Played around a little. Here is a solvable case.Take mu = constant and sigma(x) = x^(1/2), where x = S(t)/I(t)(1) dS = mu S(t) dt + V(t)^(1/2) S(t) dW, where V(t) = [sigma(S(t)/I(t))]^2 = S(t)/I(t)If I haven't made any mistakes, Ito =>(2) dV(t) = [(mu+lambda) V(t) - V(t)^2] dt + V(t)^(3/2) dW,so (1) and (2) combined are the 3/2 sv-model with rho=1, solved in my book, Ch11 for any |rho| <= 1 [Warning, this is again very likely an example of a stricly local martingale when mu=0, so put-call parity fails, etc.Maybe it is possible to flip the sign of rho, which would be better; see my p.s.] p.s.Just tried sigma(x) = x^(-1/2) , which seems to lead to dV(t) = [1 - (mu + lambda) V(t) + V(t)^2] dt - V(t)^(3/2) dW. Now, pretty sure you have a perfectly good martingalefor S(t) when mu=0; also pretty sure S(t) hits 0 as V(t) explodes, which is fine and expected in this case as itis analogous to cev with beta=1/2. Is this one exactly solvable? I don't know -- but suspect the slight change in the drift is a spoiler. The natural next one to try is sigma(x) = x^(-1), but I will stop here.
Last edited by Alan
on March 14th, 2012, 11:00 pm, edited 1 time in total.