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.

Update: 

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?