The
random behavior of the economic and political worlds governs the financial
world. Being able to measure and estimate financial risks is thus
very important. This ability interests traders, but also researchers,
in particular a team at the CNRS "Laboratoire de probabilités
et modèles aléatoires" (Laboratory of probabilities
and random models), who study quantitative methods to minimize risks,
especially those relating to options.
Holders
of options can buy or sell assets at a pre-determined time. At the
time of sale, two issues must be addressed:
1-the option price
2-the strategy to minimize risks in light of the random evolution
of the financial market.
Classical
theory suggests making a hedging portfolio that perfectly models the
behavior of the option. This effectively eliminates the risk associated
with issuing the option, and the price is fixed by arbitrage. With
the Black-Scholes model (see below), options no longer exist because
they become the exact equivalent of their hedging portfolio.
The
realistic models are based on the random character of volatility and
variations in assets. Moreover, the models consider the imperfections
of stock markets. A perfect portfolio does not exist. There is also
a residual risk which represents the difference between the option
value and the portfolio value. This difference cannot be eliminated.
Researchers
in financial mathematics have developed the idea of minimizing the
residual risk. The optimal portfolio strategy depends on a chosen
assumption by the option issuer. The first methods consisted of minimizing
the variance of the residual risk. New surroundings define the risk
measurement as a loss probability, called Value at Risk (VaR), which
minimizes the probability that the residual risk is above a fixed
limit. Mathematical research must offer the best understanding and
the best control of risks in an increasingly complex financial world.
The Black-Scholes model for the stock price St is as follows:
dSt = (µ dt + 
dWt)
St,
where the parameters µ and
are the so-called stock drift and volatility, and Wt is a Brownian
motion.
