V(0) ≡ mB (0,3) + nA (0) — the caplet Example construct synthetic using three-period zero and mma has strike — Form portfolio shares three- period zero- coupon bond with price

In the caplet Example 23.6, construct the synthetic caplet using the three-period zero and mma (the caplet has a strike rate of k = 0.04 or 4 percent) Show that the cost of construction is 0.003, the same as that with the two-period zero- coupon bond.

SOLUTION

Form a portfolio using m shares of the three- period zero- coupon bond with price B(0,3) and n units of the money market account (mma) with value A(0). The cost of construction is

V(0) ≡ mB (0,3) + nA (0).

At time one, find (m, n) to match the portfolio’s value to the traded caplet’s value in each state.

In the “up state”: V(1)u = mB(1,3)u + nA(1) = c(1)u,

or m0.9167 + n1.0309 = 0.

In the “down state”: V(1)d = mB(1,3)d + nA(1) = c(1)d,