We provide an analytic model for valuing continuous barrier options. The model is manifestly arbitrage-free, exactly fits the implied volatility smile, and produces prices consistent with an underlying stochastic volatility dynamic

Derivative contracts on multiple foreign exchange rates must be priced to avoid arbitrage by contracts on the cross-rates. Given the triangle of smiles for two underlyings and their cross, we provide an analytic formula for a joint probability density such that all three vanilla markets are...

To properly value a basket option, one should construct a joint probability density correctly repricing all asset smiles and correlation smiles. At first sight, the task seems formidable. However, by reformulating the problem, we can develop a model that is simple and fast, admitting analytic or...

It is now possible to numerically solve partial differential pricing equations so that vanilla option values are exactly recovered. The finite difference scheme is not hard to implement, but it must be done exactly correctly to achieve results to machine precision. We provide a step by step...