Option pricing methods in the City of London during the late 19th century

Abstract

City of London traders in the late nineteenth century had a much more advanced understanding of option pricing than previously thought

Acknowledgements

I would like to thank Bob Mainprize, Stavros Thomadakis, George Constantinides, Emanuel Derman, Geoffrey Poitras, Chris Brooks, Jerry Coakley, Carlo Rosa, Raphael Markellos, participants at the Essex Finance Centre (EFiC) 2016 Conference in Banking and Finance: A conference in memory of Nick Constantinou (1960–2015) and two anonymous referees for helpful comments and suggestions. This paper was previously circulated under the title “Option Pricing Methods in the Late 19th Century”. Useful resources on the history of option pricing are available here: https://sites.google.com/view/hisopt.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

† Mixon (2008) also provides further evidence on the rational pricing of options during the 19th century.

‡ I would like to thank an anonymous referee for suggesting this analogy between modern option trading and nineteenth century option trading.

§ The book is available online here: https://archive.org/details/putandcall00higgiala.

¶ According to Poitras (2009) both Josef de la Vega (1688) and Isaac de Pinto (1771) had an intuitive grasp of put-call parity, since they both knew how to covert call prices to put prices and vice versa. Knoll (2008) argues that put-call parity was used historically in order to engineer synthetic loans to evade usury prohibition and has its origins in Ancient Israel and the mortgage market of Medieval England. However, to the best of my knowledge, Higgins’ book is the first historical source with an explicit mathematical formulation that links call prices to put prices. Other important references on nineteenth century option trading and pricing are Moser (1875), Lefèvre (1873) and Castelli (1877).

‖ Although not often mentioned in modern derivative textbooks, the payoff of a straddle position is identical to the function of the absolute deviation (see Section 2.1 in this paper).

† Sotiropoulos and Rutterford (2014) also discuss the approach of Higgins with respect to the ATMF straddle, and correctly point out that the straddle is used as an ‘anchor’ for pricing other options. However, they do not show how Higgins’s method can be viewed through the lens of modern option pricing techniques. Haug and Taleb (2011) and Haug (2009) also refer to Higgins’s book and his knowledge of the put-call parity but they do not analyze his pricing methodology.

‡ At the beginning of the 20th century exact option pricing formulae were discovered by Bachelier (1900) and Bronzin (1908). Hafner and Zimmermann (2007, 2009) analyze the option pricing methods of Bronzin. Both Bachelier’s and Bronzin’s pricing methodologies are quite technical and are based on specific distribution functions. To the best of my knowledge, there is no any historical evidence that option traders at the beginning of the twentieth century were actually using these pricing formulas.

§ For a short introduction to the history of option pricing see also Goetzmann (2016, chapter 16) and Lipton (2017).

† Derivative trades were often called ‘time trades’, ‘time bargains’ or ‘jobbing trades’ (Harrison 2003).

‡ In Higgins’s book the market price of a stock is the forward price, which is also called the ‘right price’. ‘It is usual to speak of the ordinary market price for a forward bargain in firm stock as the "right" price for the period in question’, Higgins (p. 5). The fact that option contracts at that time pertained to forward prices of the underlying and the payments were made at expiry, was probably a convenient way to dispense discounting and not having to consider a (riskless) interest rate. I thank an anonymous referee for this useful observation.

§ ‘Options are frequently done at prices considerably above or below the actual prices for the period in question, and in the examples given of these “fancy” Options, we shall describe the difference between the market price and the price fixed as the “distance”’, Higgins (p. 4–5).

† The term risk-adjusted is used because Higgins is adding a markup to the historical estimate of absolute deviation in order to determine the price of the ATMF straddle.

† The book of Nelson is available for downloading here: https://archive.org/details/abcofoptionsarbi00nelsuoft. Nelson’s description is similar to a strategy that is profiting from the volatility risk premium through delta-hedged option positions (see, for example, Bakshi and Kapadia 2003). Using toady’s option pricing models (e.g., Black and Scholes), it can be shown that the delta of ATMF options is approximately 0.5 for short term maturities (see also footnote 10 in section 3.2 of this paper). Suppose that the price of the underlying is F, the option is an ATMF call or put and there is one-period until expiration. During the remaining one-period the underlying can go up by ΔF and the call is exercised or down by ΔF and the call option expires worthless. The opposite holds for the put option. The delta of the ATMF call is equal to ΔF/(2ΔF) = 0.5 and the delta of the put is equal to −ΔF/(2ΔF) = −0.5. A reasonable conjecture to make is that option traders in the nineteenth century used the one-period delta and inductively concluded that it also holds approximately for all short-term period ATMF for establishing static-delta hedged positions.

† In the academic finance literature the put-call parity first appeared in Stoll (1969).

‡ I skip time subscripts in order to be consistent with Higgins’s notation.

† According to the Black and Scholes model, the price sensitivity with respect to the strike (traders call it dual delta) of an ATMF call is equal to:$dual{\mathrm{\Delta }}_{call}^{ATMF}=-N\left(-\frac{1}{2}\sigma \sqrt{\mathrm{T}-t}\right)\approx -\left(N\right(0)-N^{\mathrm{\prime }}(0\left)\frac{1}{2}\sigma \sqrt{\mathrm{T}-t}\right)\approx -0.5+0.2\sigma \sqrt{\mathrm{T}-t}$, where N is the cumulative standardized normal distribution. For short-term options and low volatility, the delta is approximately equal to −0.5. The relationship between dual delta and delta when the option is ATMF is, $dual{\mathrm{\Delta }}_{call}^{ATMF}=-(1-{\mathrm{\Delta }}_{call}^{ATMF})$.

† MacKenzie and Millo (2003) use as a case study the Black-Scholes model and the development of the modern derivative markets to study the ‘performativity’ thesis formulated, in economic sociology, by Callon (1998). According to the performativity thesis, economics do not just describe an external reality but construct that reality, e.g., academic research ‘performs’ market practices.

† The modern approach to option pricing shows that the non-linear nature of option payoffs provides the ability to trade not only volatility but in fact any moment of the future distribution of the underlying asset.