Technically Speaking, December 2016

Editor’s note: this paper is available at SSRN.com and highlights the history of options pricing models. It demonstrates the long history of quantitative models in the markets and the relevance of history to market analysts. The complete paper, with all citations, can be read at https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2831362.

Abstract: This paper examines option pricing methods used by investors in the late 19th century. Based on the book called “PUT-AND-CALL” written by Leonard R. Higgins in 1896 and published in 1906 it is shown that investors in that period used routinely the put-call parity for option conversion and static replicating of option positions, and had developed no-arbitrage pricing formulas for determining the prices of at-the-money and slightly out-of-the-money and in-the-money short-term calls and puts. Option traders in the late 19th century understood that the expected return of the underlying does not affect the price of an option and viewed options mainly as instruments to trade volatility.

Introduction

Although trading and product innovation in derivative markets increased almost exponentially over the last four decades, option markets existed for centuries prior to the modern era. Mixon (2009) examines equity option data from an over-the-counter market in US during the 1870s and concludes that: “Traders in the nineteenth century appear to have priced options the same way that twenty-first-century traders price options. Empirical regularities relating implied volatility to realized volatility, stock prices, and other implied volatilities (including the volatility skew) are qualitatively the same in both eras…… empirical regularities regarding implied volatility are qualitatively the same in the nineteenth and twenty-first centuries. In both eras, implied volatility typically exceeded realized volatility, was substantially serially correlated, featured significant comovement among stocks, and was higher for stocks with relatively high realized volatility. An implied volatility skew is present and displays significant common movement in both eras. The empirical regularities and pricing behavior are clearly not a function of modern theoretical advances”. This is a fascinating result that leads naturally to the following question: What type of pricing techniques did investors in the late 19th century use?

Kairys and Valerio (1997) use the same data as Mixon (2009) and argue that options were expensive relative to the Black and Scholes (1973) model. However, even with today’s standards, certain options (e.g., out-of-the-money puts) may seem expensive relative to the Black and Scholes model. Moore and Juh (2006) examine daily data for warrants traded on the Johannesburg Stock Exchange between 1909 and 1922 and find that options were fairly priced. They argue that investors had an intuitive grasp of the determinants of derivative pricing. Historical evidence suggests that trading in standalone equity options first begun in the Amsterdam bourse in the 17th century (see Gelderblom and Jonker (2005) and Poitras (2009)). According to Moore and Juh (2006) books such as Higgins (1906), which was available in the early twentieth century, provided a theory of derivatives pricing based mainly on an intuitive understanding of the determinants of option prices, but option traders at that time did not have available an exact option pricing formula.

In this paper I show that option traders in the late 19th century not only had an intuitive grasp of the main determinants of option prices but they have also developed no-arbitrage pricing formulas for determining their prices. The option pricing formulas are described in a book called “PUT-AND-CALL” written by Leonard R. Higgins in 1896 and published in 1906. Higgins was an option trader in London and in his book he describes option pricing methods and option strategies used in the late 19th century in the City of London. In the preface of the book Higgins writes: “The writer of the following pages feels that, in publishing this little book on Options, he may be telling many of his professional friends what they already know perhaps better than the author”. Therefore, one can safely conclude that the methods described in Higgins’s book were common knowledge amongst professional option traders in London. In the chapter “OPTION DEALING ABROAD” Higgins writes (p. 58) that “London is the par excellence the option market of the world” and mentions that large transactions of option trading are also done in the French (Paris) and German (Berlin) markets. In this chapter he provides a vocabulary in English, French and German with terms related to options trading. For example, he says that (p. 59). “In Germany the put-and-call is treated somewhat differently from the London method. It is called Die Stellage for which no English equivalent can be found, but which corresponds with the American expression “straddle “”. The term “put-and-call” which is used repeatedly in Higgins’s book and it is also the title of the book corresponds to an at-the-money-forward (ATMF) straddle. From Higgins’s book it appears that in the late 19the century there was an active option market in Paris, London, Berlin and New York and a network of sophisticated option traders both in Europe and US who probably shared the same methodologies and pricing techniques. The book is available on line here: https://archive.org/details/putandcall00higguoft.

Based on Higgins’s book it appears that option traders in the late 19th century understood perfectly well the put-call parity relationship, used methods of static replication, understood that the expected return of the underlying does not affect the price of an option and viewed options mainly as a vehicle for trading volatility. The pricing approach described in Higgins book could be summarized as follows: First, traders were pricing short-term ATMF straddles (30, 60 or 90 days to maturity. The prices of the ATMF straddles were set equal to the risk-adjusted expected absolute deviation (Higgins uses the term average fluctuation) of the underlying price from the strike price at expiration. The expectation of the absolute deviation was based on historical estimates plus a risk premium for future uncertainty as well as some other markups. Given the ATMF straddle prices as reference points Higgins is using a linear approximation formulae based on put-call parity to price slightly out-of-the-money (OTM) and slightly in-the-money (ITM) put and call options (in Higgins’s book the ITM and OTM options are called “fancy options”). I show that the approximation used by Higgins is analogous to a first order Taylor expansion around the ATMF straddle price. Higgins acknowledges (p. 36) that the pricing methodology described in his book could not be be used when the strike price was set well above or well below the price of the underlying. Higgins also uses the same approximation formula to price “repeat contracts”. The holder of a repeat contract had the right to repeat several times the 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 lenses 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. transaction with respect to the option’s underlying. I show that Higgins’s approximation for pricing repeat contracts works quite well when compared to prices obtained from a modified version of the Black and Scholes model.

Modern theories of option pricing based on deductive reasoning and frictionless perfect market hypothesis provide exact pricing mechanisms for the full cross section of call and put option prices. From Higgins’ book it appears that option traders in the late 19th century used inductive reasoning and interpolation techniques and were able to determine quite accurately the prices of ATM and slightly OTM and ITM short-term calls and puts. The pricing methodology described in Higgins’s book shares many similarities with the pricing methods used today and that can explain the findings of Mixon (2009) that empirical regularities and behavior of options prices are qualitatively the same in the nineteenth and twenty-first centuries. Higgins’s book shows that practitioners had developed methods for determining option prices, well before the introduction of the continuous-time stochastic processes paradigm by Black-Scholes-Merton that revolutionized the area of contingent claim analysis.

The remainder of the paper proceeds as follows. In the next Section I provide a brief history of the development of option contracts from antiquity until the late 19th century. In Section 3 the option pricing methods described in Higgins’s book are presented and the last section concludes.

A short history of the development of option contracts from antiquity until the late 19th century

A widely cited example of an early development of the concept of a contingent claim is the story of Thales in Aristotle’s Politics. In Book I, Chapter 11 of Politics, Aristotle tells the story of Thales of Miletus (624-547 BC), one of the seven sages of the ancient world.

“Thales, so the story goes, because of his poverty was taunted with the uselessness of philosophy; but from his knowledge of astronomy he had observed while it was still winter that there was going to be a large crop of olives, so he raised a small sum of money and paid round deposits for the whole of the olive-presses in Miletus and Chios, which he hired at a low rent as nobody was running him up; and when the season arrived, there was a sudden demand for a number of presses at the same time, and by letting them out on what terms he liked he realized a large sum of money, so proving that it is easy for philosophers to be rich if they choose, but this is not what they care about. Thales then is reported to have thus displayed his wisdom, but as a matter of fact this device of taking an opportunity to secure a monopoly is a universal principle of business; hence even some states have recourse to this plan as a method of raising revenue when short of funds: they introduce a monopoly of marketable goods.”

According to Aristotle’s description Thales had bought a call option on the rental price of oil-presses; the right to rent oil pressures at predetermined price. However, Aristotle’s description is vague as to whether Thales would still have to rent the oil pressures should the favorable oil crop had not materialized. Poitras (2009) correctly argues that if Thales had the obligation to rent the oil pressure regardless of rental market conditions the agreement could also be interpreted as a forward contract with down payment. Aristotle’s story may well be fictitious, however he used the parable of Thales to emphasize that those who possess superior information can establish monopolies and produce large profits.

A real example of a contingent claim that was used widely in antiquity and especially in ancient Greece, was the sea loan or bottomry loan. These loans were early forms of structured products with embedded options. Bottomry loans were loans made to merchants and ship owners for financing the transportation of goods using the vessel and/or the cargo as collateral. The payment of the loan was contingent upon the successful arrival of the ship at the destination harbor. In the event of a shipwreck, the borrower did not bear any liabilities with respect to the repayment of the loan. Bottomry loans were not plain loans since the lender was exposed to risk of the voyage nor a type of partnership since the return of the loan was fixed and know in advance. Using Merton’s (1974) formulation for the valuation of corporate debt the bottomry loan can be viewed as risky debt on vessel/cargo value. The creditors had implicitly sold a put option on the value of the vessel and the cargo. The interest rate charged on a bottomry loan embedded a premium that reflected the probability of a shipwreck event. Due to the implicit put option, bottomry loans are often refereed as early forms of ship insurance. Although there are no explicit records regarding the actual yields on bottomry loans, Cohen (1992) examines sources from Demothsenes and Lysias and argues that yields probably varied between 12.5% and over 100%. Hoover (1926) describes in detail the contractual characteristics and the evolution of bottomry loans from ancient Greece to the Roman era and to medieval Genoa. Bottomry loans assume that at the maturity of the loan in the event of the shipwreck the creditor takes over any value left from the ship and the cargo (V) and if the vessel reaches its destination successfully the creditor receives the face value of the loan (B). The payoff to the creditor is min(V,B) = B-max(B-V,0), a risk-free bill and a short position in a put option. Options were an important financial innovation that contributed significantly to the development of commerce and trade.

Basking and Miranti (1997) and Cohen (1992) provide a compelling explanation based on asymmetric information and information costs as to why these contingent claim contracts were so widely used in maritime trade. They argue that these type of arrangements minimized investor’s information requirements. If the bottomry loans did not embed the cancellation provision, lenders would have to spend time and effort to collect accurate information with respect to the financial viability of the borrowing merchants. Cohen (1992) also argues based on the Lakritos case (Demosthenes Against Lacritus) that in the event of shipwreck, lender’s compensation would require costly, complex, and time-consuming legal efforts rendering the collection of debt quite uncertain. The bottomry loan is an early example of security for maritime financing whose design minimized information costs and asymmetric information.

According to the historical evidence (see Gelderblom and Jonker (2005), Poitras (2009)) exchange trading in standalone equity options first begun in the Amsterdam bourse in the 17th century. In the Amsterdam bourse there was an active market of derivative contracts on the stock of the Dutch East Indies Company (VOC from the Dutch Vereenigde Oost-Indische Compagnie). The Dutch East Indies Company was founded in 1602 (and dissolved in 1799) after the Dutch government granted the company a 21-year trade monopoly on spice trade. The Dutch East Indies Company was an early form of a limited liability company and the first public company in world’s economic history to perform an IPO with issuing negotiable shares sold to the wider public. According to Szpiro (2011), VOC shares were used routinely as loan collaterals. VOC shares were highly valued as collaterals and that contributed significantly to the reduction in interest rates with beneficial effects in commerce financing. However, lenders were exposed to default risk in the event of a significant drop in the value of the collateral. To insure themselves against such losses, they would enter put option positions to have the right to sell the VOC shares at a predetermined price. Again, the introduction of stock option contracts can, at least partially, be attributed to the existence of information asymmetries between borrowers and lenders and information costs. Should lenders did not have the ability to hedge collateral value they would have to collect additional information with respect to other assets owned by the borrower. Petram (2011) examines actual options transactions data on VOC shares and finds that call options were also routinely used for speculative reasons. Josef de la Vega (1688) and Isaac de Pinto (1762) are the primary sources regarding option trading in the Amsterdam bourse. Poitras (2009) shows that both de la Vega and de Pinto had an intuitive grasp of the put-call parity since they both knew how to covert call prices to put prices and vice versa. However, the put-call parity relationship is not explicitly formulated neither in de la Vega nor in de Pinto.

Stock and option trading gradually moved to London after the Great Revolution and in 1773 the London Stock Exchange was formally established. Houghton (1694) provides a detailed description of equity options trading in London (see Poitras (2009) for a thorough analysis of Houghton’ book). At that time derivative trades were often called “time trades” or “time bargains” or “jobbing” trades (see Harrison (2003)). Murphy (2009) examines actual option price data for the period from January 1692 to mid-1695 from the ledgers of a financial broker named Charles Blunt. Call and put options (call options were called refusals in the late seventeenth century) could be exercised at any time before expiration (American-style) with physical delivery of the underlying security. The majority of the options were at-the-money with 6- months to expiration. Murphy (2009) finds that Blunt’s clients used options for both risk management and speculative purposes and option prices were more or less consistent with basic pricing rules. Out-of-the-money options were less expensive than at-the-money and in-the-money options, short-term options were less expensive than long-term options and option prices were highly sensitive to changes in market uncertainty.

Following the events of the South Sea Bubble in 1720, the English parliament in 1734 passed Sir John Barnard’s Act which was a set of rules aimed to prevent stock jobbing and trade in options and forwards was forbidden. Poitras (2009) reports the main provision of the Act that states: “All contracts or agreements whatsoever by or between any person or persons whatsoever, upon which any premium or consideration in the nature of a premium shall be given or paid for liberty to put upon or deliver, receive, accept or refuse any public or joint stock, or other public securities whatsoever, or any part, share or interest therein, and also all wagers and contracts in the nature of wagers, and all contracts in the nature of puts or refusals, relating to the then present or future price or value of any stock or securities, as aforesaid, shall be null and void.” In Barnard’s Act, forward trades are referred to as contracts to “deliver” or “receive” and as “wagers and contracts in the nature of wagers”, put options as contracts to put upon and call options are referred to as refusals. Harrison (2003) notes that forward trades were considered wagers because they were often contracts for difference similar to modern CFDs. According to Harrison (2003) frequent instances of short-selling banning, usually after large market drops, also occurred in the Amsterdam exchange. Following the Tulip bubble crisis in the 1630’s the Dutch government also rendered time bargains unenforceable by law.

Despite the fact that “time trades” were in general unenforceable by law and often explicitly banned, the market for these contracts continued to grow during the 18th century. Harrison (2003) notes that Barnard’s Act proved to be ineffective since Bills aim to ban time trades also were introduced in 1756 again in 1756 and in 1776. The 1746 and 1756 bills came under the title “a bill more effectually to prevent the infamous practice of stock-jobbing”. Clearly, benefits from time trades outweighed the potential costs from illegal actions or counterparty default risk. Poitras (2009) notes that in 1820 even though options trading was illegal under Barnard’s Act, members of the London Stock Exchange tried to pass a rule to explicitly forbid Exchange members from dealing in options. The rule failed to pass and by the end of the 19th century equity options were traded using the central clearing house of London stock exchange.

Equity option trading in US probably started in the late 18th century but according to Poitras (2009) historical recourses are scarce. An active over-the-counter market for equity options developed in New York after the civil war. Mixon (2009) notes that option trading was generally perceived as socially undesirable and a cause of extreme speculation and volatility in agricultural commodity prices and sometimes was explicitly banned by legislation. During the second half of the 19th century option transactions in Paris started to grow and in 1885 derivative contracts became legally enforceable in France (see Weber (2009)). During the 19th century options trading also spread from France to Germany. According to Higgins (p. 58) in the late 19th century the London market was par excellence the option market of the world since “nowhere the same facility for giving and taking, for operating in long and short options, and for hedging against a favourable put or call in the firm stock as that which exists in London. It rarely happens that an option is done in the Paris market for more than one month ahead, and in Berlin too the majority of such dealings are arranged for a similar period. In London two and three months’ calls are easily negotiated in the active stocks.”

THE PUT-AND-CALL by Leonard R. Higgins

The book PUT-AND-CALL was written by Leonard R. Higgins in 1896 and published in 1906. The books consist of 11 chapters and presents in a systematic way definitions of various options contracts, examples of speculation and hedging strategies and methods for pricing option contracts. From Higgins’s book it appears that in the late 19th century there was an active options market in the City London. The underlying assets described in Higgins’s book are British and American shares, British government bonds (Consols), and Spanish government bonds. Options could be exercised only at maturity (European style), they were written on the forward price of the underlying (the forward contract had the same expiry date as the option), they were dividend protected, and time to expiration ranged from 15 days up to 90 days. According to Higgins (p.1) the Committee of the London Stock Exchange recognized the legality of option dealing only when the maturity of option ranged from 15 days to 45 days. For longer maturities the Committee “will not legislate upon any dispute arising from an option transaction”. In Higgins’s book the options writer is called the “the taker” and the option buyer is called “the giver”. Premiums were paid at expiration and standard options were ATMF. ITM and OTM options and repeat contracts are called “Fancy Options” in Higgins’s book.

I begin the analysis from the last chapter of the book, Chapter XI, (The Value of the Putand-Call) where Higgins explains how to price an ATMF straddle. Then I study Chapter V (The Conversion of Options), Chapter VI (The Principles Formulated), Chapter VII (The Call o’ more, Put o’ more), and Chapter VIII (The Call of twice more, three times more, etc,) where the author describes option conversion and static replication based on the put-call parity and uses as a starting point the price of the ATMF straddle to price fancy options.

The Value of the Put-and-Call

In Chapter VIII called “The Value of the Put-and-Call”” Higgins describes his approach for pricing ATMF straddles. According to Higgins (p. 70-71) the basic rules for finding the price of the ATFM straddle are:

“Firstly, to ascertain the past average fluctuations over a considerable period of time of the stock to be operated in.

Secondly, to consider whether there is any special influence at work calculated to modify that average result in the immediate future (such as a particular scarcity of the stock for delivery, financial strain, or probability of political complications).

Thirdly, to accept risks on approximately the same amounts of stock at regular intervals of time.

Fourthly, to add to the “average value” of the put and call an amount which will give a fair margin of profit and allowance for working expenses.

Fifthly, to make provision for possible default on the part of the giver (since the option money only becomes payable at the end of the option period), and for special contingencies, such as large differences or bad debts on option stock carried over through buying one half of the stock to convert the call into a put-and call or loss through an unexpected rise in the money rate, none of these mischances being provided for in the “average value” tables, which have been calculated simply from the average fluctuations.

Sixthly and lastly. To be careful that, having once accepted a risk, the option shall be allowed to run to the end of the option period without being tampered with by hedging operations in the firm stock or “cutting the loss ” before its time, and that at the expiry of the option the profit or loss shall be taken as final and the position be absolutely closed.”

Higgins calculates past average fluctuations for various stocks. For example, in the table below, he calculates the average two months and three months’ fluctuations of Louisville and Nashville shares:

Although not explicitly stated, Higgins calculates the average past fluctuation as the past mean absolute return which is the same as the mean absolute deviation (MAD) when the mean return is set to zero.

Under Higgins’s approach the price of the put-and-call, in the book it is denoted as p.a.c., is set equal to the past average fluctuation, plus a risk premium for future uncertainty, plus an additional markup for a profit margin, working expenses and compensation for the default probability of the option buyer (premiums were paid at expiration). Note that in Higgins’s book option prices are reported as percentages relative to the price of the underlying at the imitation of the trade.

Using modern derivatives notation, the payoff of the p.a.c. at expiration (T) can be written as:

where FtT, is the forward price (with delivery at time T) of the underlying stock at the initiation of the trade at time t. Therefore, at the initiation of the trade at time t the price of the put-and-call is equal to the expected absolute deviation:

    (2)

where Q is the risk-neutral probability measure. Brenner-Subrahmanyam (1988) show that using a Taylor expansion of the normal CDF about 0 the ATMF call option according to the Black-Scholes formula is approximately equal to:

    (3)

Using the approximation in (3) and adjusting for the fact that premiums are paid at expiration, the value of the p.a.c. is equal to:

    (4)

where σ is the annualized standard deviation. Relationship (4) holds approximately even when the underlying asset follows a process with stochastic volatility:

(5)

where is conditional risk-neutral expectation of return volatility over the time interval [t, T]. Carr and Lee (2007) show that the approximation in (5) is quite accurate under general market setting, especially when the correlation between volatility and returns is zero.

For the case of a normal distribution with standard deviation σ, a well know result in statistics is that the standard deviation is equal to s p = ´2 MAD . Under the normal distribution, the Black and Scholes approximation in (4) and the pricing equation in (2) become identical, up to an approximation error. Option traders in the late 19th century and option traders today both view the prices of ATMF straddles as risk-adjusted estimates of future stock fluctuation. The difference being that modern option traders measure stock return fluctuation using the concept of variance and standard deviation while option traders in the late 19th century measured stock return fluctuation using the mean absolute deviation.

From Higgins’s description it is evident that option traders did not consider the expected return of the underlying assert as a factor that had a direct effect on the price of the option. Using the modern option pricing approach, the expected return of the asset is eliminated from the pricing formula through dynamic delta hedging. Option traders in the 19th century arrived at the same conclusion using an approach based on intuition and experience. Τraders in the late 19th century assumed that the sensitivity of the ATMF option prices with respect to the underlying was 0.5 and therefore considered ATMF straddles as “delta neutral” assets whose value depended only on the future fluctuation of the underlying. Huang and Taleb (2011) refer to Nelson (1904) who says that “Sellers of options in London as a result of long experience, if they sell a Call, straightway buy half the stock against which the Call is sold; or if a Put is sold, they sell half the stock immediately, finding that in the long run this method usually works out a profit.” Note that Nelson (his book called the “The A B C of Options and Arbitrage) refers extensively to Higgins’s book and whole sections from Higgins’s book are also included in Nelson’s book (for example the chapter “The Value of the Put-and-Call”).  In the next section, it is shown how Higgins priced ITM and OTM options using a linear approximation around the price of the ATMF straddle. Through inductive reasoning option traders in the late 19th century concluded that since the expected return does not affect the price of the ATMF straddle it also does not affect the price of OTM and ITM options, since both were priced relative to the ATMF straddle.

Similar to modern equity markets, stock returns in the 19th century were also characterized by fat tails and time-varying volatility (see Mixon (2009)). Higgins does not postulate any particular distribution for stock returns. He is using a pricing method based on the empirical distribution of stock returns and any departures from normality are implicitly embedded into the estimate of future absolute deviation.

Higgins (p.73) also provides an example on the pricing of a two month ATMF straddle whose underlying is the stock of Louisville. Higgins’s computations are shown in the table below:

According to Higgins’s calculations, given an estimate of future absolute deviation a markup of approximately 20% is added to derive the price of the ATMF straddle. A difference of 20% between historical and implied absolute deviation is in line with the size of the variance risk premium of individuals stocks in modern option markets (see, for example, Carr and Wu (2009)). Higgins (p.68) also comments on the realization of the premium and writes that the option taker “must “take the money” very many times to make it pay. It is just so with the taker of option money. Not only must he ascertain the average past behaviour of the stock he is about to deal in, but he must be careful that he can sell this risk a sufficient number of times during the year to establish the average upon which his premium is based.”

Pricing the “Fancy Options”

In many parts of the book Higgins is using the put-call parity for the forward values of the put and the call. Given the risk-free interest rate (r), option expiration (T-t) in years, the price of the underlying (S), the price of the call (C) paid at time T, the price of the put (P) paid at time T and strike price (X) the put-call parity at time t is given below:

(6)

Adding the call price or put price in both sides of (6), Higgins expresses the put-call parity in terms of the put-and-call as follows:

(7)

where D = F-X, is called the “distance”. The put-and-call plays a central role in Higgins’s approach since fancy options are priced relative to the ATMF straddle. In order to price OTM and ITM call and put options Higgins is using the put-call parity relationship in (7) as follows:

(8)

In Chapter V (The conversion of Options) Higgins (p.26) provides a set of 8 rules based on the put-call parity:

1. “That a call of a certain amount of stock can be converted into a put-and-call of half as much by selling one-half of the original amount.
2. That a put of a certain amount of stock can be turned into a put-and-call of half as much by buying one-half of the original amount.
3. That a call can be turned into a put by selling all the stock.
4. That a put can be turned into a call by buying all the stock.
5. and 6. That a put-and-call of a certain amount of stock can be turned into either a put of twice as much by selling the whole amount, or into a call of twice as much by buying the whole amount.

7. To sell half the stock at the option price against a call is equivalent to giving twice the amount of money for the put-and-call of half the quantity of stock at the same price.
8. If option money is given for the put, and half the amount of stock is bought against it at the option price, the dealer has practically given twice the option money for the put-and-call of half the stock – at the same price.”

It is obvious that Higgins had an insightful understanding of the put-call parity and derives various replication strategies involving long/short positions in forward contracts, long/short positions in a call or a put and long/short positions in ATMF straddles. Rules 1 and 2 are based in expression (8). Rules 3 and 4 is a standard replication strategy using forward contracts and options based on the put-call parity in (6). Rules 5, 6 are based in expression (7) and rules 7 and 8 in expression (8).

In Chapter VI (The Principles Formulated) Higgins describes his methodology for pricing ITM and OTM options. The example below (p.37) describes the pricing of an ITM and OTM call.

Higgins is using the put-call parity to price slightly OTM and ITM options by assuming that the price of the ATMF straddle remains the same in strikes slightly above/below the forward price. When the option is an ITM call half the distance is added to half the price of the ATMF straddle and when the option is an OTM call half the distance is subtracted from the half the price of the ATMF straddle. Note that this approximation is equivalent to a first order Taylor expansion along the strike dimension:

and assuming that the price sensitivity with respect to the strike is -0.5.

Higgins (p. 38) understands that the approximation works well only for small deviations from the forward price (forward price is also called “the right price”) and writes “ It is quite usual for options to be done for long periods, e.g., three months ahead, where the price fixed is considerably above or below the “right” price for the period ; in these cases the call money is arranged on the basis of a put-and-call increased by an arbitrary amount calculated to cover the additional risk involved in taking option money with so much of the money already ” run off” in one direction.”

In Chaperts VII and VIII, Higgins examines the pricing of option contracts known as call o’more, put o’more, call of twice more, call of three times more etc. These were type of “repeat contracts” that were traded in options markets until the early decades of the 20th century (see Zimmermann (2009)).

The holder of a repeat contract had the right to repeat the transaction of the option n times. For example, in the case of a call option the holder had the right to buy once more (called call o’more in Higgins’ book), twice more (called the call of twice more), three times more (called the call of three time more) the amount of stock etc. In the repeat contracts the premium was paid at maturity but the premium was also added to strike price in the case of a call option and subtracted from the strike price in the case of a put option.

To fix ideas, denote as Rn the premium of a n-times repeat contract. If FtT, is the strike price of an ATMF option, c F R t T n + is the strike price of an n-times repeat call option and  – is the strike price of an n-times repeat put option. The premium of an n-times repeat call option that expires at time T is equal to and the premium of n-times repeat put option is equal to the price of European call option when the price of the underlying is FtT, and the strike price is , is the price of European put option when the price of the underlying is F and the strike price is , p F R t T n – .

Higgins writes (p.41): “From its very nature the call o’ more or put o’ more must of necessity be an optional bargain at a price other than the right price, seeing that the option money is included in the price at which the stock is bought or sold. Consequently, transactions of this description are not often done for long periods ahead, for the “distance” would become so large that the option would have to be very dear to compensate the taker’s risk in commencing a “long shot” operation with a considerable amount of the money “run off”. Call o’ more bargains are therefore more freely done from day to day or for a week, one account or one month ahead.”

In Higgins’s book the premium of the n-times repeat contract corresponds to the “distance” D relative to the forward price of the underlying, and its price is equal to the price of the ATMF straddle. In the example below Higgins (p. 51) presents his approach for pricing the repeat contract.

Conclusions

This papers examines the option pricing methodologies describe in the book called “PUT-AND-CALL” written by Leonard R. Higgins in 1896 and published in 1906. The central object in Higgins’s approach is the price of the ATMF straddle. In the chapter “The Value of the Put-and-Call” Higgins prices the ATMF straddle as the risk-adjusted future absolute deviation. The prices of ITM, OTM options and repeat contracts are determined relative to the price of the ATMF straddle using a linear approximation based on the put-call parity, which is analogous to a first order Taylor expansion. Higgins also provides set of rules for option conversion and static replication based on the put-call parity.

From Higgins’s book it appears that option traders in the late 19th century had developed no-arbitrage pricing formulas for determining the prices of at-the-money and slightly out-of-the-money and in-the-money short-term calls and puts. Higgins’s book is an important reference in the history of option pricing because it provides a pricing framework based on empirical rules and approximation methods for determining option prices. Higgins’s method could be taught in introductory derivatives valuation courses before the Black and Scholes and the binomial model to help students appreciate the historical development of option pricing methods and the contribution of option market practitioners. In future research it would be interesting to examine how the option market in the City of London evolved during the early decades of the 20th century and try to understand why the pricing methods developed by the practitioners in the late 19th century was largely lost and forgotten in the post 1960s era when trading in option markets and academic research in option pricing begun to flourish.