# How to model asset prices for a very short time period

Geometric Brownian motion is the most common model for asset price evolution. Is it still viable for modeling asset prices in a very short time period? For example, I have time series of length 3600 which is as asset price for each second in an hour, is GBM adequate for modeling this random process?

If not, what model should I use instead?

My primary interest is in exchange rates.

• Which type of assets? Stocks, exchange rates, interest rates...? Please add those details to your question. Jul 8 '20 at 12:00
• This seems like a high-frequency data topic. I am unsure whether modelling is different in those circumstances, it's not my area. On top of my head, for a single hour time-span, even if FX rates are positive, you could allow yourself to use a normal distribution (instead of the lognormal GBM), given under such short duration no large moves should be expected. However it would be best if someone familiar with market microstructure and high-frequency data was able to provide an asnwer. Jul 8 '20 at 15:13
• Hmm... I can accept that longer horizons normalise as some kind of Central Limit Theorem effect... but I struggle to think (for say any FX dollar cross) what model (ie singular) could possibly hold for your samples that include an FOMC meeting or payrolls as well as those that don't. Unless you fancy building a "weird stuff happens at 1330 GMT on the first Friday of every month" model, but go down that path and it's then a short path to sunspots, full moons, and retrograde-Mercury ;-) Feb 23 at 10:24

Is GBM still viable for modeling asset prices in a very short time period ? I assume by asset price you mean the mid-price of the asset. Ito diffusions are unable to capture stylized facts of market microstructure: in particular, you can't get volatility clustering, negligible autocorrelation in return series, the signature plots or the Epps effect. We can also note that the GBM is Markovian, which is not coherent with empirical observations at this scale. For a brief review, you can have a look at Gould et al. (2013). For a longer introduction to market microstructure I recommend the excellent work in Abergel et al. (2012) or the insightful list of recommendations in the answer to this question.

If not, what model should I use instead? First, It depends what you want to use your model for, on which time scale, how liquid the specific asset you consider is. For example if your objective is to interact with the LOB by placing orders (to acquire or liquidate a position, for market making,...), you will also need to either model a market impact function or model the first levels of the LOB. Eventually, it depends on what data the modeler has: is the modeler an academic with only market data (the public data streams), or is the modeler a hedge fund with access to alternative data trying to build an investment strategy ?

In the academic litterature, one way to model midprices at the high frequency level is through point processes, and Hawkes processes in particular: please have a look at Bacry et al. (2015) for an extensive review.

Pure Jump models

One way is to consider a bi-dimensional linear Hawkes process where $$N^1$$ models the upward jumps of the midprice and $$N^2$$ models the downward jumps. Obviously the midprice then writes $$p_t=p_0+N^1_t-N^2_t$$. The authors in Bacry et al. (2013a) force the mean reversion by imposing that the kernels $$\phi_{11}$$ and $$\phi_{22}$$ are null. They get a closed form equation for the quadratic variation estimator of this mid-price and show its re-scaled version converges to a Brownian motion. In the same sort of idea, they show in Bacry et al. (2013b), that this model verifies the lead-lag effect. In the groundbeaking contribution of Jaisson et al. (2015), the authors consider a sequence of stable linear MHPs that converges to an unstable MHP, and show it converges to a CIR model.

Jump diffusion models

Jump-diffusion models (also refered to as Ito-Levy diffusions) are sometime used. Poisson jumps were the first introduced for their tractability, soon joined by linear Hawkes processes: you can have a look at the MIH model in Alfonsi and Blanc (2016) where the authors consider an external agent interacting with this market by placing market orders with his own linear price impact function. It is well known that the market is better modeled by slowly decaying kernels (notably in the work of Bacry and Muzy), nonetheless the vast majority of applications in the litterature rely on exponential kernels, because there Markovian nature also allows to derive easier solutions to control problems.

No, you can't use GBM for this. The underlying assumptions of the GBM/Blackschooles framework is that logreturns are normally distributed. This holds true for daily returns but it can't be assumed for shorter time periods. One explanation I've seen for this (I think in Paul Wilmott Introduces Quantitative Finance) is that the daily returns are the cumulative returns of infinitesimally small time period returns throughout the entire day, and so by the central limit theorem the daily returns become normally distributed (regardless of the distribution of the infinitesimally small time period returns). So we have no reason to believe or assume that the return series for smaller time periods, like minutes or seconds, are normally distributed and hence can't rely on the Black-Schooles models.

I'm not familiar with the topic but I believe what you need to look up is Market Microstructure effects. Good luck.