Continuous distributions and time: Equivalent martingale measures.
详细信息
文摘
The problem considered is the pricing of contingent claims with continuous distributions and continuous time, which can be applied to stock trading, insurance contracts, future contracts, etc. There are some classical results about contingent claims pricing, such as the famous Black-Scholes equation, but the assumptions are often very restrictive and related to stochastic differential equations. By formulating this kind of problems as stochastic optimization problems and resorting to stochastic optimization techniques, some similar results can be obtained for significant more general cases.;The main result is an 'operational' duality theory, which plays a key role in deriving some important properties such as no-arbitrage, hedging, equilibrium equation, etc. Moreover, by virtue of this theory we can gather some very useful information from the dual problem to discretize the model with continuous distributions efficiently. The convergence of such an approximation scheme is guaranteed by some epi/hypo convergence theorem.;In practice, we need to estimate the distributions of prices from historical price data. A novel approach, quite different from the standard least square method, yields some very 'reasonable' results. Finally, this new estimation method combined with the discretization method are tested numerically on applications related to stock trading.