Editor’s Letter

PROVOCATIVE BLACK-SCHOLES

In an earlier Editor’s Letter I discussed briefly the many articles The Journal of Derivatives receives and considers for publication. The submitted articles span a wide spectrum as mapped to several characteristics. One particular category I appreciate are those that are “likely wrong but fascinating.”

A considerable number of such “wrong but fascinating” articles are those that reject Black-Scholes. The earlier letter probed the objection based on the meaning, interpretation, and application of the principle of risk neutrality. As there is no need here to re-trace that discussion, I will say more broadly that every theory and model has elements that approximate reality. Hence, questioning and criticizing models is fair game. But one must also understand what models can and cannot do.

ALL MODELS ARE WRONG, SOME MODELS ARE USEFUL

The great statistician George E. P. Box stated this models-are-wrong philosophy in many colorful ways.¹ Every model embeds assumptions and idealizations. All models betray reality in some manner for the sake of tractability. Models may be useful only if and when users comprehend the deviations from reality.

Black-Scholes is a useful model for several reasons. The model states clearly its assumptions such as: geometric Brownian motion for the underlying equity; constant values for interest rate, dividend yield, and volatility; full liquidity for long and short hedging positions in the equity; and absence of bid-offer spreads. In light of these headline assumptions, users of Black-Scholes cannot fail to understand that any results of the model must be uncertain and approximate as well.

My philosophy does not contradict Box but it veers in a different direction. As I’ve explained at greater length, models are useful primarily if builders and users act in good faith, understand that model results have limited and varying precision, and employ models primarily for learning, judgment and testing completeness and quality of input data.² That last part is fascinating. Models excel for the assessment of data integrity.

A SIMPLE BLACK-SCHOLES QUESTION

Here’s a simple Black-Scholes observation that shows how this useful model assists learning. Consider the value of a European call option on an equity with assigned parameters of: current equity price S; strike price K; zero dividends; interest rate r; volatility σ; and time to expiration T. As reference, the familiar expression for the call option value C with these stated parameters, where Φ[] denotes the standard normal cumulative distribution function, is:

In the two distinct limits of strike price K → 0 and expiry time T → ∞, the call option value C equals the equity price S. Textbook discussions frequently cite these limits to help explain why the Black-Scholes result is sensible. If strike price is zero, then the owner of the call option eventually receives the equity with no payment obligation. If the time to expiry is large, the present value of the strike price that the call owner must pay at expiry, far in the future, is small. The owner of the call option will ultimately own the equity with no or negligible payment of a strike price.

LIQUIDITY DIFFERENCE?

The question I raise is: “Should the European call option truly equal the equity price in either of these limits?” The consideration that prompts the question is the choice between the European call and the equity. If the offered prices of the two are equal, then why would an investor not choose the equity itself? The European call may have equal ultimate value (at expiry), but the equity is likely to be far more liquid. In fact, liquidity of the underlying equity for both long and short positions is a Black-Scholes tenet. Liquidity has value, the determination of which is almost entirely an unsolved problem of the financial world. The equity should, thus, trade at a premium to any arguably equivalent instrument if its liquidity is greater or more reliable.

Or perhaps there is an error in my implied opinion that European call option liquidity is necessarily inferior to that of the equity. If I own this option and there is insufficient apparent liquidity, in principle I may create the replicating hedge with the underlying equity. Hence one may argue that any derivative is liquid by this construction.

My larger point is that the Black-Scholes model permits us to ask and discuss this question and reach a seemingly new perspective that the liquidity of the derivative itself may be relevant.

DIVIDEND RISK?

Aside from liquidity, another question for the Black-Scholes model concerns the dividend dependence in the T → ∞ limit of the European call option value. Imagine that the firm, which previously paid no dividend, abruptly institutes a positive dividend. The call option’s limiting value falls from S, the equity price, to zero with the dividend initiation. An investor who owns the long-dated call option would experience huge loss of value if he/she had purchased or marked the European call option value near S. This loss is not “the model’s fault” since the model openly discloses its assumption of an unchanging dividend yield.

This conversation regarding the Black-Scholes model motivates the important topics of dividend risk and dividend derivatives. See, for example, R. Tunaru, “Equity Portfolio Trading with Volatility and Dividend Derivatives.” The Journal of Derivatives 29 (3): 46–64, 2022.

OUR NEW ARTICLES

Leading the six articles of this issue, David H. Annis and Damian F. Abasto of Vernon Capital Partners describe the earlier work of Daigler, Dupoyet, and Patterson (“The Implied Convexity of VIX Futures.” The Journal of Derivatives 23 (3): 73–90, 2016) identifying convexity of VIX futures. This new article modifies the Daigler-Dupoyet-Patterson interpolation procedure to improve accuracy and then adroitly builds the term structure for VIX convexity. In this context, “convexity” is the second moment of the VIX probability density. Annis and Abasto highlight instances of negative implied variance that are primarily due to imprecise interpolation.

Shuxin Guo of Southwest Jiaotong University and Qiang Liu of Southwestern University of Finance and Economics begin their article by complimenting the ingenious Wu-Zhu (2016) static hedge of one target option with a portfolio of three different options. Guo and Liu find an enhancement to this hedging technique by virtue of deducing a generalization of the Dupire equation that Wu-Zhu employs. Through simulation and a proposed alternative error measure, the authors find improvement in hedge performance.

Young Shin Kim of Stony Brook University, Hyangju Kim of Citigroup, Jaehyung Choi of Goldman Sachs, and Frank J. Fabozzi of EDHEC Business School propose a “normal tempered stable” (NTS) stochastic model to value multi-asset options. As the authors explain, Barndorff-Nielsen developed NTS more than twenty years ago as a Brownian motion in which the time scale has its own stochastic process. In lieu of imposing constant correlation among the assets of the underlying portfolio, this article permits time-dependent asset correlation with an Ornstein-Uhlenbeck process—prompting the model name “NTS-OU.”

Wujiang Lou of HSBC describes what he calls the pricing dilemma of total return swaps (TRS). In the real world of practitioners, valuation of derivative contracts such as TRS evolved seismically at the LIBOR disruption period beginning 2007–8. The author describes the interplay of TRS pricing, funding value adjustment (FVA), and the repo market. He resolves the dilemma and shows how collateralization, credit risk, and other aspects alter TRS valuation.

Apoorva Koticha and Joseph M. Marks of Northeastern University and Chen Li of the University of Massachusetts investigate empirically the relationship between equity index sector correlation premiums and aggregate market correlation premium with models and trading strategies. The authors find that implied volatilities of index sectors evolve in a manner that maintains a stable relationship between sector correlation premiums and the larger market correlation premium. If valid as the authors assert, this knowledge enables trading strategies that dominate those based only on sector volatility premiums.

Chen Tong of Xiamen University and Zhuo Huang of Peking University build on recent research to apply “realized semivariance” to the calculation of VIX and the pricing of VIX futures. Instead of a single variance (square of volatility), the authors employ the distinct components of volatility for positive and for negative equity index returns. Perhaps not surprisingly, this asymmetric treatment of variance provides improved models for observed asymmetric phenomena such as equity returns.

Joseph M. Pimbley

Editor