Thanks to Cuch.P

A real-valued process X(t), with X(0) = 0, is called a Lévy process if: (i) it has independent increments; that is, for any choice of n >= 1 and 0 <= t0 < t1 < t2 < .... tn the random variables X(t0), X(t1)-X(t0), ..., X(tn)-X(t_n-1) are independent;(ii) it is time-homogeneous; that is, the distribution of X(t+s) - X(s); s >= 0 does not depend upon s.(iii) it is stochastically continuous; that is, for any eps > 0, Pr{|X(t+s) - X(s)| > eps} -> 0 as t -> 0. The only property that needs elaboration, in my opinion, is number (iii), which may seem counter-intuitive for a process that often models jumps. The main thing about (iii) is thatthe jumps, when they occur, are not predictable (such an an ex-dividend day for a stock).Examples are Brownian motion, Poisson process, compound Poisson processes, and many others. regards, p.s. fixed a typo in (iii)

Alan,You presented a common definition for the Levy process. What does this definition have to do with finance? Have you done measurements to confirm that actual financial time series have properties (i), (ii), and (iii)?N

For finance, this process should be treated more as a slight generalizationof Brownian motion to include jumps. But it's counter-factual in the sense thatit omits stochastic volatility, which is a statistical property with a lot of empirical support.So, I'm not going to argue for it that way. I don't do a lot of empirical work, but have done a fair number of calibrationsto equity option prices. When you generalize the process to include stochasticvolatility, the resulting fits can be quite plausible. Pushed to the limit, all financial time series will fail stationarity tests. regards,

> But it's counter-factual in the sense that it omits stochastic volatility, which is a statistical property with a lot > of empirical support. So, I'm not going to argue for it that way.AlanWas it not so thatBates model = Heston model (stovol) + jumps?In general, we see that the equation for the spot exchange rate (to take an example) is governed now by a Sto. Intergo Diff. Eq. (SIDE) and this is different from normal SDE. Then we get a PIDE in 3 underlyings, including mixed derivatives and integral terms. So we have correlation, jumps and sto. vol. to be modelled. This is a good model?I have seen something on this in general but I do not know the details. We have a different kettle of fish now it would seem!

QuoteOriginally posted by: Cuchulainn> But it's counter-factual in the sense that it omits stochastic volatility, which is a statistical property with a lot > of empirical support. So, I'm not going to argue for it that way.AlanWas it not so thatBates model = Heston model (stovol) + jumps?In general, we see that the equation for the spot exchange rate (to take an example) is governed now by a Sto. Intergo Diff. Eq. (SIDE) and this is different from normal SDE. Then we get a PIDE in 3 underlyings, including mixed derivatives and integral terms. So we have correlation, jumps and sto. vol. to be modelled. This is a good model?I have seen something on this in general but I do not know the details. We have a different kettle of fish now it would seem!Yes, Bates combined processes as you indicated in his paper "Jumps and Stochastic Volatility".This is getting off-topic, but there are many modeling choices: choice of s.v. process, choiceof Levy jump component; whether to include a Brownian component (I would). Carr/Wu combine s.v. and Levy processes with a time-change; this is a little different. The general (markov) setup leads to an (N+1) dimensional PIDE, where the "1" is time,and the other N components follow what are called local Levy processes (diffusions + jumps).Local Levy processes do not have globally identical increments (Property (ii) below), but only "locally identical" ones. A simpler way to say it is that the PIDE coefficients are not constants (as in a true Levy process),but variable. Each component has both diffusion and jump behavior in some domain D. The diffusioncomponents are correlated (mixed derivatives). The jumps can be independent or simultaneous (and correlated) -- there is an N-dimensional local Levy measure. (I would latex it, but it would take forever). If some components can reach boundaries, there is a whole new kettle associated with what can go on there, too. Perhaps 99+% of all (continuous) QF models fit into this larger "local Levy process" framework. Are these good models? In my opinion, the game for quantitative finance is to find what I would call "the minimal markov process"for whatever you are describing; i.e., the smallest size, (mostly) time-homogeneous, markov process thatcaptures all of the important behavior. But the number of things that are important is probably inversely related totransaction costs, so QF may never develop its `theory of everything'. regards,

AlanThanks for he excellent reply. Your summary is very clear.> If some components can reach boundaries, there is a whole new kettle associated with what can go on there, too.And that's the most difficult part Daniel

As to finding the minimal model that captures the important empirical behaviours, I suggest the paper by Bedendo and Hodges (2004, I think). The authors use a data analysis approach to build up a continuous-time stochastic volatility jump diffusion model. The good thing of the approach is that the model is suggested by the (high frequency) data step by step, so that the modelling error is reduced to minimal.

A right-continuous process X_t, t E R+, with stationary independent increments is called a Levy process. A Levy process may in general include drift, diffusion and jumps and is characterised by the Levy triplet . It has an infinitely divisible distribution whose cumulative characteristic function satisfies the Levy-Khintchine equation. A subordinator is a positive nondecreasing Levy process with finite variation. The Levy measure specifies the jump size and intensity. Levy processes (especially when combined with stochastic volatilty) give a more precicse representation of asset-price processes than the Black-Scholes equation. One of the first times that jump processes (discontinuities) appeared in the finance literature was in the seminal paper "Option Pricing When Underlying Stock Returns are Discontinuous." Journal of Financial Economics 3 (January-February 1976), by Robert Merton.