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Options: Greeks Contd

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Rho is sensitivity of Option Price to changes in the riskless rate ... Intel at $20, with riskless rate at 3% and time to maturity of 3 months. ... – PowerPoint PPT presentation

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Title: Options: Greeks Contd


1
Options Greeks Contd
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(d1)
  • ? ---------------
  • S ? ? t
  • Where
  • e-(x2)/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(d1) ?
  • ?c - ---------------------- - r X
    e-rt N(d2)
  • 2? t
  • Note
  • S N(d1) ?
  • ?p - ---------------------- r X
    e-rt N(-d2)
  • 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(d1)
  • 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 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(d2)
  • Note
  • ?p - X t e-rt N(-d2)

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 options parameter to another
    options 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/ X20) at 1.47
  • Puts (w/ X20) 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
  • (10100.5519) (10100 -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
  • (8100.5519) (10100 -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
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