The Black-Scholes operator as the generator of a \(C_0\)-semigroup and applications (Q2910782)

In this article, the authors consider the Black-Scholes equation as a Cauchy problem. In particular, the \(C_0\)-semigroup generated by the Black-Scholes operator is established.NEWLINENEWLINELet \((V_t f)(x) = f(xe^t)\) for all \(f \in S_0 \) (Schwartz space on \(C^\infty(0,\infty))\), \(x \geq 0\) and \(t \in \mathbb R\). Then it is shown that \(\left\|V_t \right\| = 1\) for all \(t\) and \( \left\{V_t\right\}_{t\in\mathbb R}\) is a \(C_0\)-group on \(S_0\) with generator \(Bf := xDf\), \(D(B) := S_0\).NEWLINENEWLINEThe article finishes by applying these results to vanilla and Asian options.