1. Financial Engineering
Lecture 5

2. Greeks Delta Gamma Theta Vega Rho
The Greeks measure the various risks of an option.
Every option has risk related to the variables contained in the price of the option.
Knowing these risks allows us to create strategies and select the best option to include in the strategy.

3. Greeks
Delta - D
the rate of price change for an option, relative to the price change in the underlying asset
Also called the Hedge Ratio
D = N(d1) = DC / DP

4. Greeks
Gamma -G
The rate of change in delta, relative to the price change of the asset
G = N(d1)p - N(d1)p+1

5. Greeks
Theta - Q
The rate of change in the option price relative to a one day change in expiration
Also called Time Decay
Q = Ct - Ct-1

6. Greeks
Vega - L
The rate of change in the option price relative to a 1% change in the volatility
L = Cv - Cv+.01

7. Greeks
Rho
The rate of change in the option price relative to a 1% increase in the discount rate
Rho = Cr - Cr+.01

8. Greeks Example - original data Call = 1.70 r = 10%
Stock = 36 time = 90/365 days
Strike = 40 volatility = .40
Delta = N(d1) = .3794
Gamma = N(d1)p - N(d1)p+1
= =

9. Greeks = 1.695-1.725 = .0300 example - continued Theta = Ct - Ct-1 =
= = (daily)
= x 260 = (annual)
Vega = Cv - Cv+.01
= =
Rho = = Cr - Cr+.01 (using NUMA Web Option Calculator)
= = .0300

10. Delta Spread “volatility spread” or “playing the deltas”
The “most neutral” position you can create
Goal - To capture the time premium, with minimum market risk
Neutral Ratio = Delta Long Position
Delta Short Position

11. Delta Spread example price = 44 delta Long Apr 40C = 5 .40
Short Apr 45C = 3 .25
Assume 2:1 ratio
Max Profit = Net Credit + (# long x (S2-S1)
= (5) = 6
Break Even = (MP / # naked ) + high strike
= (6 / 1) + 45 = 51

12. Delta Spread example - continued
Neutral Ratio = .40 /.25 = 1.60 or 8 to 5
Long 5 = 5x5 = - 25
Short 8= 3x8 = Net Debit = -1
MP = x5 = 24
BE = 24/ = 53

13. Delta Spread 24
2:1 6 1 -1

14. Delta Spread 24
8:5 6 1 -1