A Probabilistic Look at Options and at the Underlying Randomness

Abstract

Consider an option based on a financial asset whose price fluctuation is modelled by a stochastic process (St) over some probability space (Ω, F, P). Let H denote the resulting pay-off; for a call option with exercise price c and terminal time T we would have H = (S t - c)+. Instead of computing the fair price of such an option as the expected value E[H] with respect to P, possibly modified by a risk premium, the Black-Scholes formula computes the price as the expected value E* [H] with respect to a new measure P* which turns (S t ) into a martingale. We review the basic argument, both froma probabilistic and an economic point of view. It involves the Itô representation of H as a stochastic integral of the price process and the interpretation of the integrand as a dynamical hedging strategy. Such hedging strategies induce a technical demand for the financial asset. Thus the question arises: What is the impact of such strategies on the underlying price process?KeywordsRisk PremiumFinancial AssetCall OptionImage EncodeGeometric Brownian MotionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.