Option Risk Mangement The Greeks

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Effect of Variables on Option Pricing

Call Put Variable Higher

S0

T

r D

+ + + + + –

+ –

– +

DELTA

THETA

VOLATILITY

INTEREST RATE

DIV or INCOME

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Option Risk Sensitivities “The Greeks”

• Following the definitions of each greek are two charts: – How that greek evolves as underlying price changes – How that greek evolves over time

• The base case option for all greek values: – $65 asset price – $65 stock price – 60 days to expiration – 3.50% cost to carry – 31% volatility

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Delta

• Delta is rate of change in the price of the option for a one unit change in the price of the underlying asset

• The SECOND definition of delta is the probability the option expiring IN THE MONEY – Therefore ATM = .50 delta – As call option moves in the money the delta approaches

+1.00 – As put option moves in the money the delta approaches -

1.00

• Analogous to duration for a bond

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Delta

5

Delta

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Gamma

• Gamma is the rate of change of delta for a one unit change in underlying asset

• Gamma is GREATEST ATM – As asset trades away from ATM, Gamma depreciates

• Gamma is GREATEST for short dated options

– As option approaches expiration if the option is close to ATM,

– The curvature becomes VERY extreme • Gamma addresses delta hedging errors caused by curvature

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Gamma Over Price

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Gamma Over Time

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Theta

• Theta is the rate of change of the price of an option with respect to passage of time

• known as Time decay; typically shown as one day decay

• Theta and Gamma are related – Longer dated ATM option = low theta & low gamma – Shorter dated ATM option = high theta & high gamma

• Special note on Theta: Option decay is adjusted coming into the weekend.

Depends on when that market is closing and what price action is expected

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Theta Over Price

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Theta Over Time

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Vega

• Vega (n) is the rate of change of the value of an option (or derivatives portfolio) with respect to a 1% change in volatility

• Vega is highest when option is at-the-money

• Vega is greater for longer term options

• Vega is smaller for shorter term options

• Note that Vega risk is opposite to Theta/Gamma risk – As expiration approaches, Vega risk declines and Theta/Gamma

risk rises

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Vega Over Price

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Vega Over Time

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Rho

• Rho is the change in the price of the options for a 1% increase in interest rates (cost to carry)

• Rho is the change in the net cost to carry the underlying asset to expiration – cost to borrow money (borrow stock) minus dividend

(Ex date)

• Synthetic forward price will change to reflect new actual forward price of asset – i.e., long call + short put = long synthetic asset – Call & put will change by the delta weighted amount of

change in cost to carry

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