# How to determine the no arbitrage price of following claim? (change of numeraire)

How do I determine the no arbitrage price for claims such as $$min(S_1(T),S_2(T))$$ or $$max(S_1(T),S_2(T))$$? We can consider a standard Black Scholes model. Hence $$S_i(T)=S_i(t)e^{(r-\sigma_i^2/2)(T-t)+\sigma_i(W(T)-W(t))}$$ and that W is brownian motion.

After some reading I saw that I need to perform change of numeraire when valuating the NA price for these kind of claims. Could anyone guide me? Thank you.

\begin{align} &\max(S_T^1,S_T^2)=S_T^2+\max(S_T^1-S_T^2,0) \\ &\min(S_T^1,S_T^2)=S_T^2-\max(S_T^2-S_T^1,0) \end{align} Terms of the form $$\max(S_T^i-S_T^j,0)$$ can be evaluated with the Margrabe’s formula in a Black-Scholes framework.
Regarding the change of numéraire, I give a sketch of how one would proceed. Letting $$\xi_T$$ be the payoff at $$T$$, we have in all generality, with $$B_T$$ the money market account: \begin{align} E^\mathcal{Q}\left(\frac{\xi_T}{B_T}\right)&=E^\mathcal{S}\left(\frac{B_TS_0}{B_0S_T}\frac{\xi_T}{B_T}\right) \\ &=S_0E^\mathcal{S}\left(\frac{\xi_T}{S_T}\right) \end{align} where $$\mathcal{Q}$$ is the risk-neutral measure, with the money market account as numéraire, and $$\mathcal{S}$$ the measure with the stock price as numéraire (note that $$B_0=1$$). Here, $$\xi_T=\max(S_T^i-S_T^j,0)$$ hence if you choose $$S^j$$ as numéraire you get: $$E^\mathcal{Q}\left(\frac{\xi_T}{B_T}\right)=S_0^jE^{\mathcal{S}^j}\left(\max\left(\frac{S_T^i}{S_T^j}-1,0\right)\right)$$ The ratio $$S^i/S^j$$ will be lognormally distributed in a Black-Scholes framework. Keep in mind you need to ensure the ratio is martingale, as any traded asset rebased by a numéraire is a martingale under the measure associated to that numéraire. For that, you have to apply Itô’s Lemma to the ratio and define the Brownian Motion $$W^{\mathcal{S}^j}_t=W^\mathcal{Q}_t+\theta_t$$ by setting $$\theta_t$$ such that $$S^i/S^j$$ has no drift.