Risk-Neutral Drift for an Ornstein–Uhlenbeck Asset Process
Summary
The discussion asks how to change from a physical probability measure to a risk-neutral measure when an asset follows an additive-noise Ornstein–Uhlenbeck-style process. It distinguishes this specification from geometric Brownian motion, where the diffusion term is proportional to the asset price, and addresses the mistaken idea that the volatility coefficient must contain a reciprocal price term.
For a tradable asset, the answer states that its risk-neutral expected return is the risk-free rate, even when its dynamics have additive diffusion. The Brownian motion changes by a drift adjustment proportional to the difference between the risk-free and physical drifts, scaled by the diffusion coefficient; in the setup shown, the asset price also appears in the drift adjustment. This is a concise answer rather than a full derivation. It does not discuss conditions for an equivalent martingale measure, whether the proposed process remains economically sensible across zero, or complications from incomplete markets and state-dependent coefficients.
Key ideas
- A tradable asset's risk-neutral expected return is set to the risk-free rate.
- Additive diffusion does not require rewriting volatility with a reciprocal asset price.
- The measure change shifts Brownian motion by a drift adjustment scaled by volatility.
- The brief answer does not establish the conditions under which the measure change exists.
Tags
Full text
# Change of measure when the underlying dynamic is Ornstein-Uhlenbeck
# Change of measure when the underlying dynamic is Ornstein-Uhlenbeck
Let the $r$ riskless rate to be constant. Let's consider the following underlying dynamic under the $\mathbf{P}$ “physical measure”
$$dS_{t}=\mu_{t}S_{t}dt+\sigma_{t}S_{t}dW_{t}^{\mathbf{P}},$$
where $W^{\mathbf{P}}$ is a Wiener process under $\mathbf{P}$. In many cases this underlying dynamic under the $\mathbf{Q}$ risk neutral measure is simply
$$dS_{t}=rS_{t}dt+\sigma_{t}S_{t}dW_{t}^{\mathbf{Q}},$$
so just the $\mu_{t}$ term is replaced with $r$.
Does the same hold if the underlying dynamic under $\mathbf{P}$ is basically an Ornstein-Uhlenbeck process:
$$dS_{t}=\mu_{t}S_{t}dt+\sigma_{t}dW_{t}^{\mathbf{P}}?$$
Or are these two processes have the same form, just the $\sigma_{t}$ contains an $\frac{1}{S_{t}}$ term?
Even the market price of risk process is so strange in this case and I'm not really sure how to change measure...
Consider the following market, where
$$dS_{t}=\mu_{t}S_{t}dt+\sigma_{t}S_{t}dW_{t}^{\mathbf{P}}$$
is the dynamic of the risky asset,
$$dB_{t}=rB_{t}dt$$
is the dynamic of the riskless asset.
The dynamic of the self-financing replicating portfolio that containts $\beta$ riskless asset and $\gamma$ risky asset is
$$dX_{t} =\beta_{t}dB_{t}+\gamma_{t}dS_{t}=\beta_{t}rB_{t}dt+\gamma_{t}\mu_{t}S_{t}dt+\gamma_{t}\sigma_{t}S_{t}dW_{t}^{\mathbf{P}} =\beta_{t}rB_{t}dt+\gamma_{t}\mu_{t}S_{t}dt+r\gamma_{t}S_{t}dt-r\gamma_{t}S_{t}dt+\gamma_{t}\sigma_{t}S_{t}dW_{t}^{\mathbf{P}} =r\left(\beta_{t}B_{t}+\gamma_{t}S_{t}\right)dt+\gamma_{t}S_{t}\sigma_{t}\left(\frac{\mu_{t}-r}{\sigma_{t}}dt+dW_{t}^{\mathbf{P}}\right) =rX_{t}dt+\gamma_{t}S_{t}\sigma_{t}\left(\frac{\mu_{t}-r}{\sigma_{t}}dt+dW_{t}^{\mathbf{P}}\right).$$
In this case we know how to change measure if it is possible, and the market prcie of risk is $\frac{\mu_{t}-r}{\sigma_{t}}$. But if the dynamic of the risky asset is the Orsntein-Uhlenbeck process as discussed above, then we can't “pull the $S_{t}$ term out of the bracket”, i.e.:
$$dX_{t} =\beta_{t}dB_{t}+\gamma_{t}dS_{t}=\beta_{t}rB_{t}dt+\gamma_{t}\mu_{t}S_{t}dt+\gamma_{t}\sigma_{t}dW_{t}^{\mathbf{P}} =\beta_{t}rB_{t}dt+\gamma_{t}\mu_{t}S_{t}dt+r\gamma_{t}S_{t}dt-r\gamma_{t}S_{t}dt+\gamma_{t}\sigma_{t}dW_{t}^{\mathbf{P}} =r\left(\beta_{t}B_{t}+\gamma_{t}S_{t}\right)dt+\gamma_{t}\sigma_{t}\left(\frac{\mu_{t}S_{t}-rS_{t}}{\sigma_{t}}dt+dW_{t}^{\mathbf{P}}\right) =rX_{t}dt+\gamma_{t}\sigma_{t}\left(\frac{\mu_{t}-r}{\sigma_{t}}S_{t}dt+dW_{t}^{\mathbf{P}}\right).$$
Is there any proper method to change measure in this case? I guess there is, but I'm not sure that in this case it means that “we just have to change the $\mu_{t}$ to $r$”.
## Answer by Frido (score 2, accepted)
https://quant.stackexchange.com/a/76869
Too long for a comment:
> Does the same hold if the underlying dynamic under 𝐏 is basically an Ornstein-Uhlenbeck process:
$$𝑑𝑆_𝑡=𝜇_𝑡𝑆_𝑡𝑑𝑡+\sigma_𝑡𝑑𝑊^𝐏_𝑡?$$
> Or are these two processes have the same form, just the $\sigma_t$ contains an $1/𝑆_𝑡$ term?
If $S$ is a tradable asset its risk-neutral rate of return is $r$, even if $S$ follows an OU process.
There is no $1/S_t$ in $\sigma_t$ for two reasons: 1. There is no $1/S_t$ in $\sigma_t$, and 2. You cannot divide by a process that can take the value $0$.
The change of measure is: $$ dW^P = dW^Q + \frac{rS_t-\mu S_t}{\sigma_t} dt $$ That's all there is to it.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.