The Cameron-Martin-Girsanov (CMG) theorem is a significant result in the fields of stochastic calculus and quantitative finance due to its ability to transform one stochastic process into another through a change of measure. This change of measure greatly simplifies problems involving expectation values involving stochastic variables, allowing us to work with shifted drift terms instead of the original path-dependent variables. In quantitative finance, the CMG theorem is particularly important in the pricing of financial derivatives, risk management, and statistical estimation.
The CMG theorem is an extension of the Girsanov theorem, which states that, given a Brownian motion Wt, one can construct a new Brownian motion W̃t under a different measure by adding a deterministic drift term. Mathematically, this can be written as:
W̃t = Wt + ∫0tusds,
where us is a deterministic function. The Cameron-Martin-Girsanov theorem generalizes this concept to a broader class of processes known as semimartingales.
In quantitative finance, the CMG theorem is commonly used to establish a "risk-neutral" measure for the pricing of financial derivatives. Risk-neutral pricing is a fundamental concept in quantitative finance, allowing for a consistent pricing of derivative contracts by removing the market participants’ risk preferences. This is achieved by assuming that the underlying asset’s drift term is replaced by the risk-free interest rate, making the discounted asset price a martingale under the risk-neutral measure.
To demonstrate this, let’s consider an asset price St modeled by the following stochastic differential equation (SDE) under the real-world probability measure P:
dSt = St[(μ − r)dt + σdWtP],
where St is the asset price, t is the time, μ is the drift term representing the asset’s expected return, σ is the asset’s volatility, WtP is a standard Brownian motion under the real-world measure P, and r is the risk-free interest rate.
Applying the CMG theorem, we can define a new probability measure, the risk-neutral measure Q, in relation to the real-world measure P through the Radon-Nikodym derivative, denoted by Zt:
$$Z_t = \frac{dQ}{dP} |_{\mathcal{F}_t} = \exp \left( -\theta \sigma W_t^P - \frac{1}{2} \theta^2 \sigma^2 t \right),$$
where $\theta = \frac{\mu - r}{\sigma}$ and ℱt is the filtration (the set of available information) at time t. Under this new measure Q, the Brownian motion WtQ is related to the real-world Brownian motion, WtP, as follows:
dWtQ = dWtP + θdt.
Substituting this modified Brownian motion into the SDE for the asset price in equation (2), we obtain:
dSt = St[(μ − r)dt + σ(dWtQ − θdt)].
Simplifying, the asset price dynamics under the risk-neutral measure Q become:
dSt = St(rdt + σdWtQ).
Notice that under this new measure, the asset price drift has been replaced with the risk-free interest rate r, and the process is now a martingale when discounted by the risk-free interest rate. Consequently, the CMG theorem allows us to derive option pricing formulas, such as the famous Black-Scholes-Merton model, and value other financial derivatives in a risk-neutral manner.
In summary, the Cameron-Martin-Girsanov theorem is of fundamental importance in both stochastic calculus and quantitative finance. By transforming the drift terms of stochastic processes, the CMG theorem enables the establishment of risk-neutral measures, greatly simplifying problems in derivative pricing, risk management, and statistical estimation.