## Exercise 8.5

This forum is to discuss the book "C++ design patterns and derivatives pricing."

### Exercise 8.5

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
pzling

Posts: 41
Joined: Sat Jan 14, 2012 8:52 pm

### Re: Exercise 8.5

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.
mj

Posts: 1341
Joined: Fri Jul 27, 2007 7:21 am

### Re: Exercise 8.5

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

Posts: 41
Joined: Sat Jan 14, 2012 8:52 pm

### Re: Exercise 8.5

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

Posts: 41
Joined: Sat Jan 14, 2012 8:52 pm