  We consider the problem of pricing discretely monitored Asian options over
$T$ monitoring points where the underlying asset is modeled by a geometric
Brownian motion. We provide two quantum algorithms with complexity
poly-logarithmic in $T$ and polynomial in $1/\epsilon$, where $\epsilon$ is the
additive approximation error. Our algorithms are obtained respectively by using
an $O(\log T)$-qubit semi-digital quantum encoding of the Brownian motion that
allows for exponentiation of the stochastic process and by analyzing classical
Monte Carlo algorithms inspired by the semi-digital encodings. The best quantum
algorithm obtained using this approach has complexity
$\widetilde{O}(1/\epsilon^{3})$ where the $\widetilde{O}$ suppresses factors
poly-logarithmic in $T$ and $1/\epsilon$. The methods proposed in this work
generalize to pricing options where the underlying asset price is modeled by a
smooth function of a sub-Gaussian process and the payoff is dependent on the
weighted time-average of the underlying asset price.
