Option Greeks

• The value of an option depends on its strike, the stock price, the time to maturity, the stock volatility, and the interest rate.
• For example, the Black-Scholes formulas are formulas for the value of European options in terms of these inputs.
• Derivatives of the option value with respect to the inputs are called Greeks.

• Derivative in the stock price is delta (\(\Delta\))
• Second derivative in the stock price is gamma (\(\Gamma\))
• Derivative in interest rate is rho (\(\rho\))
• Minus derivative in time to maturity is theta (\(\Theta\))
• Derivative in volatility is vega (\(\mathcal{V}\))

Delta

• Definition of delta mirrors binomial model:

Replication and hedging

• Delta is a replication ratio. The replicating portfolio is a hedge for someone who has written the option.
• To replicate a call,
  • invest value of call and buy delta shares of the stock, using your investment and borrowing the rest
• To replicate a put,
  • invest value of put and short \(|\text{delta}|\) shares, investing proceeds plus your investment at risk-free rate

Call option and replicating portfolio

Put option and replicating portfolio

Gamma

• Gamma is how much the delta changes when the stock price changes.
• Notice option value lies above replicating portfolio. This is called convexity. More convexity \(\Leftrightarrow\) higher gamma.
• If you write an option and hedge with the replicating portfolio, you will be “short gamma” or “short convexity.” You will lose if there is a big change in the stock price.
• Offsetting this possibility is the fact that you win from time decay (theta).

Theta and vega

• Theta \(<0\) because option values fall as time passes, holding everything else constant.
• Vega \(>0\) because option values rise when volatility increases
• Volatility risk (exposure to changing volatility) is very important in options trading, though volatility is assumed to be constant in the Black-Scholes model.