Girsanov Changes of Measure from Real-World to Risk-Neutral Probabilities

Summary

The document asks how to obtain a risk-neutral probability measure from a real-world model under a no-arbitrage assumption. It gives a geometric Brownian motion example in which an asset has drift and volatility under the real-world measure, while a deterministic bond serves as numeraire. The stated change of measure uses a Doléans–Dade exponential with the market price of risk, the drift in excess of the risk-free rate divided by volatility.

Girsanov’s theorem and the martingale characterization of Brownian motion are identified as tools behind the construction. The author then asks whether the measure change can be expressed directly from the asset process without explicitly parameterizing drift and volatility, whether the method extends to semimartingales, and what the largest class of models admitting an explicit transformation is. The excerpt poses these questions but provides no answers or conditions, so it serves as a theoretical prompt rather than a general recipe.

Key ideas

• A risk-neutral measure can be related to a real-world measure using a change of measure under a no-arbitrage setup.
• For geometric Brownian motion, the proposed density uses excess drift divided by volatility.
• The bank account is used as the numeraire in the stated setup.
• Girsanov’s theorem and martingale characterization are cited as proof tools.
• The document leaves open how broadly the explicit construction extends beyond the specified diffusion model.

Tags

# Largest class of real world probability models admitting explicit risk-neutral change of measure


# Largest class of real world probability models admitting explicit risk-neutral change of measure












Assume we have two assets, a random asset $A_t$ and deterministic risk-free bond $B_t = e^{rt}$. Let $P$ be a model of the real-world probabilities of $S$ and $Q$ the unique associated risk-neutral measure associated to the numeraire $B$ under the no arbitrage assumption.

If $P$ is determined via $A$ solving the SDE $dA = \mu A dt + \sigma A dW$ then $Q$ can be computed explicitly from $P$ via $dQ = \mathcal{E}(-\int_0^{\cdot} \frac{\mu-r}{\sigma} dW)dP$ where $\mathcal{E}$ is the Doleans-Dade exponential.

The main tools in the proof of this are Girsanov's theorem, which not SDE-specific (https://en.wikipedia.org/wiki/Girsanov_theorem#Statement) and proving the "demeaned" process is a martingale (via Levy's characterization of Brownian motion).

Question 1 $\mu$, $\sigma$, and $r$ are the parameters of the model of the system ($S$, $B$). Can $Q$ be computed directly from $P$ and $S$ without first explictly parameterizing $P$ (eq. $S$) through $\mu$, $\sigma$, and $r$?

Question 2 When $A$ is a semimartingale, such as when $A$ solves an SDE, can we still obtain an explicit change of measure?

Question 3 What is the largest class of real-world measures for which we can compute $Q$ explicitly in terms of $P$?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.