Referee’s Report on
Auto-Static for the People: Risk-Minimizing Hedges of Barrier Options
By Johannes Sivén and Rolf Poulsen
General comments:
The paper discusses a static hedging approach for barrier options based on the minimization
of convex risk measures. The problem is formulated as a convex optimization problem subject
to a compact set and solved via a sample average approximation method. After providing
numerical results for the hedge in the Bates model the authors address the question of model
risk in a simulation study by analyzing the performance of the Bates hedge in the NIG model
as well as the NIG hedge in the Bates model.
Although the idea of identifying static hedges by means of simulation and optimization is not
new, the extension to general risk functionals complements well to the existing static hedging
literature such that it is worthwhile publishing the results. However, I cannot recommend the
acceptance of the article for publication in the special issue “Computational Methods in
Finance” of Finance and Stochastics before the severe issues listed below are addressed.
Specific comments: Replies in red.
1. By moving from symmetric risk measures like E(P-Hc)2 (see Dupont [2]) to asymmetric
ones, it is necessary to introduce a risk/cost tradeoff or risk/return tradeoff in order for
the hedge to economically make sense. To be more precise, these tradeoffs parameterize
an efficient frontier (see e.g. Krokhmal et al. [4]) with various risk/return profiles. While
the authors recognized this necessity in their numerical study (page 10, second
paragraph from below), the formal Section 2 of the paper completely ignores this issue.
Hence (2) has to be replaced with the minimization of cost subject to risk constraints or
vice versa. In this framework it is also not necessary to restrict the presentation to
convex risk measures (which excludes the practically important value at risk). We
agree. We are now considerably more explicit about the risk/cost-tradeoff. We
present results in the form of efficient frontiers.
2. Page 4, third paragraph: The solution of (2) in its current form may not even be unique if
u is strictly convex. For example, the hedge portfolio might contain linear dependent
instruments. But even linear independent instruments may lead to identical E[u(P-Hc)] if
suitable probability measures are considered. We agree. These statements have been
removed.
3. The proof of Proposition 1 is wrong. The assertion E[u(P-Hc*-a)] ≤ E[u(P-Hc-a)] does
not follow from E[u(P-Hc*)] ≤ E[u(P-Hc)] and u increasing. If the expectation is taken
with respect to Dirac measures, counterexamples can be constructed easily. Similarly,
the equality E[u(P-Hc*-a1)] = E[u(P-Hc-a2)] =  cannot be assumed to hold in a setting
with discrete distributions/measures. However, if point 1. above is addressed, the whole
Section 2.2 can be omitted anyways. Yes! The proof is wrong; in fact the proposition
is false. We have – obviously – removed it, but now give a more detailed
description of the relation between the choice of risk measure and the form of the
optimization problem.
4. In Section 2.3 the authors cite stochastic programming results of Kleywegt, Shapiro and
Homem-de-Mello [3]. Since these results were derived for stochastic discrete
optimization problems, they do not apply in the setting of the paper (the set D does not
consist of a finite number of points). Since similar results can be derived for the case of
non-discrete variables, the authors should replace the reference and the presented results
5.
6.
7.
8.
by a more appropriate reference. Ups, yes, in our eager to cite something that is
“journal published” and easily accessible, we pulled the wrong Shapiro paper out
of the hat! Fixed.
Section 3 (numerical results): The numerical results as well as the interpretation of these
results needs to be improved to better clarify the contribution of the paper. First of all, as
a comparison it is important to list the optimal hedge portfolio weights for the different
approaches listed in Table 2. Secondly, Table 2 should also report the maximum loss
encountered by the portfolios. Although it is definitely worth listing the standard
deviation of the hedges, the authors should not choose this number to judge whether a
hedge is better or not, because symmetric risk measures cannot fully assess the
asymmetrical hedge error distributions produced by u(x)=x+ or the super-replication
hedge. Which of the hedges is “better” basically depends on the utility function of the
hedger. In particular statements that one or the other approach “works well” (see page
10, last paragraph) are subjective statements. However, if the numbers mentioned above
are included in the paper, the results will show that the proposed risk-minimizing hedge
will nicely fit between the aggressive Nalholm/Poulsen and the conservative
Giese/Maruhn approach. The table now gives a larger number of descriptive
statistics and consider four different risk-measures; quadratic, “positive part”,
V@R and expected shortfall. Furthermore, we give the portfolios in the forms of
plot of the pay-off profiles. This makes the comparison to N&P and G&M explicit.
Generally, portfolios optimal portfolios will also include shorter expiry options, but
as we demonstrate in Table … little is lost by excluding these. (This is in line with
N&P and G&M.)
It also seems natural to include an efficient frontier in Section 3 since risk/cost profiles
are the main contribution of the paper. Ideally, numbers could be provided for the risk
measures value at risk or conditional value at risk. We do that now.
At several points the authors mention that u(x)=x2 is a “strange” risk measure from a
hedging point of view, because it penalizes gains and losses symmetrically. It is
surprising, then, that this risk function was chosen as the basis for the results presented
in Section 4. Regarding the focus of the paper it would be more appealing to present the
results for an asymmetrical risk measure. We now work with four risk measures (u =
x^2, x^+, V@R and expected shortfall), three of which are “one-sided”.
As the authors indicate in footnote 2 on page 2, the idea of static hedging by simulation
and optimization has been investigated by several authors in the literature (Pellizzari,
Giese and Maruhn, Allen and Padovani [1], to mention a few). But in particular the risk
function u(x)=x2 has already been studied extensively in Dupont [2]. The main
contribution of the present article is hence not a simulation optimization approach, but
instead the treatment of general convex risk measures (which is a contribution in itself).
However, this contribution should be reclassified in Section one and the associated
references should not be limited to a footnote. We agree; we are not the first to cast
the (static) hedging problem in a optimization context. Our references now better
reflect that.
References:
[1] Allen, S. and Padovani, O. Risk Management Using Quasi–static Hedging. Economic
Notes, Volume 31, Issue 2, pp. 277-336, July 2002.
[2] Dupont, D. Hedging Barrier Options: Current Methods and Alternatives, Economics
Series, Institute for Advanced Studies, EURANDOM – TUE, No 103, 2001.
[3] Kleywegt, A. J., Shapiro, A. and Homem-de-Mello, T. The sample average approximation
method for stochastic discrete optimization. SIAM Journal of Optimization 12, pp. 479 –
502, 2001.
[4] Krokhmal. P., Palmquist, J., and S. Uryasev. Portfolio Optimization with Conditional
Value-At-Risk Objective and Constraints. The Journal of Risk, V. 4, Number 2, pp. 11-27,
2002.