Before revision:  The result is then used to create converging families of  bounds for vanilla options, interpolate the volatility smile, and analyse the relationships between swaption and caplet prices  .
After revision:  The result is then used to create converging families of 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 .