Tài chính doanh nghiệp - Chapter 7: Option greeks

© 2004 South-Western Publishing1Chapter 7Option Greeks2OutlineIntroductionThe principal option pricing derivativesOther derivativesDelta neutralityTwo markets: directional and speedDynamic hedging3IntroductionThere are several partial derivatives of the BSOPM, each with respect to a different variable:DeltaGammaThetaEtc.4The Principal Option Pricing DerivativesDeltaMeasure of option sensitivityHedge ratioLikelihood of becoming in-the-moneyThetaGammaSign relationships5DeltaDelta is an important by-product of the Black-Scholes modelThere are three common uses of deltaDelta is the change in option premium expected from a small change in the stock price6Measure of Option SensitivityFor a call option:For a put option:7Measure of Option Sensitivity (cont’d)Delta indicates the number of shares of stock required to mimic the returns of the optionE.g., a call delta of 0.80 means it will act like 0.80 shares of stockIf the stock price rises by $1.00, the call option will advance by about 80 cents8Measure of Option Sensitivity (cont’d)For a European option, the absolute values of the put and call deltas will sum to oneIn the BSOPM, the call delta is exactly equal to N(d1)9Measure of Option Sensitivity (cont’d)The delta of an at-the-money option declines linearly over time and approaches 0.50 at expirationThe delta of an out-of-the-money option approaches zero as time passesThe delta of an in-the-money option approaches 1.0 as time passes10Hedge RatioDelta is the hedge ratioAssume 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 hedge11Likelihood of Becoming In-the-MoneyDelta is a crude measure of the likelihood that a particular option will be in the money at option expirationE.g., a delta of 0.45 indicates approximately a 45 percent chance that the stock price will be above the option striking price at expiration12ThetaTheta is a measure of the sensitivity of a call option to the time remaining until expiration:13Theta (cont’d)Theta is greater than zero because more time until expiration means more option valueBecause time until expiration can only get shorter, option traders usually think of theta as a negative number14Theta (cont’d)The passage of time hurts the option holderThe passage of time benefits the option writer15Theta (cont’d)Calculating Theta For calls and puts, theta is:16Theta (cont’d)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). 17GammaGamma is the second derivative of the option premium with respect to the stock priceGamma is the first derivative of delta with respect to the stock priceGamma is also called curvature18Gamma (cont’d)19Gamma (cont’d)As calls become further in-the-money, they act increasingly like the stock itselfFor out-of-the-money options, option prices are much less sensitive to changes in the underlying stockAn option’s delta changes as the stock price changes20Gamma (cont’d)Gamma is a measure of how often option portfolios need to be adjusted as stock prices change and time passesOptions with gammas near zero have deltas that are not particularly sensitive to changes in the stock priceFor a given striking price and expiration, the call gamma equals the put gamma21Gamma (cont’d)Calculating Gamma For calls and puts, gamma is:22Sign Relationships DeltaThetaGammaLong call+-+Long put--+Short call-+-Short put++-The sign of gamma is always opposite to the sign of theta23Other DerivativesVegaRhoThe greeks of vegaPosition derivativesCaveats about position derivatives24VegaVega is the first partial derivative of the OPM with respect to the volatility of the underlying asset:25Vega (cont’d)All long options have positive vegasThe higher the volatility, the higher the value of the optionE.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 assetVega is also called kappa, omega, tau, zeta, and sigma prime26Vega (cont’d)Calculating Vega27RhoRho is the first partial derivative of the OPM with respect to the riskfree interest rate:28Rho (cont’d)Rho is the least important of the derivativesUnless an option has an exceptionally long life, changes in interest rates affect the premium only modestly29The Greeks of VegaTwo derivatives measure how vega changes:Vomma measures how sensitive vega is to changes in implied volatilityVanna measures how sensitive vega is to changes in the price of the underlying asset30Position DerivativesThe position delta is the sum of the deltas for a particular securityPosition gammaPosition theta31Caveats About Position DerivativesPosition derivatives change continuouslyE.g., a bullish portfolio can suddenly become bearish if stock prices change sufficientlyThe need to monitor position derivatives is especially important when many different option positions are in the same portfolio32Delta NeutralityIntroductionCalculating delta hedge ratiosWhy delta neutrality matters33IntroductionDelta neutrality means the combined deltas of the options involved in a strategy net out to zeroImportant to institutional traders who establish large positions using straddles, strangles, and ratio spreads34Calculating 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? 35Calculating Delta Hedge Ratios (cont’d)A Strangle Example (cont’d) Delta for a call is N(d1):36Calculating Delta Hedge Ratios (cont’d)A Strangle Example (cont’d) For a put, delta is N(d1) – 1. 37Calculating 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. 38Why Delta Neutrality MattersStrategies calling for delta neutrality are strategies in which you are neutral about the future prospects for the marketYou do not want to have either a bullish or a bearish position39Why Delta Neutrality Matters (cont’d)The sophisticated option trader will revise option positions continually if it is necessary to maintain a delta neutral positionA gamma near zero means that the option position is robust to changes in market factors40Two Markets: Directional and SpeedDirectional marketSpeed marketCombining directional and speed markets41Directional MarketWhether we are bullish or bearish indicates a directional marketDelta measures exposure in a directional marketBullish investors want a positive position deltaBearish speculators want a negative position delta42Speed MarketThe speed market refers to how quickly we expect the anticipated market move to occurNot a concern to the stock investor but to the option speculator43Speed Market (cont’d)In fast markets you want positive gammasIn slow markets you want negative gammas44Combining Directional and Speed MarketsDirectional MarketDownNeutralUpSpeed MarketSlowWrite callsWrite straddlesWrite putsNeutralWrite calls; buy putsSpreadsBuy calls; write putsFastBuy putsBuy straddlesBuy calls45Dynamic HedgingIntroductionMinimizing the cost of data adjustmentsPosition risk46IntroductionA position delta will change asInterest rates changeStock prices changeVolatility expectations changePortfolio components changePortfolios need periodic tune-ups47Minimizing the Cost of Data AdjustmentsIt is common practice to adjust a portfolio’s delta by using both puts and calls to minimize the cash requirements associated with the adjustment48Position RiskPosition risk is an important, but often overlooked, aspect of the riskiness of portfolio management with optionsOption derivatives are not particularly useful for major movements in the price of the underlying asset49Position 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. 50Position Risk (cont’d)Position Risk Example (cont’d) ProfitStock Price51Position 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.