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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

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
• 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
• Buy 10 calls and 10 puts
• Cost (10 1.47 100) (10 1.32 100)
• Cost 2790

17
• 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
• 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