Question

3. Let K(1)., K(n) be independent identically distributed one step returns rates on a binomial tree model for a stock price, S(n). Here K(1) = u with probability p and K(1) with probability 1 p. For which values of n and what conditions on u and d can (n) S(0)

Answer 1

E(S(1))=puS(0)+(1-p)dS(0)

S(0)=E(S(1))/exp(r)

(S(1))=E(S(2))/exp(r)

S(0)=E(S(2))/exp(2r) similarly,

S(0)=E(S(n))/exp(nr)

Now if S(n)=S(0) then

S(0)=S(0)/exp(nr)

exp(r)=pu+(1-p)d

exp(nr)=1

nr=0 then only possible condition is r=0

For all values of n>=1 we would have this case of S(n)=S(0)

p=1-d/(u-d)

u>1>d