Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of Nebraska-Lincoln
Lincoln, NE 68588-0130
http://www.math.unl.edu
Voice: 402-472-3731
Fax: 402-472-8466

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Note: To read these pages properly, you will need the latest version of the Mozilla Firefox browser, with the STIX fonts installed. In a few sections, you will also need the latest Java plug-in, and JavaScript must be enabled. If you use a browser other than Firefox, you should be able to access the pages and run the applets. However, mathematical expressions will probably not display correctly. Firefox is currently the only browser that supports all of the open standards.

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Rating

Student: contains scenes of mild algebra or calculus that may require guidance.

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Section Starter Question

We have derived and solved the Black-Scholes equation. We have derived parameter dependence and sensitivity of the solution. Are we done? What’s next? How would we go about implementing and analyzing that next step, if any?

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Key Concepts

1. The Black-Scholes model overprices “at the money” options, that is with $S\approx K$. The Black-Scholes model underprices options at the ends, either deep “in the money” $S\gg K$ or deep “out of the money” $S\ll K$.
2. This is an indication that security price processes have “fat tails”, i.e. a distribution which has the probability of large changes in price $S$ greater than would be predicted by the lognormal distribution.
3. Mathematical models in finance do not have the same experimental basis and long experience as do mathematical models in physical sciences. It is important to remember to apply mathematical models only under circumstances where the assumptions apply.
4. Financial economists and mathematicians have suggested alternatives to the Black-Scholes model. These alternatives include:
   1. Models where the future volatility of a stock price is uncertain (called stochastic volatility models),
   2. Models where the stock price experiences occasional jumps rather than continuous change (called jump-diffusion models).

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Vocabulary

1. Many security price changes exhibit leptokurtosis: stock price changes near the mean and large returns far from the mean are more likely than Geometric Brownian Motion predicts, while other returns tend to be less likely.
2. Stochastic volatility models are higher-order mathematical finance models where the volatility of a security price is a stochastic process itself.
3. Jump-diffusion models models are higher-order mathematical finance models where the security price experiences occasional jumps rather than continuous change.

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Mathematical Ideas

Validity of Black-Scholes

Recall that the Black-Scholes Model is based on several assumptions:

1. The price of the underlying security for which we are considering a derivative financial instrument follows the stochastic differential equation
   $$dS=rSdt+\sigma SdW$$

   or equivalently that $S\left(t\right)$ is a Geometric Brownian Motion

   $$S\left(t\right)=z_{0}exp\left(\left(r-{\sigma }^2∕2\right)t+\sigma W\left(t\right)\right).$$

   At each time the Geometric Brownian Motion has lognormal distribution with parameters $\left(ln\left(z_0\right)+rt-{\sigma }^2∕2t\right)$ and $\sigma \sqrt{t}$. The mean value of the Geometric Brownian Motion is $E\left[S\left(t\right)\right]=z_{0}exp\left(rt\right)$.

2. The risk free interest rate $r$ and volatility $\sigma$ are constants.
3. The value $V$ of the derivative depends only on the current value of the underlying security $S$ and the time $t$, so we can write $V\left(S,t\right)$.
4. All variables are real-valued, and all functions are sufficiently smooth to justify necessary calculus operations.

See Derivation of the Black-Scholes Equation. for the context of these assumptions.

One judgment on the validity of these assumptions statistically compares the predictions of the Black-Scholes model with the market prices of call options. This is the observation or validation phase of the cycle of mathematical modeling, see Brief Remarks on Math Models. for the cycle and diagram. A detailed examination (the financial and statistical details of this examination are outside the scope of these notes) shows that the assumption that the underlying security has a price which is modeled by Geometric Brownian Motion, or equivalently that at any time the security price has a lognormal distribution, misprices options. In fact, the Black-Scholes model overprices “at the money” options, that is with $S\approx K$ and underprices options at the ends, either deep “in the money” $S\gg K$ or deep “out of the money” $S\ll K$. This indicates that the price process has “fat tails”, i.e. a distribution which has the probability of large changes in price $S$ larger than the lognormal distribution predicts. Large changes are more frequent than the model expects.

More fundamentally, one can look at whether general market prices and security price changes fit the hypothesis of following the stochastic differential equation for Geometric Brownian Motion. Studies of security market returns reveal an important fact: Large changes in security prices are more likely than normally distributed random effects would predict. Put another way, the stochastic differential equation model predicts that large price changes are much less likely than is actually the case.

The origin of the difference between the stochastic differential equation model for Geometric Brownian Motion and real financial markets may be a fundamental misapplication of probability modeling. The mathematician Benoit Mandelbrot argues that finance is prone to a “wild randomness” not usually seen in nature [9]. Mandelbrot says that rare big changes can be more significant than the sum of many small changes. That is, Mandelbrot calls into question the applicability of the Central Limit Theorem in finance. It may be that the mathematical hypotheses required for the precise application of the Central Limit Theorem do not hold in finance.

Using more precise statistical language than “fat tails”, security returns exhibit what is called leptokurtosis: the likelihood of returns near the mean and of large returns far from the mean is greater than geometric Brownian motion predicts, while other returns tend to be less likely. For example some studies have shown that the occurrence of downward jumps three standard deviations below the mean is three times more likely than a normal distribution would predict. This means that if we used Geometric Brownian Motion to compute the historical volatility of the S&P 500, we would find that the normal theory seriously underestimates the likelihood of large downward jumps. Jackwerth and Rubinstein [4] observe that with the Geometric Brownian Motion model, the crash of 1987 is an impossibly unlikely event:

The popular term for such extreme changes is a “black swan”, reflecting the rarity of spotting a black swan among white swans. In financial markets “black swans” occur much more often than the standard probability models predict [8, 6].

Recall that we explicitly assumed that many of the parameters were constant, in particular, volatility is assumed constant. Actually, we might wish to relax that idea somewhat, and allow volatility to change in time. Nonconstant volatility introduces another dimension of uncertainty and also of variability into the problem. Still, changing volatility is an area of active research, both practically and academically.

We also assumed that trading was continuous in time, and that security prices moved continuously. Of course, continuous change is an idealizing assumption. In fact, in October 1987, the markets dropped suddenly, almost discontinuously, and market strategies based on continuous trading were not able to keep with the selling panic. Of course, the October 1987 drop is yet another illustration that the markets do not behave exactly as trading history would predict. [2].

Black-Scholes as an Approximate Model with an Exact Solution

In the classification of the section on mathematical modeling Brief Remarks on Mathematical Models., the Black-Scholes model is an approximate model with an exact solution. The previous paragraphs show that each of the assumptions made in order to derive the Black-Scholes equation is an approximation made expressly to derive a tractable model. The assumption that security prices act as a Geometric Brownian Motion means that we can use the techniques of Itô stochastic calculus. Even more so, the resulting lognormal distribution is based on the normal distribution which has simple mathematical properties. The assumptions that the risk free interest rate $r$ and volatility $\sigma$ are constants simplifies the derivation and reduces the size of the resulting model. The assumption that the value $V$ of the derivative depends only on the current value of the underlying security $S$ and the time $t$ likewise reduces the size of the resulting model. The assumption that all variables are real-valued, and all functions are sufficiently smooth allows necessary calculus operations.

Once the equation is derived we are able to solve it with standard mathematical techniques. In fact, the resulting Black-Scholes formula is expressed in a single line as a combination of well-known functions. The formula is simple enough that by 1975 Texas Instruments created a hand-held calculator specially programmed to produce Black-Scholes option prices and hedge ratios. The simple exact solution puts the Black-Scholes equation in the same position as the pendulum equation and the Ideal Gas Law, an approximate model with an exact solution. While this is convenient, exact solutions hide the approximate origins.

The simple one-line solution may actually encourage some of the misuses of mathematical modeling detailed in the next section. Once the simple solution is written and even programmed in a calculator, everyone can use it. Had the Black-Scholes equation been nonlinear or based on more sophisticated stochastic processes, it probably would not have a closed-form solution. Solutions would have relied on numerical calculation, more advanced mathematical techniques or simulation. In any case, solutions would not have been easily obtained. Those end-users who persevered in applying the theory might have sought additional approximations more conscious of the compromises with reality.

Misuses of Mathematical Modeling

By 2005, about 5% of jobs in the finance industry were in mathematical finance. The heavy use of flawed mathematical models contributed to the failure and near-failure of some Wall Street firms in 2009. As a result, some critics have blamed the mathematics and the models for the general economic troubles that resulted. In spite of the flaws, mathematical modeling in finance is not going away. Consequently, modelers and users have to be honest and aware of the limitations in mathematical modeling, [9]. Mathematical models in finance do not have the same experimental basis and long experience as do mathematical models in physical sciences. For the time being, we should cautiously use mathematical models in finance as general indicators that point to the values of derivatives, but do not predict with high precision.

Actually, the problem goes deeper than just realizing that the precise distribution of security price movements is slightly different from the assumed lognormal distribution. Even if the probability distribution type is specified, giving a mathematical description of the risk, we still would have uncertainty, not knowing the precise parameters of the distribution to specify it totally. From a scientific point of view, the way to estimate the parameters is statistically evaluate the outcomes from the past to determine the parameters. We looked at one case of this when we described historical volatility as a way to determine $\sigma$ for the lognormal distribution, see Implied Volatility.. However, this implicitly assumes that the past is a reasonable predictor of the future. While this faith is justified in the physical world where physical parameters do not change, such a faith in constancy is suspect in the human world of the markets. Consumers, producers, and investors all change habits overnight in response to fads, bubbles, rumors, news, and real changes in the economic environment. Their change in economic behavior changes the parameters.

Even within finance, the models may vary in applicability. Analysis of the 2008-2009 market collapse indicates that the markets for interest rates and foreign exchange may have followed the models, but the markets for debt obligations may have failed to take account of low-probability extreme events such as the fall in house prices [9].

Models can have other problems which are more social than mathematical. Sometimes the use of the models can change the market priced by the model. This feedback process in economics has been noted with the Black-Scholes model, [9]. Sometimes special derivatives can be so complex that modeling them requires too many assumptions, yet the temptation to make an approximate model with an exact solution overtakes the solution step in the modeling process. Special debt derivatives called “collateralized debt obligations” or CDOs implicated in the economic collapse of 2008 are an example. Each CDO was a unique mix of assets, but CDO modeling used general assumptions which were not associated with the specific mix. Additionally, the CDO models used assumptions which underestimated the correlation of movements of the parts of the mix [9]. Valencia [9] says that the “The CDO fiasco was an egregious and relatively rare case of an instrument getting way ahead of the ability to map it mathematically.”

It is important to remember to apply mathematical models only under circumstances where the assumptions apply [9]. For example “Value At Risk” or VAR models use volatility to statistically estimate the likelihood that a given portfolio’s losses will exceed a certain amount. However, VAR works only for liquid securities over short periods in normal markets. VAR cannot predict losses under sharp unexpected drops which are known to occur more frequently than expected under simple hypotheses. Mathematical economists, especially Nassim Nicholas Taleb, have heavily criticized the misuse of VAR models.

Alternatives to Black-Scholes

Financial economists and mathematicians have suggested a number of alternatives to the Black-Scholes model. These alternatives include:

1. stochastic volatility models where the future volatility of a security price is uncertain,
2. jump-diffusion models where the security price experiences occasional jumps rather than continuous change.

The difficulty in mathematically analyzing these models and the typical lack of closed-form solutions means that these models are not as widely known or celebrated as the Black-Scholes-Merton theory.

In spite of these flaws, the Black-Scholes model does an adequate job of generally predicting market prices. Generally, the empirical research is supportive of the Black-Scholes model. Observed differences have been small (but real!) compared to transaction costs. Even more importantly, the Black-Scholes model shows how to assign prices to risky assets by using the principle of no-arbitrage applied to a replicating portfolio and reducing the pricing to applying standard mathematical tools.

Sources

This section is adapted from: “Financial Derivatives and Partial Differential Equations” by Robert Almgren, in American Mathematical Monthly, Volume 109, January, 2002, pages 1–11 , and from Options, Futures, and other Derivative Securities second edition, by John C. Hull, Prentice Hall, 1993, pages 229–230, 448–449 and Black-Scholes and Beyond: Option Pricing Models, by Neil A. Chriss, Irwin Professional Publishing, Chicago, 1997 . Some additional ideas are adapted from When Genius Failed by Roger Lowenstein, Random House, New York . See also M. Poovey, “Can Numbers Insure Honesty”, .

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Problems to Work for Understanding

1. A pharmaceutical company has a stock that is currently $25. Early tomorrow morning the Food and Drug Administration will announce that it has either approved or disapproved for consumer use the company’s cure for the common cold. This announcement will either immediately increase the stock price by $10 or decrease the price by $10. Discuss the merits of using the Black-Scholes formula to value options on the stock.

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Reading Suggestion:

References

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Outside Readings and Links:

1. Paul Wilmott on Quantitative Finance, Chapter 16, Fat tails.

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