论文标题
dupire公式的扩展：随机利率和随机局部波动率
Extensions of Dupire Formula: Stochastic Interest Rates and Stochastic Local Volatility
论文作者
论文摘要
我们将dupire公式的概括为一般随机漂移和/或随机局部波动率的情况。首先，我们处理一个案例，其中漂移作为两个随机短率的差异。这种设置在外汇环境中是自然的，在外汇环境中，短率对应于两种货币的短率，具有随机股息收益率的股票单货币环境或具有随机便利性收益率的商品环境。我们在呼叫表面公式以及总暗示方差公式中介绍了公式，后者避免了日历通过构造扩散套利。我们为将两个短率作为单个因子过程提供并呈现单个随机率或所有确定性短率的限制提供了派生。限制与已发表的结果一致。然后，我们得出一种公式，该公式允许更通用的随机漂移和扩散，包括一个或多个随机的局部波动率项。在一般环境中，我们的推导允许对随机局部波动率模型的杠杆功能进行计算和校准。尽管是隐式的，但可以在定点迭代方案中以数值使用广义的dupire公式。
We derive generalizations of Dupire formula to the cases of general stochastic drift and/or stochastic local volatility. First, we handle a case in which the drift is given as difference of two stochastic short rates. Such a setting is natural in foreign exchange context where the short rates correspond to the short rates of the two currencies, equity single-currency context with stochastic dividend yield, or commodity context with stochastic convenience yield. We present the formula both in a call surface formulation as well as total implied variance formulation where the latter avoids calendar spread arbitrage by construction. We provide derivations for the case where both short rates are given as single factor processes and present the limits for a single stochastic rate or all deterministic short rates. The limits agree with published results. Then we derive a formulation that allows a more general stochastic drift and diffusion including one or more stochastic local volatility terms. In the general setting, our derivation allows the computation and calibration of the leverage function for stochastic local volatility models. Despite being implicit, the generalized Dupire formulae can be used numerically in a fixed-point iterative scheme.
