Consider the SDE: \begin{equation} dv_t = k(\theta - v_t) dt + \xi \sqrt{v_t} dW^{v}_{t} \end{equation} It describes a process $v_t$ which is a strictly positive if the drift is stronger enough, i.e. if drift parameters ($k$, the speed of mean-reverting, and $\theta$, mean-reverting level) and the Vol-of-Vol $\xi$ satisfy: \begin{equation} k \theta > \frac{1}{2} \xi^2 \end{equation} which is known as Feller condition. I know this condition can be generalized to multi-factor affine processes. For example, if the volatility of the returns $\log S_t$ is made of several independent factors $v_{1,t},v_{2,t},...,v_{n,t}$, then the Feller condition applies to each factor separately (check here at page 705, for example). Moreover Duffie and Kan (1996) provide a multidimensional extension of the Feller condition.

But I still don't understand if we still need the (or a sort of) Feller condition in case of jump-diffusion. You may consider for example the simple case of a volatility factor with exponentially distributed jumps: \begin{equation} dv_t = k(\theta - v_t) dt + \xi \sqrt{v_t} dW^{v}_{t} + dJ^{v}_{t} \end{equation} where $J^{v}_{t}$ is a compound Poisson process, independent of the Wiener $W^{v}_{t}$. The Poisson arrival intensity is a constant $\lambda$ with mean $\gamma$. I observe that in this case, the long term mean reverting level is jump-adjusted: \begin{equation} \theta \Longrightarrow \theta ^{*}=\theta + \frac{\lambda}{k} \gamma \end{equation} so I suspect if a sort of Feller condition applies it must depends on jumps.

Nevertheless, from a purely intuitive perspective, even if the barrier at $v_t = 0$ is absorbent, jump would pull back from 0 again.

Thanks for your time and attention.

  • $\begingroup$ What do you mean by "need" the Feller condition. Do you have some specific proof in mind where as it is needed? the only thing it does is to assure you that you do not hit 0 a.s. As you have in your example only an upward jump part with finite jump actitvity, its presence does not change anything in this consideration. $\endgroup$ Dec 16 '14 at 23:50
  • $\begingroup$ I see, the barrier at 0 is not reached "a fortiori" if the Feller condition holds AND there are positive jumps. You're saying the Feller condition holds (as it is k theta > 1/2 csi^2) also in presence of jumps? Thank for your comment $\endgroup$ Dec 17 '14 at 0:08
  • $\begingroup$ I still do not understand what you mean by "The Feller condition holds". It is just an assumption on parameters. $\endgroup$ Dec 17 '14 at 0:49
  • $\begingroup$ And if you make this assumption, neither the diffusion nor the jump diffusion you specified reach zero a.s. $\endgroup$ Dec 17 '14 at 0:50
  • $\begingroup$ Well, I was asked to calibrate a SVJJ a-là Duffie, Pan and Singleton 2000 on a volatility surface. But, imposing the restriction ktheta>2csi^2 of the Feller condition on the parameters, the model seemed to be no way able to fit the data.. So I was suspecting that I was simply imposing a wrong condition. But I agree with your point, the presence of positive jumps dooes not affect in any way the a.s. probability of reaching the 0 barrier (I guess the finite or infinite activity doesn't play a role, or not?). So the expression of the condition does not change. $\endgroup$ Dec 17 '14 at 8:26

The Feller condition is hardly satisfied in the market, mainly because having $k\theta >\frac 12 \xi^2$ implies a higher mean reversion which reduces the variance of the stochastic volatility, this in turn reduces the convexity of the smile, and in order to increase this convexity you have to increase your vol of vol $\xi$, but you cannot increase it freely as it is bounded by $\xi^2< 2k\theta $. So in general this condition is violated. However this is not really an issue, at the moment the volatility reaches $0$ you'll have $dv_t=k\theta dt$ and the volatility will instantly go away from $0$. The only problem you can get is in discrete time. So when you discretize your process via Euler scheme for example your process can go negative, one solution of this fact which is widely used is to floor the process to $0$ or to reflect it, which means you take its absolute value. However, for the CIR process there is an algorithm which was published on 2007 by Leif Andersen which simulate the CIR process without using discretization, it is now the standard algorithm used in the whole market for the Heston model. The article is "Efficient Simulation of the Heston Stochastic Volatility Model". I believe in the final section of the article the author add jumps which are independents of the Brownian of the CIR process. In this case the simulation is straightforward, you can simulate the CIR using the algorithm and simulate the jump independently. Hope this helps.

  • $\begingroup$ This helps a lot! Thank you! (by the way, you meant dv = k theta at the time at which v=0, right?) $\endgroup$ Dec 17 '14 at 15:08
  • $\begingroup$ Yes, It's corrected now. You're welcome. $\endgroup$
    – Hicham
    Dec 17 '14 at 16:48
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    $\begingroup$ I agree with this, only that the process can become numerically negative even in the case that Feller's condition is satisfied. An alternative way to circumvent this (due to Alfonsi, cermics.enpc.fr/~alfonsi/SC_preprint.pdf) is to rewrite the Ito Integrals into Stratonovich integrals and discretize only then. This leads to an implicit FD scheme that can even be turned into an explicit one if Feller's condition is satisfied. $\endgroup$ Dec 18 '14 at 5:42
  • $\begingroup$ But this answer does not touch on the effect of jumps on the Feller condition in terms whether the probability of reaching zero is positive at all. $\endgroup$
    – Hans
    May 25 '16 at 5:43

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