超越随机连续性的隔夜利率期限结构建模
Term structure modelling with overnight rates beyond stochastic continuity
AI总结：
本文提出基于隔夜利率的期限结构建模框架，在广义HJM设定下刻画无套利条件，用仿射半鞅处理预定日期的随机不连续性，并给出高斯情形下债券与caplets的显式估值及局部风险最小化对冲方法。
摘要：
隔夜利率，如美国的SOFR（有担保隔夜融资利率），是当前利率基准改革的核心。隔夜利率的一个显著特征是由于货币政策干预和流动性约束，在预定日期出现跳跃和尖峰。这对应于其动态中的随机不连续性（即发生在事先已知时间点的不连续性）。在本工作中，我们提出了一个基于隔夜利率的期限结构建模框架，并在广义Heath-Jarrow-Morton（HJM）框架下刻画了无套利条件。我们扩展了经典的短期利率方法以适应随机不连续性，建立了一个由仿射半鞅驱动的可处理框架。在此背景下，我们证明了简单的设定能够捕捉隔夜利率跳跃行为的典型事实。在高斯设定下，我们给出了债券和caplets的显式估值公式。此外，当标的期限结构具有随机不连续性时，我们研究了局部风险最小化意义下的对冲问题。
英文摘要：
Overnight rates, such as the SOFR (Secured Overnight Financing Rate) in the US, are central to the current reform of interest rate benchmarks. A striking feature of overnight rates is the presence of jumps and spikes occurring at predetermined dates due to monetary policy interventions and liquidity constraints. This corresponds to stochastic discontinuities (i.e., discontinuities occurring at ex-ante known points in time) in their dynamics. In this work, we propose a term structure modelling framework based on overnight rates and characterize absence of arbitrage in a generalised Heath-Jarrow-Morton (HJM) setting. We extend the classical short-rate approach to accommodate stochastic discontinuities, developing a tractable setup driven by affine semimartingales. In this context, we show that simple specifications allow to capture stylized facts of the jump behavior of overnight rates. In a Gaussian setting, we provide explicit valuation formulas for bonds and caplets. Furthermore, we investigate hedging in the sense of local risk-minimization when the underlying term structures feature stochastic discontinuities.
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