Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 211 / 1 / e

Solution

Let

d = μ - (1 + r) P_{0}

. Since

V

is symmetric,

A = Im V = (ker V)^{⊥}

. If

d \in A

, every

H \in ker V

satisfies

H \cdot μ = (1 + r) H \cdot P_{0} .

Any

H \in / ker V

has a nondegenerate normal terminal value and cannot be nonnegative almost surely. Any

H \in ker V

has the displayed deterministic relation, which excludes an arbitrage because

1 + r = (η \cdot μ) / (η \cdot P_{0}) > 0

Conversely, if

d \in / A

, choose

h \in ker V

with

h \cdot d > 0

. The zero-cost portfolio

H = h - \frac{h \cdot P _{0}}{η \cdot P _{0}} η

has deterministic terminal payoff

H \cdot P_{1} = h \cdot μ - (1 + r) h \cdot P_{0} = h \cdot d > 0.

d \in A

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