Change of measure when the underlying dynamic is Ornstein-Uhlenbeck

Question

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 1

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.