Ask Dr. Risk
William Margrabe, president of the William Margrabe Group, inaugurates a new column designed to answer your questions about risk management.
Delta Hedging Problems and Solutions
Dear Dr. Risk:
How would a person mitigate risk in a portfolio using delta hedging? Please focus less on the theoretical aspects of the subject and more on the real-life practice of delta and gamma hedging.
risk management specialist,
Tennessee Valley Authority
Dear Ms. Sumner:
Thanks for your excellent question. While delta hedging is not brain surgery, it is similar in at least five ways:
- It tends to be a last resort, when nothing else works, because it's expensive and potentially fatal.
- Often, people turn to it when it's too late to help.
- A botched job is likely to leave the customer crippled or dead.
- An amateur is likely to screw it up.
- A professional expects to get paid well for doing it, whether the customer survives or not.
The theory of delta hedging is the key insight of the Black-Scholes-Merton option pricing model that earned the 1997 Nobel Prize in Economics for Robert Merton and Myron Scholes. The practice of delta hedging has spawned a derivatives industry that generates annual trades in the trillions of dollars of notional amount and profits and compensation in the billions of dollars. While any decent risk management textbook discusses delta hedging, a summary of the theory here will build the foundation for our later discussion of the practice.
If you graph the value of an option (or any derivatives product) as a function of its underlying price (see figure 1), the slope of that graph at a given point (the rate of change in option value as the underlying price changes) is the option's delta. If you graph the delta as a function of the underlying price, the slope of that graph (the rate of change in the delta as the underlying price changes) is the option's gamma.
I like to think of it this way: option value, delta and gamma are analogous to location, speed and acceleration. The option's value, less its historical cost and accrued interest, tells you where the related profit or loss is. The delta tells the speed at which P&L changes as the underlying price changes. The gamma gives the corresponding acceleration in P&L.
Delta hedging means putting together a total portfolio of derivatives and underlying instruments, so that the total portfolio's delta approaches zero. Typically, that happens in two steps. First, assembling an investment portfolio of derivatives that is worth more than it cost, or otherwise meets business goals, but is sensitive to unanticipated movements in an underlying risk factor. Second, creating a hedge portfolio (of hedge instruments) with offsetting exposure to the underlying.
If the net delta—the sum of the deltas for the investment portfolio and hedge portfolio—equals zero, you have a perfect delta hedge. Small changes in the underlying risk factor hardly change the value of the total portfolio.
When the underlying risk factor moves, the delta may change, and the delta hedger must rebalance his or her portfolio over time, a discipline called dynamic hedging. The total portfolio gamma times the small change in the underlying price provides an approximate measure of (minus one times) the corresponding, required change in the delta hedge. Gamma hedging is delta hedging, plus creating a total portfolio with a gamma that approaches or equals zero, which reduces the need for rebalancing.
In theory, dynamic delta hedging works perfectly. Otherwise, Merton and Scholes wouldn't have won their Nobel Prizes. I've simulated delta and gamma hedging dozens of times, under ideal conditions of known volatilities, and the hedges worked beautifully.
That makes delta hedging and "running a delta book” sound easy, but ignores a number of practical problems. Hiring an amateur with a textbook knowledge of option theory to run a delta book would make about as much sense as hiring the late Karla Faye Tucker to do brain surgery with her pickax. That's why annual compensation in excess of a million dollars per year is standard for traders running delta books for major derivatives dealers.
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What problems must you solve to earn seven figures? Consider the following:
Problem: Because of back-office problems, what's in your portfolio "is a riddle wrapped in a mystery inside an enigma,” so you have no hope of delta hedging.
This was a common problem, until recently. I'd bet that some lesser OTC dealers are in this sad state, even today. Almost surely, some floor traders find themselves in the dark in fast markets.
Solution: Buy the integrated front-, middle- and back-office systems you need. I think the best thing about VAR models is that they require knowledge of the user's portfolio.
Problem: You don't have the right models to compute deltas for the positions you know.
While a bright 14-year-old can program the Black-Scholes model from a textbook, it takes a bit more talent to produce working models for exotic derivatives. An inadequate Monte Carlo pricing model for exotic interest rate derivatives may produce random, dubious deltas and useless gammas. "Missing models” can be even worse. A few years ago, one dealer was "renting” pricing models it used for market-making, but lost them after a dispute over the rent. For months after that, the dealer could neither compute the P&L for its exotic option portfolio, nor measure, nor manage its risk.
Solution: Don't skimp on models.
Problem: With a fast market for the underlying risk factor, your delta's bouncing around like a six-year-old kid on a sugar high.
Delta hedging a portfolio requires constant attention. Now, just imagine that it takes two hours to value your portfolio, and two values to compute a delta. That means you get the delta four hours after you ask for it, and by that time the market has moved and the delta is obsolete. Of course, you could use a cruder approximation to the delta and get it faster. But that would mean you might be long, when you think you're short. Think this can't happen? Don't be so sure.
Solution: Trade products you can price in real time, sleep less, drink more coffee, gamma-hedge and compute overnight the hedge reports you might need the next day.
Problem: The markets are closed when you want to trade.
Maybe you learn some important news at night, on a holiday, over a weekend or when the market has tumbled and a circuit breaker has kicked in, and you want to rebalance. You can forget about trading on that news. By the time you can trade, you may be bankrupt.
Solutions: Trade in a secondary market. For instance, for Japanese equity exposure, use the Chicago Mercantile Exchange's Nikkei futures contract or the American Stock Exchange's WEBS. Accept basis risk in a related market. Gamma-hedge.
Problem: Hedging is too expensive.
In the real world, every transaction has a variety of costs, including brokerage commission, taxes and bid-ask spread. A dynamic hedging strategy that is so expensive that it threatens to eat you alive is called an alligator hedge. Cute name, but the bite can kill.
Solution: Every time you provide liquidity, tell your salespeople to offer their customers the hedge trade at an attractive price. That way, the hedge transactions earn you money, rather than cost you. If you must delta hedge, find the cheapest market for hedging. Don't rebalance your portfolio every time the underlying moves a tick. Wait until it has moved a significant amount. In particular, this helps you a avoid an expensive, worthless response to every bid-ask bounce.
Problem: Even if the main underlying risk factor stands still, your portfolio can lose value via time decay and because other key variables—particularly volatility and interest rates (for other than interest rate derivatives)—are moving.
On occasion, the interest rate risk for an equity derivatives desk can exceed that of the same firm's interest rate derivatives desk. A swap desk could be long and short hundreds of billions of dollars of nearly offsetting swaps, with net exposure of maybe $1 billion, which the desk then hedges to nearly nothing. If it is short a mammoth position in equity index put warrants and has ignored the interest rate risk, it could have greater exposure to moves in interest rates.
Keep in mind, moreover, that the Black-Scholes-Merton model assumes that volatility is constant. Of course, it isn't. Hedge fund manager Victor Niederhoffer's short position in naked Standard & Poor's 500 index put options lost so much money so fast because the market tanked and implied volatility exploded at the same time.
Solution: Hedge also against a move in interest rates (rho = 0) and volatility (kappa = 0).
Problem: As expiration approaches for a short, at-the-money European option, you face pin risk: When expiration is about a minute away, it's delta is either 0 or (plus or minus) 1, and its terminal value could end up at 0, or below. You can't delta-hedge.
Solution: Keep such exposures many and small—hence diversified—and go naked. If you can't stand the heat, close out your short option position and pay up.
|Some Alternatives To Delta Hedging
Delta-hedging isn't the only way to manage risk, and sometimes it isn't the best way.
- Sometimes, taking a wild gamble is the only rational thing to do. Back in the 1980s, the typical savings and loan was a zombie—really dead, but appearing alive because federal regulators bent the rules to avoid massive S&L closures and a financial panic. The Federal Savings and Loan Insurance Corp. sold insanely underpriced put options on deposits, so depositors would leave their money in these moribund institutions. This gave rational S&L managers all the license he needed to "play roulette” with the deposits. Only an irresponsible manager would hedge his or her S&L's risk. Few did, the bets were losers, and the taxpayers picked up the tab for hundreds of billions of dollars of losses.
- A smart market-maker will make sure he has a book of mostly offsetting positions. For example, he could be long $100 billion notional of swaps and short $100 billion, and his net exposure might approximate $500 million of long and short forward rate agreements with various maturities. He tells his sales force that he's offering these FRAs at attractive levels, and the next day he's made a profit on them, rather than adding to his hedging costs.
- Overhedging can be a way to play safe. For example, if a dealer sells a digital call option, he can buy a European call spread that pays off at least as much. If he can sell the digital call for the price of the call spread, he will at least break even and perhaps make a profit.
- Diversification may be a better way to manage independent risks. Casinos and insurance companies have managed risk this way for centuries, and it's the foundation of modern portfolio theory. Derivatives traders can use it as well. For example, suppose that at 3 p.m. on the last day of every month a dealer sold a one-month, 50-delta call option on 100 shares. Rather than dynamically hedging that position, he might decide to hold 50 shares of the underlying stock and hope that the underlying price moves less than about 30 percent of one vol. Doing this month after month allows the dealer to diversify over time.
- Static hedging consists of buying a hedge portfolio that you never have to adjust, because changes in its delta perfectly track changes in the portfolio that you're hedging. For example, a dealer who sells an inverse floater with a minimum coupon of zero, can hedge it by buying the appropriate fixed-coupon bonds, shorting the right floater and buying an interest rate cap struck at twice the fixed-coupon rate.