1. The current time is t = 1 and our framework is the Libor model. We consider a situation with four states of the world ωi at time t = 3. Suppose Li is the Libor process with a particular tenor and B(1, 3), B(1, 4), and B(1, 4) are zero-coupon bond prices with indicated maturities. The possible payoffs of these instruments in the four future states of the world are as follows: L = 6%, 6%, 4%, 4% (119) B(1, 3) = 1, 1, 1, 1 (120) B(1, 4) = 0.9, 0.92, 0.95, 0.96 (121) B(1, 5) = 0.8, 0.84, 0.85, 0.88 (122) The current prices are, respectively, 1, 0.91, 0.86, 0.77 (123) Here the 1 is a dollar invested in Libor. It is like a savings account. Finally, current Libor is 5%.

(a) Using Mathematica, determine a state price vector q1, q2, q3, q4, that corresponds to B(1, 3), B(1, 4), B(1, 5), L as a basis.

(b) Does qi satisfy the required condition of positivity? Is there an arbitrage opportunity?

(c) Let F be the 1 × 2 FRA rate. Can you determine its arbitrage-free value?

(d) Now let C be an ATM caplet (i.e., the strike is 5%) that expires at time t = 2, but settled at time t = 3 with notional amount 1. How much is it worth?