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When pricing a spread option with two different prices, one can use Kirk's approximation combined with Margrabe's formula (https://en.wikipedia.org/wiki/Margrabe%27s_formula).

But what if I am pricing an option that involves 3 or 4 different prices ? Is there a closed form formula ? Else how can I price it ?

The payoff of my strategy looks like this:

$$\mathop{\mathbb{E}} \left[\left(\alpha K+\beta F_1(t_1) +\gamma F_1(t_2)+\zeta F_2(t_1) +\lambda F_2(t_2)\right)^+\right]$$ Where $K$ is the strike, $F_1$ and $F_2$ the price of two different assets.

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For such problems, you may consider the moment matching approach. For example, you can approximate the combination of terms where the coefficients have the same sign by a log-normal random variable, and then use the approach you mentioned. If all coefficients have the same sign, you can approximate the whole combination by a shifted log-normal random variable. That is, \begin{align*} \beta F_1(t_1) +\gamma F_1(t_2)+\zeta F_2(t_1) +\lambda F_2(t_2) \approx A + B e^{C\xi}, \end{align*} where $A$, $B$ and $C$ are constants, and $\xi$ is a standard normal random variable. Here, the parameters $A$, $B$ and $C$ can be calibrated by matching the moments up to the third order. However, you need to solve a cubic equation. For simplicity, you can assume that $A=0$ to have a second order moment matching.

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