1. Chapter 7
Option Greeks
© 2002 South-Western Publishing

2. Outline Introduction The principal option pricing derivatives
Other derivatives
Delta neutrality
Two markets: directional and speed
Dynamic hedging

3. Introduction
There are several partial derivatives of the BSOPM, each with respect to a different variable:
Delta
Gamma
Theta
Etc.

4. The Principal Option Pricing Derivatives
Delta
Measure of option sensitivity
Hedge ratio
Likelihood of becoming in-the-money
Theta
Gamma
Sign relationships

5. Delta Delta is an important by-product of the Black-Scholes model
There are three common uses of delta
Delta is the change in option premium expected from a small change in the stock price

6. Measure of Option Sensitivity
For a call option:
For a put option:

7. Measure of Option Sensitivity (cont’d)
Delta indicates the number of shares of stock required to mimic the returns of the option
E.g., a call delta of 0.80 means it will act like 0.80 shares of stock
If the stock price rises by $1.00, the call option will advance by about 80 cents

8. Measure of Option Sensitivity (cont’d)
For a European option, the absolute values of the put and call deltas will sum to one
In the BSOPM, the call delta is exactly equal to N(d1)

9. Measure of Option Sensitivity (cont’d)
The delta of an at-the-money option declines linearly over time and approaches 0.50 at expiration
The delta of an out-of-the-money option approaches zero as time passes
The delta of an in-the-money option approaches 1.0 as time passes

10. Hedge Ratio Delta is the hedge ratio
Assume a short option position has a delta of –0.25. If someone owns 100 shares of the stock, writing four calls results in a theoretically perfect hedge

11. Likelihood of Becoming In-the-Money
Delta is a crude measure of the likelihood that a particular option will be in the money at option expiration
E.g., a delta of 0.45 indicates approximately a 45 percent chance that the stock price will be above the option striking price at expiration

12. Theta
Theta is a measure of the sensitivity of a call option to the time remaining until expiration:

13. Theta (cont’d)
Theta is greater than zero because more time until expiration means more option value
Because time until expiration can only get shorter, option traders usually think of theta as a negative number

14. Theta (cont’d) The passage of time hurts the option holder
The passage of time benefits the option writer

15. Theta (cont’d)
Calculating Theta
For calls and puts, theta is:

16. Calculating Theta (cont’d)
The equations determine theta per year. A theta of –5.58, for example, means the option will lose $5.58 in value over the course of a year ($0.02 per day).

17. Gamma
Gamma is the second derivative of the option premium with respect to the stock price
Gamma is the first derivative of delta with respect to the stock price
Gamma is also called curvature

18. Gamma (cont’d)

19. Gamma (cont’d)
As calls become further in-the-money, they act increasingly like the stock itself
For out-of-the-money options, option prices are much less sensitive to changes in the underlying stock
An option’s delta changes as the stock price changes

20. Gamma (cont’d)
Gamma is a measure of how often option portfolios need to be adjusted as stock prices change and time passes
Options with gammas near zero have deltas that are not particularly sensitive to changes in the stock price
For a given striking price and expiration, the call gamma equals the put gamma

21. Sign Relationships Long call + - Long put Short call Short put Delta
Delta
Theta
Gamma
Long call + -
Long put
Short call
Short put
The sign of gamma is always opposite to the sign of theta

22. Other Derivatives Vega Rho The greeks of vega Position derivatives
Caveats about position derivatives

23. Vega
Vega is the first partial derivative of the OPM with respect to the volatility of the underlying asset:

24. Vega (cont’d) All long options have positive vegas
The higher the volatility, the higher the value of the option
E.g., an option with a vega of 0.30 will gain 0.30% in value for each percentage point increase in the anticipated volatility of the underlying asset
Vega is also called kappa or lambda

25. Vega (cont’d)
Calculating Vega

26. Rho
Rho is the first partial derivative of the OPM with respect to the riskfree interest rate:

27. Rho (cont’d) Rho is the least important of the derivatives
Unless an option has an exceptionally long life, changes in interest rates affect the premium only modestly

28. The Greeks of Vega Two derivatives measure how vega changes:
Vomma measures how sensitive vega is to changes in implied volatility
Vanna measures how sensitive vega is to changes in the price of the underlying asset

29. Position Derivatives
The position delta is the sum of the deltas for a particular security
Position gamma
Position theta

30. Caveats About Position Derivatives
Position derivatives change continuously
E.g., a bullish portfolio can suddenly become bearish if stock prices change sufficiently
The need to monitor position derivatives is especially important when many different option positions are in the same portfolio

31. Delta Neutrality Introduction Calculating delta hedge ratios
Why delta neutrality matters

32. Introduction
Delta neutrality means the combined deltas of the options involved in a strategy net out to zero
Important to institutional traders who establish large positions using straddles, strangles, and ratio spreads

33. Calculating Delta Hedge Ratios (cont’d)
A Strangle Example
A stock currently trades at $44. The annual volatility of the stock is estimated to be 15%. T-bills yield 6%.
An options trader decides to write six-month strangles using $40 puts and $50 calls. The two options will have different deltas, so the trader will not write an equal number of puts and calls.
How many puts and calls should the trader use?

34. Calculating Delta Hedge Ratios (cont’d)
A Strangle Example (cont’d)
Delta for a call is N(d1):

35. Calculating Delta Hedge Ratios (cont’d)
A Strangle Example (cont’d)
For a put, delta is N(d1) – 1.

36. Calculating Delta Hedge Ratios (cont’d)
A Strangle Example (cont’d)
The ratio of the two deltas is -.11/.19 = This means that delta neutrality is achieved by writing .58 calls for each put.
One approximate delta neutral combination is to write 26 puts and 15 calls.

37. Why Delta Neutrality Matters
Strategies calling for delta neutrality are strategies in which you are neutral about the future prospects for the market
You do not want to have either a bullish or a bearish position

38. Why Delta Neutrality Matters (cont’d)
The sophisticated option trader will revise option positions continually if it is necessary to maintain a delta neutral position
A gamma near zero means that the option position is robust to changes in market factors

39. Two Markets: Directional and Speed
Directional market
Speed market
Combining directional and speed markets

40. Directional Market
Whether we are bullish or bearish indicates a directional market
Delta measures exposure in a directional market
Bullish investors want a positive position delta
Bearish speculators want a negative position delta

41. Speed Market
The speed market refers to how quickly we expect the anticipated market move to occur
Not a concern to the stock investor but to the option speculator

42. Speed Market (cont’d) In fast markets you want positive gammas
In slow markets you want negative gammas

43. Combining Directional and Speed Markets
Directional Market
Down
Neutral Up
Speed Market
Slow
Write calls
Write straddles
Write puts
Write calls; buy puts
Spreads
Buy calls; write puts
Fast
Buy puts
Buy straddles
Buy calls

44. Dynamic Hedging Introduction Minimizing the cost of data adjustments
Position risk

45. Introduction A position delta will change as
Interest rates change
Stock prices change
Volatility expectations change
Portfolio components change
Portfolios need periodic tune-ups

46. Minimizing the Cost of Data Adjustments
It is common practice to adjust a portfolio’s delta by using both puts and calls to minimize the cash requirements associated with the adjustment

47. Position Risk
Position risk is an important, but often overlooked, aspect of the riskiness of portfolio management with options
Option derivatives are not particularly useful for major movements in the price of the underlying asset

48. Position Risk (cont’d)
Position Risk Example
Assume an options speculator holds an aggregate portfolio with a position delta of –155. The portfolio is slightly bearish.
Depending on the exact portfolio composition, position risk in this case means that the speculator does not want the market to move drastically in either direction, since delta is only a first derivative.

49. Position Risk (cont’d)
Position Risk Example (cont’d)
Profit
Stock Price

50. Position Risk (cont’d)
Position Risk Example (cont’d)
Because of the negative position delta, the curve moves into profitable territory if the stock price declines. If the stock price declines too far, however, the curve will turn down, indicating that large losses are possible.
On the upside, losses occur if the stock price advances a modest amount, but if it really turns up then the position delta turns positive and profits accrue to the position.