Resources
Available Online
Introduction to the Boltzmann Equation
Probably one of the the most important mathematical tools in pure (Mathematics
and Physics) as in applied quantitative sciences (Engineering, Finance) for the
next decades. What I have done in the solution of these equations, how I see
(the stochastic reformulation) and use them in the description of uncertain
processes and what is ahead in current research.
This brief introduction and an article on the mathematical methods I have
worked out (Boltzmann Equations, Functional Analysis and Stochastic Processes)
have been published on Applied Derivatives Trading
in the May 97 issue; "Focus on Mathematical Approaches".
Boltzmann Equations, Functional Analysis and
Stochastic Processes
An introductory paper on the properties of BEs, Solution Techniques for linear
cases, applications and the use in Stochastic Processes. The article considers
continuous as well as discrete time. It deals with Probability Densities and
Vectors as modified respectively by Kernels and Probability Tensors depending
on the kind of Randomness distribution.
Solvable Cases of the
Boltzmann Equation
A brief list of the cases where the tools i worked out may be applied. The list
is, of course, not exhaustive since the methodologies of decomposition,
degenerate kernels and so on may be applied sometimes in combination.
Furthermore there is no clear path to determining all the solvable cases of
such a class of equations.
The Theory of Options Pricing
This is one of the most interesting results I have obtained: closed analytical
expressions for the price of both American and European calls and puts. Instead
of starting from the Risk-Neutral approach and aiming to Hedging I have treated
options as observables in quantum mechanics. The abstract explains what was
wrong for me in the previous theory and what I have introduced.
Successful Trading of Digital Options
A work I have decided to render public to show the gross limits of actual
option theories. I strongly disagree on both the assumptions that 1) options
prices should be supposed independent of expectations on underlying stocks and
2) it is necessary to hedge the position in options with positions on the
underlyings. Here is shown why; along with a profitable trading strategy.
Published on Applied Derivatives Trading in the May 97
issue; "Focus on Mathematical Approaches".
Articles & Publications
A list of my articles and works along with abstracts. Ranging from Applied
Mathematics to the direct applications of stochastic processes to derivatives.
Most of them can be direclty viewed online from your browser or downloaded in
PostScript, Word or even PDF format. Unfortunately, not everything may be
rendered public here because matter of advanced investment.
The Functional Equation Headquarter
A brief explanation of what are functional equations; how they come out from
the theory of operators and appear in complex problems of (see STDO for
example) quantitative sciences. With this approach, a lot of cases can be
inspected and solved. There is also a form to fill to request my advice on
special cases.
The Random-Walk
Till now, this process had to be described using PDEs and accepting some
approximations. Among these, there was the need for a superflual boundary
condition caused by the singularities in the derivation operator. I have redone
the path without introducing the approximations for a continuous-step random-walk
in both discrete and continuous-time. The result is an IDE. This has
substantial consequences in many fields of applications.
The Theory of Human Behavior
Just another application of stochastic concepts to the real world. I show a
process where human beings interact with themselves, exchange ideas and mature
decisions. Non-linear equations arise even in the simplest example. The model
can be easily adapted to forecast the decision of consumers facing the choice
of a product; the behavior of electors toward a coming two-choice or
multi-choice decision or even the attitude of investors to take buy/sell
decision under influencing circumstances. The adaptation to the
continuous-state case (ie choosing a number in a continuous set) requires a
much deeper functional calculus but can be surely written. Solving analytically
may however be very hard.
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