8.8 Limitations of the Black-Scholes Model and Extensions
8.8.4 Drifting Away: Heston and GARCH Models
When considering Brownian motion as a model for stock price movements, we assume constant volatility. We have seen in Sect.8.8.1that the volatility implied by options depends on their maturity. This could be explained by a time-varying volatility which is in fact supported by the data. Figure8.4shows a volatility index starting from 1990 up to 2007. It is obvious that the volatility is not just random, but that there are periods with high and periods with low volatility. This phenomenon is calledvolatility drift.
0 20 40 60 80
Fig. 8.4 Volatility index (VIX) from 1990 to 2016
If we want to take this effect into account, we need to describe the volatility itself by a random process. Therefore we need two stochastic differential equations:
dS.t/DS.t/dtCp
.t/S.t/dB1.t/;
d.t/D˛..t//dtCˇ..t//dB2.t/;
where˛andˇare given andB1,B2are both Brownian motions that may correlate with each other with correlation2Œ1;C1.
A typical feature (a “stylized fact”) that can be observed for the volatility is that it tends to revert to the mean, i.e., there is a long-term average!such that the volatility eventually returns to this value. Another stylized fact is that the fluctuation of the volatility tends to be larger when the volatility is large. Both observations lead to a class of standard models with˛..t//D.!.t//(where > 0is a constant) and ˇ..t// D.t/ (where > 0and > 0are also constants). While describes how strongly the process tends to return to its mean, is the (constant part of the) volatility of. Finally, is an exponent that describes how strong the volatility of increases when increases, i.e., how strongly “the volatility makes the volatility become volatile”.
There are three frequently used models that fall into this framework:
• TheHeston modelassumes D1=2. The variance process is in this case called aCIR process, named after its inventors John C. Cox, Jonathan E. Ingersoll and Stephen A. Ross [CIR85].
• TheGeneralized Autoregressive Conditional Heteroskedasticity model17(short:
GARCH) assumes D1.
• The3=2modelassumes D3=2.
It is difficult to decide which of these models is most appropriate. If we use the data on the volatility index and measure how large the standard deviation of its daily changes is (where we always collect 50 data points with similar volatility to compute the standard deviation) we obtain Fig.8.5that gives a best-fit exponent of 0:6. This simple approach is of course not reliable enough to decide which process is best in this case, but the data shows clearly that there is a strong positive correlation between the volatility and its standard deviation as proposed by all three models.
There are recent approaches to base this (and other) stylized facts on models of interacting agents on financial markets, see the survey article by Lux [Lux09] for details and further references.
There is one more interesting feature about the volatility that is not captured by the above models: volatility tends to be lower in bull markets, and higher in bear markets. This is certainly counterintuitive, since the risk-return tradeoff should
17A vector of random variables isheteroskedasticif the random variables have different variances.
8.8 Limitations of the Black-Scholes Model and Extensions 323
0 1 2 3 4
Volatility index 0
10 20 30 40 50
Standard deviation of changes of volatility
Fig. 8.5 Standard deviation of the volatility index as a function of the volatility index. For each day the standard deviation of the daily volatility changes of the 50 days with a volatility closest to the volatility of the original day is computed and plotted with respect to the average volatility of these 50 days
reward a high volatility with larger returns, but the opposite is the case. The effect (sometimes called “leverage effect”) can be observed in many markets, compare Fig.8.6. An analysis for monthly returns of the S&P 100 yields an interesting pattern: the correlation is strong in losses, but weak in gains (compare Fig.8.7).
To model this volatility asymmetry one can use more sophisticated stochastic processes, e.g. the APARCH process.
Attempts for an explanation of this effect can be found, e.g. in [BHS01]
and [HS09], but there is still no universally accepted model that explains why volatility and stock prices correlate negatively. Recent empirical research where this asymmetry has been measured in a large number of countries found that the asymmetry seems to be particularly high when there are many private investors on the market. Assuming that private investors are more prone to behavioral biases than institutional investors, this might point to behavioral factors as one cause for volatility asymmetry [TR10].
Fig. 8.6 Volatility stock market returns in the USA, Germany, Australia and Japan (com- pare [Pap04]). The strong negative correlation is in most cases evident