This exercise verifies that, as asserted in Section 15.3, condition (15.9) is sufficient for MW to be a martingale. Let M be an SDF process such that MR is a martingale. Define B∗ by (15.8). Let W be a positive self-financing wealth process. Define W∗ = W/R.

(a) Use Itô’s formula, (13.10), (13.20), and (15.8) to show that dW∗ = 1 Rφ

σ dB∗ .

(b) Explain why the condition E∗



T 0

1 R2 φ

φ dt



< ∞

implies that W∗ is a martingale on [0,T] under the risk-neutral probability defined from M, where E∗ denotes expectation with respect to the risk-neutral probability.

(c) Deduce from the previous part that (15.9) implies MW is a martingale on [0,T] under the physical probability.