So, what is your question.

Show u, d = ...

Ok，sorry I didn't realize this is a question but some straightforward derivation.

You can expand u and d by taylor series showing below. Then put u and d into (3) rearrange the terms and ignore the terms that has a higher order than dt. Then you are done.

best,

还请多指点一下。

ok

first apply taylor expansion to the exponential function in (3) and you just need to keep the term up to order dt. The left side is therefore:

(1+mu*dt)(1+sigma*sqrt(dt)+sigma^2*dt/2+1-sigma*sqrt(dt)+sigma^2*dt/2)-ud-(1+2*mu*dt)

ud=1

(1+mu*dt)(2+sigma^2*dt)-1-(1+2*mu*dt)

2+sigma^2*dt+2*mu*dt+mu*sigma^2*dt^2-1-1-2*mu*dt

dt^2 is ignored

so the left hand side is sigma^2*dt which is the same as the right hand side.

This is a straightforward way to do it. Another way is to solve two equations:

(3)

ud=1

one of my classmate has solved it and he shows that you can get the solution (perhaps you also need some approximation)

best,