两条曲线，一个价格：定价和对冲利率衍生品解耦的远期和贴现收益率曲线
Two Curves, One Price: Pricing & Hedging Interest Rate Derivatives Decoupling Forwarding and Discounting Yield Curves
摘要：
我们重新审视了单一货币利率衍生品的定价和对冲问题，使用多个不同的收益率曲线的贴现因子和远期利率与不同的基础利率期限的市场一致性估计。在2007年夏季信贷危机爆发后市场采用的这种双曲线-单一货币框架内，标准的单曲线无套利关系不再有效，可以通过适当考虑从市场基差掉期中提取的远期基差来恢复。数值结果表明，由此产生的远期基差曲线可能会显示出更丰富的微观期限结构，这可能会对利率工具的价格产生明显的影响。通过外汇类比，我们还得到了一般的无套利双曲线市场的公式，特别是基本的普通利率衍生品，FRA，掉期，上限/下限和掉期。这些表达式包括交叉货币衍生品的典型量子调整，自然源于与两条收益率曲线相关的数值之间的变化，这带来了波动性和相关性。数值情景证实了这种修正是不可忽略的，因此原则上，未经调整的双曲线价格并非没有套利。远期基础和定量调整都可以从交易对手风险的角度找到一个自然的财务解释。
英文摘要：
We revisit the problem of pricing and hedging plain vanilla single-currency interest rate derivatives using multiple distinct yield curves for market coherent estimation of discount factors and forward rates with different underlying rate tenors. Within such double-curve-single-currency framework, adopted by the market after the credit-crunch crisis started in summer 2007, standard single-curve no-arbitrage relations are no longer valid, and can be recovered by taking properly into account the forward basis bootstrapped from market basis swaps. Numerical results show that the resulting forward basis curves may display a richer micro-term structure that may induce appreciable effects on the price of interest rate instruments. By recurring to the foreign-currency analogy we also derive generalised no-arbitrage double-curve market-like formulas for basic plain vanilla interest rate derivatives, FRAs, swaps, caps/floors and swaptions in particular. These expressions include a quanto adjustment typical of cross-currency derivatives, naturally originated by the change between the numeraires associated to the two yield curves, that carries on a volatility and correlation dependence. Numerical scenarios confirm that such correction can be non negligible, thus making unadjusted double-curve prices, in principle, not arbitrage free. Both the forward basis and the quanto adjustment find a natural financial explanation in terms of counterparty risk.
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