Dynamic hedging is a technique that is widely used by derivatives dealers to hedge gamma or vega exposures. Because it involves adjusting a hedge as the underlier moves—often several times a day—it is "dynamic." This article discusses the need dynamic hedging addresses and how it is performed. It identifies an important link between dynamic hedging and options pricing theory. It also presents a sophisticated way of thinking about options (as volatility bets) that is common among derivative dealers but alien to most end users of options. Accordingly, the article is about far more that the simple mechanics of dynamic hedging. Traded instruments or positions can generally be broken down into two types:
The former includes spot positions, forward positions and futures. Their payoffs or market values as a function of some underlier are either linear or almost linear. Non-linear instruments include vanilla options, exotic derivatives and bonds with embedded options. Their payoffs or market values as a function of some underlier are highly non-linear. This distinction is illustrated in Exhibit 1.
Derivatives dealers transact in both linear and non-linear positions with clients. They tend to prefer to transact in non-linear positions because they are more difficult for counterparties to price, which means they can make fatter profits on those transactions. There is another difference in their trading of linear vs. non-linear instruments. A dealer's clients tend to want to go long or short linear positions with about equal frequency. For example, an oil company might want to sell its oil forward to lock in a price. At the same time, a power plant operator might want to buy oil forward, also to lock in a price. Such transactions are offsetting, so a derivatives dealer who handles both transactions can offset them and maintain a largely balanced book of linear positions. The same is not true of options or other non-linear positions. Clients of derivatives dealers routinely want to buy a call, buy a put, buy a cap, or buy some exotic derivative. Rarely does a client call a derivatives dealer and ask to sell an option. Dealers are left holding massive short options positions. To hedge those positions, they would like to purchase offsetting long options, but there is no one to buy them from. It makes little sense to buy them from other derivatives dealers, who are in the same boat with their own massive short options positions. The solution is to dynamically hedge the short options positions.
Dynamic hedging is delta hedging of a non-linear position with linear instruments like spot positions, futures or forwards. The deltas of the non-linear position and linear hedge position offset, yielding a zero delta overall. However, as the underlier's value moves up or down, the delta of the non-linear position changes while that of the linear hedge does not. The deltas no longer offset, so the linear hedge has to be adjusted (increased or decreased) to restore the delta hedge. This continual adjusting of the linear position to maintain a delta hedge is called dynamic hedging. Consider an example. A derivatives dealer sells a client a put option on STU Corp. stock. At the current stock price of USD 100, the short option position has a delta of 22,000 shares. This is evident in Exhibit 2, which illustrates the market value of the short option position as a function of the underlying stock's price. A tangent line has been fit to that graph, and its positive slope indicates the positive delta.
To delta hedge the short put, the dealer sells 22,000 shares of STU stock. The deltas of the short option and the short stock cancel, yielding an overall delta of zero. The market value of the hedged position as a function of the stock price is shown in Exhibit 3. A tangent line fit to that graph has zero slope, indicating zero delta.
With the underlying stock price at USD 100, the position is delta hedged, but this doesn't last long. Soon the stock price rises to USD 103. As indicated in Exhibit 4, at that stock price, the position has a slightly negative delta. It is no longer delta hedged.
At the new stock price, the derivatives dealer adjusts the delta hedge, buying back some of the underlying stock he had previously shorted. The result is a newly delta hedged position at the new stock price of USD 103. See Exhibit 5.
The position is once again delta hedged, but not for long. Soon the underlying stock price moves again, and the delta hedge is thrown off. The dealer readjusts the delta hedge. The price moves again, and the dealer readjusts again, and so on. This ongoing process of a market move throwing off the delta hedge and the dealer readjusting the delta hedge is illustrated through several cycles of the process in Exhibit 6.
The continual readjustment of the delta hedge ensures that the portfolio always has a zero delta—and that it loses only a little value each time the underlying stock price moves. Note however, that the portfolio always loses value. It never gains it. A delta-hedged negative gamma portfolio loses money irrespective of whether the underlier rises or falls. Look again at Exhibits 2 through 6. As your eyes run from one chart to the next, you see the portfolio slowly but inexorably losing money. Each time the underlier moves, the portfolio suffers a small loss. The dealer readjusts the delta hedge, locking in that loss. The underlier moves again, causing another loss. The dealer readjusts the delta hedge again, locking in that loss as well. The process continues until the option expires and the dealer can stop dynamic hedging. This is an important observation with implications that go well beyond the immediate problem of dynamically hedging a short put option. Let's briefly explore some of those implications.
First of all, the portfolio loses money with dynamic hedging because it has negative gamma—something the dynamic hedging cannot change. How that negative gamma came about is immaterial. It could have been achieved by shorting a put, or shorting a call, or shorting some exotic derivative. The fact that the portfolio has negative gamma means that the dealer is going to lose money dynamically hedging it. If the portfolio had positive gamma, the opposite would be true. The dealer would make money dynamically hedging it. Each time the underlier moved, the portfolio would make a small profit. By readjusting the delta hedge, the dealer would lock in this small profit ... and so on. To recap, you lose money dynamically hedging a negative gamma position. You make money dynamically hedging a positive gamma position. To make sense of this observation, note that negative gamma positions arise when you sell options. You receive a premium for selling the options but lose money dynamically hedging the negative gamma position. Positive gamma positions arise when you buy options. You pay a premium for the options but make money dynamically hedging the long options position. Wouldn't it be interesting if the amount of money you could expect to lose dynamically hedging a short option position to expiration is precisely the (accumulated value of the) option premium you receive for selling the option in the first place? Actually, this isn't a new idea. When Black and Scholes published their famous option pricing formula, they asserted that the price of an option should be the (discounted value of the) cost of dynamically hedging it to expiration. With this novel idea, they launched the field of option pricing theory. Black and Scholes' analysis assumed that the underlier's volatility is constant over time, but what happens if it is not? Suppose you are dynamically hedging a short options position. How would you feel if the underlier's volatility suddenly increased? Take a moment and think about this ... In fact, you would be concerned if the volatility increased. At a higher volatility, the underlier will fluctuate more, and you will need to adjust the delta hedge more frequently. You will lose money more rapidly dynamically hedging. The opposite would be true if the underlier's volatility suddenly fell. You could readjust the delta hedge less frequently, and you would lose money more slowly dynamically hedging.
These concepts are illustrated in Exhibit 7. It shows the cash position of a derivatives dealer who sells an option and then dynamically hedges it until expiration. The dealer originally prices the option at 25% volatility, but the exhibit considers three volatility scenarios:
At expiration, the dealer's profit on the transaction is the (accumulated value of the) option premium received when selling the option less the total cost of dynamically hedging the short option to expiration. The Exhibit shows how, if the option is priced at 25% volatility but the underlier experiences 20% volatility during the life of the option, the dealer ends up with a profit. If the option is priced at 25% volatility and actual volatility also turns out to be 25%, the dealer breaks even. Finally, if the option is priced at 25% volatility but actual volatility turns out to be 30%, the dealer suffers a net loss on the transaction.
When a dealer is dynamically hedging a short options position, he doesn't care whether the underlier goes up or down. Because he is always delta hedged, he is neither long nor short the underlier. He does care whether the underlier's volatility goes up or down. In a very real sense, he is short volatility. This is the same thing as having negative vega (or short vega), so the phrases negative vega, short vega and short volatility all mean the same thing. A dealer dynamically hedging a long options position is in the opposite situation. He benefits if volatility increases, so he is long volatility. Synonyms would be long vega or positive vega. This is how derivatives dealers perceive options. Because they routinely dynamically hedge their options positions, they don't think of options as bets on the direction of the underlier. They think of them as bets on the direction of volatility. This has implications for gamma and theta as well. While contrived counterexamples are possible (see if you can think of one!) it is generally true that a negative vega position is also negative gamma. A derivatives dealer is typically in the position of having sold options, and he is dynamically hedging a position that is delta neutral, short gamma, short volatility and long theta.
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Give
an example of how vanilla options might be combined in a portfolio
to achieve the atypical combination of positive gamma but negative vega.
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