Girsanov theorem and risk neutral dynamics

Definition Fix a finite horizon $T$, a filtration $(\mathcal{F}_t)_{0 \le t \le T}$ satisfying the usual conditions and a standard Brownian motion $W$ under $\mathbb{P}$ with respect to it. Let $\lambda$ be predictable with $\int_0^T \lambda_s^2 \, ds < \infty$ almost surely, let $Z = \mathcal{E}(-\int \lambda \, dW)$ be its stochastic exponential from Exponential martingales and measure changes, and assume that $Z$ is a true martingale on $[0, T]$, that is $\mathbb{E}^{\mathbb{P}}[Z_T] = 1$, for example because the Novikov condition holds. Define $\mathbb{Q}$ on $\mathcal{F}_T$ by

Girsanov’s theorem says that $\mathbb{Q}$ is equivalent to $\mathbb{P}$ on $\mathcal{F}_T$ and that the process

for $0 \le t \le T$ is a standard Brownian motion under $\mathbb{Q}$ with respect to the same filtration. The signs belong together. A density with $-\int \lambda \, dW$ in its exponent adds $\int \lambda \, ds$ to $W$, while a density written as $\mathcal{E}(\int \theta \, dW)$ makes $W_t - \int_0^t \theta_s \, ds$ the $\mathbb{Q}$-Brownian motion. In $d$ dimensions, with $W$ a $d$-dimensional Brownian motion and $\lambda$ predictable with values in $\mathbb{R}^d$, the density exponent is $-\int \lambda^{\top} dW - \tfrac12 \int \lvert \lambda \rvert^2 \, ds$ and $W + \int \lambda \, ds$ is a $d$-dimensional $\mathbb{Q}$-Brownian motion. The proof has the following structure. The added integral has finite variation, so $[W^{\mathbb{Q}}]_t = [W]_t = t$, and because $\mathbb{Q} \ll \mathbb{P}$, sums of squared increments that converge in $\mathbb{P}$-probability converge in $\mathbb{Q}$-probability to the same limit. In the product rule for $Z W^{\mathbb{Q}}$ the term $Z_t \, dW^{\mathbb{Q}}_t$ contains $Z_t \lambda_t \, dt$, and the covariation $d[Z, W^{\mathbb{Q}}]_t = -Z_t \lambda_t \, dt$ cancels it, leaving

So $Z W^{\mathbb{Q}}$ is a $\mathbb{P}$-local martingale, the conditional Bayes formula applied after localization makes $W^{\mathbb{Q}}$ a continuous $\mathbb{Q}$-local martingale, and Lévy’s characterization identifies it as a $\mathbb{Q}$-Brownian motion. The localization details and the general semimartingale version of the theorem are further study.

Definition Lévy’s characterization, stated here without proof, says the following. Let $M$ be a continuous process adapted to $(\mathcal{F}_t)$ with $M_0 = 0$ that is a local martingale under a probability measure $\mathbb{Q}$, and suppose that $[M]_t = t$ for $0 \le t \le T$. Then $M$ is a standard Brownian motion under $\mathbb{Q}$ with respect to $(\mathcal{F}_t)$, so for $s < t$ the increment $M_t - M_s$ is $N(0, t - s)$ and independent of $\mathcal{F}_s$. For a continuous local martingale the condition $[M]_t = t$ is equivalent to $M_t^2 - t$ being a local martingale. In $d$ dimensions the condition is $[M^i, M^j]_t = \delta_{ij} \, t$, where $\delta_{ij}$ is $1$ for $i = j$ and $0$ otherwise. Each hypothesis matters. Quadratic variation is the same under equivalent measures, but the local martingale property depends on the measure, and the conclusion fails for processes with jumps.

Definition Two probability measures on $\mathcal{F}_T$ are equivalent when they have the same null events. The Girsanov measure is equivalent to $\mathbb{P}$ on $\mathcal{F}_T$, because its density $Z_T$ is an exponential and hence strictly positive, and its $\mathbb{P}$-mean is $1$ by hypothesis. Equivalence makes every almost sure statement the same under both measures. Paths remain continuous, and for an Itô process $dX_t = b_t \, dt + \sigma_t \, dW_t$ the quadratic variation $[X]_T = \int_0^T \sigma_t^2 \, dt$, a limit in probability of sums of squared increments, is the same under both. A payoff that is nonnegative almost surely and positive with positive probability under $\mathbb{P}$ has the same two properties under $\mathbb{Q}$. This is why a pricing measure must be equivalent. If the discounted value of a self-financing strategy starts at $0$ and is a $\mathbb{Q}$-martingale, its terminal value has $\mathbb{Q}$-mean $0$, so it cannot be nonnegative almost surely and positive with positive probability, and the strategy is not an arbitrage. Distributions, drifts and expectations do change. Equivalence is also a finite horizon statement. For a constant $\lambda \ne 0$ the strong law of large numbers gives $W_t/t \to 0$ almost surely under $\mathbb{P}$, while under any measure that makes $W_t + \lambda t$ a Brownian motion for all $t \ge 0$ it gives $W_t/t \to -\lambda$ almost surely, so no such measure on the whole infinite horizon is equivalent to $\mathbb{P}$.

Definition Let the hypotheses of Girsanov’s theorem hold with the density exponent $-\int \lambda \, dW$, and let $X$ be an Itô process with

under $\mathbb{P}$, where $b$ and $\sigma$ are adapted and measurable with $\int_0^T \lvert b_t \rvert \, dt < \infty$ and $\int_0^T \sigma_t^2 \, dt < \infty$ almost surely. Substituting $dW_t = dW^{\mathbb{Q}}_t - \lambda_t \, dt$ gives

The drift changes by $-\sigma_t \lambda_t$, while the diffusion coefficient and the quadratic variation stay the same. The new drift is integrable, since $\int_0^T \lvert \sigma_t \lambda_t \rvert \, dt \le (\int_0^T \sigma_t^2 \, dt)^{1/2} (\int_0^T \lambda_t^2 \, dt)^{1/2}$ by the Cauchy-Schwarz inequality. With this construction a prescribed $\mathbb{Q}$-drift $\tilde b_t$ is reached only through a $\lambda$ with $\sigma_t \lambda_t = b_t - \tilde b_t$, which needs $\sigma_t \ne 0$ wherever the two drifts differ, and that $\lambda$ must still give a true martingale density. Writing a new drift into an equation does not by itself construct a measure.

Definition Consider a dividend-free stock with $dS_t = \mu_t S_t \, dt + \sigma_t S_t \, dW_t$ under $\mathbb{P}$, where $\sigma_t > 0$, and a money-market account $B_t = \exp(\int_0^t r_s \, ds)$ in a frictionless market. Under a Girsanov measure the discounted price satisfies

so it can be a $\mathbb{Q}$-local martingale only if its drift vanishes, which forces

This is the market price of risk of $W$, the expected return in excess of the cash rate per unit of volatility, measured per square-root year. For $n$ dividend-free assets driven by a $d$-dimensional Brownian motion, with $dS^i_t = S^i_t \bigl(\mu^i_t \, dt + \sum_j \Sigma^{ij}_t \, dW^j_t\bigr)$ and an $n \times d$ loading matrix $\Sigma_t$ whose rows are assets and whose columns are risk factors, the condition becomes

If it has no solution, no measure of this Girsanov form makes every discounted price a local martingale. With constant coefficients there is then a vector $w$ with $w^{\top} \Sigma = 0$ and $w^{\top}(\mu - r \mathbf{1}) > 0$, and holding the currency amounts $w$ in the assets, with the balance in cash, gives a riskless portfolio that grows faster than cash, an arbitrage. If it has several solutions, some directions of $\lambda$ are not fixed by the traded assets. A solution must still give a true martingale density, and when the coefficients are random the discounted prices must also be checked to be true $\mathbb{Q}$-martingales.

Definition With the money-market account as numeraire, risk-neutral dynamics are the dynamics of a model under an equivalent measure $\mathbb{Q}$ for which the discounted price of every traded asset is a $\mathbb{Q}$-martingale, or for an asset that pays dividends, its discounted gains with the dividends reinvested. For the dividend-free stock with $\sigma_t > 0$, Girsanov’s theorem with $\lambda_t = (\mu_t - r_t)/\sigma_t$ gives

with the same volatility as under $\mathbb{P}$. With constant $r$ and $\sigma$ on $[0, T]$,

and the discounted price $e^{-rt} S_t = S_0 \exp(\sigma W^{\mathbb{Q}}_t - \tfrac12 \sigma^2 t)$ is a true $\mathbb{Q}$-martingale. For a stock with a continuous dividend yield $q$ and ex-dividend $\mathbb{P}$-drift $\mu$, the total return has drift $\mu + q$, the market price of risk is $(\mu + q - r)/\sigma$, and the ex-dividend price has $\mathbb{Q}$-drift $r - q$. The physical drift disappears from the $\mathbb{Q}$-dynamics, which is why it does not enter option prices in the complete constant-coefficient model of the next unit. That the Girsanov measure is the only equivalent martingale measure when the filtration is generated by $W$ rests on the martingale representation theorem, which is further study.