Assume a continuous-time economy where the state-price deflator = (t) has dynamics dt = t [rt

Assume a continuous-time economy where the state-price deflator ζ = (ζt)

has dynamics dζt = −ζt [rt dt + λ dz1t], where z1 = (z1t) is a (one-dimensional) standard Brownian motion, λ is a constant, and r = (rt) follows the Ornstein–Uhlenbeck process drt = κ[r¯ − rt] dt + σr dz1t.

This is the Vasicek model so we know that the prices of zero-coupon bonds are given by Eq. (11.28) and the corresponding yields are given by (11.31).

Suppose you want to value a real uncertain cash flow of FT coming at time T. Let xt =

Et[FT] and assume that dxt = xt



μx dt + σxρ dz1t + σx

'

1 − ρ2 dz2t



, where μx, σx, and ρ are constants, and where z2 = (z2t) is another (one-dimensional)

standard Brownian motion independent of z1.

(a) Argue that x = (xt) must be a martingale and hence that μx = 0.

(b) Show that the time t value of the claim to the cash flow FT is given by Vt ≡ V(t,rt, xt) = xte

−A(T−t)−B(T−t)rt , (*)

where B(τ ) = Bκ (τ ) and A(τ ) = A(τ ) + ρλσxτ + ρσxσr

κ (τ − Bκ (τ )).

(c) Write the dynamics of V = (Vt) as dVt = Vt[μV t dt + σ V 1t dz1t + σ V 2t dz2t]. Use (*)

to identify μV t , σ V 1t, and σ V 2t. Verify that μV t = rt +

σ V t

λt, where σ V =

(σ V 1 , σ V 2 ) and λ is the market price of risk vector (the market price of risk associated with z2 is zero! Why?).

(d) Define the risk-adjusted discount rate Rt for the cash flow by the relation Vt =

Et[FT]e−Rt[T−t]. What is the difference between Rt and yT t ? How does this difference depend on the cash flow payment date T?