3
$\begingroup$

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?

$\endgroup$

1 Answer 1

2
$\begingroup$

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.

$\endgroup$

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.