8.8 Limitations of the Black-Scholes Model and Extensions
8.8.3 Jumping Up and Down: Lévy Processes
We have seen in the last section that there are better models for the return distribution of assets than the classical log-normal distribution. In this section we present a generalized class of stochastic processes, the Lévy processes, that allow for such forms of outcome distributions, but also includes the classical Brownian motion.
Our exposition follows the introductory text by Jan Kallsen [Kal06] and the book by Jean Bertoin [Ber98].
The key idea of Lévy processesXis to generalize the notion of linear functions in time to stochastic processes – in the sense that linear functions are characterized by constant increments in time and Lévy processes are characterized by constant random distributions of their increments. More precisely, we assume that for every tthe differenceXtCı Xt follows the same probability distribution. If we remind ourselves on Definition8.2of the Brownian motion, we see that this is a part of condition (b). To complete the definition of Lévy processes we need to assume
8.8 Limitations of the Black-Scholes Model and Extensions 319 more, namely that the process starts in zero (condition (a) of Definition8.2) and that increments are independent (condition (d)). All together we define:
Definition 8.17 (Lévy process) A Lévy process is a process X defined by the properties:
(a) X.0/D0a.s.
(b) For any timest0;t1 witht0 < t1, the differenceX.t1/X.t0/follows a fixed distribution.
(c) For any times 0 t0 < t1 < t2 < < tn < 1, the random variables X.t0/;X.t1/X.t0/; : : : ;X.tn/X.tn1/are independently distributed.
We see that Brownian motions are special cases of Lévy processes that are additionally continuous and have normally distributed increments.
How can we characterize Lévy processes? A central tool we will use is the Fourier transformation (see Sect.A.5). We start with considering small time step increments of the process X, i.e., the difference between X.t C t/ and X.t/.
These differences all follow a fixed distribution (according to condition (b)), thus X.tCt/X.t/Dd X.t/X.0/Dd X.t/, whereDd means that they coincide as distributions. We now consider the Forier transform ofX. Due to the independence property (c) we obtain
X.t/O D.X.t//O t=t:
If X.t/O is nowhere zero, we can write it as the exponential of some function WR!C, thus obtaining
X.t/O Dexp. t/;
in other words, we could say that the logarithm ofXO depends linearly ont.
Let us take a closer look on . First we notice that E.X.t//D E.X.t//
t t;
thusE.X.t//has to be of ordertast!0. Second, we consider the variance var.X.t//D var.X.t//
t t;
which shows that also the variance has to be of ordert.
There are now three important cases how we can chooseX such that expected value and variance of its increments are of ordert:
1. Xcan be deterministic, i.e.,X.t/D btfor someb 2 R. The characteristic function XO.t/ then becomes XO.t/.u/ D exp.iubt/, where i2 D 1 (see Sect.A.5).
2. X can be non-deterministic and continuous: if Q is the distribution of X.t/=p
t, then XO.t/.u/ D OQ.up
t/. If we assume for simplicity that the expected value ofX.t/=p
tis zero, then a Taylor expansion ofQO yields thatXO.t/.u/is of order exp.12cu2t/ast!0, wherecWDvar.Q/. 3. X can be non-deterministic and discontinuous: take > 0 and let X change
with a probabilityt according to the distribution Q. (With the probability 1tthe process remains constant.) The characteristic function is in this case XO.t/ D .1t/CtQ. Using the Taylor expansion for the exponentialO function, we can write this fort!0asXO.t/Dexp..QO.u/1/t/. A general Lévy process can now be written as a sum of these three components – plus some remainder term that is discussed in [Kal06] and [Ber98]. This yields to the general Lévy-Chintschin Formula, where we set
h.x/WD x;jxj 1;
0;jxj> 1 and the Lévy measureFWDQ:
Lemma 8.18 (Lévy-Chintschin Formula) Let X be a Lévy process, then its characteristic functionX can be described asO
X.t/.u/O Dexp
iub.h/1 2u2cC
Z
.eiux1iuh.x//F.dx/
t
: The variance of a Lévy process has therefore two main sources: the jumps which are described by the measureF (third term in the formula) and by a continuous movement described by the parameterc(second part).
If we want to design a process with a specific distribution (e.g., a NIG distribution), we can chooseFas this distribution and the above formula will yield a Lévy process for this distribution. We just need to apply the formula to obtainXO and then the Fourier transformation to getX. Lévy processes are therefore an extremely flexible and useful class of processes for modeling financial data. Sometimes they are, however, too general and complex to obtain results, e.g., on asset pricing, without further assumptions.
One of many technically simpler subclasses of Lévy processes which has recently caught attention arestable Lévy processes with exponential decay, as introduced by Boyarchenko and Levendorskiˇi [BL02]. This class of processes encompasses in particular Brownian motion, NIG processes, hyperbolic processes etc., so many
8.8 Limitations of the Black-Scholes Model and Extensions 321 important models are covered by this generalization. On the other hand, the class is small enough in order to use analytical methods for asset pricing. Unlike in the setting of the classical Black-Scholes formula they often do not lead to a PDE, but instead to an equation involvingpseudo differential operators(see Sect.A.5for a rough intuition). For details on asset pricing in this general framework and further generalizations we refer the reader also to [BL02] and [FS07].
We see that departing from the simple world of Brownian motions makes things harder and requires sophisticated mathematical tools, but, as Douglas Adams put it:
It is a mistake to think you can solve any major problems just with potatoes.