# Stock price under Bond numeraire

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 $$\int_0^T \mu(t) dt$$.

$$exp\left\{-\int_0^Tr(t)dt\right\}*constant$$
• @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) Jan 4, 2021 at 20:01
• 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. Jan 4, 2021 at 20:36