Derivation of Magrabe formula

Question

I'm going through the following note by Davis, link.

In chapter 3 he derives the Magrabe formula. I got stuck at equation $(3.16)$. We have two assets:

$$dS_i(t)=S_i(t)\sigma_idw_i(t)$$ for $i\in\{1,2\}$ and $d\langle w_1,w_2\rangle_t = \rho dt$. The payoff we are interested in is the following:

$$C(0,s_1,s_2)=E[\max{(S_1(T)-S_2(T),0)}]$$

The idea is to perform a change of measure, using $S_2(T)$ as numéraire. We end with a new measure $\tilde{P}$ with $$d\tilde{w}_2=dw_2-\sigma_2dt$$ and $$d\tilde{w}_1=dw_1-\rho\sigma_2dt$$ both brownian motion under $\tilde{P}$. Defining $Y:=\frac{S_1}{S_2}$ one can show that:

$$dY=Y(\sigma_1d\tilde{w}_1-\sigma_2d\tilde{w}_2)$$

The authors claims that this can be writen as:

$$dY=Y\sigma dw\tag{3.16}$$

where $w$ is a standard brownian motion and $\sigma = \sqrt{\sigma^2_1+\sigma^2_1-2\sigma_1\sigma_2\rho}$.

How do we get $(3.16)$ and the brownian motion $w$, especially under which measure?

Answer 1

Let \begin{align*} w_t = \frac{1}{\sqrt{\sigma_1^2+\sigma_2^2 -2\sigma_1\sigma_2 \rho}}(\sigma_1\tilde{w}_t^1-\sigma_2\tilde{w}_t^2). \end{align*} Then, using Levy's characterization, we can show that $\{w_t \mid t \geq 0\}$ is a standard Brownian motion.