Asymptotic methods in option pricing

The financial industry, but also accounting and regulatory bodies, largely uses overly simplified stochastic models; often, the 40 year old Black-Scholes model. The financial crisis has shown rather dramatically that this practice must not be continued, and that one of the main goals of academic research should be to enable practitioners to use more realistic models. The more involved a model is, the more likely it is that the prices etc. it predicts can be profitably analysed asymptotically. Asymptotic analysis gives computational and qualitative access to many otherwise untractable problems, and simplifies problems whose solution is computationally challenging. Fast calibration of advanced models to market data is a key issue for practitioners. We will apply advanced analytic techniques to this problem, with an innovative focus on American options. This has no geographic meaning, but denotes options that can be exercised by their holder at any time (and not just at maturity). While the bulk of academic literature deals with European options, options on single stocks are almost always of American type. Another area of research concerns Models based on fractional Brownian motion, which have become very popular in recent years. They acknowledge the fact that stock prices do not evolve independently from their past. We want to apply asymptotics methods to help apply these models in practice.

In this project, we mainly studied problems arising in mathematical finance. On of the central outcomes is an analysis of a new asset model, which has some practically appealing features, such as dependency modelling of past and future price movements. Such models are used for option pricing, and we developed an approximation formula for options. Currently, banks are interested in a new generation of asset price models, which are better fitted to actual price movements than the classical models. For the analysis of our model, several mathematical tools were used and modified. Such approximation formulas have several applications. Any model depends on certain numbers, the parameters, which are a priori unknown. First, a bank using the model must fit these parameters to market data. An approximation formula can speed up this process. Second, market participants are interested in qualitative statements, which give transparent connections of the option prices which the model yields and the model parameters. The results of our study were published in a good mathematics journal. Further, a new variant of a classical theorem from probability, the so-called law of the iterated logarithm, was found. The new theorem belongs to the theory of large deviations, which studies questions of the following kind: What is the probability of obtaining head only in every tenth flip, or less, when flipping a coin a lot of times (instead of half the time)? In our project, we studied the small probability that a scaled stochastic process shows an unusual behavior over short time. This work belongs to theoretical probability theory. Further research results were made in transport theory. This name comes from technical applications; from the point of view of mathematical finance, questions of the following kind are dealt with: Suppose we know the probability distribution of an asset price at two or more future time points. As a source for this, prices of options traded on the market are used. How can information about probabilities at other time points be obtained? Which conditions must they satisfy? Here, several theoretical questions have been answered, which might have an influence on the further development of financial applications of transport theory.