Tuesday, March 24, 2015

Jumps and diffusions

I learned an interesting continuous time trick recently. The context is a note, "The fragile benefits of endowment destruction" that I wrote with John Campbell, about how to extend our habit model to jumps in consumption. The point here is more interesting than that particular context.

Suppose one time series \(x\), which follows a diffusion, drives another \(y\). In the simplest example, \[dx_t = \sigma dz_t \] \[ dy_t = y_t dx_t. \] In our example, the second equation describes how habits \(y\) respond to consumption \(x\). The same kind of structure might describe how invested wealth \(y\) responds to asset prices \(x\), or how option prices \(y\) respond to stock prices \(x\).

Now, suppose we want to extend the model to handle jumps in \(x\), \[dx_t = \sigma dz_t + dJ_t.\] What do we do about the second equation? \(y_t\) now can jump too. On the right hand side of the second equation, should we use the left limit, the right limit, or something in between?

The usual answer is to use the left limit. We generalize the model to jumps this way: \[dx_t = \sigma dz_t+ dJ_t \] \[ dy_t = y_{t_-} dx_t = y_{t_-} \sigma dz_t + y_{t_-}dJ_t \] where \(y_{t_{-}}\) denotes the left limit.

That approach has some weird properties however. Suppose \(y_{t_-}=1\), and \(dJ_t=1\). Then \(y_t\) jumps to \(y_t=2\). But suppose there are two jumps of size 1/2, one at time \(t\) and one at time \(t+\varepsilon\). Now \(y\) jumps up to 1.5 after the first jump, and then jumps another \(1.5 \times 0.5 = 0.75\), ending up at \(y_{t+\varepsilon} =2.25\). Two half jumps have a different response than one full jump.

Suppose instead we extend the original model to jumps by taking the jump limit of a continuous process. Imagine that we observe realizations of \(\{dz_t\}\) that get closer and closer to a jump in \(dx_t\), and let's find what happens to \(y_t\). The general solution to the first set of equations is \[ y_{t+\Delta} = y_t e^{(x_{t+\Delta}-x_t - \frac{1}{2}\sigma^2\Delta)}\] so, in the limit \(\Delta \rightarrow 0\) that \(x_t\) takes a jump of size \(dJ_t\), the jump-limit of a continuous movement is \[ dy_{t} \equiv y_t -y_{t_-} = y_{t_-}(e^{dx_{t}}-1) = y_{t_-}\sigma dz_t + y_{t_-}e^{dJ_t}\] rather than \[ dy_t = y_{t_-} dx_t = y_{t_-} \sigma dz_t + y_{t_-}dJ_t \] So, the left-limit method produced a response to a jump that was different from the response to a continuous process arbitrarily close to a jump. For example, the left-limit approach can produce a negative \(y_t\), but this method, like the diffusion process, cannot fall below zero. This method also produces a response to two half jumps that is the same as the response to a full jump.

As you can see, the difference is whether the state variable \(y_t\) gets to change during the jump. In the left-limit approach, the same \(y_{t_-}\) gets applied to the whole jump. In the continuous-limit version, \(y_t\) implicitly gets to move while the jump in \(x_t\) is moving.

A nonlinear function of a jump is a little novel, but there's nothing wrong with it, and it exists in the continuous time literature. We don't see it that often, because when you're only studying one series it's easier to just change the distribution of the jump process instead. This question occurs when you can see both series x and y and you want to model the relationship between them.

Which is right?

Which extension to jumps is correct? Both are mathematically correct. There is nothing wrong with writing down a model in which the response to a jump is different from the response to continuous movements arbitrarily close to jumps. The answer depends on the economic situation.

For example, consider models with bankruptcy constraints. Agents who can continuously adjust their investments may always avoid bankruptcy in a diffusion setting. If we extend such a model to jumps with the continuous limit approach, implicitly preserving the investor's ability to trade as fast as asset prices change even in the jump limit, we will preserve bankruptcy avoidance in face of a jump in prices. However, if we model portfolio adjustment to jumps with the left-limit generalization, agents may be forced in to bankruptcy for price jumps.

Sometimes, one introduces jumps precisely to model a situation in which prices can move faster than agents can adjust their portfolios, so agents may be forced to bankruptcy. Then the left-limit generalization is correct. But if one wants to extend a model to jumps for other reasons, while avoiding bankruptcy, negative consumption, negative marginal utility (consumption below zero or below habits), violations of budget constraints, feasibility conditions, borrowing constraints, and so forth, then one should choose a generalization in which the jump gives the same result as the continuous limit.

Similarly, when extending option pricing models to jumps, one may want to model the jump in such a way that investors cannot adjust portfolios fast enough. Then the left-limit extension is appropriate, and investors must hold the jump risk. But one may wish to accommodate jumps in asset prices to better fit asset price dynamics while maintaining investor's ability to dynamically hedge. Then the nonlinear extension is appropriate, maintaining the equivalence between jumps and the limiting diffusion.

A little more general treatment

A little more generally, suppose \[ dx_t = g dt + \sigma dz_t \] \[dy_t = \mu(y_t) dt + \lambda(y_t)dx_t.\] We want to add \(dJ_t\) to the first equation. The left-limit approach is \[dy_t = \mu(y_{t_-}) dt + \lambda(y_{t_-})dx_t \] If there is a jump \(dJ_t\), \(y\) moves by an amount \[\frac{1}{\lambda(y_{t_-})}dy_t \equiv \frac{1}{\lambda(y_{t_-})}(y_t - y_{t_-}) = dx_t .\] The limit of a continuous movement solves the differential equation \[\int_{y_{t_-}}^{y_t} \frac{1}{\lambda(\xi)}d\xi = dx_t\] Again, you see the crucial difference, whether the state variable gets to move "during" the jump. We can write this as a differential, by writing the solution to this last differential equation as \[y_t-y_{t_-}=f(x_t-x_{t_-};y_{t_-})\] and then \[dy_t = \mu(y_{t_-}) dt + f(dx_t;y_{t_-})=\mu(y_{t_-}) dt + \lambda(y_{t_-})\sigma dz_t+f(dJ_t;y_{t_-})\]

So, you don't have to extend the model to jumps with the left-limit approach, and you don't have to swallow the idea that a jump has a different response than an arbitrarily close continuous-sample-path movement. The last equation shows you how to modify the model to include jumps in a way that preserves the property that the jump has the same effect as its continuous limit.

The point

Why a blog post on this? I asked a few continuous-time gurus, and none of them had seen this issue before. If someone knows where this has all been worked out with proper is dotted and ts crossed, I would like to know and cite it properly. (I would think the literature on option pricing with jumps had done it, but I couldn't find a reference.) Or perhaps it hasn't been done and someone wants to do it. I'm not good enough at the technical aspects of continuous time to write this with the right precision and generality.

And it's a cool trick that may be useful to someone outside of the narrow context that we had for it.


Perhaps the right application is stock prices and option prices. When stock prices jump, someone must have studied the case that option prices move by the same amount the Black-Scholes formula gives for the same size stock price movement. Does anyone have a citation to that case?


  1. I am not sure if it is useful, but you may want to check these lecture notes as they are fairly comprehensive in terms of tools for handling jumps:
    In terms of left limit approach, one needs that only to define the jump process (here denoted by J) and to have a uniquely defined probability process for the jump in the next infinitesimal small time increment.
    As long as one is careful with that, there is an Ito's lemma for point processes (point 2.5 in the lecture notes), construct a stochastic discount factor appropriately and you're in business so to speak.
    Again, not sure if this helps, because I am not sure I understand the comment "you don't have to extend the model to jumps with the left-limit approach, and you don't have to swallow the idea that a jump has a different response than an arbitrarily close continuous-sample-path movement. "
    From a mathematical point of view, you need to treat jumps differently in the analysis. The whole point is that the outcome does not have a continuous path. Otherwise, why introduce jumps and not some other type of process? It could be that the discontinuity doesn’t lead to sudden bankruptcy due to a jump, but to me, it seems the point of introducing jumps is a discontinuity, not just obtaining the same thing as with continuous paths.

  2. Usually, in asset pricing models, we assume that processes *cannot* anticipate innovations (ie are adapted to filtrations that are right continuous, blah blah blah). That's what makes the model market arbitrage free. If we can adapt our trading strategy to the jump before or while it's happening it's easy to earn excess profits.

    I would assume the same would apply to any macro model which takes advantage of complete markets (why would you use markets to hedge future exposures, if you can trade those markets for infinite gain instead?)

    I'm probably misunderstanding your post, but maybe the answers to these questions will clarify things for others as well.


  3. Isn't a typical restriction that Pr( N_{t+e} - N_t > 1 ) = o(e), where N_t is the counting process. i.e., for arbitrarily small time intervals the probability of more than one jump is arbitrarily small.

    1. no, if the Levy measure has a pole at zero you get "infinite jump activity", every interval has an infinite number of jumps.

  4. I agree with many points in the first (Anonymous) comment.
    You have mentioned a few times the idea of "a continuous process arbitrarily close to a jump". This may be a source of confusion and error in your note. Starting from an adapted process with finite variation, one can obtain a diffusion process (continuous path, infinite first variation, finite second variation) under a certain set of conditions and a jump process (countable number of discontinuities, finite first variation, infinite second variation) under another set of conditions. I think you can obtain diffusions from jump process as a limit, but not vice versa (unless you allow for time transformations? not sure).
    Philip Protter's Stochastic Integration and Differential Equation, presents a rigorous yet accessible theory of semimartingales that encompass both diffusions and jump processes.
    Finally, stochastic integration w.r.t. a diffusion is very sensitive to the predictability of the integrand as the first comment mentioned and the idea relates to the arbitrage condition.

  5. I think Reza is on the right track. As of course you know, writing f(dx) or f(dJ) is purely heuristic. What you want to think about is whether the integrand is adapted or not, and by having "the state variable 'move' during the jump", it seems you want a the integrand to be anticipative with respect to the jump process. So the usual Ito integral won't apply, something like a forward or Skorohod integral would be called for. Nualart's Malliavin calculus book or the book by Oksendahl and others (which covers Levy processes) would be the place to look. Basically, there are some additional terms which appear in the integral to capture the additional information in the integrand.

  6. All the comments are great, and the first comment sent me off on a reading assignment.
    But, try this:

    e^(−1/2 * σ^2 * Δ)

    Expand that into its Taylor series out to some finite order. The number of terms in that series is the number of trades needed to maintain the hedge, all the trades having a unique term or transaction rate (the exponent). The economy may not have all those trades available.

  7. First off, as I'm sure you know, stochastic differentials don't really exist; we only have a theory of stochastic integration. The stochastic differential notation is a convenient shorthand, but one can really go down a rabbit hole by writing things like $e^{dJ_t}$, since it is unclear what this means as a function of a stochastic integral.

    Returning to your first set of equations, $x$ is the sum of a scaled Brownian motion $z$ and a pure jump process (or is it a compensated jump process?) $J$, and $y$ is the stochastic integral of itself with respect to $x$. Since stochastic integrals are uniquely defined, this means that you don't have any choice about how to specify the dynamics of $y$ - it is what it is. In this case, $y$ is in fact the stochastic exponential (or Doleans-Dade exponent) of $x$, and it has a unique explicit representation as an ordinary exponential function of $x$ and its quadratic variation. You can see this representation, together with a nice discussion of stochastic exponentials on p. 134 of Jacod & Shiryaev, "Limit Theorems for Stochastic Processes" (2nd ed.)

  8. In your update, are you referring to the binomial tree approximation to the Brownian log price process? In that case you just have to match the variance of the Brownian motion and you get the B-S result in the limit. But with finite jumps I don't see how you can get the B-S result. If I understand correctly, you'd like to introduce trading *during* the jump, but as discussed above, I don't see how you will avoid arbitrage.


  9. What does it mean for $y$ to move "during" the jump. The point of introducing jumps is surely that they are instantaneous? They could be predictable, of course (e.g. when a stock goes ex-dividend). But would the option price respond to the stock price "during" the jump?

  10. The jump studies of Ait-Sahalia, Jean Jacod, Torben Andersen, Tim Bollerslev, Victor Todorov, George Tauchen, Neil Shephard can be helpful. By the way, in the paper that you send, why the volatility process (i.e., sigma) is constant? (equations 4 and 6) I expect my consumption fluctuations to be stochastic and time-varying so that it does not necessarily need to jump: stochastic volatility can already capture such extreme consumption ups and downs.

  11. The first few comments got it most covered. I just want to address the issue of Ito integral and jump separately.

    Even in the continuous case, Ito integral is defined using the left end point. Hence, Ito integral can be interpreted as a self-financing strategy and the integrand can be interpreted as your holdings of x. This definition admits no arbitrage opportunity as the integrand can not anticipate how the asset changes, in other words, the agent cannot adjust his holding of the stocks based on future information. It is possible to relax this assumption, but we will not have to abandon Ito integral and use Stratonovich integral for example.

    And as Hardy pointed out, the general solution to the first set of equations should be the stochastic Ito exponential if you believe in Ito integral. And hence, it is not simply y(t)exp(x(t+delta)-x(t)-0.5*sigma^2*delta)). If you don't believe in Ito integral, then it should not be this expression for the continuous time case either.

    After all the math, here's my thought: it is possible to abandon Ito integral. We don't need the integrand to be adapted to have no arbitrage condition: the real market has frictions. Which integral to use should depend on the data, not just assumptions.

    The second issue is about the definition of jumps. The jump, defined in the mathematical sense, happens instantaneously. Also, by definition, no two jumps can happen at the same time (this is easier to see if we represent Levy process by a time-changed Brownian motion). Hence, one does not have puzzle that two jumps of size 0.5 produce different effects than one jump of size 1: they are not equivalent.

    However, again, the real world acts slightly different than the mathematical definition. If you look at high frequency data, you almost never observe real jump: the perceived jump over an interval of 5-min is usually composed of many trades within the 5-min interval. Does this mean jumps can be anticipated and the agents have time to adjust their portfolio? Or shall we not interpret it as jump but as Ito process with a burst in volatility? These issues are worth investigating.

    Anyway, really interesting post!

    1. Thanks for all the comments. I wasn't clear about an important issue however. Both the two jumps business and the adjusting portfolio during the jump are heuristic. What we're doing here is not constructing a single jump process. That's in the x already. The question is is how to model the response of the series y to the series x that takes the jump. One approach is, "well, suppose we saw realizations of a diffusion that are very close to a jump. Find how much y moves in that case. And then specify that y moves by that amount if there is a jump in x." This is the continuous limit case. Another approach is, "use the same differential representation for y, and put the left limit in the function specifying how y responds to x." That's the left limit approach. This is just about two equally mathematically valid ways of constructing a new model for y, and specifying how y responds to a jump in x. Both ways of doing so respect the nonanticipation etc. constraints.

      It seems a bit of a mistake on my part to have explained it with the two small jumps or as if agents can trade analogies. Those seem to have set off some comments that I don't think we really need to worry about. If x follows diffusions, then two small jumps have the same effect as a big one and agents can trade. So if we write the y process as the limit of a big diffusion movement, then it will move as the limit of such cases. But obviously in the actual jump agents can't trade during the jump. Still, one can specify that y moves in the jump as it moves in the arbitrarily nearby diffusion realizations.

  12. I don't know if this is what you are looking for, but a very helpful paper on dividends (as jumps) and volatility is http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1141877.

  13. Maybe this will help.

    The stochastic process you have written down is a Levy process. Assuming the existence of the first moment (and a cadlag adapted process), the Levy-Ito decomposition tells us that we can write the Levy process as,

    L_t = (b)t + c W(t) + \int_0^t\int_{R} x(\mu^L - \nu^L)(ds,dt),

    where (b)t is deterministic, W(t) is the usual continuous Brownian motion, \nu^L is the Levy measure, and \mu^L is the Poisson random measure.

    Now, we have two ways we can proceed when thinking about exponentials:

    (1) S_t = S_0\exp(L_t)


    (2) dS_t = S_{t-}dL_t.

    The second approach has the Doleans-Dade exponential as a solution and can take on negative values unless jumps are larger than -1 (technically the support of the Levy measure is [-1, \infty) for a non-negative process). The first approach stays positive. In finance with Levy processes, it's usually the first specification that we usually see. So, while in the Black-Scholes model both specifications give the same result, the two specifications are different.


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