Now consider a

one$-$step trinomial model as follows: The current price is S$0.$ At time t $=1,$ it has $1$

$3$ probability to

have for i $=1,2,3,$ S$(1)=$ fiS$0,$ where $0<$ f$1<$ f$2<$ f$3,$ and the annual interest rate is r $>1.$

Show f$1<1+$ r $<$ f$3$ if and only if there is no arbitrage.

Problem $3.$ Interest rate.

$($a$)(6$ points$)$ For the following three investment opportunities, calculate the corresponding effec$-$

tive annual rate and chose which one you would invest your money in$.$

a$)20\%$ interest rate with quarterly compounding,

b$)18\%$ interest rate with monthly compounding,

c$)15\%$ interest rate with continuous compounding.

$($b$)(4$ points$)$ Consider a bond with face value F $=100$ and annual coupons C $=6$ maturing

after $5$ years, is trading at par. Find the implied continuous compounding rate.

Problem $4.(10$ points$)$: Suppose the stock prices in the following three scenarios are

Scenario S$(0)$ S$(1)$ S$(2)$

$1100110120$

$2100105100$

$310090100$

with probabilities $1$

$4,1$

$4,1$

$2,$ respectively.