Pricing an option with a certain payoff

Suppose an option with a payoff function

$$\max((1+k)S_1,kS_2)$$ where $$S_1, S_2$$ are stock prices and $$k>0$$ is a constant value.

To value such an option, one would decompose this payoff function into plain vanilla options (and possibly bonds), as they can be valued using the Black-Scholes formula.

How can this payoff function be decomposed?

Thank you very much!

$$\max(S_1, S_2) = S_1 + \max(S_2-S_1, 0)$$ In the case of GBM, the second term in the RHS can be valued using Magrabe's formula. The scalars $$k, 1+k$$ don't change this significantly.
• I got $(1+k)S_1 + max(kS_2-(1+k)S_1,0)$. Is it correct to price the part $max(kS_2-(1+k)S_1,0)$ by directly utilizing the Magrabe's formula, i.e., by simply plugging $kS_2$ and $(1+k)S_1$ into the Magrabe's formula? Thanks! Commented Oct 20, 2023 at 14:38
• You mentioned that the formula assumes GBM, how about if we have the dynamic $$S_{ik} = S_{i0}*e^{(\mu_i-\frac{1}{2}(\sigma^2_{i,1}+\sigma^2_{i,2}))k+\sigma_{i,1}Z_{i,1}+\sigma_{i,2}Z_{i,2}}$$, where $i=1,2$? Commented Oct 20, 2023 at 16:30