Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 211 / 6 / c

Solution

Let

τ = T - t

and set

V (t, z) = exp {P (τ) + Q (τ) z + R (τ) z^{2}} .

Then

\frac{V _{z}}{V} = Q + 2 R z, \frac{V _{zz}}{V} = 2 R + (Q + 2 R z)^{2}, \frac{V _{t}}{V} = - (\dot{P} + \dot{Q} z + \dot{R} z^{2}) .

For

B (z) = a - b z

and

C (z) = c

, matching constant, linear, and quadratic coefficients gives

\dot{R} = 2 c^{2} R^{2} + 2 (θρ c - b) R + \frac{1}{2} θ (θ - 1),

\dot{Q} = 2 a R + (θρ c - b) Q + 2 c^{2} QR,

and

\dot{P} = a Q + c^{2} R + \frac{1}{2} c^{2} Q^{2} .

The terminal condition becomes

P (0) = Q (0) = R (0) = 0

. The first equation is a Riccati equation; once it is solved, the second is linear in

Q

, followed by direct integration for

P

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