1. Options: 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  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

7. 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

8. 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% in value for each percentage point increase in the anticipated volatility of the underlying asset.

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

10. 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

11. 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 )

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

14. Delta – Neutral  Consider our strategy of a long Straddle:  A long Put and a long Call, both at the same exercise price.  What we are interested in is the Stock price movement, either way, and with symmetric returns.

15. Straddle Example  Intel at $20, with riskless rate at 3% and time to maturity of 3 months. Volatility for Intel is 35%.  Calls (w/ X=20) at $1.47  Puts (w/ X=20) at $1.32

16. Straddle Example  Buy 10 calls and 10 puts  Cost = (10 * $1.47 * 100) + (10 * $1.32 * 100)  Cost = 2790

17. Straddle Example  Intel  $22, C = $2.78, P = $0.63  Value = (10 * 2.78 * 100) + (10 *.63 * 100)  Value = $3410  Gain = $620  Intel  $18, C = $0.59, P = $2.45  Value = (10 * 0.59 * 100) + (10 * 2.45 * 100)  Value = $3040  Gain = $250  More Gain to upside so actually BULLISH!

18. Delta - Neutral  Delta of Call is 0.5519  Delta of Put is -0.4481  Note: Position Delta = (10*100*.5519) + (10*100* -0.4481) = +103.72  BULLISH!  Delta Ratio is: 0.4481 / 0.5519 = 0.812 which means we will need.812 calls to each put (or 8 calls and 10 puts).

19. Delta - Neutral Straddle Example  Buy 8 calls and 10 puts  Cost = (8 * $1.47 * 100) + (10 * $1.32 * 100)  Cost = 2496 Note: Position Delta = (8*100*.5519) + (10*100* -0.4481) = -6.65  Roughly Neutral

20. Delta - Neutral Straddle Example  Intel  $22, C = $2.78, P = $0.63  Value = (8 * 2.78 * 100) + (10 *.63 * 100)  Value = $2854  Gain = $358  Intel  $18, C = $0.59, P = $2.45  Value = (8 * 0.59 * 100) + (10 * 2.45 * 100)  Value = $2922  Gain = $426  Now Gains roughly symmetric; delta-neutral