Milky Eggs

Reading the classics: Black and Scholes (1973) on options pricing

I’d like to announce the first post in what I hope will be an ongoing series of short but fascinating expositions of classic papers.

In general, I think there’s value in reading classic papers that isn’t easily captured from modern treatments. Although textbooks, for example, typically have a measured pedagogical approach, they often fail to capture the historical context that preceded and motivated foundational work. Furthermore, older papers typically have a more concrete (and therefore more easily understood) approach, as opposed to modern approaches, which tend toward the theoretical. It is my earnest hope that you, the reader, will also come to appreciate the unique benefits of perusing classic literature.

My intention is not to provide exhaustive derivations or up-to-date surveys of the literature. Instead, I am reading these papers to learn subjects which are new to me, and I anticipate that the process of drafting these posts will itself be part of that learning process. I hope that the patient reader may also derive some modicum of value from my struggles.

In the following, I will give a brief overview of the derivation of the Black-Scholes formula for pricing of European options, following the argument given by Black and Scholes (1973). This content is largely new to me, and I chose to read this paper because it seemed like a good idea to be familiar, at some level, with such an important model. I will then briefly touch upon extensions of the Black-Scholes model and applicability to cryptocurrency trading.

Introduction

At a basic level, an option is “a security giving the right to buy or sell an asset, subject to certain conditions, within a specified period of time.” The asset under consideration is typically denoted the “underlying.” In an European option, the option (or, precisely, the associated right to buy or sell the underlying) may only be exercised at a specified time, typically called the “expiry.” The underlying is bought or sold at a fixed price associated with the option, known as the strike or exercise price. A call option gives you the right to purchase the underlying, whereas a put option gives you the right to sell the underlying.

In the early 1900s, the accurate pricing of options was an open research question. Despite efforts to identify precise pricing criteria, approaches were generally unsatisfactory. For example:

• Sprenkle (1961) gives a formula for an option’s price in terms of free parameters k and k*, but is unable to empirically determine the value of those parameters
• Samuelson (1965) prices an option as the time value-discounted expected value of the underlying’s return; however, this approach does not take into account market equilibria related to the price of the option itself
• Samuelson and Merton (1969) extend the above approach in various ways, but although they recognize that discount rates depend on an investor’s willingness to hold an option relative to the investor’s individual utility function, they do not properly consider that investors may construct hedged portfolios using options in ways that do not necessarily fit within their approach
• Thorp and Kassoup (1967) determine an empirical fit for options prices and explicitly construct a hedged portfolio using an option and its underlying, but fail to use the ability to construct a hedged portfolio as a starting basis to derive a theoretically sound pricing formula

Building upon this previous work, Black and Scholes published The Pricing of Options and Corporate Liabilities in 1973, a foundational paper which established a model of options pricing still used in the present day.

The basic intuition behind the Black-Scholes model is quite simple. Suppose that options are correctly priced by the market, so that you cannot combine long and short positions in options and their underlying assets such that your portfolio has a return greater than the short-term interest rate. (If not, market participants would arbitrage these profits away.) Surprisingly, and quite elegantly, it is possible to derive the price of a standard European call option from this fundamental assumption of market equilibrium alone. Notably, this is independent of the expected rate of return of the underlying, which I will discuss below.

Derivation of the Black-Scholes model

First, we begin with a number of basic assumptions:

• There is a fixed short-term interest rate
• Stock prices at expiry follow a lognormal distribution
• The stock has no dividends, there are no transaction fees, there is no premium charged for short selling, etc.

In the above scenario, the price of a call option with known expiry and strike price can be represented as w(x,t), where x is the price of the underlying stock and t is the present time. Now, suppose we construct a portfolio as follows:

• Long 1 share of the stock
• Short 1 / (∂/∂x w(x,t)) call options

We can see that the value of this portfolio is independent of x. Suppose that the stock price changes by a small amount Δx. The value of the long position increases by Δx. Similarly, the option price decreases by (∂/∂x w(x,t)) * Δx; multiplying through with the size of our short position, which is 1 / (∂/∂x w(x,t)), we find that the value of the short position decreases by Δx. Therefore our portfolio value is independent of small changes in x. Although this linear approximation only holds true for small values of Δx, it can be seen that continuous adjustment of this hedged portfolio will render its value wholly independent of x.

In general, the value of our hedged portfolio is given by

where for simplicity partial derivatives of w(x,t) in the ith position are denoted with a subscript of i. Suppose that in a short time interval Δt the value of the stock changes by Δx, meaning that our portfolio’s value changes by

Assuming that v is the standard deviation of the rate of return of x, we can expand Δw using standard results in stochastic calculus as

Therefore we can expand the previous expression for the change in value of our portfolio as

Now, suppose we assume that markets are at equilibrium; since our hedged portfolio has zero risk originating from changes in the price of x, its overall return must be equal to the short-term interest rate. If it were not, then the ability to construct such positions would be exploited by arbitrageurs until its rate of return were driven down to the short-term interest rate. Therefore, the change in value of our portfolio in a time interval Δt must also be equal to the overall portfolio value multiplied by the short-term interest rate r, such that

After appropriate rearrangements, this can be rewritten as a partial differential equation:

Suppose that t* is the expiry date and c is the strike price. This gives us boundary conditions for the differential equation, namely that w(x,t*) = 0 when x < c and that w(x,t*)= x – c when x ≥ c. This uniquely determines the option price w(x,t). With some substitutions, the above differential equation may be formulated as a standard heat transfer equation, which has a known solution from classical physics. Writing the cumulative normal density function as N(d), the Black-Scholes model for the price of an European call option is therefore given as:

Comments on the Black-Scholes model

One can straightforwardly show that the above model takes on expected values when appropriate limits are taken; for example, w(x,t) converges to x as each of t*, r, or v increases (sensibly so, considering that the value of a call option cannot rationally be greater than the price of its underlying stock).

Another immediate observation is that the option price w(x,t) is independent of the expected rate of return of the stock, even though one might typically expect a stock price to drift upward or downward with time. This is typically considered to be a surprising and unintuitive result of the Black-Scholes model. However, it is possible to supply the appropriate intuition (credit: quant.SE).

Suppose that the market price of a driftless stock is 105 and that you have a call and a put, both with strike price 100. Now, suppose that the stock begins to exhibit upward drift. In this situation, one might naively expect the price of the call to increase, because it is more likely to expire profitably. Conversely, one might also naively expect the price of the put to decrease, because it is more likely to expire unprofitably. However, were this the case, you would be able to construct a portfolio (including calls, puts, the underlying, and some cash) where you have riskless profit. Because this is not possible in equilibrium conditions, the price of these options must be independent of the drift of the underlying.

The above argument is effectively given by Derman and Taleb (2005), The Illusions of Dynamic Replication, but although it yields a good intuition for why the Black-Scholes model is independent of drift, the astute reader would do well to also examine Ruffino and Treussard (2006), Derman and Taleb’s ‘the Illusions of Dynamic Replication’: A Comment, which shows that Derman and Taleb (2005) is not in and of itself a complete derivation of the Black-Scholes model.

Broadly speaking, a very large body of work over the last several decades has extended the Black-Scholes model to more general settings as well as studied its applicability to real-world options pricing. Typically, these generalizations focus on relaxation of one or more of its assumptions, e.g. the lognormal distribution of stock returns, the presence of a constant interest rate, the lack of stock dividends, American-style options allowing exercise prior to expiry, etc. This work, which is both extensive and detailed, will not be discussed here.

Options in cryptocurrency trading

Trading standard European options on cryptocurrency prices has become increasingly popular over time. Initially, these options became available on centralized exchanges, most notably Deribit, which commands the majority of the cryptocurrency options trading volume. Recently, a number of decentralized and fully on-chain options have also gained traction, such as Ribbon Finance or Dopex, which utilize “vault” mechanisms to circumvent the difficulties associated with standard options market making on-chain.

Interestingly, Alexander and Imeraj (2021), Inverse Options in a Black-Scholes World points out that Deribit options are really “inverse products,” in that the contract size is denominated in USD and the settlement is provided in bitcoin or ether. This is “inverted” relative to a more typical setup where the contract size is denominated in bitcoin or ether and the option is settled in USD (analogously with traditional options on prices of equities, commodities, etc.). These “inverse options” have unique characteristics which are not captured by the traditional Black-Scholes model. Perhaps surprisingly, upon examination of the implied volatilities quoted on Deribit, Alexander and Imeraj suggest that Deribit traders are not accurately valuing the options they are trading — potentially a profitable opportunity for the reader to capitalize upon.

To quickly touch upon the decentralized “vault” model: in general, it appears to me that option sellers deposit collateral into various “vaults” to underwrite calls or puts at their selected strike prices, after which option buyers have the ability to purchase those options. The premiums collected from buyers are then returned to sellers after expiry. Note that this means that sellers may not know the size of the premiums that they will receive until after options purchases are closed in any given epoch. Typically, these vaults are marketed as “yield” for depositors, but that is generally not an accurate description as the underwriting of options exposes sellers to various risks associated with adverse upward or downward price movements.

I would like to leave readers with a question for further inquiry. Dopex has announced plans to introduce “Atlantic options.” In essence, these appear to allow the buyer of an option to post the underlying in return for collateral put up by the seller. It is not clear to me how this is intrinsically different from standard American options, which allow “early exercise” prior to expiry. In general, any exchange of underlying for collateral, assuming it does not leave the position undercollateralized, should be equivalent to early exercise; I assume that the funding rate paid to option sellers then somehow corresponds to the difference in premia between American and European options (the former should always have a higher premium as it gives buyers strictly greater optionality). I would be interested to see explanations of these “Atlantic options” which clarify their value-add above standard early exercise.

Final thoughts

Perhaps because I am a newcomer to quantitative finance, it was quite interesting to read through Black and Scholes (1973) and follow their derivation, which was thankfully far simpler than I had feared. It gives me renewed appreciation for the power of approaching long-standing problems from the correct angle — tackling options pricing from the perspective of market equilibrium conditions seemed to nearly trivialize it.

I hope to follow this post with similar expositions in the future. Readers are welcome to provide reading suggestions.