WILMOTT Magazine July 2023 – Black-Scholes 50th Anniversary Issue Part 2

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The July 2023 issue of WILMOTT Magazine completes our celebration of the publication of the seminal paper, ‘The Pricing of Options and Corporate Liabilities’ by Fischer Black and Myron Scholes on the valuation of options.

Nassim Taleb begins the proceedings, proffering an unpublished gem from the vaults (somewhere in the mountains of Extremistan, I think) with ‘Tail Option Pricing Under Power Laws’ Taleb builds a methodology that takes a given option price in the tails with strike K and extends (for calls, all strikes > K, for puts all strikes <K) assuming the continuation falls into what he defines as “Karamata Constant” over which the strong Pareto law holds. The heuristic produces relative prices for options, with for sole parameter the tail index α, under some mild arbitrage constraints.

Usual restrictions such as finiteness of variance are not required.

The heuristic allows Taleb to scrutinize the volatility surface and test various theories of relative tail option overpricing (usually built on thin tailed models and minor modifications/fudging of the Black-Scholes formula).

In ‘71,’87’ Aaron Brown tells us why this celebration should lead us to think about the history and the most important idea that no one really appreciated at the time.

Andrei Lyashenko and Fabio Mercurio and present ‘The Generalized Forward Market Model: Summary of Results and Extensions’ In this paper, the authors review the generalized Forward Market Model (FMM) and its subsequent extension introduced by Lyashenko and Mercurio (2019, 2020). Lyashenko and Mercurio then introduce new analytical results and discuss the issue of discontinuity of the implied instantaneous rates. The authors conclude by suggesting a new generalization that decouples the volatilities of the short rate from those of the FMM rates.

Peter Hauser, Andrew Kumiega, Gary Lahey and Greg Sterijevski invite us to consider’ A financial Möbius strip: Black-Scholes and Technology’. The authors discuss how Black-Scholes and the later binomial tree calculations started many foundational technologies used in Fintech today.

Anatoliy Swishchuk contributes ‘Romancing with Black-Scholes Model: An Overview’ This paper is an overview of the author’s previous and recent results associated with the modifications of Black-Scholes model and formula. There were and are attempts to extend the classical Black-Scholes model by introducing the randomness into the coefficients r, μ, σ (interest rate, drift and volatility, respectfully) or to describe them by SDEs, and so on, and then to find explicit or close-form formulas for the hedging strategies and option prices. If coefficients r, μ, σ depend on some parameter x ∈X, where X is some state space, i.e., r(x), μ(x), σ(x) then Swishchuk calls it (B, S, X)-security markets or (B, S)-security markets in random environment X. The author considers different (B, S, X)-securities markets, including Markov, semi‑Markov cases, jumps, and analogues/generalizations of Black-Scholes formulas for them. Black-Scholes formula with delay is also presented. Two types of telegraph process are considered, classical/symmetric and asymmetric, and two applications of those telegraph processes in finance are given by presenting European call and put option prices. Moreover, Swishchuk introduces a new type of Hawkes process, namely, the exponential one-dimensional general compound Hawkes process (E1DGCHP), consider its limit theorems (LLN and FCLT) and present an analog of  Black‑Scholes formula in this case. Numerical examples are presented for both telegraph processes and E1DGCHP.

‘Once upon a time there was a magic formula …’ writes Laura Ballotta about an illuminating example of cross-pollination, which has led to a new field of study, mathematical finance, and a deeper understanding of the pricing mechanisms for any security in the market.

Daniel Bloch presents ‘A Review Of “The Pricing of Options and Corporate Liabilities” Bloch chronicles and reviews the  literature predating and post-dating the publication of this work in order to identify precisely which new concepts they introduced to the field.

In ‘The Black-Scholes Magic Trick, Explained!’ David Orrell writes that while it is really a predictive formula, which estimates option prices based on a probabilistic price distribution, its trick is to present itself as a prescriptive formula (which somehow defines the correct option price — like a mentalist who predicts the future by making it happen. By using it as a calculating device, investors only seem to confirm its predictions. What kind of higher-level voodoo is this?

In ‘Hedging in the Age of Statistical Learning’ Jörg Kienitz presents a data driven and model free approach termed Proxy GMM Hedge for hedging.

To wrap things up for this special section ‘There is No I in Black-Scholes but There is Me in Merton’ Looking at these seminal, Nobel-prize winning contributions, that arguably founded the field of quantitative finance half a century on, the natural question is: How can I make this about me? Asks Rolf Poulsen.

Elsewhere this issue, ‘Shorts Should not be Worn too Long’ writes Uwe Wystup. If indices fall by 10 percent, shorts will rise by the same amount, right? Buckle up, we’re in for a ride

‘Thinking Differently About Asset Allocation’ Graham Giller asks what aspects of nature are omitted from the mean-variance optimization framework that would lead to the observation that different strategies outperform in the real world?

In ‘Explain Yourself’ Jesper Andreasen considers techniques to reduce the computational complexity of computing high-order P&L explains. Specifically, Jesper shows that you only need first-order risk to compute third-order accurate P&L explains. P&L explains often go bad in practice. Why and what you can do about it is discussed.

Rudi Bogni contemplates ‘Language and the LLMs’. Concerns abound as the world cuddles up to ChatGPT, not least the impact that outsourcing the natural development of language might have on the haves and the have-nots

Luigi Ballabio learns from other banks’ mistakes and introduces us to ‘Assessing duration risk with QuantLib’

‘Machine Learning In Finance’ by Michael Kelly looks at machine learning (ML) or as it is more popularly called, artificial intelligence (AI), through the Wolfram lens.