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Risk-neutral pricing

Derivative pricing in modern quantitative finance is primarily an arbitrage story, not a forecasting story. The key idea is that (under standard assumptions) we can price without ever estimating the real-world drift \(\mu\).

The martingale pricing principle

Fix a strictly positive traded asset as a numéraire. In the simplest single-currency setting we use the money-market account

\[ B_t = \exp\!\left(\int_0^t r_u\,du\right), \]

and with constant \(r\),

\[ B_t = e^{rt}. \]

In an arbitrage-free market, there exists an equivalent martingale measure \(\mathbb{Q}\) such that every traded asset price, when discounted by the numéraire, is a \(\mathbb{Q}\)-martingale. For a stock \(S_t\),

\[ \frac{S_t}{B_t} \text{ is a } \mathbb{Q}\text{-martingale.} \]

Equivalently, for \(0\le t\le T\),

\[ \frac{S_t}{B_t} = \mathbb{E}^{\mathbb{Q}}\!\left[\left.\frac{S_T}{B_T}\right|\mathcal{F}_t\right]. \]

Pricing formula

Let a claim pay \(H_T\) at time \(T\). If the market is arbitrage-free (and complete in the classic Black–Scholes setup), then its price process \(V_t\) satisfies

\[ V_t = B_t\,\mathbb{E}^{\mathbb{Q}}\!\left[\left.\frac{H_T}{B_T}\right|\mathcal{F}_t\right] = \mathbb{E}^{\mathbb{Q}}\!\left[\left.D(t,T)\,H_T\right|\mathcal{F}_t\right], \]

where \(D(t,T)=\frac{B_t}{B_T}=\exp\big(-\int_t^T r_u\,du\big)\) is the stochastic discount factor.

That is the “one-line” recipe behind most models: simulate / compute under \(\mathbb{Q}\) and discount.

Change of measure and the Radon–Nikodym derivative

A change of measure from \(\mathbb{P}\) to \(\mathbb{Q}\) is described by a Radon–Nikodym derivative (density) process \((Z_t)_{t\ge 0}\) such that

\[ \left.\frac{d\mathbb{Q}}{d\mathbb{P}}\right|_{\mathcal{F}_t} = Z_t,\qquad Z_t>0,\qquad \mathbb{E}^{\mathbb{P}}[Z_t]=1. \]

Intuitively: \(Z_t\) reweights paths so that the discounted traded assets lose their drift and become martingales.

From \(\mu\) to \(r\) in Black–Scholes (Girsanov in one page)

Under the real-world measure \(\mathbb{P}\), the Black–Scholes model assumes

\[ dS_t = \mu S_t\,dt + \sigma S_t\,dW_t^{\mathbb{P}},\qquad S_0>0, \]

with constant \(\sigma>0\).

Define the (constant) market price of risk

\[ \lambda := \frac{\mu-r}{\sigma}. \]

Then the exponential martingale

\[ Z_t = \exp\!\left(-\lambda W_t^{\mathbb{P}} - \tfrac{1}{2}\lambda^2 t\right) \]

defines an equivalent measure \(\mathbb{Q}\) via \(d\mathbb{Q}=Z_T\,d\mathbb{P}\) (for each fixed horizon \(T\)). Under \(\mathbb{Q}\), Girsanov’s theorem implies

\[ dW_t^{\mathbb{Q}} = dW_t^{\mathbb{P}} + \lambda\,dt \]

is a Brownian motion, and substituting into the SDE yields

\[ dS_t = r S_t\,dt + \sigma S_t\,dW_t^{\mathbb{Q}}. \]

So the drift becomes \(r\) under the pricing measure.

Practical notes

  • Dividends: with a continuous dividend yield \(q\), the risk-neutral drift is \(r-q\).
  • Different numeraires: changing numéraire (e.g., to a forward measure) changes \(\mathbb{Q}\) but keeps prices invariant.
  • Curves: in real markets you typically use the appropriate discount curve for \(D(t,T)\); the martingale idea remains the same.

Where to go next

  • The change-of-measure step uses Brownian motion; see Brownian motion and Itô.
  • The resulting stock dynamics under \(\mathbb{Q}\) are solved explicitly in GBM.