Black-Scholes Option Pricing and Greeks

1. Black-Scholes Formula

For a European call and put option, the Black-Scholes formulas are:

Call Option:
\[ C = S_0 \Phi(d_1) - K e^{-rT} \Phi(d_2) \]

Put Option:
\[ P = K e^{-rT} \Phi(-d_2) - S_0 \Phi(-d_1) \]

Where:

\( d_1 = \frac{\ln(S_0 / K) + (r + \sigma^2 / 2)T}{\sigma \sqrt{T}} \)
\( d_2 = d_1 - \sigma \sqrt{T} \)
\( \Phi(\cdot) \) is the cumulative standard normal distribution.

Assumptions:

Underlying follows geometric Brownian motion: \( dS = \mu S dt + \sigma S dW_t \)
No dividends, constant interest rate \( r \)
No arbitrage, frictionless markets, European-style options

2. Derivation Sketch

Using risk-neutral valuation and Ito's Lemma on a portfolio hedged with \(\Delta = \frac{\partial C}{\partial S}\), we derive the Black-Scholes PDE:

\[ \frac{\partial C}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} + rS \frac{\partial C}{\partial S} - rC = 0 \]

Solving this PDE with terminal condition \( C(S,T) = \max(S - K, 0) \) gives the Black-Scholes formula.

3. Greeks (Sensitivities)

Delta ( \( \Delta \) )

Sensitivity to underlying asset price.

\[ \Delta_{\text{call}} = \Phi(d_1), \quad \Delta_{\text{put}} = \Phi(d_1) - 1 \]

Gamma ( \( \Gamma \) )

Sensitivity of delta to price changes.

\[ \Gamma = \frac{\phi(d_1)}{S_0 \sigma \sqrt{T}}, \quad \text{same for calls and puts} \]

Vega

Sensitivity to volatility.

\[ \text{Vega} = S_0 \phi(d_1) \sqrt{T} \]

Theta ( \( \Theta \) )

Time decay.

\[ \Theta_{\text{call}} = -\frac{S_0 \phi(d_1) \sigma}{2\sqrt{T}} - rK e^{-rT} \Phi(d_2) \] \[ \Theta_{\text{put}} = -\frac{S_0 \phi(d_1) \sigma}{2\sqrt{T}} + rK e^{-rT} \Phi(-d_2) \]

Rho ( \( \rho \) )

Sensitivity to interest rate.

\[ \rho_{\text{call}} = K T e^{-rT} \Phi(d_2), \quad \rho_{\text{put}} = -K T e^{-rT} \Phi(-d_2) \]

4. Notes

Greeks are used for hedging: Delta for directional risk, Gamma for convexity, Vega for vol risk.
Put-Call parity: \( C - P = S_0 - K e^{-rT} \)
Black-Scholes fails for American options with early exercise or dividends.