Set Vt = logt, where t is the volatility of a non-dividend-paying asset with price S. Assume

Set Vt = logσt, where σt is the volatility of a non-dividend-paying asset with price S. Assume dSt St

= μt dt +σt dB1t , dVt = κ(θ −Vt)dt + γ



ρ dB1t + 

1− ρ2 dB2t



, where μ, κ, θ, γ , and ρ are constants and B1 and B2 are independent Brownian motions under the physical probability measure. Assume there is a constant risk-free rate.

(a) Show that any SDF process must satisfy dMt Mt = −r dt − μt − r σt dB1t −λt dB2t +
dεt εt (17.19)
for some stochastic process λ, where ε is a local martingale uncorrelated with B1 and B2.

(b) Assume that λ in the previous part is a constant. Show that dSt St = r dt +σt dB∗
1t dVt = κ∗
(θ ∗ −Vt)dt − γρ(μt −r)
σt dt + γ

ρ dB∗
1t + 
1− ρ2 dB∗
2t 
for some constants κ∗ and θ ∗, where B∗
1 and B∗
2 are independent Brownian motions under the risk-neutral probability corresponding to M.

(c) Let W(t,St,Vt) denote the conditional probability probR t (ST > K) for a constant K. Show that W must satisfy the PDE Wt +rSWS +
κ∗
θ ∗ −κ∗
V −γρ(μ −r)e−V
WV +
1 2 e2VS2 WSS +
1 2 γ 2 WVV + γρeVSWSV = 0.