2018-04-23 开胃学习数学系列 - 最大熵 Curve Building

Curve building revisited

我们已经学习了如何使用以下方法，可以建立良好的CDS / IR曲线：
(咳咳，我其实是一点都不记得，之前也不理解的，所以还是好好重新学习一次吧)

• Bootstrap and iteration Bootstrap和迭代

• Tension spline interpolation 样条插值
  但是，有一些未解决的问题：

• which is a better state variable, zero rate r(t) =−1/t log(b(t)) or cumulative yield y(t)=−log(b(t))?

• what tension parameter makes more sense?

• what if market prices are arbitrageable? 如果市场价格是可套利

• 最大熵方法可以帮助回答这些问题

IR curve building

Typical instruments to build USD IR curves 建立美元IR曲线的典型工具：

• deposits: 1D, 1M, 2M, 3M
• 3M IR期货（或FRA）: 3M, 6M, 12M, ..., 33M, 36M
• Swaps: 4Y, 5Y, ..., 19Y, 20Y, 25Y, 30Y, 35Y, 40Y, 50Y

these instruments have different liquidity and their prices are not necessarily compatible with each other 工具有不同的流动性，价格不一定相互兼容

• it is not necessary to exactly match all mid market prices, usually it is adequate to be within the bid/ask 没有必要完全匹配所有的中间市场价格，通常在Bid/Ask 内就足够了
• it is not uncommon for market prices to be arbitrageable, thus impossible to exactly fit all market mids 市场价格可以套利的情况并不少见，因此不可能完全适合所有市场中立??

Instruments for building IR curves

b(t) 是在t季度到期的零息债券的市场价格
then the market observable prices can be expressed as a linear system of Cb=v:
则市场可观察价格可以表示为线性表达： Cb = v