Assume Mt = exp(X1t +(X2t a)2)is an SDF process, where dX1 = dt + dB1t

Assume Mt = exp(X1t +(X2t − a)2)is an SDF process, where dX1 = μdt + σ dB1t , dX2 = −κX2t dt + φ dB2t , with μ, σ, κ, and φ being constants and with B1 and B2 being independent Brownian motions under the physical probability.

(a) Derive dM/M, and deduce that r is a quadratic function of X1 and X2.

(b) Given the prices of risk calculated in the previous part, find Brownian motions B∗

1 and B∗

2 under the risk-neutral probability and show that the dX satisfy (18.5) for an affine μ and a constant σ (thus, the model is quadratic).