Assume ertMt is a martingale. (a) Using Girsanovs theorem, show that dD D = ( )dt

Assume ertMt is a martingale.

(a) Using Girsanov’s theorem, show that dD D = (μ− σ λ)dt + σ dB∗ , where B∗ is a Brownian motion under the risk-neutral probability associated with M.

(b) Calculating under the risk-neutral probability, show that the asset price is Pt def

= Dt r + σ λ −μ .

Verify that the expected rate of return of the asset under the risk-neutral probability is the risk-free rate.

(c) Define δ = r + σ λ −μ, so we have D/P = δ (in other words, δ is the dividend yield). Verify that dP P = (r − δ)dt + σ dB∗ .

(d) Write down the fundamental PDE—which is here actually an ODE—for the asset value Pt = f(Dt). Verify that the ODE is satisfied by f(D) = D/(r +σ λ − μ).