Jump diffusion models can be useful in assessing how catastrophic risk should be taken into account in financial decisions such as – for example – determination of the dividend distribution policy of a firm or when to make an irreversible investment. This subject is studied in a doctoral thesis written by Teppo A. Rakkolainen, Essays on Optimal Control of Spectrally Negative Lévy Diffusions in Financial Applications, which shall be examined at Turku School of Economics on Friday, 6 February 2009. The doctoral study belongs to the field of quantitative methods in management.
In the traditional diffusion models of mathematical finance, which are based on normal distribution, the value of the underlying asset changes continuously in relation to time. In Teppo Rakkolainen’s doctoral study, models based on jump diffusions – or Lévy diffusions – are investigated. In such models, the value of the underlying asset can change relative to time not only continuously but also in a discontinuous manner, i.e. by jumping instantaneously to a new level.
The research on Lévy diffusions can be applied in, for example, examining the pricing of real options or a firm’s optimal dividend distribution policies. From the mathematical point of view, the purpose is to stipulate steering policies for stochastic (random) processes which will lead to the best possible result.
A reply to quantitative critique
Particularly from the perspective of risk management, jump diffusion models are in many ways superior to traditional models used in mathematical finance. Contrary to the models based on normal distribution, distributions which are skewed and ”heavy-tailed” can be generated by means of jump diffusion models. In these kinds of distributions, the probabilities of extreme events are greater than in normal distribution situations.
”Empirical observations indicate that the distributions of actual financial quantities are typically skewed and heavy-tailed. From the perspective of risk management, extreme events and tails of distributions are interesting – in fact, of paramount importance. With jump diffusion models, it is also possible to respond partly to the critique presented in connection with the financial crisis, in which the serviceability of quantitative methods in finance has been questioned,” Mr Rakkolainen says.
Taking the risk of catastrophe into account
The models examined in this study are asymmetrical in the sense that the jumps are presumed to occur in only one direction, lowering the underlying value. Asymmetry can be explained by means of empirical observations, according to which markets react to negative signals more strongly than to positive ones: the decline of rates in a stock market crash is generally sharper than the rise in a stock market boom. In applications connected with risk management, asymmetry can be explained by reference to the principle of prudence: preparations are made for possible but unlikely losses, but no allowance is made for similarly unlikely profits.
The model class used enables the modelling of a catastrophic risk, i.e. one that is unlikely but disastrous if realized.
“In the light of the financial crisis, modelling based on jump diffusions would appear to have some uses. By such modelling, investment risks can be assessed more adequately. A typical question could be: ‘If it is possible that the value of my investment suddenly drops 80 per cent, how this possibility should be reflected in my investment decision?’” Mr Rakkolainen illustrates.
Contact information:
Teppo A. Rakkolainen
+358 2 481 4340, teppo.rakkolainen(a)tse.fi
The doctoral thesis is available online: http://info.tse.fi/julkaisut/vk/Ae1_2009.pdf