Jump-Diffusion Models for Valuing the Future: Discounting under Extreme Situations

Round 1

Reviewer 1 Report

Summary

In this paper, the authors derive a general formula of the discount function for mean-reverting Ornstein-Uhlenbeck processes in the presence of jumps. The jumps are modeled as finite and thus through a Poisson process. The authors present and discuss some results.

 

Comment #1

In finance, models with jumps typically make use of Poisson processes. These models consider a finite number of jumps on a finite time interval.  Nevertheless, there are models that instead consider infinitely jumps in finite time intervals (e.g., Madan and Seneta (1990); Eberlein and Keller(1995)). These latter models allow both frequent and infrequent jumps. Since, in this paper, the authors assume, as traditionally, that the discontinuities are finite, I recommend discussing this choice a little bit further. Namely, the practical advantages/disadvantages of assuming a finite/infinite number of jumps.

 

References:

(1) Madan, D. B. and Seneta, E. (1990). The Variance Gamma (V.G.) model for share market returns. Journal of Business, 63, pp. 511-524.

(2) Eberlein, E. and Keller, U. (1995). Hyperbolic distributions in finance. Bernoulli, 1, pp. 281-299.

 

Comment #2

The authors motivate the choice of an Ornstein-Uhlenbeck process by the need to model real rates (that can be negative). Nevertheless, it is important to highlight in the paper that this is a mean-reverting process. Some discussion about the limitations of mean-reverting processes should be done.

 

Comment #3

Related with the structure of the paper, I suggest adding in the end of Section 1 an outline with the next sections. This to inform the reader what to expect from the next sections.

 

The discussion of the results, in Section 4, is very dense. I recommend do add a new Section 5 and transfer from Section 4 to this new section the main conclusions of the paper. In Section 4, the authors should just discuss the results presented in the paper.

 

Minor comments

The authors should be coherent with the decimal separator (e.g, line 33 “0,3%”; line 277 $\lambda = 0.02$).

 

Some detected typos:

Line 146 – repetition of the article “a”;

Line 178 – the reference is missing “[?]”;

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

The paper studies interest rates (IR) models governed by jump-diffusion processes; the authors particularly focus on an Ornstein-Uhlenbeck process with discontinuities governed by Poisson jumps with various distributions. They discuss the existence and behavior a long term interest rate/discount factor, and how it is impacted by the existence of discontinuities.

The paper is well-written and, in my opinion, it can be accepted after the minor revisions listed below.

• P.1 l.18 please specify that this holds true only if r is assumed to be continuously compounded;
• In introduction, please improve the bibliography: a reference to the HJM framework (which contains most IR models) should be made; also, I am surprised that no mention is made to C. M.  Ahn and H.E. Thompson "Jump-diffusion processes and the term structure of interest rates" (The Journal of Finance 1988). Also, please refer to the classical jump-diffusion models used in quantitative finance such as the Merton model or the Kou model;
• P.3 eq. (2) please specify the measure under which the expectation is taken;
• P.6 l. 178 there is an empty reference;
• In approximations P.8, I would like to see a precise estimate for the remaining terms (notably for eq. 40)
• P. 9 l. 245 the authors conclude that "for finite and symmetric jumps where ups and downs in return are equally likely discontinuities always reduce the long-run discount rate"; I think this should be put into the perspective with the discussion of Ahn and Thompson who essentially observe the same results (the bond prices are strictly higher under jump-diffusion).
• In subsection 3.2, the authors study 2 types of distributions of jumps. I would like to see a short discussion on the benefit of using non-random (Dirac) vs. exponential (Laplace) jumps in terms of modelling and calibration.
• I think a conclusive section with some potential future works would be welcome. In particular, the condition for the characteristic function to be finite in -i/alpha appears to be rather close to the condition for the existence of a martingale measure (which is equivalent to the finiteness of the characteristic exponent in -i), which may make for an interesting extension of the work towards option pricing for instance.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf