  We develop quantum algorithms for pricing Asian and barrier options under the
Heston model, a popular stochastic volatility model, and estimate their costs,
in terms of T-count, T-depth and number of logical qubits, on instances under
typical market conditions. These algorithms are based on combining
well-established numerical methods for stochastic differential equations and
quantum amplitude estimation technique. In particular, we empirically show
that, despite its simplicity, weak Euler method achieves the same level of
accuracy as the better-known strong Euler method in this task. Furthermore, by
eliminating the expensive procedure of preparing Gaussian states, the quantum
algorithm based on weak Euler scheme achieves drastically better efficiency
than the one based on strong Euler scheme. Our resource analysis suggests that
option pricing under stochastic volatility is a promising application of
quantum computers, and that our algorithms render the hardware requirement for
reaching practical quantum advantage in financial applications less stringent
than prior art.
