$dS=μSdt+σSdB$
$P(S,T)=[(1/n)∑S(t\{i\})-K]⁺$
is the asian option payoff. Which is also clearly pathwise continuous. How can i mathematically show that it is continuous?
$dS=μSdt+σSdB$
$P(S,T)=[(1/n)∑S(t\{i\})-K]⁺$
is the asian option payoff. Which is also clearly pathwise continuous. How can i mathematically show that it is continuous?
The question does not make much sense. You can prove however that $T \mapsto \left( \frac{1}{T-t}\int_t^T S_u du - K\right)_+$ is pathwise continuous, which is just a composition of continuous functions, if you work on the set where $t \mapsto S_t(\omega)$ is continuous.