Selected applications of differential equations in Vanilla Options valuation

  In financial models one of the basic assumptions about investors is that they want to gain as much as it is possible but they have aversion taking the risk. Each investing strategy can be considered as a compromise between willing of profit and fear of losses - usually possible profit increase with the probability of loss. An option can be considered as some kind of insurance - a more prudent speculator might want to reduce the maximal loss by a quantity K>0. He thus will buy an option which would correspond to the strike price K. For him (option holder) it is the way to protect himself against the risk, for option issuer, it is the possibility to profit by selling this financial product. The fundamental question is what is the value of this security? The answer has essential meaning in the financial world and the global economy. The first record of an option contract can be found in the "Politics" of Aristotle. According to the story, the Greek philosopher Thales profited by option-type agreement around the 6th century B.C. The problem of fair valuing this kind of financial instrument was not formalized until 1900. At this year L.Bachelier by his pioneering thesis began the theory of option pricing. In the same work he initiated the study of diffusion processes five years before recognized as the groundbreaking works of A.Einstein, M.Smoluchowski and decades before famous works of K.Itô, P. Lévy and N.Wiener. Bachelier as the first developed the theory of Brownian motion and found practical application of this concept in financial engineering. The culminating event in developing theory of option pricing was 1973 when Black, Scholes and Merton found consistent formulas for the fair prices of European options. The discovery was of such great importance that the autors were awarded the Nobel Prize for Economics in 1997. Very interesting is fact that for short time to maturity formulas of Bachelier are very close to results of Black, Merton and Scholes. Until today, there is no knowledge of any analytical formula of American option fair price, which could have any practical application. In order to determine this value, as a rule it is given main importance to Monte Carlo methods. Usually they are easier to implement, but require more time or are related to higher numerical errors than deterministic methods.