How to Avoid Bankruptcy?: Monte Carlo Simulation of Three Financial Markets, using the Multifractal Model of Asset Returns

Abstract: This paper has been an effort to apply fractal mathematics to understanding the general behaviour of financial markets. Fractals are special shapes that look similar at various scales. The specific model used is called the Multifractal Model of Asset Returns (MMAR) - the first ever model used for multifractal financial analysis. I apply the MMAR-model to analyse approximately 30 years of data for three distinct financial markets: (1) the US dollar/Norwegian krona (USD/NOK) currency market; (2) the Swedish stock market index - the OMXS30; and (3) the 12-month LIBOR-rate in British pounds. First, I outline the basic concepts behind the application of fractals to finance. The point is that (multi-)fractal models should be able to capture three stylized facts about financial markets: high-kurtosis in price changes, non-independent price movements and the tendency for highly volatile days to come in clusters. The advantage of the MMAR is that the same parameters can generate simulations that fit different timescales (months, years etc.), unlike the GARCH-family or random-walk models. Next, I applied the model to three markets and ran 10,000 Monte Carlo simulations on each market to see what the model predicts. I compared these to simulations from an ordinary random-walk model. I found that all three markets appear to exhibit multifractal characteristics. To the naked eye, the simulations look quite realistic and exhibit the behaviour outlined above. However, to my knowledge there is no statistical method to check the model's accuracy (although here we run into a philosophical issue - the characteristics of a "realistic simulation" are not well-defined). Some simulations pass the Kolmogorov-Smirnov test for equal distributions, but the number is fewer than half. The model also appears to underestimate (or "under-simulate") kurtosis if the kurtosis of the original data was very high. Nevertheless, it is still capable of generating simulations with a wide range of kurtosis values, and these are often consistent with the data. A simple simulation suggested that, under high-kurtosis, high leverage could lead to very rapid bankruptcy, even in one day. Finally, I discuss some implications of the MMAR-model, including some methodological, practical and philosophical considerations. I conclude that the model certainly looks better than the simple random-walk, and can be used for stress-testing portfolios. Despite this, I am unsure of its general utility for risk management, mainly because the model takes a long time to calculate, because it requires a lot of human judgement, and because its predictions are unclear and its interpretability uncertain. Also, it seems that the model can "break" after a high-kurtosis event actually occurs.