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									Options: Greeks Cont’d
      Hedging with Options
 Greeks (Option Price Sensitivities)
   delta, gamma    (Stock Price)
   theta (time to expiration)
   vega (volatility)
   rho (riskless rate)
                  Gamma
 Gamma is change in Delta measure as
 Stock Price changes
              N’(d1)
      = ---------------
            S**t
Where
                    e-(x^2)/2
        N’(x) =   -------------
                     (2)
              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!
           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.
                   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
                               Theta
 Call Theta calculation is:
Note: Calc is Theta/Year, so divide by 365 to get option value loss per day
       elapsed

          S * N’(d1) * 
  c = - ---------------------- - r * X * e-rt * N(d2)
               2 t

Note:
            S * N’(d1) * 
  p = - ---------------------- + r * X * e-rt * N(-d2)
                2 t
                     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.
                  Vega

    = S *  t * N’(d1)


For a given set of option model inputs,
the call vega equals the put vega!
                     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
                           Rho
 Like vega, measures % change for each
  percentage point increase in the
  anticipated riskless rate.

        c = X * t * e-rt * N(d2)

Note:
        p = - X * t * e-rt * N(-d2)
      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.
   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
           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.
         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
            Straddle Example
 Buy 10 calls and 10 puts
     Cost = (10 * $1.47 * 100) + (10 * $1.32 * 100)
     Cost = 2790
          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!
                    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).
                      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
                   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

								
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