  Option pricing is the most elemental challenge of mathematical finance.
Knowledge of the prices of options at every strike is equivalent to knowing the
entire pricing distribution for a security, as derivatives contingent on the
security can be replicated using options. The available data may be
insufficient to determine this distribution precisely, however, and the
question arises: What are the bounds for the option price at a specified
strike, given the market-implied constraints?
  Positivity of the price map imposed by the principle of no-arbitrage is here
utilised, via the Gelfand-Naimark-Segal construction, to transform the problem
into the domain of operator algebras. Optimisation in this larger context is
essentially geometric, and the outcome is simultaneously super-optimal for all
commutative subalgebras.
  This generates an upper bound for the price of a basket option. With
innovative decomposition of the assets in the basket, the result is used to
create converging families of price bounds for vanilla options, interpolate the
volatility smile, price options on cross FX rates, and analyse the
relationships between swaption and caplet prices.
