Risk-Neutral Pricing and Numeraire Changes for Multi-Asset Claims

# Risk neutral measure & change in numeraire


# Risk neutral measure & change in numeraire












There are two questions about risk neutral and change in numeraire I am not so sure if my answer is correct.

Question 01: Risk neutral

Let says I have 2 risky asset A and B. Each has stochastics process as follow:

\begin{align} \frac{dA}{A}=\mu_Adt + \sigma_Adw \\ \frac{dB}{B}=\mu_Bdt + \sigma_Bdw \end{align}

Under risk neutral process $\mathbb{Q}$ we have:

\begin{align} \frac{dA}{A}=r_tdt + \sigma_Adw^{\mathbb{Q}} \\ \frac{dB}{B}=r_tdt + \sigma_Bdw^{\mathbb{Q}} \end{align}

Let say I have some financial product f which is a function of A and B. For simplicity, let f = A + B. Then I wondering if the following is correct or not?

\begin{align} df & = dA + dB\\ df & = r_tAdt + \sigma_AAdw^{\mathbb{Q}} + r_tBdt + \sigma_BBdw^{\mathbb{Q}} \\ & = r_t(A+B)dt + (\sigma_A + \sigma_B)dw^{\mathbb{Q}} \\ \end{align}

Divide both RHS and LHS for f, we have: \begin{align} \frac{df}{f} = r_t \frac{A+B}{f}dt+ \frac{\sigma_A + \sigma_B}{f}dw^{\mathbb{Q}} \end{align}

By now, because $\frac{df}{f}$ stochastics process are under risk neutral measure $\mathbb{Q}$ hence, we can derive PDE that:

\begin{align} r_t \frac{A+B}{f}dt & = r_t dt \\ A + B - f & = 0 \end{align}

I don't know if my argument above is correct. If not please help me indicate if there is anything wrong anywhere?

Question 02: Change in numeraire

I use financal product $f$ above for continuing example. Let say, if I want to derive the stochastic process of $\frac{df}{f}$ under new measure $\mathbb{R}$ in which, the stochastic process of $R$ is:

$$dR = \mu_R dt + \sigma_R dw$$

Then I must change $dR$ into form $\frac{dR}{R}$ which is:

\begin{align} \frac{dR}{R} & = \frac{\mu_R}{R} dt + \frac{\sigma_R}{R} dw \\ & = r_t dt + \frac{\sigma_R}{R} dw^{\mathbb{Q}} \end{align}

Then back to our goal, the price of risk for stochastic process of $\frac{df}{f}$ under new measure $\mathbb{R}$.

\begin{align} \Theta^{\mathbb{R}} & = - \frac{\sigma_R}{R} \\ \frac{df}{f} & = \left[ \frac{r_t(A+B)}{f} + \frac{\sigma_R}{R} \frac{\sigma_A + \sigma_B}{f} \right]dt + \frac{\sigma_A + \sigma_B}{f} dw^{\mathbb{R}} \\ \end{align}

And if my arugment in Question 01 is correct then, stochastic process can be simplified into:

$$\frac{df}{f} = \left[ r_t + \frac{\sigma_R}{R} \frac{\sigma_A + \sigma_B}{f} \right]dt + \frac{\sigma_A + \sigma_B}{f} dw^{\mathbb{R}} $$

Are my two arguments for two questions above are right? If there is anything wrong I hope I can get help from you guys

Thank you