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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/iii/paper-35/5/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 35 / 5 / a / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

P (T, T) = 1

. Its continuously compounded zero-coupon yield to maturity, instantaneous forward rate, and short rate are

R (t, T) = - \frac{lo g P ( t , T )}{T - t}, f (t, T) = - \partial_{T} lo g P (t, T), r (t) = T ↓ t lim R (t, T) = f (t, t) .

(1)

The last equality assumes the appropriate maturity differentiability at the diagonal. Integration of the forward-rate definition and differentiation of the yield definition give

P (t, T) = exp [- \int_{t}^{T} f (t, u) d u] = e^{- (T - t) R (t, T)}, R (t, T) = \frac{1}{T - t} \int_{t}^{T} f (t, u) d u, f (t, T) = R (t, T) + (T - t) R_{T} (t, T) .

(2)

At fixed observation time

t

, the yield curve plots this yield against maturity

T

; it describes today's discount prices at different horizons. Forward rates are extracted from these prices, not identified with the subsequent realized path of the short rate.

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