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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’(d 1 )

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Hedging with options l.jpg
Hedging with Options

  • Greeks (Option Price Sensitivities)

    • delta, gamma (Stock Price)

    • theta (time to expiration)

    • vega (volatility)

    • rho (riskless rate)


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Gamma

  • Gamma is change in Delta measure as Stock Price changes

    N’(d1)

     = ---------------

    S *  *  t

    Where

    e-(x^2)/2

    N’(x) = -------------

     (2)


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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!


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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.


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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


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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


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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.


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Vega

 = S *  t * N’(d1)

For a given set of option model inputs, the call vega equals the put vega!


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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


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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)


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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.


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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


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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.


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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


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Straddle Example

  • Buy 10 calls and 10 puts

    • Cost = (10 * $1.47 * 100) + (10 * $1.32 * 100)

    • Cost = 2790


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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!


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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).


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Delta - NeutralStraddle 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 example20 l.jpg
Delta - NeutralStraddle 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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