Option Pricing with the Logistic Return Distribution

Round 1

Reviewer 1 Report

This paper sets out to derive an option pricing formula based on the logistic distribution rather than the normal.  This is a worthy pursuit because the logistic distribution is more convenient analytically than the normal, and the it is also quite possible that the logistic assumption fits the data better.

However, I regret to say there is a serious flaw with the paper.  The derivation of the option pricing formula in Section 3 is based on the assumption that the value of the underlying asset at maturity follows a logistic distribution.  This is illogical because the logistic distribution is symmetric and has range -infty to +infty.  What is required is a distribution that is positively skewed, and defined only on the range 0 to +infty.  In other words, you should be assuming that the price at maturity follows a LOG-LOGISTIC distribution.  This is obvious when you consider the analogous situation of the BS model which is based on the assumption that the price at maturity follows not a normal distribution but a LOGNORMAL distribution.

In the paper, there are early signs of this confusion.  The first line of the Abstract reads "The BS model ... implies a log-normal distribution of stock returns".  WRONG.  Returns must be taken to mean proportional changes in the price over a fixed time period.  The BS model assumes a NORMAL distribution for returns, and consequently a LOGNORMAL distribution for the price at maturity.

I have no doubt that this paper has promise on the basis that this serious flaw is addressed. 

Author Response

Please see the attached revision report.

Author Response File: Author Response.pdf

Reviewer 2 Report

I am not convinced by the authors' arguments in section 2 for the use of a logistic price distribution. (See attached file for specific comments.)

Development of the option pricing formula following the Black-Scholes-Merton method (section 3) is straightforward.

The empirical results in section 4 use options on the S&P 500. It would have been interesting (though I don't insist) to see results on individual stocks. (See the attached file for other suggestions on this section.)

The authors' comments in section 5 regarding the fact that trading is NEVER continuous are very germane and need to be emphasized more often in the literature. However, again the authors need to resort to distribution fits to argue that finite transaction times necessarily imply logistic price distributions. 

The paper requires some revision. I'm never clear on a threshold between "minor" and "major" - but ask the authors to address my comments in the attached file.

Comments for author File: Comments.pdf

Author Response

Please see the attached revision report.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments for author File: Comments.pdf

Author Response

Please see the attached revision report.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Thanks for responding to the comments.

Author Response

Thank you for your review.

Reviewer 2 Report

I thank the authors for responding to my previous comments regarding "editorial" issues with the manuscript. With those out of the way, I can concentrate on what the authors' work is.  I'm afraid I retain two serious issues regarding this work.

1. The title and sections 1 and 2 argue that the price return distribution is best-fit (compared to 10 competitors) by a logistic distribution. I'm willing to accept their argument. However, in section 3 they assume the price distribution (at a fixed time - corresponding to a maturity date) is logistic. I simply cannot see that.

Whether using continuous (i.e. log-) daily returns or discrete (arithmetic) daily returns, if the return distribution R ~ logistic (u, s), then the price P=exp(R) is not logistic. This makes "perfect sense", as the logistic distribution has support (-\infty, \infty) while P=exp(R will have support (0, \infty).

(This is clearly the case in the standard theory based upon geometric Brownian motion where the returns are normally distributed and the prices are log-normal.)

So I believe that the discussion in section 2 invalidates the fundamental assumption in section 3, that the price distribution at maturity can be logistic.

The authors attempt to address this on page 4 of the introduction ("This unorthodox approach has pros and cons ..."), however the fundamental inconsistency between sections 1,2 and 3 is very problematic.

While the assumption of a logistic price distribution may lead to "nicer" Black-Scholes-Merton type option pricing formula, "niceness" is not an acceptable substitute for consistency and accuracy.

2) I am not a fan of continued use of the Cox-Ross-Rubenstein (CRR) and Jarrow-Rudd (JR) type approaches where option pricing is performed only in the risk-neutral world. In my mind, any useful approach to option pricing

a) must begin in the natural world and

b) pass to the risk neutral world through a clear no-arbitrage, market complete formulation,

c) leading to the identification of an equivalent martingale measure.

The approach a),  b) and c) maintains a clear map between quantities in the risk-neutral and natural worlds, something neither the CRR or JR formulations preserve.

In summary, I retain two serious concerns with this work. I will recommend a "rejection" and let the editors make a final decision.

Author Response

The fundamental approach taken in the paper, which is employing the logistic distribution of end-of-period values to derive option pricing.

This approach has pros and cons, which are openly discussed in the paper. The paper makes the case that this approach is valuable, as it yields an elegant pricing formula that empirically outperforms Black and Scholes.

The referee's second objection is very strange: he is willing to accept that the end-of-period value distribution is logistic, but is unwilling to accept that the distribution of total returns is logistic. This is a puzzling objection, because the total return is just the end-of-period value divided by the price at time 0. We explained this somewhat trivial point in our revision report.