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Chapter 17

Chapter 17. Option Greeks. Greeks. The “Greeks” are sensitivities of option price to model inputs: Delta , denoted ∆ : Measures impact of a "small" change in asset price. Gamma , denoted Γ :

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Chapter 17

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  1. Chapter 17 Option Greeks

  2. Greeks • The “Greeks” are sensitivities of option price to model inputs: • Delta, denoted ∆: • Measures impact of a "small" change in asset price. • Gamma, denoted Γ: • Measures option curvature, and can be used to estimate impact of a "large" change in asset price. • Theta, denoted : • Measures impact of the passage of time. • Vega, denoted : • Measures impact of a change in volatility. • Rho, denoted : • Measures impact of change in the interest rate.

  3. The role of the option Γ will be described shortly. A Broad Overview

  4. Simple Examples

  5. The Principal Option Pricing Derivatives • Delta – three uses: • Measure of option sensitivity • Hedge ratio • Likelihood of becoming in-the-money • Delta is an important by-product of the Black-Scholes model

  6. Measure of Option Sensitivity • Delta is the change in option premium expected from a small change in the stock price • For a call option: • For a put option:

  7. Measure of Option Sensitivity • 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 • The delta of a call is always positive: ∆C≥ 0. • That is, call values increase when the price of the underlying increases. • The delta of a put is always negative: ∆P≤ 0. • That is, put values decrease when the price of the underlying increases. • Moreover, the delta of an option is always less than one in absolute value: ∆C≤ 1 and ∆P ≥ —1, but the sum of the |deltas| = 1 • In the BSOPM, the call delta is exactly equal to N(d1)

  8. Mo Delta • 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 • If an option is deep out-of-the-money, its value is close to zero and not very responsive to changes in the price of the underlying. • Deep OTM options have deltas close to zero. • If an option is deep in-the-money, its value responds almost one-for-one to changes in the price of the underlying. • Deep ITM options have deltas close to one (in absolute value). • In general, as an option moves from deep OTM to deep ITM, its delta (in absolute value) moves from zero towards one.

  9. Using the Option Delta If the stock price changes by a small amount dS, the estimated change in an option price is the option delta × dS: dC = ∆C × dSdP = ∆P × dS For example, suppose a put is trading at $11.45 and has a delta of −0.70. Suppose the price of the underlying increases by $0.50. Then: dS = +0.50 and ∆P = −0.70, so the estimated change in the put value: (−0.70) × (+0.50) = −0.35. Estimated new put value: 11.45 − 0.35 = 11.10.

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

  11. 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 • The passage of time hurts the option holder • The passage of time benefits the option writer

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

  13. Theta (cont’d) Calculating Theta 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).

  14. 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

  15. Gamma • Gamma measures the curvature of the option pricing function. • Curvature is the change in the slope of the function: a function whose slope never changes has no curvature (it is a straight line). • Since the slope is measured by the delta, gamma is the rate of change of the delta. • Intuitively, Γ = Change in Option Delta per $1 change in S

  16. Sign of the Gamma First, consider the sign of the gamma. Call and put deltas both increase as S increases. This means the gammas of both puts and calls are positive. This just says that option prices have positive curvature, or, in mathematical terms, option prices are convex in S.

  17. 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 • 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

  18. Gamma (cont’d) Calculating Gamma For calls and puts, gamma is:

  19. Vega • Vega is the first partial derivative of the change in OP with respect to the volatility of the underlying asset:

  20. Vega • 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, omega, tau, zeta, and sigma prime

  21. Vega (cont’d) Calculating Vega

  22. Rho • Rho is the first partial derivative of the OP with respect to the riskfree interest rate: • 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

  23. The Greeks of Vega • Two derivatives measure how vega changes: • Vommameasures how sensitive vega is to changes in implied volatility • Vannameasures how sensitive vega is to changes in the price of the underlying asset

  24. Great Greeks

  25. Delta Neutrality • 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

  26. Calculating Delta Hedge Ratios 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?

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

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

  29. 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.

  30. 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 • 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

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

  32. 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

  33. 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

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

  35. Combining Directional and Speed Markets

  36. Dynamic Hedging • Assume a dealer sells 1,000 DCRB June 125 calls at the Black-Scholes-Merton price of 13.5533 with a delta of 0.5692. Dealer will buy 569 shares and adjust the hedge daily. • To buy 569 shares at $125.94 and sell 1,000 calls at $13.5533 will require $58,107. • We simulate the daily stock prices for 35 days, at which time the call expires.

  37. Dynamic Delta • The second day, the stock price is 120.4020. There are now 34 days left. Using BSM, we get a call price of 10.4078 and delta of 0.4981. We have • Stock worth 569($120.4020) = $68,509 • Options worth -1,000($10.4078) = -$10,408 • Total of $58,101 • Had we invested $58,107 in bonds, we would have had $58,107e0.0446(1/365) = $58,114. • We must adjust to the new delta of 0.4981. We need 498 shares so sell 71 and invest the money ($8,549) in bonds.

  38. DD • At the end of the second day, the stock goes to 126.2305 and the call to 13.3358. The bonds accrue to a value of $8,550. We have • Stock worth 498($126.2305) = $62,863 • Options worth -1,000($13.3358) = -$13,336 • Bonds worth $8,550 (includes one days’ interest) • Total of $58,077 • Had we invested the original amount in bonds, we would have had $58,107e0.0446(2/365) = $58,121. We are now short by over $44. • At the end we have $59,762, a excess of $1,406.

  39. 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 • Homework: 7, 8, 21, 22, 23, 25, 30

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