Before revision:  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,  and analyse the relationships between swaption and caplet prices . Beyond these practical applications, the approach is liberated from the requirement that the assets are commutative, accommodating an extension to the framework for mathematical finance that is equally applicable in noncommutative geometries .
After revision:  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 . Beyond these practical applications, the approach is liberated from the requirement that the assets are commutative, accommodating an extension to the framework for mathematical finance that is equally applicable in noncommutative geometries .