# Ortiz Publishes on Delta Hedging Mortgage Servicing Rights Portfolio

By Purnell T. Cropper | November 2, 2010

Dr. Carlos Ortiz, Professor of Computer Science and Mathematics, published a paper on “Delta hedging a portfolio of servicing rights under gamma and vega constraints with optimal fixed income securities” in the Journal of Risk Finance (Vol. 11 Iss: 4, pp.377 – 400) with co-authors Anne Zissu of Citytech and Charles Stone of Brooklyn College.

“The aim of this paper is to develop the optimal delta hedge for a portfolio of mortgage servicing rights (MSR) under the constraint of a zero-gamma in order to avoid costs related to the rebalancing of such portfolio,” according to the abstract.

“Design/methodology/approach—The paper develops the optimal delta hedge ratio with gamma and vega constraints for an MSR portfolio by finding the different combinations of coupon/maturity bonds (c, n) that satisfy the constraints under different yields (y). Rather than search for the available fixed income securities and see by trial and error which one should be added to the portfolio of MSR such that when market rates go up and down, the value of the portfolio is not affected, the model finds the optimal pairs of coupon/maturity bonds over a range of yields, that satisfy the constraints. It develops the conditions for a delta-hedged portfolio of bonds and MSR under an investor’s or a portfolio manager’s value constraint K. The share a of the MSR’s value and the share ß of the bond’s value had to be such that a zero-delta portfolio that satisfies the constant value of the portfolio can be created. The paper ‘optimized’ a by requiring simultaneously that a’=0 and also that a?=0, for a fixed y given by the market that will guarantee that the function a satisfies simultaneously the two conditions: a’=0 and a?=0. These two conditions will yield equations in the parameters of c and n. The ‘optimization’ problem for arbitrary y’s is solved, and n and c for the appropriate bond are found to be added to the MSR portfolio.

“Findings—Maple software is used to simulate a portfolio of MSR that is delta hedged with bonds, whose appropriate coupon and maturity are found with the model developed under the constraint of a zero-gamma, in order to avoid costs related to the rebalancing of such portfolio. The optimal hedge ratio with gamma and vega constraints for an MSR portfolio is developed by finding the different combinations of coupon/maturity bonds that satisfy the constraints under different yields (the ‘triples’).”