Black-Scholes Model

The cornerstone of modern quantitative finance. This section explores the pricing of European options and the fundamental stochastic calculus concepts that make it possible.

1. Martingales and Risk-Neutral Pricing

The entire Black-Scholes pricing framework relies on changing the probability measure (via Girsanov's Theorem) so that the discounted stock price becomes a Martingale.

A Martingale is a stochastic process where the conditional expectation of the next value, given all prior information, is equal to the present value. In mathematical terms:

E^{Q} [e^{- r T} S_{T} ∣ F_{t}] = e^{- r t} S_{t}

Under the real-world measure $P$ , the stock grows at a drift rate $μ$ . However, under the risk-neutral measure $Q$ , the drift is replaced by the risk-free rate $r$ . This allows us to price any derivative as the discounted expected value of its future payoff:

C (S_{t}, t) = e^{- r (T - t)} E^{Q} [max (S_{T} - K, 0) ∣ F_{t}]

2. Ito Isometry

When calculating the variance of the payoff or the variance of the hedging P&L, we frequently encounter the expectation of squared stochastic integrals. Ito Isometry is a crucial theorem that simplifies these calculations.

It states that the expected value of the square of an Ito integral with respect to Brownian motion is equal to the expected value of the standard Riemann integral of the squared integrand:

E (\int_{0}^{T} Δ_{t} d W_{t})^{2} = E [\int_{0}^{T} Δ_{t}^{2} d t]

This property is essential for proving that the Black-Scholes delta hedge perfectly replicates the option variance, ensuring that the portfolio is truly risk-free in continuous time.