Derivative Valuation

• We developed new algorithms for risk neutral valuation of derivative financial instruments
• Theoretical prices of derivative instruments are obtained by discounting their expected payoffs under the equivalent martingale measure using money market interest rate.
• The core algorithm is Path Integral Monte Carlo which used to generate arbitrary distributions of underlying risk factors (stocks, bonds, short interest rates, commodities, indices etc.)
• The advantage of the new algorithm is that sensitivities of derivative prices with respect to changes in all model parameters are computed in a single simulation. This is crucial for effective hedging.
• Parallel version of the algorithm is written in C and MPI and relies on task parallelism and functional decomposition
• Monte Carlo samples are generated on multiple processors in embarrassingly parallel fashion
• Pricing modules can either run in lock-step with the Monte Carlo module which generates histories of risk factors or asynchronously perform valuation functions on the histories which are broadcast as they are generated by the Monte Carlo module
• We are linking this flexible algorithm with a novel scheme based on Maximum Entropy method which generates implied probability distributions from reported option prices. The implied distributions can be used within the Path Integral Monte Carlo module to price exotic contracts consistently with exchange-traded contracts and they can also be used to search for arbitrage opportunities
• Estimation of implied distributions requires large scale global optimizers. We are developing two parallel stochastic optimizers based on mean field approximation (Laplace formula) and Langevin equation