Exercise 8.5

[pzling]

Hi

How should the expected value of the Spot prices be taken as a martingale?

I'm using the same risk neutral probabilities as for the payoffs and the Spots are converging to another value other than that used to build the final time layer.
eg. If I use a Spot of 100 to build the final time layer, and then take the expected present values of the Spots all the way back to time 0, I get an initial Spot of say, 71 (which is not equal to 100).

This also results in the American options having the same price as the European options.

Many thanks

[mj]

in this book, i've taken p=0.5 for simplicity and done the Jarrow Rudd tree. However, normally people take

p = (e^{r\delta t) - d)/(u-d)

to ensure E(S_t+\delta t) = e^{r \delta t} S_t

you'll find that the value is close to 0.5 but NOT exactly equal to it.

See More mathematical finance for extensive discussion.

[pzling]

I see thanks. Will definitely order that in once I've gone through this one.

In the meantime, I forgot to mention I did this on the trinomial tree (and used the standard risk neutral probabilities ... looks like [(exp(A)-exp(B))/(exp(C)-exp(B)]^2).

So at the risk of turning this into a math question now, do the trinomial risk neutral probabilities simulate geometric brownian motion like the binomial probabilites do? Or were they derived specifically for a trinomial model?

[pzling]

Actually, sorry, I just noticed my implemented probabilities are off, which would probably explain why it wasn't converging to the right value