Mathematics / Review Essay

Vol. 4, NO. 4 / July 2019

Emanuel Derman

#### Letters to the Editors

The Black–Scholes model describes the value of a stock option as a function of the underlying stock price and its volatility. The model suggests an analogy between the random fluctuations of a stock price and a concept from physics, the Brownian motion of randomly diffusing particles. The analogy is not entirely new. More than 100 years ago, the French mathematician Louis Bachelier developed the theory of Brownian stock-price fluctuations to analyze stock options.1 The model has become the foundation for valuing options of all kinds. The Black–Scholes equation allows traders to treat volatility as an asset and trade it by buying or selling portfolios of options. An impalpable property of a financial system—volatility—has become concrete enough to be bought and sold. The result has been a transformation in mathematical finance and an explosion of new financial products. Although the Black–Scholes model is genuinely useful, it fails to capture the reality of the market’s behavior. This failure has triggered further ingenious extensions of the model.

These extensions, too, eventually fail. Markets and prices are social phenomena, not physical ones, and it is unlikely that there is an accurate predictive theory of human behavior.

## Stocks and Options

A  call option on a stock is a contract between a buyer A and a seller, or counterparty, B, that gives A the right, but not the obligation, to purchase the stock for a specified price on some future date. If one share of a stock, S, is selling for $200 today, a call option with a strike of$250 and an expiration of one year, gives A the right to buy S one year from today for $250. A is long the option, and B short. If S trades in the market at$300, A can exercise the option, purchasing one share at $250 from B and selling it for$300, obtaining a payoff of $50. If S trades below$250, A need do nothing. A put option is the corresponding right to sell the stock on a future date for a specified price.

What is the rational price of a one-year call option on S? Before 1973, it seemed obvious that the current value of a call option should reflect the expected future price of its stock. Different people would have different expectations about that price. What was obvious turned out to be false. In 1973, Fischer Black, Myron Scholes, and Robert Merton demonstrated that if stock prices fluctuated randomly as described by the theory of Brownian motion, the future payoff of a stock option could be replicated by a portfolio comprising borrowed money and a fractional share of the stock.2 Suppose the solution to the Black–Scholes equation yields a value today of one dollar for the call option on S described above. The equation then also shows that this one-dollar call option can be replicated by investing one dollar, borrowing 15 dollars, and then using their sum to purchase 16 dollars’ worth of S. If the current stock price of S is $200, and if it fluctuates randomly, this portfolio will behave precisely like an option in the future, with one year to expiration and a strike of$250. The amount of money borrowed, and the amount of stock in the portfolio, must be continuously adjusted as the stock price changes. At expiration, the portfolio and the call option have the same payoff. The adjustments represent a manufacturing cost. The greater the volatility of the stock, the more rapidly the stock price changes, and the greater the manufacturing cost.

An option’s value does not depend on the expected future price of its stock.

## Risk and Return

Investors traditionally measure their success by their return, that is, the percentage change in the value of an investment. Since future stock prices are unknown, returns are always risky. In the early 1950s, Harry Markowitz introduced a more refined way of judging investments, considering not only the expected return on an investment, but its volatility as well.3 Markowitz characterized every stock S by its expected annual return μS, the standard deviation σS of its annual returns, and the covariance matrix ρI,J between the returns of different stocks I and J.4 It is σS that measures the volatility of S. Instead of thinking about individual stocks, Markowitz argued, an investor should consider a range of portfolios containing all possible stocks, with arbitrary weights for each. Every portfolio P has a composite expected return μP and volatility σP, both calculated from the statistics of their individual stocks. The volatility σP represents the risk of the portfolio. An investor should buy the portfolio with the highest return for the risk he is willing to accept.

For Markowitz, both the expected return and its standard deviation were cross-sectional statistics; these served to define the distribution of future returns. He assumed that the future statistics of all stocks were known, and that, for any future risk, a range of portfolios P and their corresponding expected returns μP was available. In reality, only past market statistics are known. Still, the greater the risk σ of an investment, the greater the desired return μ. Given that returns on stocks are uncertain, the rational investor needs to know the appropriate relation between risk and return—the expected value of μ for a particular value of σ.

There is one privileged security whose return has no uncertainty. It is the riskless bond, which serves as a touchstone for measuring all other returns. The return obtainable on a short-term loan to the US Treasury, for instance, is denoted by the guaranteed return r and carries essentially no risk. The value of r varies from moment to moment as economic conditions vary. An investor can now characterize the return and risk of any security S by (μσ), and compare it to (r, 0) in the case of the riskless bond. Knowing that the return r corresponds to zero risk allows one to find the expected return for other, riskier securities. Consider a security S = (μσ). Given its volatility σ, what is the expected value of μ? One strategy for finding the expected return is to hedge away the risk of the security (μσ) by combining it with other available securities into a portfolio P consisting of at least two securities such that P has zero volatility. In that case, P replicates a riskless bond and should produce the guaranteed return r.5 That a riskless portfolio P has a guaranteed riskless return r, despite containing the risky security (μσ) determines the relation between μ and σ for the security S = (μσ).6

Suppose we apply this strategy to a call option C on a stock S. Embedding the call option in a portfolio with its stock such that the portfolio has zero risk will produce the Black–Scholes equation that allows us to determine the value of the option.

## The Black–Scholes Equation

During the 1960s, Paul Samuelson described the evolution of a stock price S by modeling its differential return dS/S = μdt + σdZ where μ represents drift, σ, volatility, and dZ is a random increment of a standard Wiener process during time dt. All of the uncertainty in the future price is driven by dZ. Increments dZ generate a normal distribution of returns, with mean value μt and standard deviation $\sigma \sqrt{t}$. As time increases, uncertainty in the return grows as the square root of the elapsed time.

Some ten years later, Black, Scholes, and Merton enlarged on Samuelson’s technique by using hedging to construct a riskless portfolio out of positions in a call option C and its underlying stock S. Ito’s lemma was then used to calculate the differential return of the riskless portfolio as a function of the random stock price. Since the hedged portfolio is riskless, this differential return must equal r.

The road to the Black–Scholes equation now lay open.

Imagine a stock S whose current value is S. At a future time T it could have a range of values ST from zero to infinity. A call option C on a stock with strike K and expiration T is a security that contractually pays the buyer MAX(ST – K, 0) at T. If ST > K, the seller must pay (ST – K); if ST ≤ K, the seller pays nothing. The contract is asymmetric; the buyer receives potential gains, but need not cover potential losses. The payoff is a nonlinear convex function of the terminal stock price ST. The option C is sometimes called a contingent claim, or derivative security, because its value is contingent on the price of the underlying security S. What is the appropriate price of C at an earlier time t?

Assume now that the stock price undergoes Brownian motion between times t and T with an expected return μ and a volatility σ. Since the call value depends on the stock price, C = C(S, t), where the function C( ) is unknown.

Like S, C is risky. Embedding it in a portfolio serves to eliminate the risk. Consider a portfolio P consisting of a long position in one call option C, and a short position in an arbitrary number of Δ shares of the stock S, so that P = C – ΔS, with value P = C(S, t) – ΔS. P is a function of t and of a Brownian variable S. Ito’s lemma yields the differential change in the value of P in terms of dt and the random change dZ. Thus

$dP=\left[\frac{\partial C}{\partial t}+\left(\frac{\partial C}{\partial S}-\Delta \right)\mu S+\frac{1}{2}\frac{{\partial }^{2}C}{\partial {S}^{2}}{\sigma }^{2}{S}^{2}\right]dt+\left(\frac{\partial C}{\partial S}-\Delta \right)\sigma SdZ$.

The differential dP contains the random element $\left(\partial C/\partial S-\Delta \right)\sigma SdZ$. Suppose that Δ, the number of shares shorted at time t, is chosen to be $\partial C/\partial S$, so that the random element is zero. Then all exposure to the dZ term vanishes in the expression for dP, and

$dP=\left[\frac{\partial C}{\partial t}+\frac{1}{2}\frac{{\partial }^{2}C}{\partial {S}^{2}}{\sigma }^{2}{S}^{2}\right]dt$.

The parameter μ drops out of the equation. The differential change in the value of the portfolio is independent of the expected return of the stock.

Since this expression for dP has no random component, P is a riskless investment and replicates a riskless bond. It must therefore grow at the riskless rate r over time dt. The riskless increase in P is

$dP=rPdt=r\left(C-\frac{\partial C}{\partial S}S\right)dt$.

Combining these equations yields the Black–Scholes equation for C:

$\frac{\partial C}{\partial t}+rS\frac{\partial C}{\partial S}+\frac{1}{2}{\sigma }^{2}{S}^{2}\frac{{\partial }^{2}C}{\partial {S}^{2}}=rC$.

If C(ST, T) = MAX(ST – K, 0), its solution is

C(S, t, K, T, r, σ) = SN(d1) – Ker(Tt)N(d2)

${d}_{1,2}=\frac{\mathrm{l}\mathrm{n}\left(S/{Ke}^{-r\left(T-t\right)}\right)±0.5{\sigma }^{2}\left(T-t\right)}{\sigma \sqrt{T-t}}$

where N( ) is the cumulative normal distribution function. A call option C is instantaneously equivalent to the replicating portfolio

C = N(d1)SB,

where S denotes the stock at time t and B denotes at t the riskless bond worth N(d2)K dollars at T. The value of the hedge ratio Δ of the option is

$N\left({d}_{1}\right)=\frac{\partial C\left(S,t,K,T,r,\sigma \right)}{\partial S}$,

the number of shares of stock that have the same instantaneous risk as the option. It varies with the stock price and the time to option expiration.

Though the risky stock evolves with average growth rate μ, only the riskless growth rate r appears in the option formula. The actual drift μ is irrelevant if the call option’s risk is eliminated by hedging. The hypothetical economic world in which stocks, no matter how risky they are, have an average growth rate equal to the riskless rate r is called a risk-neutral world.7 Indeed, the expression N(d2) is the probability that the stock exceeds the strike at expiration in the risk-neutral world.

## Using Black–Scholes

The Black–Scholes formula C = SN(d1) – Ker(Tt)N(d2) may be used to value, replicate, or hedge an option. Options market makers use the model to accommodate their bets without losing money. A market maker can sell a call option on a stock to a counterparty who is betting that the stock will rise. If it does, the buyer profits. Had the market maker done nothing but sell the option, he would have lost money. But the market maker can use the model to hedge the risk of the option and eliminate exposure to changes in the stock price, thereby accommodating the buyer without risking a loss. The market maker has dynamically replicated a call option for himself to offset the risk of having sold one to his client.

The development of these tactics has led to a flourishing of options market makers who accommodate risk takers while, in theory, continually hedging away their own risk. The derivative of Δ with respect to the stock price S,

$\mathrm{\Gamma }\left(S,t,K,T,r,\sigma \right)=\frac{\partial \Delta \left(S,t,K,T,r,\sigma \right)}{\partial S}=\frac{{\partial }^{2}C\left(S,t,K,T,r,\sigma \right)}{{\partial S}^{2}}$,

indicates how often the hedge must be adjusted. The other derivatives of the option value allow market makers to hedge changes in market parameters beyond the stock price itself. The derivative $\partial C/\partial r$, for example, estimates how much the value will change as interest rates change. Typical market makers buy or sell thousands of options and hedge them collectively; they apply the model to their entire book and compute the net number of shares they need to buy or sell to hedge the whole portfolio. Ideally, they aim for a portfolio with low Γ so that they do not need to rehedge often.

In theory, options market makers who use Black–Scholes are eliminating the influence of changes in the stock price on their portfolio. If they charge a bit more than the fair Black–Scholes price for their service, they can make a riskless profit. To do so, they must know the future volatility of the stock. In life, no one knows what level of volatility will prevail. Market makers who hedge options are therefore taking a more sophisticated kind of risk—volatility risk. In order to hedge, they need to calculate the hedge ratio

,

whose value requires an estimate of the future volatility σ. The estimate is most often made by fitting the current price of the option they want to hedge to the Black–Scholes formula and backing out the value of volatility σI that equates the market price to the formula. This process is called calibration of the model, and σI is called the implied volatility, since it is the future volatility of the stock implied by the option price filtered through the model. σI is a calibration parameter entered into the Black–Scholes formula, and not a statistic measured from a time series of returns.

An option at its market price determines σI. By shorting Δ shares of stock, computed using σI as the value of the future volatility, a trader may instantaneously hedge his bet, earning the riskless return with no profit or loss. This is the theory. In practice, estimates of future volatility will often be wrong. Suppose that the realized volatility of the stock turns out to be σR. In that case, the value of the hedged portfolio will still be relatively insensitive to changes in the stock price, but strongly sensitive to the value of future volatility. One can calculate the profit or loss from continuously hedging an option, as t, S, and the volatility change, to obtain

$P&L={\int }_{t}^{T}\frac{1}{2}\mathrm{\Gamma }\left(S,t,K,T,r,{\sigma }_{\mathbf{I}}\right){S}^{2}\left[{\sigma }_{R}^{2}-{\sigma }_{\mathbf{I}}^{2}\right]dt$,

where the second derivative of the Black–Scholes formula is

$\mathrm{\Gamma }\left(S,t,K,T,r,{\sigma }_{\mathbf{I}}\right)=\frac{{\partial }^{2}C\left(S,t,K,T,r,{\sigma }_{\mathbf{I}}\right)}{{\partial S}^{2}}$.

Γ measures the rate of change of the hedge ratio and is calculated using σI, the hedger’s estimate of the future volatility. Profit and loss depend on the difference $\left({\sigma }_{R}^{2}-{\sigma }_{\mathbf{I}}^{2}\right)$, where σR is measured at every instant as the stock price evolves from t to T. σI is a parameter entered into the Black–Scholes formula, whereas σR is a true statistic, the standard deviation of the returns on the stock. At every instant, the model must be recalibrated to find the new prevailing σI in order to continue hedging and trading.

Hedging an option therefore generates a bet on the difference between future realized and previously estimated volatility. If at each instant the realized volatility is equal to the estimated volatility, there is no profit or loss and the replication works perfectly. Options market makers typically hedge against or bet on changes in future volatility, whereas the retail investors they deal with usually betting on stock price movements alone. The failure of Black–Scholes, which requires a constant future volatility, and the consequent need for recalibration, has paradoxically provided a method for trading volatility itself as an asset.

The implied volatility σI of a particular option CK,T with a strike K and expiration T is an estimate of the future volatility of the underlying stock S. The stock itself knows nothing about the option contract written on it, and so, if the model describes markets correctly, options of all strikes and the same expiration on a particular stock should have the same implied volatility. σI should be independent of K.

The most widely traded options are options on equity indexes such as the S&P 500. Prior to the 1987 market crash, implied volatilities of S&P 500 options were approximately independent of K. After the crash, in which market participants witnessed a decline of more than 20 percent in a single day, the behavior of implied volatilities changed. Since then, graphs of implied volatilities with a particular expiration display a pronounced dependence on the strike K, a phenomenon called the “volatility smile.” Each S&P 500 option with a particular strike implies a different, and therefore inconsistent, future volatility for the underlying index. This inconsistency occurs for options on equity market indices in other countries, and, indeed, for individual stocks. This behavior invalidates the model and the accuracy of its hedge ratios, because the future volatility of an underlying index, if the model’s description is correct, cannot depend on the option above it.8

This paradox has intensified over the past 30 years as almost every options market has developed its own uniquely shaped volatility smile. Options modelers have responded by elaborately extending Black–Scholes to account for implied volatilities that vary with strike. The simplest extension allows the stock’s volatility σ to vary with stock price and time. More complex models allow σ itself to be a random Brownian motion. In some models the stock price S and the volatility σ can make random Poisson jumps. Not all of these models allow the replication of a riskless hedge. When stocks experience random jumps of arbitrary size, a hedge with a finite number of securities is impossible. Though many of these models are used productively, none has been accepted as a unique replacement for Black–Scholes.

The Black–Scholes formula depends on a variety of assumptions: that the stock follows a Brownian process with known volatility, that the risk-free rate is known, that the stock pays no dividends, that the short position in the stock can continuously be adjusted with no transaction cost, and that the market allows for no riskless arbitrage opportunities. Although some of these assumptions are incidental, others are problematic. As a result, the model is not a truthful description of the mechanics of actual markets. Observed stock prices do not diffuse continuously according to Brownian motion. Both prices and volatility often jump discontinuously. The existence of the volatility smile directly reflects the inadequacy of the Black–Scholes model. Financial models, which are models of social phenomena, are always metaphors and therefore ultimately inadequate. Nevertheless, Black–Scholes, with its focus on volatility and hedging, provides an immensely useful way of thinking about valuations.

By buying and hedging options with an imperfect model, traders can speculate on the imperfect model’s parameters. Suppose that we adopt an extended version of the Black–Scholes model in which volatility itself is stochastic. To use it, one must then specify the future volatility of the stock’s volatility. But no one truly knows what value that will take. At any instant one can similarly calibrate the value of the implied future volatility of volatility from market option prices in the extended model. But then, a moment later, as market options prices change, the implied volatility of volatility will change too and the model will require recalibration. Paradoxically, again, the failure of this extended model will allow markets to trade volatility of volatility as an asset.

There is likely no correct model of markets and options. Social behavior changes as people learn from changes in market behavior, acquire new technologies, and develop new theories. The perfect replication of a riskless bond is not possible. Whenever a trader buys or sells an option, or an option on an option, he or she is creating rather than replicating a price. There is a substantial amount of truth to Elie Ayache’s view that

[p]robabilistic models and … derivative valuation tools are only internal episodes that we require in order locally, and always, imperfectly, to hedge something. We have to keep in mind … that those episodes are present and useful only insofar as they will be recalibrated.9

1. An English translation of Bachelier’s 1900 thesis is available at Louis Bachelier, Theory of Speculation, trans. D. May (2011).
2. Fischer Black and Myron Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy 81, no. 3 (1973): 637–54; Robert Merton, “Theory of Rational Option Pricing,” Bell Journal of Economics and Management Science 4, no. 1 (1973): 141–83.
3. Harry Markowitz, “Portfolio Selection,” The Journal of Finance 7, no. 1 (1952): 77–91.
4. Variance is the square of the standard deviation.
5. If it had a greater return, it would be possible to earn a riskless profit by borrowing money at the riskless rate r and buying P more cheaply than the riskless bond, and then paying off the loan with the proceeds of P when it matures. A similar argument holds if P has a lesser return. The principle of “no riskless arbitrage,” on which most of modern finance depends, demands that such riskless profits be impossible, under the assumption that if they existed, traders in the market would buy the cheaper security and sell the richer one, altering prices by supply and demand until their returns equilibrated.
6. The strategy of replication was introduced by Franco Modigliani and Merton Miller, “The Cost of Capital, Corporation Finance and the Theory of Investment,” American Economic Review 48, no. 3 (1958): 261–97, in their analysis of the value of stock in a levered firm.
7. That Black–Scholes option values can be written as expected value of their future payoffs in a risk-neutral world has led to the development of very formal “martingale” methods in options valuation. These methods focus on the probabilistic interpretation of the solution even though the probabilities, being probabilities in a hypothetical risk-neutral world, are pseudoprobabilities rather than genuine ones.
8. Analogously, the actual height of a person being photographed cannot depend on the camera being used to take the photo.
9. See Elie Ayache, The Medium of Contingency (London: Palgrave Macmillan, 2015), 92.

Emanuel Derman is a Professor of Professional Practice and Director of the Financial Engineering Program at Columbia University.