1. The Greek Letters Chapter 14

2. The Greeks are coming!
Parameters of SENSITIVITY
Delta = 
Gamma = 
Theta = 
Vega = 
Rho = 

3. Example:
S=100; K = 100; r = 8%; T-t =180 days;
 = 30%.
Call Put
Premium
Delta = 
Theta = 
Vega = 
Rho = 
Gamma = 

4. The GREEKS are measures of sensitivity
The GREEKS are measures of sensitivity. The question is how sensitive the position’s value is to changes in any of the variables that contribute to the position’s market value.These variables are S, X, T-t, r and the return’s standard deviation, s.
DELTA  =
The Delta of any position measures the $ change/share in the position’s value that ensues a “small” change in the value of the underlying.

5. Delta (See Figure 14.2, page 302)
Delta (D) is the rate of change of the option price with respect to the underlying
Option
price A B
Slope = D
Stock price

6. The stock: (S) = S/S = 1
In mathematical terms DELTA is the first derivative of the option’s price with respect to S. As such, Delta carries the units of the option’s price; I.e., $ per share.
A Call: (c) = c/S
A Put: (p) = p/S
The stock: (S) = S/S = 1

7. (c) = c/(T-t) (p) = p/(T-t)
THETA 
Theta measures the sensitivity of the option’s price to a “small” change in the time remaining to expiration:
(c) = c/(T-t) (p) = p/(T-t)
Theta is given in terms is $/1 year. Thus, if (c) = - $ /year, it means that if time to expiration increases (decreases) by one year, the call price will increase (decreases) by $ Or, /365 = 3.34 cent per day.

8. (S) = (S)/S = 2S/ S2 = 0.
GAMMA 
Gamma measures the change in delta when the market price of the undelrlying asset changes.  =
Gamma is the second derivative of the option’s price with respect to the underlying price.
(c) = (c)/S = 2c/ S2
(p) = (p)/S = 2p/ S2
(S) = (S)/S = 2S/ S2 = 0.

9. Thus, Vega is in terms of $/1% change in S.
Vega measures the sensitivity of the option’s market price to “small” changes in the volatility of the underlying asset’s return.
Vega =
(c) = c/s
(p) = p/s
Thus, Vega is in terms of $/1% change in S.

10. Rho is in terms of $/%change of r.
Rho measures the sensitivity of the option’s price to “small changes in the rate of interest. (c) = c/r
(p) = p/r
Rho = 
Rho is in terms of $/%change of r.

11. DELTA-NEUTRAL POSITIONS Vposition = Sn(S) + cn(c)
We just sold n(c) overvalued calls and we wish to protect the profit against possible adverse move of the underlying asset price. To do so, we intend to purchase share of the underlying in a quantity that GUARANTIES that a small price change will not have any impact on the short call and long shares position.
Vposition = Sn(S) + cn(c)
(V) = n(S) + (c)n(c)

12. DELTA-NEUTRAL POSITIONS Vposition = Sn(S) + cn(c)
(V) = n(S) + (c)n(c)
In order to have DELTA-neutral position we solve the above equation for (V) = 0.
The solution is: n(S) = - n(c) (c)
The negative sign of the RHS of the solution indicates that the calls and the underlying asset must be held in opposite direction.

13. EXAMPLE:
We just sold 10 CBOE calls whose delta is $.50/shares. Each call covers 100 shares. How many shares of the underlying stock we must purchase in order to create a delta-neutral position? n(s) = - n(c)(c).
(c) = 0,50 and n(c) = 10 but every call covers 100 shares. Therfore: n(c) = - 1,000 shares.
n(s) = - [ - 1,000(0,50)] = 500.
The DELTA-neutral position consists of of 500 long shares and 10 short calls.

14. DELTA NEUTRAL POSITIONS
AGAIN, A DELTA-NEUTRAL PORTFOLIO IS DEFINED BY ITS
DELTA(PORTFOLIO) = 0.
IN OUR CASE OF CALLS AND SHARES:
(V) = 0  n(S) + (c) n(c) = 0.
THE NUMBER OF SHARES OF THE UNDERLYING TO BE PURCHASED IS:
n(S) = - n(c)(C).

15.  = n(S) + (c)n(c) + (p)n(p).
EXAMPLE: We just sold 20 calls and 20 puts whose deltas are $.7/share and -$.3/share, respectively. Every call and every put covers 100 shares. How many shares of the underlying stock we must purchase in order to create a delta-neutral position?
the position’s delta is:
 = n(S) + (c)n(c) + (p)n(p).
n(S)+(.7)(-2,000)+(-.3)(-2,000)=0
n(S) = 800.

16. Portfolio: The portfolio consisting of 20 short calls, 20 short puts and 800 long shares is
delta- neutral.
Price/share: $1 -$1
shares +$800 -$800
calls -$1,400 +$1,400
Puts +$600 -$600
Portfolio $0 $0

17. 2. Using the Black and Scholes formula, it is possible to show that:
Results:
The deltas of a call and a put on the same underlying asset, (with the same time to expiration and the same exercise price) must satisfy the following equality:
(c) = 1 + (p)
2. Using the Black and Scholes formula, it is possible to show that:
(c) = n(d1)  < (c) < 1
(p) = n(d1) - 1  -1 < (p) < 0

18. EXAMPLES
1. Long 100 shares of the underlying stock, long one put and short one call on this stock is always delta-neutral:
(position) = [n(d1) – 1](100)
+ n(d1)(-100) = 0.
2. The delta of a long STRADDLE long 15 puts and long 15 calls (same K and T-t) with (c) = .64 and (p) = is: (100)[.64 + (- .36)] =$ 420/share.

19. 3. A financial institution holds:
5,000 CBOE calls long; delta .4,
6,000 CBOE puts long; delta -.7,
10,000 CBOE puts short; delta -.5,
Long 100,000 shares
(portfolio) = (.4)500,000 + (-.7)600,000
- (-.5)1,000, ,000
= $380,000/share
S  $1 => V(portfolio)  $380,000.

24. (S) = (S)/S = 2S/ S2 = 0.
GAMMA 
Gamma measures the change in delta when the market price of the undelrlying asset changes.
In mathematical terms Gamma is the second derivative of the option’s price with respect to the underlying price.
(c) = (c)/S = 2c/ S2
(p) = (p)/S = 2p/ S2
(S) = (S)/S = 2S/ S2 = 0.

25. GAMMA 
In general, the Gamma of any portfolio is the change of the portfolio’s delta due to a “small” change in the underlying’s price.
As the second derivative of the option’s price with respect to S, Gamma measures the sensitivity of the option’s price to “large” underlying asset’s price changes.

26. Interpretation of Gamma
For a delta neutral portfolio,
dP » Q dt + ½GdS 2 dP dP dS dS
Positive Gamma
Negative Gamma

27. Clearly, the derivatives of these deltas with respect to S are equal.
Result: The Gammas of a put and a call are equal. Using the Black and Shcoles model:
(c) = n(d1) (p) = n(d1) – 1.
Clearly, the derivatives of these deltas with respect to S are equal.
EXAMPLE: (c) = .70, (p) = and let gamma be
Holding a short call and a long put has:  = (- .30) = and
 = – = 0.

28. simultaneously! This portfolio is
EXAMPLE:
(c) = .70, (p) = and let gamma be
Holding the underlying asset long, a long put and a short call yields a portfolio with:
= (- .30) = 0 and
 = , ,2345 = 0,
simultaneously! This portfolio is
delta-gamma-neutral.

32. Thus, Vega is in terms of $/1% change in s.
Vega measures the sensitivity of the option’s market price to “small” changes in the volatility of the underlying asset’s return.
(c) = c/s
(p) = p/s
Thus, Vega is in terms of $/1% change in s.

37. THETA 
Theta measures the sensitivity of the option’s price to a “small” change in the time remaining to expiration:
(c) = c/(T-t)
(p) = p/(T-t)
Theta is given in terms is $/1 year. Thus, if (c) = - $20/year, it means that if time to expiration increases (decreases) by one year, the call price will increase (decreases) by $20. Or, 20/365 = 5.5 cent per day.

42. Rho is in terms of $/%change of r.
Rho measures the sensitivity of the option’s price to “small changes in the rate of interest. (c) = c/r
(p) = p/r
Rho is in terms of $/%change of r.

47. SUMMERY OF THE GREEKS
Position Delta Gamma Vega Theta
LONG STOCK
SHORT STOCK
LONG CALL
SHORT CALL
LONG PUT
SHORT PUT

48. The sensitivity of portfolios
1. A portfolio is a combination of securities - assets and options.
All the sensitivity measures are mathematical derivatives.
Theorem(Calculus): The derivative of a combination of functions is the combination of the derivatives of these functions: The sensitivity measure of a portfolio of securities is the portfolio of these securities’ sensitivity measures.

49. Example:The DELTA of a portfolio of 5 long calls, 5 short puts and 100 shares of the stock long:
(portfolio) = (5c - 5p + 100S)
= (5c - 5p + 100S)/S
= 5c/S - 5p/S + 100
= 5c - 5p + 100
This delta reveals the $/share change in the portfolio value as a function of a “small” change in the underlying price

50. = $4.33/barrel and  = - $0.46/barrel.
Example: the price of oil is S = $28.57/barrel. Call Delta Gamma
A $0.63/bbl $0.22/bbl
B $0.45/bbl $0.34/bbl
C $0.82/bbl $0.18/bbl
Portfolio: Long:3 calls A; 2 calls C; 5 barrels of oil. Short: 10 calls B.
 = 3(0.63)+ 2(0.82) + 5(1) – 10(0.45)
= 3(0.22)+ 2(0.34) + 5(0) – 10(0.18)
= $4.33/barrel and  = - $0.46/barrel. .

51. The portfolio’s Delta and Gamma are
= 4.33 means that a “small” change of the oil price, say one cent per barrel, will change the value of the above portfolio by 4.33 cents in the same direction.
 = means that a “small” change in the oil price, say one cent per barrel, will change the delta by about half a cent in the opposite direction.

52. GREEKS BASED STRATEGIES
Greeks based strategies are opened and maintained in order to attain a specific level of sensitivity. Mostly, these strategies are set to attain zero sensitivity. What follows, is an example of strategies that are:
Delta-neutral
Delta-Gamma-neutral
Delta-Gamma-Vega-Rho-neutral

53. EXAMPLE: The underlying asset is the S&P100 stock index
EXAMPLE: The underlying asset is the S&P100 stock index. The options on this index are European.
S = $300; K = $300; T = 1yr;  = 18%; r = 8%; q = 3%.
C = $28.25.
= .6245
= .0067
= .0109
 = .0159

54. DELTA-NEUTRAL
Short the call. W0 = - 1. Long WS = of the index.
Case A1: S increases from $300 to $301.
Portfolio Initial Value New value Change
Call - $ $ $.63
(.6245)S $ $ $.62
Error: - $.01
Case A2: S decreases from $300 to $299.
Portfolio Initial value New value Change
Call - $ $ $.63
(.6245)S $ $ $.62
Error: + $.01

55. Case B1: S increases from $300 to $310.
Portfolio Initial Value New value Change
Call - $ $ $6.56
(.6245)S $ $ $6.24
Error: - $.32
The point here is that Delta has changed significantly and does not apply any more.
S = $300 $301 $310
 =
We conclude that the delta-neutral portfolio must be adjusted for “large” changes of the underlying asset price.

56. Call #0 Call #1
S = $300 S = $300
K = $300 K = $305
T = 1yr T = 90 days
= 18% r = 8% q = 3%
c = $ c = $10.02
 =  = .4952
 =  = .0148
 =  = .0059
 =  = .0034

57. A DELTA-GAMMA-NEUTRAL PORTFOILO
 = 0: WS + W0(.6245) + W1(.4952) = 0
= 0: W0(.0067) + W1(.0148) = 0
Solution:
W0 = -1
W1 = - (.0067)(-1)/.0148 = .453
WS = - (.6245)(-1) – (.453)( = .4
Short the initial call : W0 =
Long .453 of call #1 W1 =
Long .4 of the index WS = .400

58. THE DELTA-GAMMA-NEUTRAL PORTFOLIO
Case A1: S increases from $300 to $301.
Portfolio Initial value New value Change
1.0 # $ $ $.63
(.453)# $ $ $.23
(.4)S $ $ $.40
Error: 0
Case B1: S increases from $300 to $310.
Portfolio Initial value New value Change
1.0 #0 - $ $ $6.56
(.453)#1 $ $ $2.57
(.4)S $ $ $4.00
Error: + $.01

59. the volatility and the risk-free rate
If we examine the exposure level to all parameters, however, we observe that:
Risk
.4000
.400S
.0015
.0027
.0067
.2245
.453(#2)
-.6245
-1.00(#0)
Rho
Vega
Gamma
Delta
Portfolio
The above numbers reveal that the Delta-Gamma-neutral portfolio is exposed to risk associated with
the volatility and the risk-free rate

60. Delta-neutral portfolio with volatility: 12%, 18% Y 24%,
- $8.82
- $2.67
$1.72
$330
- $7.69
- $1.90
$2.89
$325
- $7.24
- $1.24
$3.94
$320
- $6.89
- $0.71
$4.84
$315
- $6.67
- $0.32
$5.57
$310
- $6.56
- $0.08
$6.09
$305
0.00
$6.40
$300
- $6.70
- $0,08
$6.47
$295
- $6.97
- $0,35
$6.29
$290
- $7.38
- $0,79
$5.82
$285
- $7.92
- $1,42
$5.08
$280
- $8.61
- $2,24
$4.05
$275
- $9.45
- $3,26
$2.73
$270
24%
18%
12%
S s

61. The Delta-Gamma-neutral portfoli0 Volatility 12%, 18% Y 24%,
- $5.56
- $0.01
$4.38
$330
- $5.78
$0.01
$4.80
$325
- $5.99
$5.19
$320
- $6.17
$5.56
$315
- $6.34
$5.89
$310
- $6.48
0.00
$6.17
$305
- $6.56
$6.40
$300
- $6.62
$6.55
$295
- $6.63
$6.62
$290
- $0.04
$6.57
$285
- $0.12
$6.38
$280
- $0.25
$6.04
$275
- $6.64
- $0.45
$5.54
$270
24%
18%
12%
S s

62. Case C1: S increases from $300 to $310 and simultaneously,
r increases from 8% to 9%.
Portfolio Initial value New value Change
1.0 (#0) - $ $ $4.80
(.453) # $ $ $2.37
(.4)S $ $ $4.00
Error: - $1.57

63. Delta-Gamma-Vega-Rho-neutral portfolio
CALL K
T(days)
Volatility 18% 18% % 18%
r % % % %
Dividends 3% % % %
c $ $ $ $18.59

64. Delta-Gamma-Vega-Rho-neutral portfolio
CALL     S

65. The DELTA-GAMMA-VEGA-RHO-NEUTRAL-PORTFOLIO Rho =  = 0 simultaneoulsy
In order to neutralize the portfolio to all risk exposures, following the sale of the initial call, we determine the portfolio’s proportions such that all the portfolio’s sensitivity parameters are zero simultaneously.
Delta =  = 0 and Gamma = = 0 and
Theta =  = 0 and Vega =  = 0 and
Rho =  = 0 simultaneoulsy

66. Delta =  = 0
WS+W0(.6245)+W1(.4952)+W2(.6398)+W3(.5931) = 0
Gamma =  = 0
W0(.0067)+W1(.0148)+W2(.0138)+W3(.0100) = 0
Vega =  = 0
W0(.0109)+W1(.0059)+W2(.0055)+W3(.0080) = 0
Rho =  = 0
W0(.0159)+W1(.0034)+W2(.0044)+W3(.0079) = 0

67. We short the call 0, I.e.,
W0 = - 1 and we solve the simultaneous equations. The solutions is:
W0 = short call #0
W1 = long .840 call #1
W2 = short call #2
W3 = long call #3
WS = long .212 of the index

68. To see what this solution means in practical terms, multiply all the weights by 10,000. The portfolio becomes:
Short 100 CBOE calls #0;
Long 84 calls #1;
Short 190 calls #2;
Long 204 calls #3;
Long 2,120 units of the index.
Every index unit is $100, so buy $212,000 worth of the index.

69. THE DELTA-GAMMA-VEGA-RHO- NEUTRAL PORTFOLIO
Case D: S increases from $300 to $ r increases from 8% to 9% s increases from 18% to 24%
Portfolio Initial value New value Change
1.0(#0) $ $ $14.56
(.212)S $ $ $2.12
(840)# $ $ $8.02
(-1.9)# $ $ $19.92
(2.04)# $ $ $24.25
Error: $.09