Delta hedging a portfolio of servicing rights under gamma and vega constraints with optimal fixed income securities

Anne Zissu (Citytech, Brooklyn, New York, USA)
Carlos Ortiz (Arcardia University, Philadelphia, Pennsylvania, USA)
Charles Stone (Brooklyn College, Brooklyn, New York, USA)

Journal of Risk Finance

ISSN: 1526-5943

Publication date: 17 August 2010

Abstract

Purpose

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.

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 α of the MSR's value and the share β 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” α by requiring simultaneously that α′=0 and also that α″=0, for a fixed y given by the market that will guarantee that the function α satisfies simultaneously the two conditions: α′=0 and α″=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”).

Practical implications

The optimal triples (c, n, y) are obtained in order to optimize α and simultaneously α′=0 and α″=0. For example, for a yield of 16.1 percent, a bank with a portfolio of MSR based on the data used, should add to it, bonds with a 6.33 percent coupon that matures in 7.55 years.

Originality/value

This paper is the first to the authors' knowledge to derive the different triples (yield, coupon, maturity) of bonds that when combined with MSR create a portfolio that is dynamically hedged against interest rate risk and prepayment risk, and therefore eliminates the need of periodic rebalancing of a portfolio of MSR, which is costly.

Keywords

Citation

Zissu, A., Ortiz, C. and Stone, C. (2010), "Delta hedging a portfolio of servicing rights under gamma and vega constraints with optimal fixed income securities", Journal of Risk Finance, Vol. 11 No. 4, pp. 377-400. https://doi.org/10.1108/15265941011071511

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Publisher

:

Emerald Group Publishing Limited

Copyright © 2010, Emerald Group Publishing Limited

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