Submitted by
Assigned_Reviewer_7
Q1: Comments to author(s).
First provide a summary of the paper, and then address the following
criteria: Quality, clarity, originality and significance. (For detailed
reviewing guidelines, see
http://nips.cc/PaperInformation/ReviewerInstructions)
This paper considers the problem of option pricing in
finance. Using a minimax approach, the authors construct a game between
Nature and an Investor, prove that the game value of this game converges
to the classic BlackScholes option price, and give an explicit hedging
strategy that achieves this value.
Clarity: This is a very
mathheavy paper. Unfortunately, I am not very knowledgeable in the area
of stochastic calculus, so I am unable to verify the correctness of the
proofs in the paper and in the 14page supplementary material. The authors
do provide a reference to a standard book in stochastic calculus, but
unfortunately I do not have the time to familiarize myself with the
material. This is not a critisism of the paper. In fact, the authors
explain the results in words very well and outline proof steps clearly for
anyone who may look to build off these results further in the future.
Quality: As mentioned above, I could not verify the correctness of
the results, but rigorous proofs are provided for all of the theorems. The
authors clearly state the strengths of their results (explicit strategy,
weaker assumptions required) over previous work.
Originality: The
Nature vs. Investor game approach is not new, which the authors make clear
in the introduction. This paper is basically an extension of Abernethy et
al. [1] in two ways. Firstly, the assumptions required for the main result
are strictly weaker than the assumptions used by Abernethy et al.
Secondly, the authors provide an explicit hedging strategy for achieving
the game value, whereas such a strategy was absent from [1]. While the
mechanism may not be new, the analysis appears to be sufficiently
original.
Significance: Given the statement above, I would
describe the contributions here as incremental as the results are not
extremely gamechanging compared to previous work. However, the analysis
appears quite difficult and novel, and the problem does appear to be
addressed in a better way than previous research.
On the other
hand, I am somewhat concerned about this paper's relevance to the NIPS
community. After a quick scan through the technical areas listed in the
call for papers, I had a hard time placing this paper in any of the given
categories (although there is a "but not limited to" clause). Given the
paper is purely analytical with no empirical results, perhaps this paper
is more appropriate for a mathematics / ecommerce journal or conference.
On that note, I was wondering if there are any other lessons to be learned
here for a practitioner (Investor) beyond the fact that we now have more
evidence that BlackScholes hedging is robust.
Typo:  line
227: "to to"
RESPONSE TO AUTHOR FEEDBACK:
While option
pricing does not seem to fit the typical NIPS audience, I did fail to make
the connection with stochastic optimization and sequential decision making
algorithms. These are certainly appropriate areas for
NIPS. Q2: Please summarize your review in 12
sentences
A very mathheavy paper that improves upon previous
work, is clearly explained with words, and contains many rigorous proofs.
However, the main contributions appear incremental and the relevance to
NIPS is questionable.
Q1:Author
rebuttal: Please respond to any concerns raised in the reviews. There are
no constraints on how you want to argue your case, except for the fact
that your text should be limited to a maximum of 6000 characters. Note
however that reviewers and area chairs are very busy and may not read long
vague rebuttals. It is in your own interest to be concise and to the
point.
We would like to thank the reviewers for their helpful
and thorough reviews of our work. Please see specific responses below.
== REVIEWER 5: ==
 Regarding the two different types of
constraints, and how they are presented notationally, it is true that we
could have presented both the jump constraint and the budget constraint as
inputs to the function V. But we chose to make the jump constraint a
parameter of V described by a subscript because it's not changing through
one run of the investor/hedging game. The budget constraint is something
that shrinks as the asset's price fluctuates. We decided to distinguish
the changing variables (price, budget) from the nonchanging ones (jump
constraint, total number of rounds of game) this way.
 Regarding
the scale of the constraint \zeta on price jumps: First, a jump of more
than \sqrt{c} would exceed the variance budget in a single trading round.
Second, recall that in the standard BlackScholes analysis the GBM
assumption effectively bounds the typical fluctuation to be something like
\sqrt{T * \sigma^2 / n}, where T is the time to expiration, \sigma^2 is
the variance parameter, and n is the number of trading rounds remaining.
In our work, one can think of T \sigma^2 as the variance budget constraint
c. So, in particular, a fluctuation on the order of \sqrt{c} is much
larger than a fluctuation of \sqrt{c/n} which is essentially what is
assumed in the Black Scholes analysis. Our bounds show how the regret
depends on a fixed value of the fluctuation constraint \zeta. For our
asymptotic results, we do need the fluctuation bound \zeta to tend to 0,
but it can do so at a very slow rate, such as 1/\log n, as opposed to
1/\sqrt{n}. This allows for much larger jumps than would typically be seen
in a GBM.
 Indeed, it would improve the paper to highlight where
the convexity of g arises; it is actually a critical part of our analysis.
The details are in the appendices, most notably Appendix C and D. We will
work to clarify this in the final version.
== REVIEWER 6 ==
 Regarding your point (1): Yes this is a fair criticism, we
should be more careful with our language here. We note that the constraint
zeta <= 1/16 in Theorem 4 is only to make the constant (18c +
8/sqrt{2\pi}) explicit. A larger vale of zeta would result in a larger
constant.
 Regarding your point (2): Yes this is an important
observation. We should definitely say this in the paper.
==
REVIEWER 7 ==
 Regarding the point on 'Clarity': We certainly
don't expect the average NIPS reader to focus heavily on the stochastic
calculus, and try to understand that entire literature. We simply wanted
to include enough of a sketch so the reader can at least appreciate the
connection we are making. The proofs in the appendix are somewhat
mathheavy, but it's mostly calculations that use standard real and convex
analysis. We hope to shorten the presentation for a future version.
 Regarding the point on 'Relevance': All methods for online
learning (an area that has been wellrepresented at NIPS) can be viewed as
hedging strategies that make predictions which insure against future
choices of the process generating the data. We would agree that the area
of "option pricing" doesn't seem an obvious fit for the NIPS audience. But
this is perhaps somewhat surprising given that the BlackScholes model
relies on concepts like stochastic optimization and sequential decision
algorithms for the hedging strategy design. One of the goals of this work
is to draw interest in this topic, especially by showing how tools such as
"online learning" and "minimax analysis"  both very much a part of the
NIPS toolkit  give new insights into this classical framework. We very
much hope that this will spur new research in this direction.
