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 Incoming links: Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2017 / 1 / b

Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2017 / 1 / c / Solution 2026-10-25

The differential operator in part b is the infinitesimal generator of the auxiliary diffusion process

d S_{u} = r S_{u} d u + σ d W_{u}

, with killing rate

r

. The Feynman-Kac formula therefore gives

V (S, t) = e^{- r (T - t)} E^{Q} [P_{o} (S_{T}) ∣ S_{t} = S] .

Equivalently, Itô's formula makes

e^{- r (u - t)} V (S_{u}, u)

a local martingale under these dynamics; under appropriate growth and integrability assumptions it is a martingale, whose terminal conditional expectation yields the same formula. The expectation uses the drift-

r

law, rather than the physical drift-

α

law in part a. This is risk-neutral pricing.

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