The analysis and management of options requires that one be able to digest a certain Greek alphabet soup. The options analyst encounters Delta, Gamma, Theta, Rho, and Vega, and must not only understand what these variables mean, but must know how to utilize the information to achieve desired risk and return results. This article will explain what the Greek letters represent, and will describe a trick for remembering the basic properties of the most important of these variables: Delta.
According to the Black-Scholes formula, an option's value depends on five variables: the current stock price, the strike price, the time to expiration, the risk-free interest rate, and the volatility of the underlying stock. One of these variables, the strike price, does not really vary. During the life of any particular option, the strike price remains fixed. The other variables are apt to change. The Greek letter variables represent the sensitivities of an option's price to changes in the Black-Scholes parameters.
Delta is the sensitivity of an option's price to a change in the underlying stock price. Theta is the sensitivity of the option price to a change in time to expiry. Rho is the sensitivity of the option price to changes in the interest rate. And Vega, which is not truly a Greek letter, but sounds like one, is the sensitivity of the option price to a change in volatility. Some purists refer to Vega as Lambda or Kappa.
Note that Rho begins with the English letter r, which is usually the symbol used to represent the interest rate. Theta begins with a t, which stands for time. Vega and volatility both begin with v, which helps explain its canonization as an honorary Greek letter in the options soup, as well as the lesser popularity of Lambda and Kappa.
Suppose an option has a Theta of -20 GBP/year. This means that if .1 year passes while all other variables remain constant, the option will fall GBP2.00 in price. Suppose an option has a Rho of .30 GBP/%. This means that if the interest rate were to rise instantaneously by 1%, from 5% to 6%, for example, the option price would rise by GBP0.30. Suppose an option has a Vega of 1.4 GBP/%. This means that if the underlying stock's volatility rose 1%, from 30% to 31%, for example, the option price would rise by GBP1.40.
Now come Delta and Gamma. If an option has a Delta of .6, then if the underlying stock were to rise by GBP1, the option would rise by GBP0.60. Gamma is the second derivative (from calculus). It is the change in Delta for a unit change in the stock price. Delta is the link between the option price and the stock price. Delta indicates the risk exposure in an option position, and also the effectiveness of an option hedge. Speculators need to know Delta in order to anticipate likely option price fluctuations. Hedgers need to know Delta in order to maintain the right balance between offsetting option and stock positions.
Delta has three definitions. The trick to dealing with Delta is to learn them all. The three definitions taken together provide greater insight into option dynamics than any one of the definitions alone.
1) Delta is a probability. It is not exactly the probability that a
call will expire in the money, but it is conceptually close.
Thus, Delta must always be between 0 and 1. Also, as the
stock price rises, Delta increases. If the stock price falls,
Delta decreases.
2) Delta is the number of shares required to replicate a call
option. If the Delta is .6, then .6 shares are required to
replicate one call. 60 shares are required to replicate 100 calls.
Of course, the replication of a call also requires borrowed funds.
3) The third definition is the original definition given above.
Delta is the change in an option's price resulting from a
change in the underlying stock price. As such, it is the slope
of the option price function shown in the exhibit.
Suppose we synthesize a call option with shares of stock and borrowed money. Definitions 1 and 2 tell us that if the stock price were to increase, so that the probability of the call expiring in-the-money were to rise, the replicating portfolio must be rebalanced by buying more stock. Thus, option replication requires that we buy shares from a rising market, and sell into a falling market.
Definition 2 provides insight into definition 3. The greater the Delta, the greater is the replicating portfolio's stake in the underlying stock. Consequently, the greater the Delta, the greater is the sensitivity of an option to the stock price.
Putting definitions 1 and 3 together, we learn that the more likely an option is to expire in-the-money, the closer will be the pound-for-pound movements in the option and stock prices. That is why deep in-the-money calls behave so much like stocks. Their Deltas are close to one.
Delta is the key to option management. Whether speculating or hedging, Delta is the crucial information needed to gauge option dynamics.
This week's Learning Curve was written by Steven P. Feinstein, assistant professor of finance at Babson College, Babson Park, MA.
