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 unfamiliar 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  are either linear or almost linear functions of their underliers. Non-linear instruments include vanilla options , exotic derivatives  and bonds  with embedded options. Their payoffs or market values are non-linear functions of their undeliers. Exhibit 1 illustrates with two examples.

Exhibit 1: Market values as a function of some underlier value are illustrated for a long future and a long call option. The future is a linear position. The option is a non-linear position.

Derivatives dealers transact in both linear and non-linear instruments with clients. They tend to prefer to transact in non-linear instruments because these are more difficult for clients to price, which means they can make larger profits on those transactions. There is another difference in their trading of linear vs. non-linear instruments: A dealers clients tend to want to go long or short linear instruments 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 instruments. 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. After selling to multiple clients, dealers are left holding large short  options positions. To hedge  those positions, they would like to purchase offsetting long  options, but there is no one to buy these from. It makes little sense to buy them from other derivatives dealers, who are in the same boat with their own large short options positions. The solution is to dynamically hedge the short options positions.

Dynamic hedging is delta hedging  of a non-linear position using 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 underliers 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 price. A tangent line has been fit to that graph, and its positive slope indicates the positive delta.

Exhibit 2: A derivative dealer sells a put option on STU stock. Its market value is indicated above as a function of the underlying stock price. The current stock price (indicated by the triangle) is 100. A tangent line has been fit to the graph at that value. Its positive slope indicates that the position has 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.

Exhibit 3: The dealer delta hedges the short put option by selling stock short. The market value of the hedged position (short put plus short stock) is shown as a function of the underlying stocks price. A horizontal tangent line indicates that the hedged position has a zero delta.

With the underlying stock price at USD 100, the position is delta hedged, but this doesnt last long. Soon the stock price rises to, say, USD 103. As indicated in Exhibit 4, at that stock price, the position has a slightly negative delta. It is no longer delta hedged.

Exhibit 4: When the underlying stock price rises, the position is no longer delta hedged. This is indicated by the tangent line fit to the graph at the new stock price. It has a negative slope, indicating a negative delta.

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.

Exhibit 5: The dealer adjusts the delta hedge by buying back some of the underlying stock he previously shorted. The position is now delta hedged again at the new underlying stock price.

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.

Exhibit 6: With dynamic hedging, the dealer readjusts the delta hedge for each move in the underlier, either buying or selling shares to achieve a net delta of zero at each new underlying stock price. This exhibit illustrates three iterations of that process.

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, dynamic hedging of a negative gamma position loses money. Dynamic hedging of a positive gamma position makes money. 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.

Wouldnt 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 isnt 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 underliers 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 underliers 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 underliers 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:

• The underlier experiences 20% volatility.
• The underlier experiences 25% volatility.
• The underlier experiences 30% volatility.

At expiration, the dealers 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.

Exhibit 7: A dealer sells an option priced at 25% volatility and then dynamically hedges the position until expiration. This exhibit considers how the dealers cash balance evolves over time under three scenarios. Under all scenarios, we assume an initial cash balance of zero. When the option is sold, the dealer receives a premium, so the cash balance jumps. Next, the dealer dynamically hedges the short option, gradually losing cash as he does so. Under the first scenario, the underlier experiences 20% volatility. Dynamic hedging costs less than it would have had the underlier experienced the 25% volatility used to price the option. The dealer ends up with a profit. Under the second scenario, the underlier experiences 25% volatility. This is the volatility at which the option was priced, so the dealer breaks even on the transaction. Finally, under the third scenario, the underlier experiences 30% volatility. This is higher than anticipated, and the dealer ends up with a loss.

When a dealer is dynamically hedging a short options position, he doesnt 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 underliers 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 dont 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.