1. © Stefano Grazioli - Ask for permission for using/quoting: grazioli@virginia.edu Gamma Hedging Wilhelm's Warriors No Hedgetation The Gobs of Money Machine The Ira Harris Experience

2.  Easy meter  Last homework is due on Friday

3. N AME : O VERALL CONTRIBUTION TO TEAM SUCCESS T EAM W ORK Q UALITY OF WORK A MOUNT OF WORK K NOWLEDGE OF THE PROJECT AREA I DEAS CONTRIBUTED TO THE PROJECT O RGANIZATION OF TEAM WORK ( ADMIN.) T EAM R APPORT L EADERSHIP E NTHUSIASM, ATTITUDE, INITIATIVE R ESPECT FOR OTHERS D EPENDABILITY, GOOD TEAM PLAYER M EETING ATTENDANCE, PUNCTUALITY May affect your grade

5.  Name  A couple of things that you are learning from the class  Things that you like/that can be improved  Is the class getting you to think on your own?  Concerns about the HT

6. © Stefano Grazioli - Ask for permission for using/quoting: grazioli@virginia.edu Gamma, Vega, Theta & Rho

7. Stock price Payoff Put portfolio Call portfolio Source: Delta Gamma Hedging and the Black-Scholes Partial Differential Equation - S. Raju, JEFE 2012

8.  Delta (  ) measures the change in portfolio value as the underlier’s price S changes (~speed).  Gamma (  ) measures the rate of change in portfolio value as S changes (~acceleration).

9.  Gamma = N’(d1) S  t  N’(d1) = e –(d1) 2 /2  (2  )  d1 as in Black Scholes Strike Stock price S Gamma

10.  portfolio =  qty i *  i

11.  During small periods of time t 2 -t 1  portfolio = ½   portfolio  (S 2 -S 1 ) 2 Example: If you do not rebalance, the underlier price (e.g., GE) drops from $52 to $50, the change of value in a Delta-Neutral Portfolio is approximately = ½   (S 2 -S 1 ) 2 = 0.5 * (-10,000) * (52-50) 2 = -$20,000 ~

12.  If Gamma is small (abs.), small changes in S will not affect much Delta (and your portfolio value), so there is less need to take immediate rebalancing action.  If Gamma is large, small changes in S will affect Delta (and your portfolio value) significantly, so there is a stronger need to take immediate rebalancing action.

13. Stock price Payoff Put portfolio Call portfolio Source: Delta Gamma Hedging and the Black-Scholes Partial Differential Equation - S. Raju, JEFE 2012

14.  More stable than a delta neutral-only.  Cannot use the stock to reach Gamma neutrality because the stock has Gamma = 0 1)  portfolio <> 0 2)  portfolio +  x qty x = 0 3) qty x = -  portfolio /  x  Warning: Acquiring qty x will disturb Delta neutrality. You will need to rebalance.

15. It’s a financial strategy, not a sorority.  Find out what you need to achieve Delta neutrality.  Find out what you need to achieve Gamma neutrality.  Find out what you need to re-achieve Delta neutrality. Stock is ideal because it will not affect Gamma.

16. 1. Simultaneous Delta Gamma 2. Conditional Gamma 3. Extreme transaction costs minimization Come and see me – not on the last day!

17.  portfolio +  x1 qty x1 = 0  portfolio +  x2 qty x2 = 0 {

18. Conceptually similar to Delta  Theta = change in portfolio value when time changes  Vega = change in portfolio value when the volatility changes  Rho = change in portfolio value when the rate of interest changes

19. © Stefano Grazioli - Ask for permission for using/quoting: grazioli@virginia.edu What Is New In Technology?

21. SP500SP500 beta + noise STOCKS Bids and Asks OPTIONS Bids and Asks B.S.+ noise real g, real volatility real initial prices real beta, real volatility NOT random!