I want to model energy prices. I have two markets, lets say market 1 and 2.
Market 1 is continuously traded, and I will assume it follows brownian motion. So the value of the asset could be defined like this on this market alone.
$$dS = \mu S dt + \sigma S dW$$
with $dW = \epsilon \sqrt{dt}$ and $\epsilon \sim \mathcal{N}(0,1)$
Now market 2 is a bit different. It is auction based, and the arrival times of these auctions are geometrically distributed. The duration of them aswell, but lets ignore them for now. So if I understand correctly this second market is a Poisson point process.
Lets also assume that the price of the asset differs from one market to another. So my idea was to model the two markets together as a jump-diffusion or spike diffusion process: The brownian motion is provided by market 1, and the occasional jumps are provided by market 2. So we could write the two markets together like:
$$dS = \mu S dt + \sigma S dW + Sd\left(\sum_{i=1}^{N(t)} (V_i -1)\right)$$
With $N(t)$ a poisson process with rate $\lambda$, as defined by the jump diffusion paper by Kou. (See: http://www.columbia.edu/~sk75/MagSci02.pdf)
The exact way the jumps are defined is not really important. What I care more about is the fact that these jumps dont occur all the time, instead they are stochastically distributed, so they appear once in a while.
However I dont want to model the 2 markets with 1 model, I would like to have separate models. My main intention for separating the models of market 1 and market 2 is because I have additional markets, and would like to have a separate process for each market in order to combine them at will.
So lets say I just want to model the price of an asset only on the second market. Could I then isolate the jump part from the diffusion part?
In that case we would have something like this I assume:
$$dS = S d\left(\sum_{i=1}^{N(t)} (V_i -1)\right)$$
If yes, then the rate of change of $S$ would be 0 without the jumps. I guess one could add some process into that such that the value of S would be mean reverting (I did that in the below picture). And then ideally I have a model that, when simulated looks like this:
Questions:
- does this make sense? Is it 'allowed' to separate the jump and diffusion parts or could this have undesired consequences?
- does anyone know of something similar in literature or another way one can use a stochastic process to model an stochastic-auction based market?