Delta Hedging Example Pdf Format

Delta hedging example pdf format

Dynamic hedging is a technique that is widely used by derivative dealers to hedge gamma or vega exposures. 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.

Futures can reduce delta exposure without touching any of the other greeks. Read on to learn more.

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.

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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. 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.

Delta Hedging Explained (Visual Guide w/ Examples)

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. Rarely does a client call a derivatives dealer and ask to sell an option. 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.

Delta hedging example pdf format

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.

Delta Hedging

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 A tangent line has been fit to the graph at that value.

The Gamma and the Delta

Its positive slope indicates that the position has positive delta. To delta hedge the short put, the dealer sells 22, 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.

Dynamic Hedging

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. A horizontal tangent line indicates that the hedged position has a zero delta.

Soon the stock price rises to, say, USD As indicated in Exhibit 4, at that stock price, the position has a slightly negative delta.

Delta hedging example pdf format

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.

Delta hedging example pdf format

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 See Exhibit 5.

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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.

Delta hedging example pdf format

Soon the underlying stock price moves again, and the delta hedge is thrown off. The dealer readjusts the delta hedge.

Delta hedging example pdf format

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.

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.

P&L profile

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.

How to Hedge Option Delta Using Futures

To make sense of this observation, note that negative gamma positions arise when you sell options. Positive gamma positions arise when you buy options.

Delta Neutral Hedging

You pay a premium for the options but make money dynamically hedging the long options position. Suppose you are dynamically hedging a short options position. 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. 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. 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.

Basic Delta Hedging Example

Next, the dealer dynamically hedges the short option, gradually losing cash as he does so. The dealer ends up with a profit.

This is the volatility at which the option was priced, so the dealer breaks even on the transaction. This is higher than anticipated, and the dealer ends up with a loss. Because he is always delta hedged, he is neither long nor short the underlier. In a very real sense, he is short volatility. A dealer dynamically hedging a long options position is in the opposite situation.

Synonyms would be long vega or positive vega. This is how derivatives dealers perceive options. They think of them as bets on the direction of volatility. While contrived counterexamples are possible see if you can think of one!

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. Dynamic Hedging by Glyn Holton Jun 2, Traded instruments or positions can generally be broken down into two types: linear , and non-linear.