Chapter 7 Option Greeks © 2002 South-Western Publishing

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

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

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

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

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

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

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)

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

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

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

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

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

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

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

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).

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

Gamma (cont’d)

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

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

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

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

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

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

Vega (cont’d) Calculating Vega

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

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

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

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

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

Delta Neutrality Introduction Calculating delta hedge ratios Why delta neutrality matters

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

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?

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

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

Calculating Delta Hedge Ratios (cont’d) A Strangle Example (cont’d) The ratio of the two deltas is -.11/.19 = -.58. 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.

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

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

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

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

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

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

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

Dynamic Hedging Introduction Minimizing the cost of data adjustments Position risk

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

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

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

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.

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

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.