Semigroup theory applied to options

Abstract

Black and Scholes (1973) proved that under certain assumptions about the market place, the value of a European option, as a function of the current value of the underlying asset and time, verifies a Cauchy problem. We give new conditions for the existence and uniqueness of the value of a European option by using semigroup theory. For this, we choose a suitable space that verifies some conditions, what allows us that the operator that appears in the Cauchy problem is the infinitesimal generator of a $C_0$-semigroup $T\left(t\right)$. Then we are able to guarantee the existence and uniqueness of the value of a European option and we also achieve an explicit expression of that value.