Changing the Numeraire to a Stock and the Stock-Ratio Martingale

# Stock Price as Numeraire, Two Stocks & One Money Market Account


# Stock Price as Numeraire, Two Stocks & One Money Market Account












We have two uncorrelated Stock price processes and the classical Money-Market (MM) account. Under the MM Numeraire, both stocks are Martingales when discounted by the MM, as usual.

Question: I would like to change the Numeraire to one of the stocks and apply the Fundamental Theorem of Asset pricing to show that the stock, when discounted by the other stock as Numeraire, is still a martingale.

I fail to show that, here goes my attempt:

Assume that we have two stocks, $X(t)$ and $Y(t)$ and the usual deterministic Money Market Account $M(t)$. Under the risk-neutral measure $Q$ associated with $M(t)$, let the processes for $X(t)$ and $Y(t)$ be as follows:

$$ X(t)=X(0)+\int^{t}_{0}r X(h)dh+\int^{t}_{0}\sigma_x X(h)dW_x(h) $$

$$ Y(t)=Y(0)+\int^{t}_{0}r Y(h)dh+\int^{t}_{0}\sigma_y X(h)dW_y(h) $$

Above, $W_x$ and $W_y$ are two uncorrelated standard Wiener processes under $Q$. Changing the Numeraire to $X(t)$ gives the following Radon-Nikodym:

$$ \frac{dQ_{X(t)}}{dQ}=\frac{M(t_0)}{M(t)}* \frac{X(t)}{X(t_0)} = \frac{e^{rt-0.5\sigma_x^2t+\sigma_xW_t}}{e^{rt}} = e^{-0.5\sigma_x^2t+\sigma_xW_t} $$

We see that the Radon-Nikodym can be directly applied via the Cameron-Martin-Girsanov theorem to both $W_x$ and $W_y$, whereby it will add drift of $\sigma_xt$ to both Brownians under the measure $Q_{X(t)}$. Consequently, the processes for $X_t$ and $Y_t$ under $Q_{X(t)}$ become:

$$ X_t = X_0exp{(rt+0.5\sigma_x^2t+ \sigma_x \tilde{W_x}(t)}) $$

$$ Y_t = Y_0exp{(rt-0.5\sigma_y^2t+ \sigma_x \sigma_y t+ \sigma_y \tilde{W_y}(t)}) $$

Above, $\tilde{W_y}(t)=W_y(t)-\sigma_xt$ is a standard Wiener process under the measure $Q_{X(t)}$ and also $\tilde{W_x}(t)=W_x(t)-\sigma_xt$ is another (uncorrelated) standard Wiener proces under the same measure $Q_{X(t)}$.

Now will the ratio of $X_t$ and $Y_t$ be a Martingale under $Q_{X(t)}$?

Let's see. First note that the ratio $\frac{Y(t)}{X(t)}$ itself will require the application of Ito's Lemma. Let $F(X_t,Y_t)=\frac{Y(t)}{X(t)}$, application of two-dimensional Ito's lemma yields:

$$ F(t)=F(0)+\int^{t}_{0}\left(r_y - r_x + \sigma_x^2-\rho \sigma_x\sigma_y\right) F(h)dh+\int^{t}_{0}\sigma_x F(h)d\tilde{W}_x(h)+\int^{t}_{0}\sigma_y F(h)d\tilde{W}_Y(h) =\\F(0)+\int^{t}_{0}\left(\sigma_x\sigma_y\right) F(h)dh+\int^{t}_{0}\sigma_x F(h)d\tilde{W}_x(h)+\int^{t}_{0}\sigma_y F(h)d\tilde{W}_Y(h) = \\ F_0exp\left( \left[\sigma_x\sigma_y-0.5\left(\sigma_x^2 + \sigma_y^2 \right) \right]t+ \sigma_x \tilde{W}_x + \sigma_y \tilde{W}_y \right) $$

Now:

$$\mathbb{E}^{Q_{X(t)}}{ \left[ \frac{Y(t)}{X(t)}|\mathcal{F}_0 \right] } = \mathbb{E}^{Q_{X(t)}}{ \left[ F(t)|\mathcal{F}_0 \right] } = F_0exp \left( \sigma_x \sigma_y \right) = \frac{Y(0)}{X(0)} exp \left(\sigma_x \sigma_y \right) $$

(above, I used the fact that: $\sigma_1 W_1(t)+\sigma_2W_2(t)=(distribution)={W_3}(t)\sqrt{\sigma_1^2 + \sigma_2^2}$ where $W_1$ and $W_2$ are two independent Brownian motions and $W_3$ is just another Standard Brownian motion. The equality holds in distribution, so can be used inside an expectation operator).

Clearly $\frac{Y(0)}{X(0)} exp \left(\sigma_x \sigma_y \right) \ne \frac{Y(0)}{X(0)} $ so the Martingale condition is NOT satisfied.

Any help on this would be hughly appreciated.