprising that option markets based on other securities take relatively little interest in them. They clearly have other fish to fry, however. “Fixed-income traders are much more concerned with aspects of the yield curve, which distracts them from options,” says Empirica’s Taleb. “Their lack of understanding may not matter, since the option dimension of things is not that important when compared with the yield curve.”

Foreign exchange marketers have also had the benefit of experiencing a huge growth in discontinuous types of payoff products—digitals, reverse knockouts, double knockouts and now partial barriers, which have effectively enforced an understanding of these second-order derivatives. (The alternative is going broke.) Although discontinuous-payoff products obviously exist in equities and fixed income, they are far less prevalent. “The foreign exchange, interest rate and equity markets have all grown at different paces,” says DerivaTech’s Savery. “The foreign exchange market was in the right position to advance into these products and therefore had to learn the intricacies of DVega/dVol in order to survive.”

Comparing the scale and complexity of typical trading books in the respective markets is also instructive. Although equity markets are often regarded as more sophisticated in terms of options than fixed income, their books (except for indices) usually consist of a lot of small, not particularly complex positions in a wide range of underlying securities. The net result has been that the market has grown more in the direction of correlation-based, rather than discontinuous-payoff, products. By contrast, in currency options people tend to have a small number of underlying securities and massively complex positions, often broken into only two or three big option books with a huge number of strikes. “That effectively compels them to study the intricacies of options,” says Taleb.

In addition, the foreign exchange market, by virtue of its huge liquidity and price transparency, is in a strong position when it comes to taking a precise approach to these matters. “In foreign exchange, you can have a much sharper view. You’re not saying, ‘I think vomma/vanna will increase my price by roughly 30 basis points to 40 basis points; instead, you know it will change it between 32 basis points and 36 basis points,’” says Deutsche’s Hoosenally. “You can’t do that in a market where liquidity in the underlying vanilla instruments is patchy because market prices aren’t as sharp. Fuzzy inputs beget a fuzzy result.”

How It All Started—And How It Nearly All Ended For
When people started getting the models to do reverse knockouts and double no-touches around 1993–95, simply being able to do the pricing put them way ahead of the pack. While people were aware in the most general sense of second-order effects, in most cases that didn’t translate into accurately allowing for the costs of hedging them. “Early analytic versions of models were widely available to price up ranges, knockouts and so on in a simple way that didn’t take second-order derivatives into account,” says CSFB’s Craig Puffenberger. “People in a hurry to get into the barrier options business started using them without real understanding.”

The net result was an expensive learning curve in 1995 and 1996. In simplistic terms, 1995 was the year that vanna (DVega/dSpot) came home to roost. 1996 was vomma’s (DVega/dVol’s) moment of notoriety. In both instances, U.S. dollar/Deutsche mark was probably the worst underlying culprit. In both cases, a thorough appreciation of these second-order Greeks wouldn’t have actually averted the market movement (or lack of it). But it would at least have ensured that market-makers were pricing trades realistically enough to cover the costs of their hedges—and the costs of unwinding them when trades knocked out. While these trades were actually being priced above theoretical value, the premium being charged was still completely insufficient to cover the costs of the consequences.

March 8, 1995, is a date that is etched in the minds of many seasoned exotics traders. Having opened at around 1.37, the U.S. dollar/Deutsche mark cross rate suddenly spiked down to below 1.3450. The net result was that a large number of reverse knockouts were tripped. While that meant that dealers, who were mostly positioned in the same direction, no longer had to pay out on the original trades, it also left them massively short of options that they had been selling as a hedge. In the ensuing scramble to buy back these positions, one-month volatility rocketed to around 15 percent. Since the costs of this endeavor had not been factored into prices by taking proper account of vanna at the outset, huge losses resulted.

The flip side occurred in 1996, since the U.S. dollar/Deutsche mark spot rate remained largely static for much of the year and one-year volatilities collapsed from 12.5 percent to around 8 percent—a massive drop. Unfortunately for market-makers, the buy-side piled into a huge number of double no-touch (range) trades in U.S. dollar/Deutsche mark (and also U.S. dollar/yen) at the start of the year, which left market-makers extremely long of volatility. With the need to make up the difference between the premium received and the possible payout (which became an increasingly probable payout as the year wore on and the spot rate remained range-bound), market-makers started heavy selling of ATM forward options. (At least if the spot didn’t move and they had to pay out on the range trade, they’d get the premium on the short options.)

Unfortunately for market-makers, the longer that spot stayed in range, the more volatility their models told them to sell. As they continued to do so and spot remained stationary, the market kept marking the implied volatility in the ATM forward options lower. Because the range trades had negative convexity, as the volatilities continued to fall the market-makers became longer and longer vega—and so had to sell even more options. Since the majority of market-makers were once more positioned in the same direction, the net result of all this selling was a big black hole. Once again, although taking proper account of the second-order Greek wouldn’t have averted the move, it would at least have meant that more realistic pricing at the outset would have cushioned the blow somewhat.

—A.W.

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