Under the assumptions of the Black--Scholes model, I read that the market price of risk of two assets $S_1$ and $S_2$ are the same, if they both follow Geometric Brownian motion driven by the same Brownian motion.
The claim is that if \begin{align*} dS_1(t)&=\mu_1S_1(t)dt+\sigma_1S_1(t)dW(t),\qquad\text{and} \\ dS_2(t)&=\mu_2S_2(t)dt+\sigma_2S_2(t)dW(t) \end{align*} then $$\frac{\mu_1-r}{\sigma_1}=\frac{\mu_2-r}{\sigma_2}$$ where $r$ is the risk-free rate. The 'proof' of this relies on constructing a portfolio of $\sigma_2S_2$ units of $S_1$ and $-\sigma_1S_1$ units of $S_2$ and assuming that this portfolio is self-financing, then using Ito's formula on the value of this portfolio to show that it only has a drift term. I don't believe the assumption that this portfolio is self-financing holds.
Does the claim hold, and if so is there a proof of this result?
EDIT:
Thought about this a bit more and realised it falls out of the Second Fundamental Theorem of Asset Pricing where the risk-neutral measure is unique if and only the market is arbitrage-free and complete.
Assuming that the market is arbitrage-free and complete, we can construct measures $\mathbb{Q}_1$ and $\mathbb{Q}_2$ such that $$W_1(t)=W(t)+\frac{\mu_1-r}{\sigma_1}t,\qquad W_2(t)=W(t)+\frac{\mu_2-r}{\sigma_2}t$$ are $\mathbb{Q}_1$ and $\mathbb{Q}_2$ Brownian motions respectively. Both these measures give rise to a measure such that discounted asset prices are martingales. By uniqueness, $\mathbb{Q}_1=\mathbb{Q}_2$ and so $$\frac{\mu_1-r}{\sigma_1}=\frac{\mu_2-r}{\sigma_2}.$$