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# Options

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