8.8 Limitations of the Black-Scholes Model and Extensions
8.8.1 Volatility Smile and Other Unfriendly Effects
The geometric Brownian motion assumes that volatility is constant in time. Is this a reasonable assumption? A standard way to figure this out is to compute theimplied volatility, i.e., the volatility that an asset should have, assuming that the price of the call option traded on this asset obeys the Black-Scholes formula. For standard assets like the S&P 500 we can compute the implied volatility for various values of strikeKand maturityT, thus we obtain a function in the two variablesKandT that is usually represented as a three-dimensional plot called the(implied) volatility surface.
If the assumptions of the Black-Scholes model were perfectly right, this surface would be flat. However, like the earth, this surface is not flat:
• At-the-money options tend to have lower implied volatility than options with a strike far away from the current price of the underlying. This curved shape reminded some researchers of a smile, thus the namevolatility smile.
• There is also a time-dependence of the implied volatility: different maturities lead to different implied volatilities, an effect which is calledterm structure of volatility.
Both effects are not as friendly as a smile usually is, because they exhort us to improve the underlying assumptions. We could say with Huxley that “the great tragedy of science is the slaying of a beautiful hypothesis by an ugly fact”. It is in particular not correct to assume that the volatility is constant in time, and that returns are normally distributed. Indeed, that stock returns have fat tails (i.e., that the probability for extreme events is larger than for a normal distribution) is a well- established empirical fact that we will explain in the next section. These fat tails can explain the volatility smile: options with strikes far away from the current price of the underlying are priced in a way that seems to imply a larger volatility, since only such a larger volatility could explain the high probabilities associated to strong price movements, given the assumption that returns are normally distributed. There
is support for this explanation from the US market: before the crash in 1987 there was no smile, as if investors had not been aware of the potential for large price changes. Since then, a volatility smile can be observed on the US market as well.
One could say that investors have learned their lesson. . .
The term structure, on the other hand, can often be explained by the expectation of news. A typical example are earning reports: stock options with maturity shortly after the earning report show higher implied volatility than options with a later maturity.
8.8.2 Not Normal: Alternatives to Normally Distributed Returns We have already mentioned that standard assets like stocks and bonds do not usually have normally distributed returns (compare Fig.8.1). This was already observed in the early twentieth century by various researchers [Mit15,Oli26,Mil27]. There are various ways to confirm this observation empirically. A simple method is to measure whether returns of real assets have significant skewness or excess kurtosis (compare Sect.A.2). To this aim one can use a kind of Monte Carlo method: let us assume you have the data ofN past returns of an asset. First simulateNrandom returns under the assumption of normality, then compute skewness and excess kurtosis of this (finite) distribution, finally iterate thisntimes to get a large sample of approximated normal distributions, each withNdata points. If the measured skewness of the data
−0.080 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08
5 10 15 20 25 30 35 40
Returns
Density
MSCI World Index Normal FIt NIG Fit
Fig. 8.1 The distribution of daily returns of the MSCI World from January 1, 1970 to April 30, 2007. The best fit with a normal distribution shows that the fat tails are underestimated.
A better fit can be obtained by using NIG distributions as we explain below
8.8 Limitations of the Black-Scholes Model and Extensions 315 sample exceeds the skewness of the vast majority of simulated distributions, then it is very unlikely that the data follows a normal distribution. More precisely, ifmof nsimulated values are below the data value, then the probability is approximately pD.nm/=n, which is very small ifmn. The probabilitypis calledsimulated p-value. Similarly, excess kurtosis can be tested. This method is calledsimulated p-test. Some results for stocks, bonds and hedge funds that show that most asset classes show excess kurtosis and skewness can be found, e.g., in [RSW].
The fact that assets are not normally distributed has several important con- sequences: first, asset pricing based on the Black-Scholes formula needs to be corrected (remember this unfriendly volatility smile!). Second, judging investments by mean-variance which would be correct under the assumption of normally distributed returns is also not correct, since it underestimates the risk of large losses (fat tails in the distribution!).
But how could one improve the model? There are in fact many possible ways to do this. In this section we will outline some of them. In order to apply them for option pricing, we need also a stochastic process that generates non-normal distributions that allow for skewness and fat tails. This will be possible in the general framework of Lévy processes, see Sect.8.8.3.
As an example for a class of very versatile distributions we introduce thenormal inverse Gaussian distributions (short:NIG) that has been introduced to financial applications by Barndorff-Nielsen [BN97].14They are specified by four parameters that roughly correspond to the first four moments. In other words, they allow to model distributions with skewness and excess kurtosis, as we encounter them in the data. A NIG distribution is defined by the following probability density function
NIG.xI˛; ˇ; ; ı/WD ˛ı eı
p˛2ˇ2Cˇ.x/K1.˛p
ı2C.x/2/ pı2C.x/2 ;
wherex; 2 R; 0 ı; 0 jˇj ˛andK1 is the modified Bessel function of the third kind with index 1 (see [BN97] and [Sch08]). The mean, the variance, the skewness and excess kurtosis15ofXNIG.˛; ˇ; ; ı/are given by
E.x/DC ı .12/1=2;
var.x/D ı
˛.12/3=2;
14Originally, NIG distributions have been used in physics, more precisely in the modeling of turbulence and sand grain distributions. Only years later they made it into finance.
15Excess kurtosis refers to the amount of kurtosis that exceeds that of the normal distribution.
S.x/D 3
.ı˛/1=2.12/1=4; K.x/D3 4 2C1
ı˛.12/1=2; whereDˇ=˛.
While the normal distribution has zero skewness and a kurtosis equal to three, we see that aNIG.˛; ˇ; ; ı/distributed variable has parameter-dependent moments, which are interacting with one another. The four parameters˛; ˇ; ; ıhave natural interpretations relating to the overall shape of the density function: the parameter
˛ controls the steepness of the density, in the sense that the steepness increases monotonically with an increasing˛. This also has implications for the tail behavior:
large values of˛imply light tails, while smaller values of˛imply heavier tails as illustrated in Fig.8.2. The parameterˇis a skewness parameter, in the sense that ˇ < 0implies a density skew to the left andˇ > 0implies a density skew to the right, i.e., the skewness of the density increases asˇincreases. In the symmetric case where the parameterˇis equal to 0, the density is symmetric around. Figure8.3 shows the dependency onˇ. Finally, the parameterıis akin to the standard deviation of the normal distribution and represents a measure of the spread of returns.
NIG distributions have a nice property: when we combine two independent NIG distributionsNIG.˛1; ˇ1; 1; 1/andNIG.˛2; ˇ2; 2; 2/, the result is again a NIG distribution,providedthat˛1D˛2andˇ1 Dˇ2.
As an example for a fit with NIG, we consider the daily returns of the S&P 500 from January 4, 1988 to May 4, 2007.16 Using a likelihood estimate gives the parameters ˛ D 183:7, ˇ D 8:0, D 0:000 and D 0:0033 for the NIG distribution. This corresponds to a mean of 0:02%, a variance of 0:002%,
Fig. 8.2 The effect of different values of˛on a NIG distribution
16Other data gives very similar results. For illustration we concentrate on one particular case.
See [RSW] for details.
8.8 Limitations of the Black-Scholes Model and Extensions 317
Fig. 8.3 The effect of different values ofˇon a NIG distribution
a skewness of 0:17 and an excess kurtosis of 5:03. It is interesting to notice that an likelihood estimate with only the two parameters of the normal distribution gives a much larger variance of0:42%. In other words: most of the variance that we observe when approximating returns by a normal distribution is quite likely only an artifact of higher moments. The likelihood of the estimate using NIG increases by about 2%, so besides theoretical reasons in favor of a distribution capable of modeling skewness and fat tails, there is also a quantitative gain in the approximation accuracy. On the other hand, there is of course a cost in dealing with a more complicated model.
What other approaches are there to replace normal distributions? There are in fact several other models, maybe the best-known is the Lévy skew alpha-stable distribution, named after the French mathematician Paul Lévy who invented them in 1925 [Lév25] and introduced to finance by the French-American mathematician Benoợt Mandelbrot in 1963 [Man63], who later gained popular fame for his
“Mandelbrot set” and scientific fame for a number of important contributions to fractal geometry. The central difference to the NIG distributions is that Lévy skew alpha stable distributions (LSASD) do usually not have finite variance. This property might frighten us a little, after all we have spent substantial time with mean-variance approaches, where the assumption of a finite variance was as natural as it was crucial. But maybe we want to risk the revolution and forego the finiteness of the variance if we can gain something for it in exchange. In fact, there is not everything bad about LSASD: in particular, combining two independent LSASDs yields again a LSASD. (This property is meant when we talk about “stability”.)
How are LSASDs defined? Like the NIG distrubution, the LSASD depends on four parameters. They are denoted by˛,ˇ,cand. The distribution is defined as theFourier transformation(see Sect.A.5) of a characteristic function', i.e.
f.xI˛; ˇ;c; /D 1 2
Z C1
1 '.t/eitxdt:
The characteristic function is defined as
'.t/Deitjctj˛.1ˇsign.t/˚.˛;t//;
where sign.t/gives the sign oft(i.e.,C1or1) and˚.˛;t/is given by
˚.˛;t/WD
(tan.˛=2/ ;for˛¤1;
.2=/logjtj ;for˛D1:
The parameters can be interpreted similar to the NIG distribution:ˇis a measure of asymmetry, where the distribution is symmetric around the shift parameterif ˇD0. The parametercis a scale factor that describes the width of the distribution and˛specifies the asymptotic behavior of the distribution. (Like in the case of the NIG distribution there are other parameterizations in the literature.)
It can be proved that the variance of the LSASD is infinite if˛ < 2and that several prominent distributions are special cases: for˛ D 2we obtain the normal (Gaussian) distribution, for˛ D 1 andˇ D 0 we obtain theCauchy distribution and for˛D1=2andˇD1theLévy distribution. Finally, forc!0or˛!0we obtain the Dirac distribution, i.e., a certain outcome at(compare Sect.A.4).
The most notable property of Lévy distributions is that the sum of arbitrary random variables with tails following the power-lawjxj.˛C1/ with˛ > 0 (and hence with infinite variance) tend to the stable Lévy distributionf.˛; 0;c; 0/.
Empirical values for ˛ that describe actual price movements of stocks and commodities quite well have been found already in [Man63] as˛1:7. Applying this model to asset pricing is, however, a non-trivial task. The foundation for this will be laid in the next section, where we show how processes can be constructed that generate non-normal return distributions.