Consistent curves in the -world: optimal bonds portfolio

We derive arbitrage-free conditions for a parametric yield curve in the $ \mathcal {P} $ P-world, where market prices of risk are present. As in the Heath–Jarrow–Morton (HJM) theory, we impose the bonds evolution to be free of arbitrage opportunities. However, in the classical HJM theory, first volatilities are specified, and subsequently the drift of the forward curve is obtained up to the market prices of risk that must be specified exogenously. Here, the problem is inverted: we first impose a family of shapes upon the yield curve and subsequently derive market prices of risk in a self-consistent manner and hence the market prices of risk follow a stochastic differential equation obtained directly from the curve dynamics. Leveraging this framework, we formulate a bonds-portfolio optimization as a stochastic control problem with the Hamilton–Jacobi–Bellman (H-J-B) equation. We discover that in this case, the H-J-B equation is essentially different than the classical obtained in optimal investment problems.