Taming Risks: Financial Models and Numerics
by Jiri Hoogland and Dimitri Neumann
The increasing complexity of the financial world and the speed
at which markets respond to world-events requires both good models
for the dynamics of the financial markets as well as proper means
to use these models at the high speed required in present-day
trading and risk-management. Research at CWI focuses on the development
of models for high-frequency data and applications in option-pricing,
and tools to allow fast evaluation of complex simulations required
for option-pricing and risk-management.
The modeling of equity price movements already started in 1900
with the work of Bachelier, who modeled asset prices as Brownian
motion. The seminal papers by Merton, Black, and Scholes, in which
they derived option prices on assets, modeled as geometric Brownian
motions, spurred the enormous growth of the financial industry
with a wide variety of (very) complex financial instruments, such
as options and swaps. These derivatives can be used to fine-tune
the balance between risk and profit in portfolios. Wrong use of
them may lead to large losses. This is where risk-management comes
in. It quantifies potentially hazardous positions in outstanding
contracts over some time-horizon.
Option pricing requires complex mathematics. It is of utmost importance
to try to simplify and clarify the fundamental concepts and mathematics
required as this may eventually lead to simpler, less error-prone,
and faster computations. We have derived a new formulation of
the option-pricing theory of Merton, Black, and Scholes, which
leads to simpler formulae and potentially better numerical algorithms.
Brownian motion is widely used to model asset-prices. High-frequency
data clearly shows a deviation from Brownian motion, especially
in the tails of the distributions. Large price-jumps occur in
practice more often than in a Brownian motion world. Thus also
big losses occur more frequently. It is therefore important to
take this into account by more accurate modeling of the asset-price
movements. This leads to power-laws, Levy-distributions, etc.
Apart from options on financial instruments like stocks, there
exist options on physical objects. Examples are options to buy
real estate, options to exploit an oil-well within a certain period
of time, or options to buy electricity. Like ordinary options,
these options should have a price. However, the writer of such
an option (the one who receives the money) usually cannot hedge
his risk sufficiently. The market is incomplete, in contrast with
the assumptions in the Black-Scholes model. In order to attach
a price to such an option, it is necessary to quantify the residual
risk to the writer. Both parties can then negotiate how much money
should be paid to compensate for this risk. We explore ways to
partially hedge in incomplete markets.
A relatively new phenomenon in the financial market has been the
introduction of credit risk derivatives. These are instruments
which can be used to hedge against the risk of default of a debitor.
It is obvious that this kind of risk requires a different modeling
approach. The effect of default of a firm is a sudden jump in
the value of the firm and its liabilities, and should be described
by a jump process (for example, a Poisson-process). In practice,
it is difficult to estimate the chance of default of some firm,
given the information which is available. For larger firms, credit-worthiness
is assessed by rating agencies like Standard and Poors. We are
looking at methods to estimate and model the default risk of groups
of smaller firms, using limited information.
The mathematics underlying financial derivatives has become quite
formidable. Sometimes prices and related properties of options
can be computed using analytical techniques, often one has to
rely on numerical schemes to find approximations. This has to
be done very fast. The efficient evaluation of option prices,
greeks, and portfolio risk-management is very important.
Many options depend on the prices of different assets. Often they
allow the owner of the option to exercise the option at any moment
up to the maturity of the (so-called) American-style option. The
computation of prices of these options is very difficult. Analytically
it seems to be impossible. Also numerically they are a tough nut
to crack. For more than three underlying assets it becomes very
hard to use tree or PDE methods. In that case Monte Carlo methods
may provide a solution. The catch is that this is not done easily
for American-style options. We are constructing methods which
indirectly estimate American-style option prices on multiple assets
using Monte Carlo techniques.
Monte Carlo methods are very versatile as their performance is
independent of the number of underlying dynamic variables. They
can be compared to gambling with dice in a casino many, many times,
hence the name. Even if the number of assets becomes large, the
amount of time required to compute the price stays approximately
the same. Still the financial industry demands more speedy solutions,
ie faster simulation methods. A potential candidate is the so-called
Quasi-Monte Carlo method. The name stems from the fact that one
gambles with hindsight (prepared dice), hence the ‘Quasi’. It
promises a much faster computation of the option-price. The problems
one has to tackle are the generation of the required quasi-random
variates (the dice) and the computation of the numerical error
made. We try to find methods to devise optimal quasi-random number
generators. Furthermore we look for simple rules-of-thumb which
allow for the proper use of Quasi-Monte Carlo methods.
For more information see
http://dbs.cwi.nl:8080/cwwwi/owa/cwwwi.print_themes?ID=15
Please contact:
Jiri Hoogland or Dimitri Neumann - CWI
Tel: +31 20 5924102
E-mail: {jiri, neumann}@cwi.nl
