Minimum-relative-entropy calibration of asset-pricing models (Q2703108)

A continuous-time economy is represented by a state-vector \(X(t)=(X_1(t), \dots, X_\nu(t))\) which follows a diffusion process under the prior probability measure \(P_0\): NEWLINE\[NEWLINEdX_i(t)= \sum^\nu_{j=1} \sigma_{ij}^{(0)} dZ_i(t)+ \mu_i^{(0)} dt,\quad i=1, \dots,\nu.NEWLINE\]NEWLINE Here \((Z_1,\dots, Z_\nu)\) are independent Brownian motions, and \(\sigma_{ij}^{(0)}\) are functions of \(X\) and \(t\). Suppose that there are \(N\) benchmark securities, with prices \(C_1,\dots, C_N\). An algorithm is presented to find a risk-neutral probability measure \(P\) such that \(P\) is consistent with these prices and \(P\) minimizes the relative entropy w.r.t. \(P_0\). The algorithm involves solving a Hamilton-Jacobi-Bellman PDE and minimizing the value of the solution over a finite-dimensional space of Lagrange multipliers. It is shown that the algorithm is a special case of algorithms for calibrating models based on stochastic control and convex optimization. The sensitivities of contingent claims prices to variations in input prices are studied. They can be interpreted as regression coefficients of the payoffs of contingent claims on the set of payoffs of the benchmark instruments, in the risk-neutral measure. As an illustration, forward rate curves from US LIBOR data are constructed.