DERIVATIVES 'R US/V1N10/It's All Greek to Me
VOLUME 1, NUMBER 10, April 24, 1995
Not long after the Black-Scholes option pricing model was developed, someone decided that a natural next step was to determine the amount by which an option price changes when one of the inputs changes. This operation, known in economics as "comparative statics," involves taking the first derivative (the calculus derivative, that is) with respect to the stock price, exercise price, risk-free rate, time to expiration and standard deviation. One can then also move to the second derivative and so on. Of course, the calculus derivatives tell us only what happens to the option price if the input variable changes by a very small amount.
What seemed primarily like an exercise in mathematics has now developed into an extremely important banking concept. Banks have become vendors of options, forwards and swaps. When a bank sells an option, it takes a naked short position. It then typically buys an option or other derivative to offset the risk, thereby earning the bid-ask spread. The correct number of positions in this offsetting instrument is given by the Greeks.
For example, let's assume the underlying asset is a stock. When the stock price changes, by how much does the option price change? This is approximately given by the first derivative with respect to the stock price and is called the Delta. From the B/S model, it is the value N(d1). If you're short 1,000 call options (that's individual options, not 100-lot contracts at the CBOE) with a delta of .5, you'll need to hold 500 shares of the stock. If the stock goes down by $1, you'll lose $500 on the stock and each option will fall by $0.50. With 1,000 options, you'll gain $500 to offset. This is called Delta hedging and you are said to be Delta-neutral.
The delta is correct, however, only for very small stock price changes. An actual option price increase will be more than 50 % of the stock price increase. In fact, the larger the stock price increase, the more the call price change will deviate upwards from 50 %. On the downside the call price change will be less than 50 % and the difference will be greater the larger the stock price change. This is the second order effect coming from the second calculus derivative, which is known as the Gamma and results from the convex shape of the call price curve with respect to the stock price. The Gamma is the change in the Delta for a change in the stock price. Large Gammas mean that the Delta is very sensitive and this means that Delta hedging is much more difficult. Think of the Gamma as a measure of the uncertainty about whether the option will expire in- or out-of- the money. For an expiring option either in- or out-of-the money, then Gamma will move toward zero. For an expiring option that is approximately at-the-money, the Gamma will become very large. That's because at expiration the Delta is going to jump to either one or zero. To protect against a large Gamma, the bank can Gamma hedge as well as Delta hedge but this requires a position in at least one additional instrument.
Another measure of uncertainty is Vega, the first calculus derivative with respect to the volatility (standard deviation). Actually Vega is not a Greek letter and this term is sometimes referred to as Kappa or Lambda. An option price is very sensitive to the volatility and this can induce substantial risk to an otherwise hedged position. We should note, however, that the B/S model is actually inconsistent with changing volatility. It assumes volatility is constant so taking the derivative really means nothing more than saying what the option price would be if you plugged a different volatility in the model. But the model presumes that the volatility will remain constant for the full life of the option. I like to compare this to what you get in Einstein's famous E = mc^2. You can take the derivative with respect to c, the speed of light, and you get 2mc. Energy will change by 2 times the mass times the speed of light for every change in the speed of light. But the speed of light does not change! Now in this case, the model fits reality. In the B/S case, the model doesn't completely fit reality. It doesn't allow volatility to change but in reality it does. Other more sophisticated models can account for changing volatility though the added complexity of the models is not always worth it.
The time decay of the option is the Theta. It is the first calculus derivative with respect to time. A negative theta indicates time value decay. The effect of a change in the risk- free interest rate is called Rho. It is the first calculus derivative with respect to the risk-free rate. Standard options have very low Rhos, meaning that interest rates do not impart much of an effect on their prices. However, like the volatility, the B/S model assumes away any interest rate changes. Other more sophisticated models do permit interest rate changes but still often fail to capture the complex interaction between interest rates, stock prices and the volatility.
Finally, while there is a calculus derivative with respect to the exercise price, standard options do not have changing exercise prices so no one pays much attention to this concept. I've never even heard it given a Greek name. I should note that it is possible to construct options with changing exercise prices. Perhaps I can cover that in a later issue.
In conclusion a Delta-hedged, Gamma-hedged, Vega-hedged, Rho- hedged position is at best an attempt to immunize against changes in stock prices, volatility and interest rates. It is impossible to be fully hedged so I leave you with a famous derivatives adage, "The only perfect hedge is in a Japanese garden."
BACK ISSUES: Back issues are available by anonymous ftp from fbox.vt.edu/filebox/business/finance/dmc/DRU or can be accessed using a Web browser at