1
$\begingroup$

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 $r$”.

$\endgroup$

1 Answer 1

2
$\begingroup$

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.

$\endgroup$
2
  • 2
    $\begingroup$ 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. $\endgroup$
    – Kevin
    Oct 2, 2023 at 10:42
  • 2
    $\begingroup$ @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. $\endgroup$
    – Frido
    Oct 2, 2023 at 11:14

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.