Superreplication of Financial Derivatives via Convex Programming

We give a method based on convex programming to calculate the optimal super-replicating and sub-replicating prices and corresponding hedging strategies of a financial derivative in terms of other financial derivatives. Our method finds a model that matches the superreplicating (or sub-replicating) price within an arbitrary precision and is consistent with the other financial derivatives prices. Applications include robust replication in terms of call prices with various strikes and maturities of forward start options, volatility and variance swaps and derivatives, cliquets calls, barrier options, lookback and Asian options. Numerical examples show that, in some cases, the super-replicating and/or sub-replicating prices are within 10% of the Black-and-Scholes price but considerably differ from it in other cases. Our method can take into account bid-ask spreads, interest rates and dividends and various limitations to the diffusion model. An alternative method to optimally super-replicate and sub-replicate forward start options using semi-definite and linear programming is presented.