The earliest recipients of the option pricing formula weren’t Fischer Black and Myron Scholes, but Paul Samuelson and Robert Merton. Why, then, did Paul Samuelson not receive a second Nobel Prize for option pricing, and why is it that when people think of option pricing, they only remember the Black-Scholes model?

It all boils down to a small detail: the irrelevance of expected returns in the pricing of options.

Allow me to simplify this issue so that we can all ponder it together. When the price of the underlying asset changes by a certain degree, the price of the option itself changes even more. From this perspective, an option is essentially a leveraged stock. This has led to the development of many different investment strategies, not necessarily related to option pricing models.

If we consider an option as a leveraged stock, then its pricing should theoretically be related to the expected returns of the stock. However, as we all know, the Black-Scholes stock option pricing model does not include the expected return of the stock; it is only related to the following five factors:

• The current price of the underlying asset

• The strike price

• Interest rates

• The time to expiration

• The volatility of the underlying asset

What is the reason for this? Is it possible that the Black-Scholes model is flawed?

To understand this, we need to look at the history of option pricing.

The End of Tortuous Economic Intuition

The research related to option pricing began as early as 1827 when Robert Brown discovered the random motion of particles in a fluid, which became known as Brownian motion. Seventy-five years later, a doctoral student named Louis Bachelier used Brownian motion to describe the randomness of stock prices and subsequently derived the price of options based on the expected difference between the stock price and the strike price. However, his work was not recognised by the academic community at the time, and he was unable to secure a position at a top research institution. Following this, in 1905, Einstein made contributions to Brownian motion by deriving the heat equation. Later, in 1918, Norbert Wiener’s doctoral thesis mathematically defined the assumptions of Brownian motion rigorously, and in his honour, Brownian motion is also referred to as the Wiener Process.

The next significant development was the famous Modigliani-Miller (MM) theorem, which stated that a company's structure and dividend policy are irrelevant to its valuation. The arbitrage-based explanation provided by Franco Modigliani and Merton Miller inspired Fischer Black’s thinking on the irrelevance of expected returns in option pricing.

This leads us to Paul Samuelson’s most direct attempt at option pricing.

Paul Samuelson had actually been thinking about the issue of option pricing for over a decade, but his progress was slow due to his lack of a strong mathematical background. This changed when he met his future protégé, Robert Merton. They assumed that the expected returns for stocks and options were alpha and beta, respectively, and that these remained constant until the option expired. Based on a model balancing investors’ risk preferences and supply-demand relationships, they estimated alpha and beta. Their approach was to use alpha to estimate the stock price at the option's expiration and then discount the difference using beta to estimate the option’s price.

The formula they derived was indeed the correct one for option pricing, but the values of alpha and beta depended on investors' risk preferences, which should not be a property of the option itself. The larger problem, which was later proven, is that option pricing is independent of the expected returns of both the option and the stock. The entire model’s derivation did not require alpha and beta.

Paul Samuelson and Robert Merton were just one step away from the correct pricing of options—a distance that Samuelson referred to as an epsilon and Merton as a mathematical limit.

From CAPM to the Black-Scholes Model

Fischer Black didn’t enter the field of finance until he was 30 years old, and one of his earliest encounters was with Jack Treynor’s version of the Capital Asset Pricing Model (CAPM). For him, option pricing wasn’t a dynamic programming problem aimed at maximising utility, but rather an issue of managing real-time risk exposure. Therefore, he viewed options as a leveraged stock, only related to the stock price and time. He then used CAPM and simple calculations to derive a differential equation, which turned out to be the heat equation published by Einstein in 1905, leading to the Black-Scholes model we know today.

Afterwards, Fischer Black and Robert Merton engaged in a prolonged debate. To Merton, the CAPM model was fundamentally incomplete, prompting him to search for flaws in the Black-Scholes model.

Meanwhile, Fischer Black was unhappy with Merton’s criticism and decided to collaborate with Myron Scholes to derive the Black-Scholes model using a different approach, this time without relying on the CAPM framework. This led to the development of the Black-Scholes risky hedge.

Although the Black-Scholes risky hedge didn’t satisfy Merton, it did inspire him to realise that options could be dynamically replicated by shorting cash and buying stock, where the only required information was the hedge ratio w1. Thus, the essence of an option is indeed a leveraged stock. Merton successfully completed the replication of options, confirming that Black’s derivation was correct. He then called Myron Scholes and said, ‘You are right.’

So, the Black-Scholes-Merton model we know today actually consists of two parts: the model proposed by Fischer Black and Myron Scholes, and Robert Merton’s subsequent dynamic replication of options.

The relationship between Fischer Black and Robert Merton was quite subtle. Although their disputes over option pricing were intense, later on, when Merton was working as an academic advisor at Goldman Sachs, he asked Black if any of his students were suitable to join Goldman. Black responded that he himself was suitable, and subsequently left MIT to join Goldman Sachs. At Goldman, Emanuel Derman discovered a slight discrepancy between the replicated options and the theoretical options when using Robert Merton’s dynamic replication method. When Fischer Black heard about this, he was excited and said, ‘You know, I always thought there was something wrong with the replication method.’ Except it was later proven to be just a programming error, and Robert Merton’s replication was indeed perfect.

Why the Risk-Free Rate in the Cox-Ross-Rubinstein Model?

You might recall the risk-neutral valuation in option pricing, which involves discounting the terminal option price using the risk-free rate. If you don’t remember, think about the binomial tree method of option pricing.

An important question arises: why use the risk-free rate to discount? Are we to believe that the final cash flow is risk-free? Not necessarily, and we don’t need to believe that the cash flow is risk-free either. Interestingly, no matter what discount rate we use, the option price remains the same, which further demonstrates the irrelevance of expected returns to option pricing.

Let me give an example:

The stock price may rise by 30% or fall by 10% over the next year

Current stock price: 100

Risk-free rate: 4%

Based on this information, under the market-neutral assumption, the stock has a 65% probability of falling by 10% and a 35% probability of rising by 30%. A call option would be worth 30 if the stock price rises and would be worthless if the stock price falls. Using risk-neutral pricing, the option price can be calculated as follows:

Cp = (0.35 * 30 + 0.65 * 0) / 1.04 = 10.0962

If we use a different expected return, say we assume the stock has a 10% expected return, implying a 50% chance of rising and a 50% chance of falling, the leverage for the option relative to the stock would be:

(30 - Cp) / (Cp * 0.4) = 75 / Cp

Then the discount rate for the option would be:

(75 / Cp) * (0.1 - 0.04) + 0.04 = 4.5 / Cp + 0.04

Finally, the option price would be:

Cp = (0.5 * 30 + 0.5 * 0) / (4.5/Cp + 0.04)

Cp = 10.0962

If we express everything mathematically, it becomes clear that our assumptions ultimately cancel out, leaving us with the market-neutral valuation found in the Cox-Ross-Rubinstein model. This is one way to understand the irrelevance of expected returns to option pricing. Therefore, using the risk-free rate to discount doesn’t mean we believe the final cash flow is risk-free; it simply indicates that expected returns are irrelevant.

The same logic applies in the continuous case.

Even though this article leans towards theoretical concepts, I believe that theory is very important because it can offer practical insights for real-world investing, and sometimes these insights can lead to astonishing results in investing, especially when they are violated.

Reference

Derman, E., 2004. My life as a quant: reflections on physics and finance. John Wiley & Sons.

Kritzman, M.P., 2002. Puzzles of finance: six practical problems and their remarkable solutions (Vol. 89). John Wiley & Sons.

Mehrling, P. and Brown, A., 2011. Fischer Black and the revolutionary idea of finance. John Wiley & Sons.

Disclaimer: The data and information mentioned are from third-party sources, and accuracy is not guaranteed. This article shares information and views, not professional investment advice. Consult professional advice before making investment decisions.

This article was originally written in Chinese and posted on my WeChat platform in 2016. The Chinese link can be found here