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The Radon-Nikodym derivative going from the bank-acount Numeraire $N(t)$ to the bond numeraire $P(t,T)$ is:

$$\frac{dP}{dN}(T|\mathcal{F}_t)=\frac{1}{N(T)P(t,T)}$$

Suppose I now want to price an option on stock $S_t$, but under non-deterministic rates, i.e. $N(t)=e^{\int_0^tr(t)dt}$. I am interested in $C(t_0,T)=\mathbb{E}_{t_0}^N\left[e^{-\int_0^Tr(t)dt}(S_T-K)^+\right]$. Suppose I want to use the Radon-Nikodym from above, I could do the following:

$$C(t_0,T)=P(t_0,T)\mathbb{E}_{t_0}^N\left[P(t_0,T)^{-1}e^{-\int_0^Tr(t)dt}(S_T-K)^+\right]=P(t_0,T)\mathbb{E}_{t_0}^P\left[(S_T-K)^+\right]$$

But now I'd need to figure out the process $S_t$ under $P(t,T)$. Supposing that under $N(t)$, $S_t$ follows the process:

$$S_t=S_0e^{\int_0^tr(t)dt-0.5 \sigma^2 t +\sigma W_t}$$

How can I use the Radon-Nikodym:

$$\frac{1}{N(t) P(t,T)}=\frac{1}{e^{\int_0^tr(t)dt} \mathbb{E}^N[e^{\int_0^tr(t)dt}]}$$

To apply the Cameron-Martin-Girsanov Theorem to the process for $S_t$ under $N_t$?

Here is my attempt:

We want Radon-Nikodym that looks like $exp\left\{\int_0^T \mu (t) dW_t-0.5 \int_0^T \mu^2 (t) dt\right\}$. By CMG Theroem, applying this Radon-Nikodym to a Brownian $W_t$ will result in a new measure under which the same Brownian motion will acquire a drift $\int_0^T \mu(t) dt$.

Instead, we have:

$$exp\left\{-\int_0^Tr(t)dt\right\}*constant$$

Is there a way to proceed further?

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    $\begingroup$ You need a model for the interest rate, such as the Hull-White short rate model. $\endgroup$
    – Gordon
    Commented Jan 4, 2021 at 18:40
  • $\begingroup$ @Gordon I see. In that case $$exp\left\{-\int_0^Tr(t)dt\right\}*constant=exp\left\{-\int_0^T\left(\int_0^t\alpha(\mu-r(s))ds+\int_0^t\sigma dW_s\right)dt\right\}*constant$$ and I need to work out a way to turn that into $$exp\left\{\int_0^T \mu (t) dW_t-0.5 \int_0^T \mu^2 (t) dt\right\}$$ is that how I would need to proceed? Before I give it a shot, is it hard? :) (so I am mentally prepared for either a "long struggle" or I know it's relatively easy) $\endgroup$ Commented Jan 4, 2021 at 20:01
  • $\begingroup$ It is not hard. You basically need to derive the dynamics for $\frac{B_t}{P(t, T)}$ so that you use CMG, where $B_t$ is the bank-account value and $P(t, T)$ is the bond value. $\endgroup$
    – Gordon
    Commented Jan 4, 2021 at 20:36

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