Option Pricing Theory

The publication of the option pricing work of Black, Scholes and Merton in 1973 ranks amongst the most famous and influential developments of financial economics. Their work threaded together three concepts that had individually emerged earlier in the twentieth century: Brownian motion as a description of the behaviour of security prices; valuation by the law of noarbitrage; and the use of risk-neutral probabilities in valuation.

The earliest known example of the modelling of financial assets as a type of Brownian motion was produced by the French mathematician Louis Bachelier in his doctoral thesis, which was accepted at the Sorbonne in 1900.[1] In fact, the stochastic modelling of Bachelier’s thesis is now generally recognised as the first mathematical modelling of Brownian motion in any context, beating

Albert Einstein’s famous particle motion paper by a handful of years (Einstein was unaware of Bachelier’s work).

Bachelier’s motivation for developing a stochastic model of security prices was not merely to obtain probability distributions of future prices of fundamental assets like equities or bonds or their path behaviour. Such developments would in themselves be revolutionary for the time, but to Bachelier the stochastic description of assets was merely a means to an end: his thesis was not on how to model equities or bonds, but on how to value options (specifically options on French government perpetuities). He had the conceptual insight that the valuation of such options would require a continuous-time stochastic process to be specified for the path of the underlying asset. This is an astute insight given that no such form of mathematical model existed at the time. Bachelier wasn’t merely identifying new applications for existing mathematical frameworks, he was identifying the need for a mathematical apparatus that did not exist. So Bachelier proceeded to develop it and attempted to apply it to the problem of option pricing. As well as introducing the mathematical modelling of Brownian motion, he also developed the Chapman-Kolmogorov equation and made other important developments that showed how the mathematics of physics could be transferred to the emerging field of stochastic analysis. His work was not mathematically rigorous, but it was recognised by the leading stochastic theoreticians of the following half-century such as Kolmogorov as they developed the formal mathematics of probability theory.

Despite being well-known to probability theorists (and being cited in the most important stochastic analysis papers of the century), Bachelier’s work, and in particular, his attempts at option valuation, went completely unnoticed by the economics profession for over 50 years. This changed sometime in the mid to late 1950s, when a leading American statistician of the time, Jim Savage, sent a note to a few leading economists asking if they had read Bachelier. Paul Samuelson, the MIT professor who can be viewed as the leader of the new breed of quantitative financial economists that emerged in the third quarter of the century, was one of the recipients of the note. Samuelson had not heard of Bachelier or his work, but he browsed the MIT library and found a copy of his 1900 thesis. Inspired by what he read, he circulated Bachelier’s work amongst his peers and students. Bachelier had not solved option pricing, but he was able to make some accurate statements about option pricing behaviour—for example, showing that an at-the-money option price would increase in proportion to the square root of its term to maturity (this is not strictly true under more realistic stochastic processes for the underlying asset than the arithmetic Brownian motion assumed by Bachelier, but is nonetheless usually a good approximation). He was also able to derive many probability statements for the behaviour of the underlying asset such as the probability that it would reach a particular price over a given time period (again under the assumption that it followed an arithmetic Brownian motion).

Bachelier had done enough to show Samuelson that his application of continuous-time stochastic processes to derivative pricing opened up many possibilities for further research. And the work that had been done by stochastic mathematicians in the intervening fifty years to develop Bachelier’s modelling intuitions into rigorous mathematics provided a ready-made modelling toolbox for a new generation of financial economists.

The next notable contribution to follow Bachelier on the application of Brownian motion as a description of the stochastic path behaviour of asset prices took a full 59 years to appear. M.F.M. Osborne, a researcher at the US Naval Research Laboratory in Washington DC, published a paper, ‘Brownian Motion in the Stock Market’,[2] which broke new ground in several ways. Osborne proposed that stock prices followed a geometric Brownian motion— that is, it is the natural logarithm of the asset price that is normally distributed, rather than the asset price itself. Osborne does not appear to have been aware of Bachelier, but the use of a geometric rather than arithmetic process was fairly intuitive: the possibility of negative prices was removed; and the process could be viewed as a multi-period generalisation of the single-period mean/variance description of security returns introduced by Markowitz a few years earlier. From this point on, geometric Brownian motion would be the standard modelling assumption for the stochastic process of an asset price. Secondly, Osborne performed some statistical tests of how well empirical asset price behaviour (US stock prices from various sources going as far back as 1831) conformed to the geometric Brownian motion model. Osborne tested for this in two ways: by considering the cross-sectional dispersion of many stock price changes on a given date; and by analysing how the variance of stock prices increased with the length of the measurement interval (up to ten years). Rubenstein has noted that Osborne was the first researcher to perform this second type of test on asset price data.[3] Osborne concluded that the empirical data accorded reasonably well with the geometric Brownian motion model (though he noted that the empirical data produced tails that were fatter than those predicted by the model).

In 1965, Paul Samuelson, now armed with Bachelier’s option pricing thesis and Osborne’s geometric Brownian motion proposal and supporting empirical stock price analysis, synthesised these ideas in a paper that examined option pricing under the assumption that the underlying asset follows a geometric Brownian motion.[4] Samuelson, with the assistance of the world-leading stochastic processes professor Henry McKean, who was also at MIT, was able to derive a call option pricing formula under the assumption that the option had some constant required return P and the underlying stock had a constant expected return of a (and followed a geometric Brownian motion with volatility a). The theory of stochastic analysis was now sufficiently developed to have permitted a closed-form solution to be found for the discounted present value of the option pay-off. But, critically, Samuelson had no theory for how to determine the option discount rate p. He recognised that the call option was a ‘leveraged’ exposure to the underlying asset in the sense that a given change in the stock price would have a bigger proportional impact on the call price than the stock price. In the CAPM terminology, the call option’s beta was bigger than that of the underlying asset. So he argued it must be the case that P was greater than a. But the size of the option’s leverage (and hence beta) also evidently varied with the underlying asset price—it was a stochastic process itself. Samuelson had no strategy for allowing for that. He was close. He had perfected the application of stochastic processes to determine the option’s expected pay-off, but he had no way of transforming it into an economically justified present value.

1973 saw the publication of two papers—the Black-Scholes paper[5] and a paper by Merton[6]—that used a crucial economic insight to determine how Samuelson’s formula could be parameterised to deliver an arbitrage-free option pricing formula for the first time. Collectively, the publication of these papers represented a watershed moment: the era of modern option pricing theory starts here. Scholes and Merton were awarded the Nobel Prize in Economics in 1997, primarily for the contributions in these papers (Fisher Black died in 1995 and so could not receive the award).

The Black-Scholes paper arrived a few months before Merton’s. Although the papers were written contemporaneously and cover similar ground, the Black-Scholes one is the natural starting point for the reader as it is somewhat more direct and less elaborate (at least for the non-mathematician). Merton is more general and more formal. The critical breakthrough in Black and Scholes’s treatment of the option pricing problem came with the recognition that changes in the option value were entirely driven by changes in the stock price—over an instantaneous period, the option and the underlying asset were therefore perfectly correlated assets. This implied that a continuously rebalanced portfolio of the option and the underlying asset could be constructed that would generate a certain rate of return. That is, a portfolio could be constructed that included one unit of the underlying asset and an appropriate short position in the call option that provided an equal and offsetting exposure to movements in the underlying asset. The short position would be determined by the rate of change of the call option value with respect to the underlying asset. In a footnote in their paper, Black and Scholes attribute this critical idea of the riskless dynamic portfolio to Robert Merton.

Mathematically, the portfolio would generate an instantaneous certain return if it was of the form:

The certain return earned by the hedge portfolio must be the risk-free rate in order to avoid arbitrage—if it was greater (lower) than the risk-free rate, an investor could borrow (invest) at the risk-free rate and create unlimited profits by investing in (shorting) the hedge portfolio. Black and Scholes assumed the stock price followed a geometric Brownian motion. As the call option price is a function of the stock price, stochastic calculus (in particular, Ito’s lemma) can be used to express a change in the call option price as a function of the derivatives of its value with respect to the underlying asset and time:

where p and о are the drift and variance terms of the underlying assets’ geometric Brownian motion and dZ(t) is the underlying assets’ Brownian motion or Weiner process.

From this and the assumption that the hedge portfolio must earn the riskfree rate, together with the elementary boundary conditions of the option pay-out at maturity, a partial differential equation can be derived which is familiarly known as the heat-transfer equation of physics (or simply the heat equation), and for which there was a well-established pre-existing solution.

Just like that, the arbitrage-free option pricing equation was obtained (under various technical assumptions such as investors’ having the ability to continuously rebalance asset positions without transaction costs). The observation that dynamic hedging could replicate the option’s pay-offs (or equivalently, generate a risk-free portfolio) was pivotal and had transformed the option pricing challenge into a strictly relative problem: it did not require a grand, unified explicit model of how the economy priced all risky securities; instead, it took the current value of the underlying asset as an exogenous input; that input, together with a description of the underlying asset price’s stochastic process and the risk-free interest rate, was enough to define the arbitrage-free price (at least for their case where the stochastic process was ‘well-behaved’ and followed a geometric Brownian motion). Moreover, not all characteristics of the underlying asset’s stochastic process had to be specified in their option pricing equation. The expected return of the underlying asset (a in Samuelson’s notation) did not even feature in the Black-Scholes pricing equation.

Black and Scholes were not satisfied with providing a single route to their arbitrage-free pricing equation: they also provided a second, alternative approach to its derivation that was based on the CAPM. The CAPM was a single-period model. It had no direct answer to the problem of valuing a security whose beta changed continuously and stochastically. However, Black and Scholes showed that the CAPM could be used to determine the return required on the option at any given instant. They did this by first showing that the option’s beta can be related to the underlying asset’s beta by the equation:

This allowed the required return of the option to be written in terms of the underlying asset beta, and by equating this with the expected return implied by the Ito expansion of the change in the value of the call option as a function of the underlying asset and time, they were again able to obtain the heat equation that they obtained in the dynamic hedging derivation (and the various P terms cancel out). In the historical context of the development of financial economics of the previous 20 years, this CAPM approach to the option pricing formula would perhaps have been the most natural to a contemporary economist, but it was the dynamic hedging argument that proved to be most revolutionary in its broader applicability to derivative pricing and hedging.

Black and Scholes did not use the concept of risk-neutral valuation in either of their two derivations of the option pricing formula. They noted, almost in passing, that the option price does not depend on the expected return on the underlying asset (though the expected return on the option does depend on the expected return on the underlying asset through the above beta

relationship). They did not note that their option pricing formula gave the risk-neutral expected option cashflow discounted at the risk-free rate (i.e. the Samuelson formula with a = в = r ).

It was Merton who generalised and extended the Black-Scholes result in his 1973 paper, and arguably he provided a more mathematically rigorous treatment of the derivation of the pricing formula. Merton highlighted that the Black-Scholes formula was equal to the risk-neutral cashflow expectation discounted at the risk-free rate, though this was not a central focus of his paper—the power of the generality of risk-neutral valuation was not yet appreciated or anticipated. Merton also generalised the Black-Scholes formula by introducing a stochastic (rather than fixed) interest rate into the model. He showed that in this case, the dynamic hedge portfolio needed to include a third asset alongside the underlying asset and call option: a zero-coupon riskfree bond that matures at the option maturity date. He also extended the formula to allow for the impact of dividends paid on the underlying asset during the term of the option. As a final flourish, he even included an option pricing formula for a ‘down-and-out’ exotic call option (where the option expires if the underlying asset falls to a specified level at any time before maturity).

With these two papers, the permanent and rigorous foundations of arbitrage-free option pricing theory had been laid. The vast research programme on option pricing theory that their work inspired over the following decade would focus on a few key themes: applying option pricing theory to the theory of capital structure; understanding the fundamental valuation ideas that could be identified in the arbitrage-free option pricing formula (in particular, risk-neutral valuation); fully developing the mathematical rigour of the arbitrage-free valuation so as to apply it to a broader range of problems (such as yield curve modelling). We discuss each of these threads of development briefly below (yield curve modelling is discussed in greater detail in the next section).

The financial economics academic profession around this time was almost obsessed with option pricing. Exchange-traded option contracts were, especially in the 1970s, relatively unimportant, niche financial instruments (though trading volumes increased significantly in the years following Black- Scholes-Merton). The academic interest in option pricing theory was primarily driven by the recognition that option-type pay-offs arose in many forms of more widely relevant contingent claims. Perhaps most importantly, the Modigliani-Miller capital structure proposition could be viewed as a form of put-call parity, where the equity of the firm was a call option on the value of the firm’s assets, and the debt of the firm was a risk-free bond less a put option on the value of the firm’s assets. For Black and Scholes, this was the ultimate area of interest, as is reflected in the title of their paper, which refers to the pricing of corporate liabilities. Modigliani and Miller showed that, in wellfunctioning markets, the sum of the claims on the firm always added up to the same total. But they did not say anything about how to value each of the different claims. Option pricing theory opened up the possibility of answering this question.

The application of the Black-Scholes option pricing formula to the valuation of corporate liabilities was further developed by Robert Merton in a 1974 paper.[7] The paper did not introduce significant new, fundamental theories. Rather, it formally clarified the conditions required for the use of the Black- Scholes formula in the valuation of corporate liabilities, and it tabulated and discussed the corporate bond prices, yields and spreads implied by the formula for various assumptions for firm asset volatilities, and the term and level of debt (all assuming the value of the underlying assets of the firm follows a geometric Brownian motion). This paper has resulted in the terminology of the ‘Merton model’ being used to refer to corporate bond pricing models that use the insight that a corporate bond is a risk-free bond less a put option on the value of the firm’s assets. Such models have been widely used in modern risk management practice in banking and credit risk modelling.

Curiously, the analogous insight for equities—that equities can be viewed as a call option on the value of the firm’s assets—has not been so widely used as the basis for the stochastic modelling of equity returns and the measurement of equity risk. Such a modelling approach can provide a logical economic explanation for the well-documented failings of modelling equities directly as a geometric Brownian motion. Since the 1970s, academics and practitioners have taken geometric Brownian motion as the starting point for equity modelling, and have then extended it in various complex ways to capture the fat tails and negative skew that are observable in empirical equity return data and that are implied by equity option market prices. By starting with the assumption that it is the value of the underlying assets of the firm that follow the geometric Brownian motion, and then recognising that equity is a call option on the value of the firm, similar dynamics can be produced and arguably in a more economically coherent way. Under this approach, an option on the equity of the firm becomes a ‘compound’ option—an option on an option. Robert Geske, a financial economics professor at UCLA, developed this logic and obtained the valuation formula for the compound option when the underlying firm asset value follows geometric Brownian motion in a paper published in 1979.[8] In his paper he argues that this approach to valuing equity options is a better theory that produces better empirical results:

The introduction of these leverage effects adds a new dimension to theoretical option pricing. Any change in the stock price will cause a discrepancy between the compound option value and the Black—Scholes value [when applied directly to the equity of the firm]. The qualitative discrepancies between these two formulas correspond to what practitioners and empiricists observe in the market — namely, that the Black—Scholes formula underprices deep-out-the-money options and near-maturity options, and it overprices deep-in-the-money options.[9]

We now move onto the next thread of post-Black-Scholes-Merton research: the general application of risk-neutral valuation as a method of valuing options. Cox and Ross explored this topic in an important paper published in 1976,[10] where they argued:

The fact that we use a hedging argument to derive [the option pricing equation] and the fact that P(S, t) [the price of the option] exists uniquely means that given S and t the value of the option, P, does not depend directly on the structure of investors’ preferences. Investors’ preferences and demand conditions in general enter the valuation problem only in so far as they determine the equilibrium parameter values. No matter what preferences are, as long as they determine the same relevant parameter values, they will also value the option

identically____A convenient choice of preferences for many problems (although

one can envision problems where another preference structure might be more suitable) is risk neutrality. In such a world equilibrium requires that the expected returns on both stock and option must equal the risk free rate.[11]

By this time, the concept of risk-neutrality in valuation was not new. As we saw above, Merton noted in his 1973 paper that the Black-Scholes formula was equivalent to the risk-neutral expected cashflow of the option discounted at the risk-free rate. Furthermore, Samuelson and Merton had derived an early glimpse of the risk-neutral valuation equation in a paper published in 1969.[12] Nonetheless, Cox and Ross highlighted the generality of its applicability as a computational technique for option valuation. They illustrated this point by showing how it could be used to value options under a range of assumed stochastic processes for the underlying asset that were alternatives to the now ubiquitous geometric Brownian motion. Perhaps most importantly, they established the link between risk-neutral valuation and arbitrage-free prices. As Merton himself has written, ‘They [Cox and Ross] were the first to recognise that this relation [risk-neutral valuation] is a fundamental characteristic of “arbitrage-free” price systems in continuous-trading environments.’[13]

Cox and Ross collaborated with Mark Rubinstein to publish in 1979 another seminal paper in the history of option pricing theory: ‘Option Pricing: A Simplified Approach’.[14] The paper attempted to cut through the increasingly sophisticated stochastic calculus of the option pricing theory of the previous few years and clarify the underlying economics. They did this by abandoning the use of continuous-time stochastic processes to describe the underlying asset behaviour. Instead, they considered option pricing theory in the presence of the simplest possible stochastic process for the underlying asset: a single discrete time-step where the underlying asset can take two possible end-period values with specified (non-zero) probabilities (often referred to as a binomial tree). They then extended this tree structure to multiple time-steps and used a backward recursive process (like that used by Pascal in 1654 in his solution of the Problem of Points discussed in Chap. 1) to find option values at every ‘node’ of the tree.

Cox, Ross and Rubinstein showed that the key option pricing theory results—risk-neutral valuation produces the arbitrage-free price and the same pricing function can be obtained by a dynamic hedge strategy; the option price is not a function of the underlying asset’s expected return but does depend on its volatility—applied in this simpler setting. They then showed that the binomial tree option pricing formula converged with the Black-Scholes formula as the number of time-steps in the binomial tree increased towards infinity. This method was very useful for two quite different reasons: it provided a highly intuitive approach to teaching option pricing theory that allowed the fundamental economic logic to be accessed without resorting to advanced mathematics; and the backward recursive tree process provided an advanced analytical approach for valuing path-dependent option features.

A remarkable decade in the history of option pricing theory was rounded off by two papers co-authored by Michael Harrison, a Stanford Professor in operations research.[15] These papers took the risk-neutral valuation logic of the 1976 Cox and Ross paper, and further developed it into a more general and mathematically rigorous valuation theory. These papers showed that an ‘equivalent martingale measure’ could be used to define the arbitrage-free prices of all forms of securities (under appropriate technical conditions). This meant that, in an arbitrage-free market, there existed a probability measure such that the discounted prices of all securities (stocks, bonds, options, other derivatives) must follow a martingale: that is, the expected value of the ratio of the security price to the risk-free account must be ‘driftless’ over time. The papers’ mathematical rigour came at the expense of accessibility. As noted by Davis and Etheridge, ‘This paper [Harrison and Pliska] has turned financial economics into mathematical finance’.[16] It marked the completion of an extraordinary stream of economic research that found immediate and extensive application in the financial sector.

In stark contrast to the binomial trees of Cox, Ross and Rubinstein, the complete and generalised mathematical description of dynamic hedging, replication, arbitrage, martingales and risk-neutral valuation was formidable. A quantitative revolution in finance was first heralded by Redington and Markowitz in 1952. Their work required a limited mathematical tool-kit to understand it—essentially little more than elementary differential calculus and, in Markowitz’s case, also basic mathematical statistics. By 1981, work at the cutting-edge of financial theory required the post-doctoral quantitative skills of the rocket scientist. Unfortunately, this inaccessibility could get in the way of the ‘big ideas’ that were highly intuitive and yet still with profound implication for the management of financial risk. We will revisit this topic in later discussions of how actuarial thinking evolved alongside these developments in the second half of the twentieth century.

Option pricing theory was, and is, just that—a theory. The development of the arbitrage-free pricing models of Black, Scholes, Merton, Cox, Ross and the rest all rely on some assumptions that are quite evidently unrealistic. Of these, perhaps the most fundamentally important is that the underlying asset of the derivative follows a form of stochastic process that permits dynamic hedging to deliver perfect replication of the option’s pay-off. If, for example, the assumption of geometric Brownian motion was relaxed to also permit discontinuous jumps to occur in the underlying asset price, it would not be possible to perfectly replicate the option pay-off through continuous rebalancing of a portfolio of the underlying asset and cash. The logic of arbitrage-free pricing would therefore no longer hold. Given the empirical tendency of markets to suffer short-term jumps in price that are difficult to reconcile with a continuous form of stochastic process for the price, this has and should prompt caution in applying the model in ‘real-life’ applications (including actuarial ones).

As with any model, it is crucially important that the model user understands its assumptions and limitations. These limitations have generally been well-understood by market practitioners, arguably since the 1970s and particularly since the equity market crash of 1987. Market option prices are not fully explained by the Black-Scholes model—as is evidenced by the ‘smile’ in Black-Scholes’ implied volatilities that has long been observed. Nonetheless, these implied volatilities are also the currency in which traders quote option prices—such is the ubiquity of the Black-Scholes model. The academic developments in option pricing theory of the 1970s have had a most profound impact on the practices of structuring, pricing and hedging complex financial market securities; and also in the theoretical valuation of forms of non-linear contingent claim that can occur in an immensely wide array of economics and business—from executive compensation to the valuation of oil fields. As we shall see later, eventually it played an important role in British actuarial thought and practice, particularly in the life sector.

  • [1] Bachelier (1900). For a usefully annotated English translation and accompanying historical discussion,see Davis and Etheridge (2006).
  • [2] Osborne (1959).
  • [3] Rubinstein (2006), p. 135.
  • [4] Samuelson (1965b).
  • [5] Black and Scholes (1973).
  • [6] Merton (1973).
  • [7] Merton (1974).
  • [8] Geske (1979).
  • [9] Geske (1979), p. 76.
  • [10] Cox and Ross (1976).
  • [11] Cox and Ross (1976), p. 153.
  • [12] Samuelson and Merton (1969). There are also other competing claims for the earliest derivation of risk-neutral valuation, but the Samuelson and Merton work was most closely related to the option valuationliterature.
  • [13] Merton (1990), p. 335.
  • [14] Cox, Ross and Rubinstein (1979).
  • [15] Harrison and Kreps (1979); Harrison and Pliska (1981).
  • [16] Davis and Etheridge (2006), p. 114.
 
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