No-Arbitrage Principle and Risk-Neutral Valuation › Risk-Neutral Valuation in Continuous Time

The discrete binomial framework extends naturally to continuous time, where calculus replaces algebra and Brownian motion replaces discrete jumps. The core insight remains the same: we can change probability measures to simplify derivative pricing.

In continuous time, the transition from real-world to risk-neutral probabilities involves changing the drift of the Brownian motion. This change of measure, which generalizes what we did in the discrete binomial model, is accomplished through the Girsanov theorem. Under the real-world measure $P$, the stock follows:

$$dS_t = \mu S_t \, dt + \sigma S_t \, dW_t^P$$

where:

• $dS_t$: change in stock price
• $S_t$: stock price at time $t$
• $\mu$: expected return under measure $P$
• $dt$: small time increment
• $\sigma$: volatility
• $dW_t^P$: Brownian increment under measure $P$

Here the drift $\mu$ reflects the stock's actual expected return, which typically exceeds the risk-free rate due to the equity risk premium.

The Girsanov theorem, a fundamental result in stochastic calculus, tells us we can change to an equivalent measure $Q$ under which:

$$dS_t = r S_t \, dt + \sigma S_t \, dW_t^Q$$

where:

• $dS_t$: change in stock price
• $r$: risk-free interest rate
• $S_t$: stock price at time $t$
• $dt$: small time increment
• $\sigma$: volatility
• $dW_t^Q$: Brownian increment under risk-neutral measure $Q$

The drift changes from $\mu$ to $r$, but the volatility $\sigma$ remains unchanged. The change of measure affects the expected return but not the magnitude of the random fluctuations. Under this risk-neutral measure, the stock earns the risk-free rate, and we can price any derivative as:

$$f_0 = e^{-rT} E^Q[f_T]$$

where:

• $f_0$: price of the derivative
• $r$: risk-free interest rate
• $T$: time to expiration
• $E^Q[\cdot]$: expectation under risk-neutral measure $Q$
• $f_T$: payoff at expiration

The relationship between the two Brownian motions involves the market price of risk $\lambda = (\mu - r)/\sigma$:

$$dW_t^Q = dW_t^P + \lambda \, dt$$

where:

• $dW_t^Q, dW_t^P$: Brownian increments under measures $Q$ and $P$
• $\lambda$: market price of risk
• $dt$: small time increment

This quantity measures the excess return per unit of volatility that investors demand for bearing stock price risk. A high market price of risk indicates that investors require substantial compensation for bearing uncertainty. The Girsanov change of measure essentially absorbs this risk premium into the probability measure itself, transforming a world where risky assets earn excess returns into a world where everything earns the risk-free rate.