Bonds with volatilities proportional to forward rates

TL;DR

This paper investigates the existence of solutions to the Heath-Jarrow-Morton equation with linear volatility driven by jump processes, providing conditions for existence or non-existence based on Levy-Khinchin exponents.

Contribution

It establishes new criteria for the existence of solutions in bounded fields for the HJM equation with jump-driven volatility, including models with Levy measures of stable type.

Findings

Existence when Levy-Khinchin exponent's first derivative grows slower than logarithm

Non-existence when it is bounded below by a fractional power

Numerous examples including Levy stable models

Abstract

The problem of existence of solution for the Heath-Jarrow-Morton equation with linear volatility and purely jump random factor is studied. Sufficient conditions for existence and non-existence of the solution in the class of bounded fields are formulated. It is shown that if the first derivative of the Levy-Khinchin exponent grows slower then logarithmic function then the answer is positive and if it is bounded from below by a fractional power function of any positive order then the answer is negative. Numerous examples including models with Levy measures of stable type are presented.