Question

1.1. Suppose that you have a stock in the one-period binomial model with fixed u, d, and r such that 0< d< 1 +r < u. Suppose that there are positive numbers pi and such that pi, qi < 1, pi + q-1, and (1 + r)So = PiSi (H) + qi Si (T). Show that pi = p ad qi = q. Hint: You know that the risk-neutral probabilities satisfy these equations as well.

Answer 1

There are two (and only two) possible prices for the underlying asset on the next date. The underlying price will either:

–Increase by a factor of u% (an uptick)

–Decrease by a factor of d% (a downtick)

The one-period interest rate, r, is constant over the life of the option (r% per period).

Let Time T is the expiration day of a call option. Time T-1 is one period prior to expiration.

T-1 T,u T-1

given

$S_{0}=p_{1}S_{1}(H)&plus;q_{1}S_{1}(T)/(1&plus;r)$

for a risk neutral scenario, the porbability is given by

The risk neutral pricing formula is

Now comparing the two models we can safely assume that the one period binomial model is modelled on a risk neutral model and hence its proability will mirror the probability behaviour of a risk neutral model.

or, $p~p*;q=(1-p)~q*$