# Separating jumps and diffusion

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?
• If the price of an asset is different in two markets is this not arbitrage? Feb 1 at 13:47
• Yes, but there are several factors that make exploiting this arbitrage hard. It is an electricity market, and electricity storage comes at a cost. Market #2 is a congestion market, and may only take place in certain locations. Lastly, market #2 is relatively new, so some arbitrage opportunities may be left, which is what I am exploring. Feb 1 at 15:29
• IMO, from a technical point of view, yes, it should be no problem. As @KurtG. pointed out, think a bit about no-arbitrage. You might also want to model a premium for jumps into the process (i.e. a drifte equal to (-1) times the jump average) or some such. Feb 2 at 20:37

More some thoughts than a definite answer.

• Even if you want to allow some form of price divergence between the two markets (because arbitrage oportunities are hard to monetise) it is probabily unlikely that the prices in the two markets are totally uncorrelated.

• If you start with a Merton 76 jump-diffusion model for each market: \begin{align} S^1_t=S^1_0\,e^{rt+\sigma_1W^1_t-\frac{\sigma_1^2\,t}{2}}\prod_{i=1}^{N^1_t}(1+U^1_t)\,e^{-\lambda^1t}\\ S^2_t=S^2_0\,e^{rt+\sigma_2W^2_t-\frac{\sigma_2^2\,t}{2}}\prod_{i=1}^{N^2_t}(1+U^2_t)\,e^{-\lambda^2t}\\ \end{align} you will either have to use identical Poisson processes $$N^1_t,N^2_t\,,$$ or they will be totally uncorrelated and uncorrelated with both Wiener processes.

• This is a similar limitation that reduced form credit risk models are facing where the default of different names is modelled with exponentially distributed default times.

• Credit modellers have found various ways to overcome this unrealistic lack of correlation between defaults of different names. You should google for stuff like copulas, or contagion. I think there is even a way to use three Poisson processes that are cleverly put togehter to have two correlated jump diffusions.

It depends in the end what exactly you want to do with that model. If you are the expert on those market mechanisms you could even invent your own.

• Thank you, these are some interesting thoughts. I appreciate your answer and thoughts on the matter. Feb 3 at 10:02