In the Heston model (17.10), define Y1 = V/ 2 and Y2 = logS V/ . (a)

In the Heston model (17.10), define Y1 = V/γ 2 and Y2 = logS− ρV/γ .

(a) Derive the constants ai, bij, and β such that



dY1 dY2



=



a1 a2



dt +



b11 b12 b21 b22Y1 Y2



dt +

√Y1 0 0 √βY1

dB1 dB2



.

(b) Consider a price of risk specification

λ1t = c10 +c11Y1t √Y1t

,

λ2t = c20 + c21Y1t √βY1t

, for constants cij. Derive c20 and c21 as functions of c10 and c11 from the fact that (17.12) must hold for all V. Note: The specification in Section 17.4 is the special case c10 = 0.

(c) Assume that M defined in terms of c10 and c11 is such that MR is a martingale. Derive constants a∗
i and b∗
ij in terms of c10 and c11 such that 
dY1 dY2 
= 
a∗
1 a∗
2 
dt +

b∗
11 b∗
12 b∗
21 b∗
22Y1 Y2 
dt +
√Y1 0 0 √βY1 dB∗
1 dB∗
2 
, where the B∗
i are independent Brownian motions under the risk-neutral probability.

(d) Assume κθ/γ 2 ≥ 1/2. Under what condition on c10 is a∗
1 ≥ 1/2?