7 Option Models, Volatility, and the Greeks
7.1 From a contingent payoff to a current price
The Black–Scholes framework prices a European option under assumptions about the underlying price process and continuously adjustable hedging. For a non-dividend-paying stock, a European call has theoretical value
\[C=S_0N(d_1)-Ke^{-rT}N(d_2),\]
with
\[d_1=\frac{\log(S_0/K)+(r+\tfrac12\sigma^2)T}{\sigma\sqrt T},\quad d_2=d_1-\sigma\sqrt T.\]
The formula is a model relationship, not proof that volatility or continuous hedging behaves as assumed.
7.2 Greeks as local sensitivities
- Delta measures first-order sensitivity to underlying price.
- Gamma measures how delta changes with underlying price.
- Theta summarizes sensitivity to the passage of time under a convention.
- Vega measures sensitivity to implied volatility.
- Rho measures sensitivity to the assumed rate.
These are local derivatives and can change materially as the market moves. Implied volatility is the volatility parameter consistent with an observed option price under a chosen model; it is not a direct measurement of future realized volatility.
7.3 Beyond Black–Scholes
Trees, simulation, finite-difference methods, and stochastic-volatility models handle different features and constraints. Models for American-style options may need to account for optimal early exercise.
Exercise: Explain why two calls with the same delta need not have the same gamma.
Living chapter. This is an initial working treatment. Derivations, examples, and references will be expanded in future editions.