Hi, all, I'm puzzled by the step from binomial tree to the trinomial tree. It's well known that binomial tree is quite intuitive-- it allows us to replicate the position at each node. From the constraint that current price is the expected price at next step discounted by risk free interest rate: S=(p*Su+(1-p)*Sd)*exp(-rdt), we could solve for the risk neutral probability at each step. When we move to trinomial tree, we just add another constraint about the variance ( the second moment). If we say trinomial tree allows us to select the state price freely (so we could take into account skewness), and gives us some freedom compared to the binomial tree.Then how about 4-nomial tree, 5-nomial tree, with those trees we could even take into account of kurtosis or even higher moments. But what additional constraints to add-- just as we do in trinomial tree, discretize the continuous process and add a constraint about a higher moment? Or trinomial tree is enough, 4-nomial, 5-nomial are redundant, and not justified for consideration? Actually, I wonder whether it could be proved theoretically that the trinomial tree will converge to the continous distribution when the time step goes to zero. For binomial tree, there is theoretic proof as in CRR's paper. But for trinomial tree, it seems that nobody cares about it and just use trinomial tree for granted. I was suggested by some one who think trinomial tree is just a special case of binomial tree (I remember M Rubinstein has a paper on this topic) -- in the case we combine two steps of recombining binomial tree into one step in trinomial tree: ---------Suu ---- SuS ---------Sud ---- Sd ----------SddBut in trinomial tree, you can't trade at node Su and Sd which are tradable nodes in binomial tree and therefore a part of the synthetical replication arguement.Anyone comes up with an explanation.Thanks!Siro

it is true that there is an extra-degree of freedom in the trinomial that you in fact use to say match a barrier exactly, it is also true that unlike binomial, trinomial does not allow a complete representation of the market (replicating the option with stock and cash) …however, if you take the Black-Scholes PDE and apply an explicit Finite Difference scheme you will end up with a trinomial tree OR binomial depending on your parameters choice …Paul and Elie are the specialists on this, I just wanted to start the ball rolling ...

Guys, have a go at Quiz 4 on the website!P

In most problems, the key is the number of branches from each node relative to the number of independent securities you can trade. There is no magic to the binomial tree unless you have exactly two underlying assets (the stock and the risk free asset). A trinomial tree with three underlying assets has the same perfect hedge possibility, as does an n-nomial tree with n underlying assets.Some models like one more branch than you have assets in order to make the derivative independent of the underlying (not statistically independent, just not completely determined). That's useful for certain exotic options.Trinomials are also handy in interest rate modeling because you seem to need a minimum of two assets to characterize the term structure (a short rate and a long rate) and some shape freedom.