## Find an upper bound for z(T) as a function of T, τ and z(T + τ ).

Forward interest rate. This problem is about forward interest rates and goes beyond the scope of equity derivatives. Consider a market with no arbitrage and infinite liquidity where investors can lend and borrow money for any maturity T at the zero-coupon rate z(T). Let z(T, τ ) denote the forward rate agreed today on a loan beginning at time t = T and ending at time t = T + τ . For instance, the ‘6-month × 1-year’ forward rate z(0.5, 1) is the rate agreed today on a loan starting in six months and ending in eighteen months.

(a) In this question the zero-coupon rate curve is given as: z(T) = 5% + T × 0.5%.

(i) Draw z(T) as a function of maturity T.

(ii) Suppose that z(1, 2) = z(3). Determine two ways to invest or borrow money over 3 years and show that there is an arbitrage opportunity.

(iii) Calculate the rate z(1, 2) which eliminates the arbitrage and generally z(1, τ ) for any borrowing or lending period τ . Draw z(1, τ ) as a function of τ .

(b) In this question the zero-coupon rate curve z(T) ≥ 0 is arbitrary.

(i) Show that the forward rate z(T, τ ) must satisfy the equation: (ii) Verify your results in question (a)(iii).

(iii) Find an upper bound for z(T) as a function of T, τ and z(T + τ ).

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