Przemyslaw Repetowicz
Department of Physics
Trinity College Dublin 2
Ireland

Financial mathematics with stable fluctuations

Abstract: The objective in financial mathematics consists in computing
distributions of derivative instruments, like options or bonds,
for a predefined probabilistic model of the underlying asset ( stock price
or rate of interest). This is used in investment decisions for hedging
risks.

The traditional mathematical approach [1] in this context bases
on the assumption the fluctuations of asset prices are Gaussian
distributed.
However, empirical facts have invalidated this assumption [2].
It appears that returns of asset prices are much better modelled as
Levy stable random variables.

Due to the lack of mathematical tools for computing distributions of
functions of stable random variables little has been done so far to
compute prices of options or bonds on stocks driven by stable
fluctuations, and to compare them to real data.
The existing methods [3,4,5] are based on abstract theorems that state
that every stochastic process is a martingale under some new probability
measure. As physicists we have a conviction that probabilities of
events in the market are settled and have to be measured rather than changed.
This conviction is confirmed in experiment; stock prices are not
martingales. The martingale approach is rewarding intellectualy but the
results do not fit the data [6].

We have derived a novel mathematical approach [7] to computing
distributions of functions of stable-variables. This gives us the means to
properly deal with Levy-stable fluctuations on the stock market in
insurance and in other areas of life. In the talk I will describe our new results
related to two models, the Heath-Jarrow-Morton model
and the Black and Scholes model with stable fluctuations.

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[1] Musiela M, Rutkowski M,
Applications of Mathematics: Stochastic Modelling and applied probability
Springer Berlin (1997)

[2] Gopikrishnan P et al, Sclaing of distributions of price fluctuations
of individual companies,
Phys. Rev. E 60 (1999) 6519--6529

[3] Schoutens W,
Levy Processes in Finance, Pricing financial derivatives,
Wiley & Sons Ltd

[4] Cont R, Tankov P,
Financial Modeling with Jump Processes,
Chapman and Hall (2004)

[5] Delbaen F and Schachermayer W,
A general version of the fundamental theorem of asset pricing,
Math. Ann. 300, 463--520 (1994)

[6] Hurst S R and Platen E,
Option pricing for a logstable asset price model,
Mathematical and Computer Modelling 29 (1999) 105--119

[7] Repetowicz P, Lucey B and Richmond P,
Modeling the term structure of interest rates a la Heath-Jarrow-Morton but
with non-Gaussian fluctuations,
preprint uk.arxiv.org/ps/cond-mat/0408292

[8] Repetowicz P and Richmond P,
Pricing options on stocks driven by multi-dimensional operator stable
fluctuations
in preparation