Mathematical Pathologies in Derivatives
Modelling
The following information is derived from the introduction of
Dr. William Shaw's new book, Modelling Financial Derivatives with
Mathematica. Dr. Shaw is head of Financial Instrument
Modelling in the Quantitative Analysis group of Nomura International
PLC, the wholly owned European subsidiary of The Nomura Securities
Co. Ltd of Japan, one of the world's largest investment banks.
In the text below, Dr. Shaw characterizes and provides some simple
examples of the hazards presented by the ill-considered use of
mathematical models--even some standard textbook models--governing the
trading of financial derivatives. The hazards generally fall into one
of two categories: flawed computer implementation of algorithms, or
flaws within the algorithms themselves.
Dr. Shaw also describes how these hazards can be overcome using the
symbolic algebra capabilities of Mathematica, the same
sophisticated technical computing system used by scientists and
researchers worldwide to perform higher mathematics.
Text from Modelling Financial Derivatives
with Mathematica
Financial analysts use often-complex mathematical models to guide
their
decisions when trading derivative financial instruments. However,
derivative securities are capable of exhibiting some diverse forms of
mathematical pathology that confound our intuition and play havoc with
standard or even state-of-the-art algorithms. The potential traps fall
into two categories. The first category contains problems arising from
the complexity of some models, leading to their being seriously
error prone in their implementation, even if not intrinsically flawed. The
second category contains algorithms that are intrinsically flawed.
A Look at Some Problems in Each Category
An obvious example of a type-one problem relates to the computation of
hedge parameters, or "Greeks." These are the partial derivatives of the
option value with respect to the underlying price and other variables
such as time and interest rates. For all but the simplest vanilla
options, the pen-and-paper computation of such entities is very complex
and therefore error prone, leading to the potential of errors in coding.
The estimation of such quantities by purely numerical methods
(differencing) leads to other types of problems associated with
inaccuracies in the estimate of the analytical derivative. Such
difficulties can be eliminated in one swoop with a system such as
Mathematica, which is able to compute the symbolic derivatives--and
hence the hedge parameters--exactly by analytical differentiation of
the option-pricing formula.
A more subtle type-one difficulty relates to the computation of
implied
volatility, which is a favorite parameter of traders. Implied
volatility makes sense only for the simplest vanilla options. In other
cases, the implied volatility may be unstable, double valued, or
triple valued or may even possess infinitely many values. The implementation
must check that the price is a strictly increasing or a strictly
decreasing function of volatility; otherwise, nonsense can and will be
obtained for the implied volatility. In Mathematica the graphical
tools can be used to test this very quickly.
Some quite well known algorithms are intrinsically flawed. Problems
which we might identify as a type-two issue can be found in the following models.
- Binomial models
- Implicit finite-difference models
- Monte Carlo simulation models
These are essentially numerical methods, and the book looks
in detail at them in comparison with exact solutions for known cases.
This is straightforward in a system such as Mathematica, where
complex, exact solutions can be expressed exactly and worked out to any degree of
precision. As numerical methods, they involve an essential
discretization of time and other relevant variables such as the
underlying asset price. A common theme is what happens when the
time-step is taken to be large, which is very tempting in an
implementation in order to obtain results quickly.
For example, several of the standard binomial models suffer from the
well-known difficulty that as the time-step becomes large, the
probabilities associated with the underlying tree model may become
negative, which is manifest nonsense. In other types of models, the asset
prices can become negative. Both of these effects are well known. What
appears not to be understood is that the reason for these difficulties
has a common root in the fact that tree models are typically underspecified from
a mathematical point of view.
A number of constraints can be written
down that should apply to a tree. The solution of a full set can be
quite hard, so in practice the authors of tree models have worked with a
subset and made up one or more missing conditions in order to solve for
the tree structure. This leads to the problems with negative
probabilities or negative asset prices. When one is armed with Mathematica's
symbolic equation-solving capabilities, the solution of a full set of
tree constraints is a straightforward matter--and in fact leads to a
model where neither the up-and-down tree probabilities nor the asset
price can become negative. Other problems with trees, discovered by
others in relation to barrier-and-cap effects, are also discussed.
One of the most surprising and deeply rooted difficulties relates to the
use of implicit finite-difference schemes. In principle, these allow a
larger numerical time-step to be used than in treelike models and are
becoming increasingly popular. When properly used, they combine accuracy
with efficiency. There is, however, a major difficulty with them that
appears not to have migrated in its appreciation from the academic
numerical analysis community to the market practitioners.
When the initial conditions for the associated partial differential equation (in
financial terms, the option payoff) are nice and smooth (in loose terms, continuous
with continuous slopes), one can get away with almost
any implicit finite-difference scheme. This is emphatically not the case
in option-pricing problems, where the payoffs are typically nonsmooth
and frequently discontinuous. Such "glitches" in the payoff will
propagate through the solution, and while they do not necessarily cause
a large error in the option value, they can cause significant errors in
the Greeks such as delta, gamma, and theta. This will occur with some of
the most common schemes in current use for larger time-steps. It can
be avoided only with a certain subset of implicit schemes. Which subset
works and which does not is in fact well known to the numerical analysis
community. In the text this is made crystal clear by
comparison with some exact solutions; and the good, but infrequently
used, schemes are contrasted with the bad, but widely used, schemes.
Monte Carlo simulation is a popular method for the valuation of European-style
but path-dependent options. The manner in which simulated
solutions converge to the correct answer is investigated for some cases
where the exact solution is known. This reveals several difficulties
with such numerical simulation methods, and in particular the very slow
convergence associated with certain classes of options. We give
suggestions for control variates in a number of useful cases but
highlight the difference between getting the variance down--but possibly
converging to the wrong answer--and getting the right answer.
Other Possibilities with a Symbolic Computation System
In addition to being able to do calculus, Mathematica has other
advantages over traditional modelling environments such as spreadsheets
and C/C++. For example, the presence of a vast library of special
functions, coupled with the ability to do differentiation and
integration, means that novel, exact solutions can be implemented with
ease. A beautiful example of this is the exact solution for the Asian
option with arithmetic averaging, which requires that one invert the
Laplace transform of a hypergeometric function. This requires just a few
lines in Mathematica and can be directly differentiated to obtain the
Greeks. Other areas in which Mathematica can be fruitfully applied
include novel analytical techniques for double-barrier options and
accurate analytical approximations for American options.
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