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I wonder, what would happen if we use the binomial tree to price exchange option, an option to exchange one asset for another at the expiry date. Payoff is $\max(S_1-S_2,0)$

For instance, I have two assets whose payoff are the following: $\begin{bmatrix}1.1&0.9\\1.1&1.1\\0.9&1.1\end{bmatrix}$, and $S_0^1=100$ and $S_0^2=95$, risk free rate is $R_f=4\%$ and $T=6$

How to deal with the one more dimension? My intuition is to use the binomial tree as usual. However, it is difficult to tell how many possible paths lead to a payoff, given that I have listed all possible combination of payoffs of asset 1 and 2.

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    $\begingroup$ Why do you keep deleting your old questions and then re-posting almost identical ones? quant.stackexchange.com/questions/43846 $\endgroup$ – LocalVolatility Feb 2 at 23:28
  • $\begingroup$ what is the "payoff" of an asset ? I don't understand your notation sorry. $\endgroup$ – Ezy Feb 3 at 16:52
  • $\begingroup$ @Ezy the matrix is the possible return.each column is one asset each row is one scenario $\endgroup$ – veryBigman Feb 3 at 16:53
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    $\begingroup$ well what you are describing starts to look more like Monte-Carlo than a binomial tree. Besides it does not make sense to simulate multi assets without taking into account the correlation structure between those. So I am not convinced that what you are trying to do is very helpful. $\endgroup$ – Ezy Feb 3 at 17:12
  • $\begingroup$ @Ezy Yeah, there are three possible return between two assets, If I just use the return given, the correlation will be -0.5, then how should I proceed from here $\endgroup$ – veryBigman Feb 3 at 22:37
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Under Black-Scholes assumption for the 2 assets $S_1$ and $S_2$ with volatilities $\sigma_{1,2}$ and correlation $\rho$ the value of this option has an explicit expression which is the Margrabe formula

To quote the result explicitly

Introducing $\sigma = \sqrt{\sigma_1^2 + \sigma_2^2 - 2 \sigma_1\sigma_2\rho}$, Margrabe's formula states that the fair price for the option at time 0 and expiry $T$ is:

$$e^{-q_1 T}S_1(0) N(d_1) - e^{-q_2 T}S_2(0) N(d_2)$$

where $q_1,q_2$ are the expected dividend rates of the prices $S_1,S_2$ under the appropriate risk-neutral measure, $N$ denotes the cumulative distribution function for a normal distribution,

$$d_1 = (\ln (S_1(0)/S_2(0)) + (q_2 - q_1 + \sigma^2/2)T)/ \sigma\sqrt{T}$$, $$d_2 = d_1 - \sigma\sqrt{T}$$

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