Before revision:  Applying techniques from the analysis of quantum systems, operator algebra methods are here used to generate an upper bound for the price of a basket option , depending only on a covariance matrix generated from the constituent assets in the basket . 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:  Applying techniques from the analysis of quantum systems, operator algebra methods are here used to generate an upper bound for the price of a basket option , depending only on a covariance matrix generated from the constituent assets in the basket , the result is used to create converging families of bounds for vanilla options, interpolate the volatility smile, and analyse the relationships between swaption and caplet prices .