# Hedging Derivative Products Using Greeks

To understand how a commercial bank dealer hedges a derivative exposure, four key terms must be identified and explained. These terms are at the center of risk control reports that should be clearly reported by trading systems to management.

The terms are Delta, Gamma, Theta and Vega.

Delta: The Delta is the dollar change in the option price relative to the dollar change in the underlying security. Those of you with a calculus background will know this as the first derivative. Values of Delta are between 0 and 1.

Going back to the example of IBM call options, the March 105 call with the stock trading at 100 will have a higher delta than the 110 call because it is closer to realizing its value. Using the example with a put option, a 100 put would have a greater delta than the 95 put.

Traders will often use the terms in the money, out of the money when discussing delta. Generally the following can be stated.

If delta < .30 then it is out of the money

If delta = .50 then it is at the money

If delta > .70 then it is in the money

The first form of hedging is known as delta hedging.

Gamma: The Gamma is the change in the delta relative to the dollar change in the underlying security. As this is the rate of change of the rate of change, then it must be the second derivative. Unlike delta, which has values bounded by 0 to 1,gamma can vary from -10,000 to 10,000.Most Gamma values are around + /- 500.Gamma is highest when an option is at the money and lowest when the option is either in or out of the money.

The interaction between delta and gamma is often overlooked. The real life example of the stock market crash in 1987 will provide insight.

Theta: The theta is the dollar change in the option price relative to the decrease in the time to expiration. It is known as time decay. Let’s go back to the IBM example.

The March 110 calls sell for 1½ and the June 110 calls sell for 2½. As each day passes, the life of the option diminishes giving a reduced possibility for a payoff. Since March comes before June the March option will decay faster than the June option.

Vega: The vega is the dollar change in the option price relative to a 1% change in the volatility. Vega is also known as omega. To understand this concept, let us look at the IBM example.

Assume that on Thursday, IBM 105 call settled at 3 with the stock at 100.Then on Friday morning, IBM schedules a press conference to introduce a revolutionary new product. Since investors might be unsure of the products viability (it could be a huge success or a cash drain),they will bid up the volatility of the stock. That is, the chance of higher or lower stock performance is now increased so the option price will rise through a higher vega, independent of the movement in the stock price.

There are the other terms that risk management systems highlight, such as colour and speed. These are secondary terms. The most crucial are the gamma and delta followed by the theta and vega. Now that these concepts have been defined, let’s examine a typical derivative product trading book.

The most important point to remember is that a derivative transaction is not hedged

In isolation but placed into a book of existing trades. Then the position’s exposure is netted against the book’s to result in a new net exposure. It is the portfolio approach that makes management so difficult as new deals enter and old ones expire almost simultaneously.

Compounding this issue is the fact that bank dealers are net sellers of derivative products so their exposure risks are further heightened. The advent of technology has greatly aided risk management but like in all things in life, the key is user intelligence. Consequently, I believe an examiner’s focus should be on the risk manager’s understanding and interaction with their technological platform, rather than the platform itself.