# 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 $$r$$”.

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.

• I guess, however, when Ornstein-Uhlenbeck is used to model the underlying asset, it's not always a traded asset (interest rates, volatility, some commodities, etc.) such that a different risk-neutralisation strategy is necessary. The tradability of the underlying is key. Oct 2 at 10:42
• @Kevin Yes exactly, good to stress that point. An example of using OU to model a tradable asset is when the OU asset is a model for the difference between two stock prices. Oct 2 at 11:14