  A derivative is a financial security whose value is a function of underlying
traded assets and market outcomes. Pricing a financial derivative involves
setting up a market model, finding a martingale (``fair game") probability
measure for the model from the given asset prices, and using that probability
measure to price the derivative. When the number of underlying assets and/or
the number of market outcomes in the model is large, pricing can be
computationally demanding. We show that a variety of quantum techniques can be
applied to the pricing problem in finance, with a particular focus on
incomplete markets. We discuss three different methods that are distinct from
previous works: they do not use the quantum algorithms for Monte Carlo
estimation and they extract the martingale measure from market variables akin
to bootstrapping, a common practice among financial institutions. The first two
methods are based on a formulation of the pricing problem into a linear program
and are using respectively the quantum zero-sum game algorithm and the quantum
simplex algorithm as subroutines. For the last algorithm, we formalize a new
market assumption milder than market completeness for which quantum linear
systems solvers can be applied with the associated potential for large
speedups. As a prototype use case, we conduct numerical experiments in the
framework of the Black-Scholes-Merton model.
