Jumps In Addition To Diffusions
I learned an interesting continuous fourth dimension fox recently. The context is a note, "The frail benefits of endowment destruction" that I wrote alongside John Campbell, almost how to extend our habit model to jumps inwards consumption. The betoken hither is to a greater extent than interesting than that item context.
Suppose i fourth dimension serial \(x\), which follows a diffusion, drives some other \(y\). In the simplest example, \[dx_t = \sigma dz_t \] \[ dy_t = y_t dx_t. \] In our example, the instant equation describes how habits \(y\) respond to consumption \(x\). The same sort of construction mightiness depict how invested wealth \(y\) responds to property prices \(x\), or how alternative prices \(y\) respond to stock prices \(x\).
Now, suppose nosotros desire to extend the model to handgrip jumps inwards \(x\), \[dx_t = \sigma dz_t + dJ_t.\] What arrive at nosotros arrive at almost the instant equation? \(y_t\) forthwith tin dismiss bound too. On the right manus side of the instant equation, should nosotros role the left limit, the right limit, or something inwards between?
The park answer is to role 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 \(y_{t_-}=1\), too \(dJ_t=1\). Then \(y_t\) jumps to \(y_t=2\). But suppose at that spot are 2 jumps of size 1/2, i at fourth dimension \(t\) too i at fourth dimension \(t+\varepsilon\). Now \(y\) jumps upwardly to 1.5 later the showtime jump, too thus jumps some other \(1.5 \times 0.5 = 0.75\), ending upwardly at \(y_{t+\varepsilon} =2.25\). Two one-half jumps receive got a dissimilar response than i total jump.
Suppose instead nosotros extend the master copy model to jumps past times taking the bound bound of a continuous process. Imagine that nosotros bring out realizations of \(\{dz_t\}\) that acquire closer too closer to a bound inwards \(dx_t\), too let's bring out what happens to \(y_t\). The full general solution to the showtime laid upwardly of equations is \[ y_{t+\Delta} = y_t e^{(x_{t+\Delta}-x_t - \frac{1}{2}\sigma^2\Delta)}\] so, inwards the bound \(\Delta \rightarrow 0\) that \(x_t\) takes a bound of size \(dJ_t\), the jump-limit of a continuous motility 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 bound that was dissimilar from the response to a continuous procedure arbitrarily simply about a jump. For example, the left-limit approach tin dismiss arrive at a negative \(y_t\), but this method, similar the diffusion process, cannot autumn below zero. This method also produces a response to 2 one-half jumps that is the same every bit the response to a total jump.
As you lot tin dismiss see, the departure is whether the acre variable \(y_t\) gets to modify 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 deed piece the bound inwards \(x_t\) is moving.
Influenza A virus subtype H5N1 nonlinear business office of a bound is a trivial novel, but there's null incorrect alongside it, too it exists inwards the continuous fourth dimension literature. We don't run across it that often, because when you're alone studying i serial it's easier to simply modify the distribution of the bound procedure instead. This inquiry occurs when you lot tin dismiss run across both serial x too y too you lot desire to model the human relationship betwixt them.
Which is right?
Which extension to jumps is correct? Both are mathematically correct. There is null incorrect alongside writing downwards a model inwards which the response to a bound is dissimilar from the response to continuous movements arbitrarily simply about jumps. The answer depends on the economical situation.
For example, consider models alongside bankruptcy constraints. Agents who tin dismiss continuously arrange their investments may e'er avoid bankruptcy inwards a diffusion setting. If nosotros extend such a model to jumps alongside the continuous bound approach, implicitly preserving the investor's powerfulness to merchandise every bit fast every bit property prices modify fifty-fifty inwards the bound limit, nosotros volition save bankruptcy avoidance inwards human face upwardly of a bound inwards prices. However, if nosotros model portfolio adjustment to jumps alongside the left-limit generalization, agents may live on forced inwards to bankruptcy for toll jumps.
Sometimes, i introduces jumps just to model a province of affairs inwards which prices tin dismiss deed faster than agents tin dismiss arrange their portfolios, thus agents may live on forced to bankruptcy. Then the left-limit generalization is correct. But if i wants to extend a model to jumps for other reasons, piece avoiding bankruptcy, negative consumption, negative marginal utility (consumption below null or below habits), violations of budget constraints, feasibility conditions, borrowing constraints, too thus forth, thus i should pick out a generalization inwards which the bound gives the same outcome every bit the continuous limit.
Similarly, when extending alternative pricing models to jumps, i may desire to model the bound inwards such a agency that investors cannot arrange portfolios fast enough. Then the left-limit extension is appropriate, too investors must concur the bound risk. But i may wishing to accommodate jumps inwards property prices to improve jibe property toll dynamics piece maintaining investor's powerfulness to dynamically hedge. Then the nonlinear extension is appropriate, maintaining the equivalence betwixt jumps too the limiting diffusion.
A trivial to a greater extent than full general treatment
Influenza A virus subtype H5N1 trivial to a greater extent than generally, suppose \[ dx_t = g dt + \sigma dz_t \] \[dy_t = \mu(y_t) dt + \lambda(y_t)dx_t.\] We desire to add together \(dJ_t\) to the showtime equation. The left-limit approach is \[dy_t = \mu(y_{t_-}) dt + \lambda(y_{t_-})dx_t \] If at that spot is a bound \(dJ_t\), \(y\) moves past times an amount \[\frac{1}{\lambda(y_{t_-})}dy_t \equiv \frac{1}{\lambda(y_{t_-})}(y_t - y_{t_-}) = dx_t .\] The bound of a continuous motility solves the differential equation \[\int_{y_{t_-}}^{y_t} \frac{1}{\lambda(\xi)}d\xi = dx_t\] Again, you lot run across the crucial difference, whether the acre variable gets to deed "during" the jump. We tin dismiss write this every bit a differential, past times writing the solution to this final differential equation every bit \[y_t-y_{t_-}=f(x_t-x_{t_-};y_{t_-})\] too thus \[dy_t = \mu(y_{t_-}) dt + f(dx_t;y_{t_-})=\mu(y_{t_-}) dt + \lambda(y_{t_-})\sigma dz_t+f(dJ_t;y_{t_-})\]
So, you lot don't have to extend the model to jumps alongside the left-limit approach, too you lot don't receive got to swallow the catch that a bound has a dissimilar response than an arbitrarily unopen continuous-sample-path movement. The final equation shows you lot how to modify the model to include jumps inwards a agency that preserves the belongings that the bound has the same outcome every bit its continuous limit.
The point
Why a spider web log postal service on this? I asked a few continuous-time gurus, too none of them had seen this number before. If somebody knows where this has all been worked out alongside proper is dotted too ts crossed, I would similar to know too shout out it properly. (I would mean value the literature on alternative pricing alongside jumps had done it, but I couldn't bring out a reference.) Or perchance it hasn't been done too somebody wants to arrive at it. I'm non proficient plenty at the technical aspects of continuous fourth dimension to write this alongside the right precision too generality.
And it's a cool fox that may live on useful to somebody exterior of the narrow context that nosotros had for it.
Update:
Perhaps the right application is stock prices too alternative prices. When stock prices jump, somebody must receive got studied the instance that alternative prices deed past times the same amount the Black-Scholes formula gives for the same size stock toll movement. Does anyone receive got a citation to that case?
Suppose i fourth dimension serial \(x\), which follows a diffusion, drives some other \(y\). In the simplest example, \[dx_t = \sigma dz_t \] \[ dy_t = y_t dx_t. \] In our example, the instant equation describes how habits \(y\) respond to consumption \(x\). The same sort of construction mightiness depict how invested wealth \(y\) responds to property prices \(x\), or how alternative prices \(y\) respond to stock prices \(x\).
Now, suppose nosotros desire to extend the model to handgrip jumps inwards \(x\), \[dx_t = \sigma dz_t + dJ_t.\] What arrive at nosotros arrive at almost the instant equation? \(y_t\) forthwith tin dismiss bound too. On the right manus side of the instant equation, should nosotros role the left limit, the right limit, or something inwards between?
The park answer is to role 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 \(y_{t_-}=1\), too \(dJ_t=1\). Then \(y_t\) jumps to \(y_t=2\). But suppose at that spot are 2 jumps of size 1/2, i at fourth dimension \(t\) too i at fourth dimension \(t+\varepsilon\). Now \(y\) jumps upwardly to 1.5 later the showtime jump, too thus jumps some other \(1.5 \times 0.5 = 0.75\), ending upwardly at \(y_{t+\varepsilon} =2.25\). Two one-half jumps receive got a dissimilar response than i total jump.
Suppose instead nosotros extend the master copy model to jumps past times taking the bound bound of a continuous process. Imagine that nosotros bring out realizations of \(\{dz_t\}\) that acquire closer too closer to a bound inwards \(dx_t\), too let's bring out what happens to \(y_t\). The full general solution to the showtime laid upwardly of equations is \[ y_{t+\Delta} = y_t e^{(x_{t+\Delta}-x_t - \frac{1}{2}\sigma^2\Delta)}\] so, inwards the bound \(\Delta \rightarrow 0\) that \(x_t\) takes a bound of size \(dJ_t\), the jump-limit of a continuous motility 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 bound that was dissimilar from the response to a continuous procedure arbitrarily simply about a jump. For example, the left-limit approach tin dismiss arrive at a negative \(y_t\), but this method, similar the diffusion process, cannot autumn below zero. This method also produces a response to 2 one-half jumps that is the same every bit the response to a total jump.
As you lot tin dismiss see, the departure is whether the acre variable \(y_t\) gets to modify 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 deed piece the bound inwards \(x_t\) is moving.
Influenza A virus subtype H5N1 nonlinear business office of a bound is a trivial novel, but there's null incorrect alongside it, too it exists inwards the continuous fourth dimension literature. We don't run across it that often, because when you're alone studying i serial it's easier to simply modify the distribution of the bound procedure instead. This inquiry occurs when you lot tin dismiss run across both serial x too y too you lot desire to model the human relationship betwixt them.
Which is right?
Which extension to jumps is correct? Both are mathematically correct. There is null incorrect alongside writing downwards a model inwards which the response to a bound is dissimilar from the response to continuous movements arbitrarily simply about jumps. The answer depends on the economical situation.
For example, consider models alongside bankruptcy constraints. Agents who tin dismiss continuously arrange their investments may e'er avoid bankruptcy inwards a diffusion setting. If nosotros extend such a model to jumps alongside the continuous bound approach, implicitly preserving the investor's powerfulness to merchandise every bit fast every bit property prices modify fifty-fifty inwards the bound limit, nosotros volition save bankruptcy avoidance inwards human face upwardly of a bound inwards prices. However, if nosotros model portfolio adjustment to jumps alongside the left-limit generalization, agents may live on forced inwards to bankruptcy for toll jumps.
Sometimes, i introduces jumps just to model a province of affairs inwards which prices tin dismiss deed faster than agents tin dismiss arrange their portfolios, thus agents may live on forced to bankruptcy. Then the left-limit generalization is correct. But if i wants to extend a model to jumps for other reasons, piece avoiding bankruptcy, negative consumption, negative marginal utility (consumption below null or below habits), violations of budget constraints, feasibility conditions, borrowing constraints, too thus forth, thus i should pick out a generalization inwards which the bound gives the same outcome every bit the continuous limit.
Similarly, when extending alternative pricing models to jumps, i may desire to model the bound inwards such a agency that investors cannot arrange portfolios fast enough. Then the left-limit extension is appropriate, too investors must concur the bound risk. But i may wishing to accommodate jumps inwards property prices to improve jibe property toll dynamics piece maintaining investor's powerfulness to dynamically hedge. Then the nonlinear extension is appropriate, maintaining the equivalence betwixt jumps too the limiting diffusion.
A trivial to a greater extent than full general treatment
Influenza A virus subtype H5N1 trivial to a greater extent than generally, suppose \[ dx_t = g dt + \sigma dz_t \] \[dy_t = \mu(y_t) dt + \lambda(y_t)dx_t.\] We desire to add together \(dJ_t\) to the showtime equation. The left-limit approach is \[dy_t = \mu(y_{t_-}) dt + \lambda(y_{t_-})dx_t \] If at that spot is a bound \(dJ_t\), \(y\) moves past times an amount \[\frac{1}{\lambda(y_{t_-})}dy_t \equiv \frac{1}{\lambda(y_{t_-})}(y_t - y_{t_-}) = dx_t .\] The bound of a continuous motility solves the differential equation \[\int_{y_{t_-}}^{y_t} \frac{1}{\lambda(\xi)}d\xi = dx_t\] Again, you lot run across the crucial difference, whether the acre variable gets to deed "during" the jump. We tin dismiss write this every bit a differential, past times writing the solution to this final differential equation every bit \[y_t-y_{t_-}=f(x_t-x_{t_-};y_{t_-})\] too thus \[dy_t = \mu(y_{t_-}) dt + f(dx_t;y_{t_-})=\mu(y_{t_-}) dt + \lambda(y_{t_-})\sigma dz_t+f(dJ_t;y_{t_-})\]
So, you lot don't have to extend the model to jumps alongside the left-limit approach, too you lot don't receive got to swallow the catch that a bound has a dissimilar response than an arbitrarily unopen continuous-sample-path movement. The final equation shows you lot how to modify the model to include jumps inwards a agency that preserves the belongings that the bound has the same outcome every bit its continuous limit.
The point
Why a spider web log postal service on this? I asked a few continuous-time gurus, too none of them had seen this number before. If somebody knows where this has all been worked out alongside proper is dotted too ts crossed, I would similar to know too shout out it properly. (I would mean value the literature on alternative pricing alongside jumps had done it, but I couldn't bring out a reference.) Or perchance it hasn't been done too somebody wants to arrive at it. I'm non proficient plenty at the technical aspects of continuous fourth dimension to write this alongside the right precision too generality.
And it's a cool fox that may live on useful to somebody exterior of the narrow context that nosotros had for it.
Update:
Perhaps the right application is stock prices too alternative prices. When stock prices jump, somebody must receive got studied the instance that alternative prices deed past times the same amount the Black-Scholes formula gives for the same size stock toll movement. Does anyone receive got a citation to that case?
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