1. Greeks Cont’d

2. Hedging with Options  Greeks (Option Price Sensitivities)  delta, gamma (Stock Price)  theta (time to expiration)  vega (volatility)  rho (riskless rate)

3. Gamma  Gamma is change in Delta measure as Stock Price changes N’(d 1 )  = --------------- S *  *  t Where e -(x^2)/2 N’(x) = -------------  (2  )

4. Gamma Facts  Gamma is a measure of how often option portfolios need to be adjusted as stock prices change and time passes  Options with gammas near zero have deltas that are not particularly sensitive to changes in the stock price  For a given set of option model inputs, the call gamma equals the put gamma!

5. Gamma Risk  Delta Hedging only good across small range of price changes.  Larger price changes, without rebalancing, leave small exposures that can potentially become quite large.  To Delta-Gamma hedge an option/underlying position, need additional option.

6. Theta  Call Theta calculation is: Note: Calc is Theta/Year, so divide by 365 to get option value loss per day elapsed S * N’(d 1 ) *   c = - ---------------------- - r * X * e -rt * N(d 2 ) 2  t Note: S * N’(d 1 ) *   p = - ---------------------- + r * X * e -rt * N(-d 2 ) 2  t

7. Theta  Theta is sensitivity of Option Price to changes in the time to option expiration  Theta is greater than zero because more time until expiration means more option value, but because time until expiration can only get shorter, option traders usually think of theta as a negative number.  The passage of time hurts the option holder and benefits the option writer

8. Vega = S *  t * N’(d 1 ) For a given set of option model inputs, the call vega equals the put vega!

9. Vega  Vega is sensitivity of Option Price to changes in the underlying stock price volatility  All long options have positive vegas  The higher the volatility, the higher the value of the option  An option with a vega of 0.30 will gain 0.30 cents in value for each percentage point increase in the anticipated volatility of the underlying asset.

10. Rho  Like vega, measures % change for each percentage point increase in the anticipated riskless rate.  c = X * t * e -rt * N(d 2 ) Note:  p = - X * t * e -rt * N(-d 2 )

11. Rho  Rho is sensitivity of Option Price to changes in the riskless rate  Rho is the least important of the derivatives  Unless an option has an exceptionally long life, changes in interest rates affect the premium only modestly

12. General Hedge Ratios  Ratio of one option’s parameter to another option’s parameter:  Delta Neutrality:  Option 1 /  Option 2  Remember Call Hedge + (1/  C ) against 1 share of stock….Number of Calls was hedge ratio + (1/  C ) as Delta of stock is 1 and delta of Call is  C.

13. Rho, Theta, Vega Hedging  If controlling for change in only one parameter, # of hedging options:   Call /  Hedging options for riskless rate change,   Call /  Hedging options for time to maturity change,  Call / Hedging options for volatility change  If controlling for more than one parameter change (e.g., Delta-Gamma Hedging):  One option-type for each parameter  Simultaneous equations solution for units