0
THE GREEKS: A MEASURE
 OF RISK FOR OPTIONS


 ALAN ANDERSON, Ph.D.
   ECI RISK TRAINING
  www.ecirisktraining.com

       ...
THE GREEKS

The Greeks are risk measures that describe the
sensitivity of option prices to changes in:


      the underl...
The Greeks are:

         • delta
         • gamma
         • theta
         • vega
         • rho
                      (...
DELTA


The delta of an option is the sensitivity of
the option’s price with respect to a change in
the price of the under...
For a call option, delta is defined as:


                ∂C
           ΔC =
                ∂S


                        ...
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 an...
A call’s delta equals the slope of its price curve:




                                        (c) ECI Risk Training 2009...
(c) ECI Risk Training 2009
8
     www.ecirisktraining.com
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 clos...
For a put option, delta is defined as:


                    ∂P
               ΔP =
                    ∂S


             ...
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 fo...
(c) ECI Risk Training 2009
12
      www.ecirisktraining.com
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 ze...
The price curve of an American put
is shown in the following diagram:




                               (c) ECI Risk Trai...
(c) ECI Risk Training 2009
15
      www.ecirisktraining.com
PORTFOLIO DELTA


Since delta is a linear measure, the delta of a
portfolio of assets is a weighted average of
the deltas ...
This is computed as follows:

                   n
          Δ π = ∑ wi Δ i
                  i =1




                   ...
where:


         π   = portfolio delta
    wi = weight of asset i

         i   = delta of asset i

                     ...
DELTA NEUTRAL


A portfolio with a delta of zero is perfectly
hedged; its value is unaffected by changes in
market prices
...
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...
CALL GAMMA

For a call option, gamma is defined as:

                ∂C
   ΓC =
        ∂ Δ( )
            =
             ...
PUT GAMMA

For a put option, gamma is defined as:

                  ∂P
     ΓP =
            ( )
          ∂ Δ
          ...
NOTE

Gamma is identical for a call and a put option
with the same strike, maturity and underlying
asset.


Gamma’s value ...
Gamma reaches its maximum value when an
option is close to being at the money, and
declines as the option moves further in...
Since the delta of the call and put differ by a
constant, the slopes of their delta functions are
equal.


In both cases, ...
Since the call and put delta function have
positive slopes throughout; therefore, gamma
is always positive.




          ...
Gamma


         0.03




        0.025




         0.02
Gamma




        0.015




         0.01




        0.005




...
NOTE

The gamma of the underlying asset is zero.
Since a forward contract is a linear instrument,
its delta is a constant;...
THETA

The theta of an option is the sensitivity of the
option’s price with respect to a change in the time
to maturity.

...
NOTE

Theta is usually negative; it can be positive
for an in-the-money European put on a non-
dividend paying stock due t...
Theta’s value declines continuously
with the option’s time to maturity.




                                 (c) ECI Risk ...
Call Theta


          0
               5   10   15   20   25   30   35   40   45    50     55   60   65   70   75     80 ...
Put Theta


          0
               5   10   15   20   25   30   35   40   45    50    55   60   65   70   75     80   ...
VEGA


The vega (sometimes known as lambda or
kappa) of an option is the sensitivity of the
option’s price with respect to...
NOTE

Vega is identical for a call and a put
option with the same strike, maturity and
underlying asset.


Vega is always ...
Vega reaches its maximum value when an
option is close to being at the money, and
declines as the option moves further int...
Vega


       20


       18


       16


       14


       12
Vega




       10


        8


        6


        4


...
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 inter...
Call Rho


      50


      45


      40


      35


      30
Rho




      25


      20


      15


      10


      ...
Put Rho


       0
            5   10   15   20   25   30   35   40   45    50   55   60   65   70   75     80    85    90...
Upcoming SlideShare
Loading in...5
×

The Greeks

1,494

Published on

The Greeks are measures of the sensitivity of option prices to changes in underlying variables. These are useful for risk management purposes.

Published in: Economy & Finance, Business
0 Comments
3 Likes
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total Views
1,494
On Slideshare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
Downloads
123
Comments
0
Likes
3
Embeds 0
No embeds

No notes for slide

Transcript of "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
  1. A particular slide catching your eye?

    Clipping is a handy way to collect important slides you want to go back to later.

×