Trinomial Tree and Martingale Stock Process

[pzling]

Hi

I just wanted to check I'm not chasing my tail here. This actually stems from an exercise in C++ Design but it's starting to become more of a math question. The intention is to model the stock process as a martingale rather than a series of up/down/mid factors from the initial spot. So as per the usual tree methodology, the final time layer of spots is obtained with the up/down/mid factors. But then instead of calculating the next time layer from the initial spot, we want to take the expected present values of the spots in the final time layer.

With a binomial tree, I think this works fairly well as the formulae for the (risk neutral) probabilities gives something close to 0.5. However when I use the trinomial tree probability formulae to take the expected present value of the final time layer spot values, I consistently undervalue the next layer spot values compared to if I were to calculate it from the initial spot value.

The result is that if I try to value an american option using a trinomial tree with martingale stock process, the american price is always equal to the european price. The undervaluing of the next layer spots means that at the next time layer, the expected present value of payoffs is always the expected present value of the payoff from the final time layer (and never say, for a call, max(Spot-Strike, 0)).

So for a trinomial tree, is it possible to take the expected present values of the final time layer spots so that it converges to the initial spot value?

Thanks

[mj]

as long as you pick the probabilities so that the first and second moments are correctly matched it should converge and give the correct expectation.

For an overview see,

http://ssrn.com/abstract=1261745

[pzling]

ah thanks. i'm sure you get this already from many others, but love your work!

[pzling]

i initially naively picked my trinomial model from Wikipedia (http://en.wikipedia.org/wiki/Trinomial_tree) and took this to be the 'standard' model. in the description (and caveat emptor taken with all things wiki), it said that the parameters were chosen to ensure a martingale.

please excuse my terrible math as I can't see how the moments match. I've implemented the same calculations on the first 2 trinomial trees in the chan, joshi, tang, yang paper, I still can't get the moments to match.

so to clarify, when we say the first and second moments match, do we mean that E[X] = E[X^2]?

With the standard tree, with arguments of (r=0.05, d=0.10, vol=0.2, Steps=552, Time=2):
X = {u, d, m} = {1.017171, 1, 0.983119}
p_i = {pu, pd, pm} = {0.242, 0.257, 0.500}
which gives
E[X] = 0.999819, E[X^2] = 0.999783??


With the EqualProb tree, with arguments of (r=0.05, d=0.10, vol=0.2, Steps=552, Time=2):
X = {u, d, m} = {1.014965, 1, 0.98547}
p_i = {pu, pd, pm} = {1/3, 1/3, 1/3}
which gives
E[X] = 1.000145, E[X^2] = 1.000435??

[mj]

moments matching means agreeing with those of the lognormal distribution.

E(X^2) > E(X)^2 is always the case.

[pzling]

thanks again. i have only just realised moments matching is a lot more involved than i thought, which i will leave as an exercise for another day and just accept that the given probabilities work!

just to see if it's because i've written the trinomial tree incorrectly, i tested the concept on the binomial tree with its corresponding probabilities (just using 0.5 for simplicity). i'm getting the same issue with american prices evaluated to be the same as european prices due to the spot values converging to a value less than the initial spot.

could i check if the correct way to apply the probabilities to obtain the expected present value of spots is
E[S(t)] = exp(-r.dt)*(pu*S(t+1, u) + pm*S(t+1, m) + pd*S(t+1, d))?

[mj]

that is correct