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# The Greeks

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The Greeks are a set of risk measures that describe the sensitivity of options prices to changes in economic factors, including the underlying asset price, volatility, time to maturity and risk-free rate of interest.

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### The Greeks

1. 1. THE GREEKS: A MEASURE OF RISK FOR OPTIONS ALAN ANDERSON, Ph.D. ECI RISK TRAINING www.ecirisktraining.com (c) ECI Risk Training 2009 1 www.ecirisktraining.com
2. 2. THE GREEKS The Greeks are risk measures that describe the sensitivity of option prices to changes in:   the underlying asset price   the volatility of the underlying asset   the risk-free rate of interest   the time to maturity of the option (c) ECI Risk Training 2009 2 www.ecirisktraining.com
3. 3. The Greeks are: • delta • gamma • theta • vega • rho (c) ECI Risk Training 2009 3 www.ecirisktraining.com
4. 4. DELTA The delta of an option is the sensitivity of the option’s price with respect to a change in the price of the underlying asset (c) ECI Risk Training 2009 4 www.ecirisktraining.com
5. 5. For a call option, delta is defined as: ∂C ΔC = ∂S (c) ECI Risk Training 2009 5 www.ecirisktraining.com
6. 6. This represents the change in C with respect to a change in S The delta of a call option can assume a value between 0 and 1 (c) ECI Risk Training 2009 6 www.ecirisktraining.com
7. 7. A call’s delta equals the slope of its price curve: (c) ECI Risk Training 2009 7 www.ecirisktraining.com
8. 8. (c) ECI Risk Training 2009 8 www.ecirisktraining.com
9. 9. Delta is close to zero when the call is deep out of the money, rises to 0.5 when the call is at the money, then moves close to one as the call moves deep into the money (c) ECI Risk Training 2009 9 www.ecirisktraining.com
10. 10. For a put option, delta is defined as: ∂P ΔP = ∂S (c) ECI Risk Training 2009 10 www.ecirisktraining.com
11. 11. The delta of a put option can assume a value between -1 and 0. A put’s delta equals the slope of its price curve; the following diagram shows a European put: (c) ECI Risk Training 2009 11 www.ecirisktraining.com
12. 12. (c) ECI Risk Training 2009 12 www.ecirisktraining.com
13. 13. Delta is close to -1 when the put is deep in the money, moves to -0.5 when the put is at the money, then moves close to zero as the put moves deep out of the money (c) ECI Risk Training 2009 13 www.ecirisktraining.com
14. 14. The price curve of an American put is shown in the following diagram: (c) ECI Risk Training 2009 14 www.ecirisktraining.com
15. 15. (c) ECI Risk Training 2009 15 www.ecirisktraining.com
16. 16. PORTFOLIO DELTA Since delta is a linear measure, the delta of a portfolio of assets is a weighted average of the deltas of the assets in the portfolio (c) ECI Risk Training 2009 16 www.ecirisktraining.com
17. 17. This is computed as follows: n Δ π = ∑ wi Δ i i =1 (c) ECI Risk Training 2009 17 www.ecirisktraining.com
18. 18. where: π = portfolio delta wi = weight of asset i i = delta of asset i (c) ECI Risk Training 2009 18 www.ecirisktraining.com
19. 19. DELTA NEUTRAL A portfolio with a delta of zero is perfectly hedged; its value is unaffected by changes in market prices This portfolio is said to be delta neutral (c) ECI Risk Training 2009 19 www.ecirisktraining.com
20. 20. GAMMA The gamma of an option is the sensitivity of the option’s price with respect to a change in the delta of the option (c) ECI Risk Training 2009 20 www.ecirisktraining.com
21. 21. CALL GAMMA For a call option, gamma is defined as: ∂C ΓC = ∂ Δ( ) = ∂( ) ∂S = ∂ C 2 ∂S ∂S ∂S 2 (c) ECI Risk Training 2009 21 www.ecirisktraining.com
22. 22. PUT GAMMA For a put option, gamma is defined as: ∂P ΓP = ( ) ∂ Δ = ∂( ) ∂S = ∂ P 2 ∂S ∂S ∂S 2 (c) ECI Risk Training 2009 22 www.ecirisktraining.com
23. 23. NOTE Gamma is identical for a call and a put option with the same strike, maturity and underlying asset. Gamma’s value is a function of the moneyness of the option: (c) ECI Risk Training 2009 23 www.ecirisktraining.com
24. 24. Gamma reaches its maximum value when an option is close to being at the money, and declines as the option moves further into or out of the money. These features of gamma can be seen by noting that gamma is the slope of the delta function for both the call and the put option. (c) ECI Risk Training 2009 24 www.ecirisktraining.com
25. 25. Since the delta of the call and put differ by a constant, the slopes of their delta functions are equal. In both cases, the slope of the curve reaches its maximum value near the strike price of the option. (c) ECI Risk Training 2009 25 www.ecirisktraining.com
26. 26. Since the call and put delta function have positive slopes throughout; therefore, gamma is always positive. (c) ECI Risk Training 2009 26 www.ecirisktraining.com
27. 27. Gamma 0.03 0.025 0.02 Gamma 0.015 0.01 0.005 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Stock Price (\$) (c) ECI Risk Training 2009 27 www.ecirisktraining.com
28. 28. NOTE The gamma of the underlying asset is zero. Since a forward contract is a linear instrument, its delta is a constant; therefore, its gamma is also zero. (c) ECI Risk Training 2009 28 www.ecirisktraining.com
29. 29. THETA The theta of an option is the sensitivity of the option’s price with respect to a change in the time to maturity. Theta is also known as the option’s time decay. (c) ECI Risk Training 2009 29 www.ecirisktraining.com
30. 30. NOTE Theta is usually negative; it can be positive for an in-the-money European put on a non- dividend paying stock due to the possibility that it is currently selling for less than its intrinsic value. (c) ECI Risk Training 2009 30 www.ecirisktraining.com
31. 31. Theta’s value declines continuously with the option’s time to maturity. (c) ECI Risk Training 2009 31 www.ecirisktraining.com
32. 32. Call Theta 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 -0.5 -1 -1.5 Theta -2 -2.5 -3 -3.5 Stock Price (\$) (c) ECI Risk Training 2009 32 www.ecirisktraining.com
33. 33. Put Theta 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 -0.5 -1 -1.5 Theta -2 -2.5 -3 -3.5 Stock Price (\$) (c) ECI Risk Training 2009 33 www.ecirisktraining.com
34. 34. VEGA The vega (sometimes known as lambda or kappa) of an option is the sensitivity of the option’s price with respect to a change in the volatility of the underlying asset. (c) ECI Risk Training 2009 34 www.ecirisktraining.com
35. 35. NOTE Vega is identical for a call and a put option with the same strike, maturity and underlying asset. Vega is always positive and is a function of the option’s moneyness. (c) ECI Risk Training 2009 35 www.ecirisktraining.com
36. 36. Vega reaches its maximum value when an option is close to being at the money, and declines as the option moves further into or out of the money (c) ECI Risk Training 2009 36 www.ecirisktraining.com
37. 37. Vega 20 18 16 14 12 Vega 10 8 6 4 2 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Stock Price (\$) (c) ECI Risk Training 2009 37 www.ecirisktraining.com
38. 38. RHO The rho of an option is the sensitivity of the option’s price with respect to a change in the risk-free rate of interest. For a call option, rho is positive; for a put option, rho is negative. (c) ECI Risk Training 2009 38 www.ecirisktraining.com
39. 39. Call Rho 50 45 40 35 30 Rho 25 20 15 10 5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Stock Price (\$) (c) ECI Risk Training 2009 39 www.ecirisktraining.com
40. 40. Put Rho 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 -5 -10 -15 -20 Rho -25 -30 -35 -40 -45 -50 Stock Price (\$) (c) ECI Risk Training 2009 40 www.ecirisktraining.com