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.

  • 1
    $\begingroup$ 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
  • 1
    $\begingroup$ 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

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,

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.