Articles & Publications
Abstract
A new approach to the Boltzmann equation yields a fairly general
corresponding solution technique which actually applies to the linear case.
I show that linear Boltzmann equations can be solved by inspecting the operatorial
properties of its members and that many new cases will be solvable as soon as some
questions posed here will be given an answer by functional analysis.
As an example, I also report, up to a quadrature, the solution of the problem of neutron
slow-down by light moderator which to my knowledge had never been solved earlier. On the
other side, by studying such equations it is natural to give a general theory of them and
have a general understanding of their meaning.
I evidence that integro-differential equations (IDEs) represent the most general
description of those phenomena in which the use of probability densities is required:
stochastic processes.
By the use of an example (many others can be easily thought) I show that the Fokker-Planck
equation, which is now considered the mathematical support of stochastic processes, can
not cope with all kind of physics and must be replaced by IDEs.
I finally show how the Boltzmann equation applies to those random phenomena (these will be
better explained in future articles) that have nothing in common with quantum physics and
is, therefore, very general.
I also illustrate how to build IDEs for uncertain processes and then eventually solve
them.
The reader realizes that Transport Theory has only been the first field to need stochastic
formulation but is far from being the last.
It is showed that "thinking stochastic" is a very powerful tool and has
immediate applications to many areas of research such as Financial Markets, Behavior Science and Game Theory.
Every time there is some uncertain process there is a Boltzmann equation governing it.
Abstract
A simple exercise shows how profitable trading of options can be easily achieved without
any hedged position and refusing the actual assumption that options prices are independent
of underlying-asset expectations.
Digital options allow to have only two events of profit and loss thus avoiding multiple
integrations in the equations.
Moreover, I give here an example of how a stochastic process can not be outlined without
the use of integral dependencies.
Accurate Pricing of American Options
Abstract
New developments in the study of random-walk lead to a
different theory of options-pricing. This is deducted by applying an integral formulation
of stochastic processes which I based on the Boltzmann equation.
It differs from the existing one (governed by the famous Fokker-Planck’s equation)
for the fact that it takes into account all orders of randomness instead of the first two.
As a result, the Black-Scholes model for options reflects this approximation giving price
estimations that can easily differ more than 10% from true ones.
I also briefly explain the difference between a discrete-time approach and a
continuos-time one by evidencing the probabilistic assumptions they imply on trading.
It is a wrong custom to actually use either one of them by merely considering
discrete-time a numerical version.
Till now, neither a discrete-time version of the Black-Scholes model could have been
worked out however since this would have required some applications of operators and functional calculus. For this reason, discrete-time
models (like the Cox-Ross-Rubinstein’s) suffer an even deeper simplification.
It is than easy to understand how these differences reflect in the earnings of major
financial institutions who daily operate on options markets and how a more accurate theory
can yield profits.
Despite apparent mathematical complexity these considerations lead to an explicit formula
for options prices which could be even implemented on pocket scientific calculators and is
given in a ready to use form.
Somewhat more complex is the derivation of this and, although it has been my aim to avoid
any unneeded formalism, many concepts from random variables apply.
Therefore, I do not focus on long and annoying demonstrations but, given the problem, I
try to always keep on the way to the solution.
Abstract
The world of derivatives is based on three main concepts: hedging, arbitrage and pricing.
Most investment banks, hedge funds and other financial institutions base their behavior in
derivatives' markets on the assumption that a position in derivatives can be replicated
and therefore balanced by a position in other equities.
As a consequence, these financial institutions pull-out derivatives prices from the prices
of equivalent positions and given an imbalance between theoretical prices and market ones
they operate an arbitrage. In such a strategy they claim to lock-in sure profits.
It is the aim of this article to redo the path from position-structuring to pricing and
show how the arguments mentioned above are absolutely false.
This is evident once the problem is treated with mathematical rigor and without the need
for any assumptions on the behavior of markets.
Moreover, I show that the impossibility of hedging even holds under more extensive
position-structuring capabilities who do not actually take place on markets.
A complete deduction is done for simple options but the method can be easily generalized
to more complex derivatives.
Some non-mathematical conclusions try to give an explanation for such a mistake and are
aimed to prevent investors from taking unneeded and uncontrollable risks.
Write me for more articles and/or publications in Italian.
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