ESTRUCTURALISMO E HISTORIA
III. La lingüística estructural: la lengua como materia
returns
2.7.1
Completely Monotone L´evy Densities
From an empirical point of view it is clear that it is reasonable to ask that, fix- ing the direction of the move of the stock price, large moves are less frequent that small moves, that is jumps of larger size have a lower arrival rate than smaller rates. Mathematically a structural property of the L´evy densities is that of monotonicity. This property amounts to asserting that for differen- tiable densities, the derivative is positive for negative jumps and negative for positive jumps. The property of monotonicity may be strengthened to complete monotonicity by requiring that derivatives of the same order have the same sign and be alternating in sign. This property links analytically the arrival rate of small and large jumps, requiring that large jumps arrive less frequently than small jumps. By Bernstein’s theorem all completely mono- tone L´evy densities are mixtures of exponential functions and are given by
the Laplace transform of positive measures ρ(da) on the positive half line and can be expressed as20
k(y) =
Z +∞ 0
e−ayρ(da)
Although this requirement of complete monotonicity seems to be quite intu- itive, we note that not all the option pricing models presented in literature satisfy it. In particular the jump diffusion model based on the reflected normal distribution for the jump size is not completely monotone as eas- ily observed by seeing that the normal density shifts from being a concave function near zero to a convex function near infinity.
Another desirable property of processes describing stock returns is that, once fixed the size of the move, down moves have an arrival rate and a risk neutral price independent and higher than those of the corresponding up moves. The independence of the down an up moves can be obtained using two non negative functions, each of them having a single argument which is a positive real. One of these functions determines the arrival rate associated with the absolute size of down move, the other determining the arrival rate of up moves. A premium for negative moves can be created by requiring that the function determining the arrival rates related with down moves has higher mean than the one determining arrival rates for up moves. Com- pletely monotone functions satisfy these requirements of independence of up and down moves and of premium for negative moves.
2.7.2
Finite Variation Processes
Consider a process x(t) defined on an interval [0, T]. If P = {0 = t0 < t1 < ... < tn = T} is a partition of [0, T], write ∆xt = x(tk)−x(tk−1), for t = 1,2, ..., n. If for any path there exists a positive number M such that
n X
i=1
|∆xt| ≤M
for all partitions of [0, T], then xis said to be of finite, or bounded, variation on [0,T]. It can be proved that a function is of finite variation if and only if it can be expressed as the difference of two increasing functions.
20See H´elyette Geman, Dilip B. Madan and Marc Yor,“Time Changes for L´evy Pro-
As Carr, Madan, Geman and Yor note21, processes of finite variations are potentially more useful than those ones of infinite variation in explain- ing the measure change from the statistical to the risk neutral process as they allow greater flexibility between the local characteristic of the martin- gale components under the two measures. In the case of infinite activity processes like the Brownian motion, the volatility, and hence the local mar- tingale component, is invariant under an equivalent change in measure. This equivalence of measure change for infinite variation jump processes implies that the difference between the risk neutral and the statistical L´evy densities is of finite variation. This requires that the two processes have the same exponent. On the other side, if the processes are themselves of finite varia- tion, then the difference in the L´evy densities will automatically be of finite variation and therefore no parametric restriction on the processes is required.
2.7.3
Finite Activity Processes
A pure jump process is defined to be of finite activity if the number of jumps in any interval of time is finite. On the other side, if the number of jump in any interval is infinite, we define the process as of infinite activity. We can think about an infinite activity process as an approximation of a highly liquid market with large activity.
2.7.4
Variance Gamma Properties
Let’s consider again the representation of the L´evy measure for the variance gamma process as difference of two gamma processes presented in equation (2.23). We can see that the L´evy density is divided by the absolute value of the jump size, therefore the L´evy density has the behavior of |x|1 in the neighborhood of zero and the resulting process is one of infinite activity, as the variance gamma L´evy measure integrates to infinity. Moreover, since |x|
is integrable with respect to the variance gamma L´evy density, the process is one of finite variation.
21Peter Carr, H´elyette Geman, Dilip B. Madan and Marc Yor, “The Fine Structure of
Asset Returns: An Empirical Investigation”, The Journal of Business, Vol. 75, No. 2, 2002, page 312.