Page 28 - CCFA Journal - Tenth Issue

数学建模 Math Modeling 加中金融

Inflation-indexed Swaps

An inflation-indexed swap is a type of financial derivative that is used to hedge against inflation risk. It is a swap contract where one
party pays a fixed rate of interest on a notional amount while the other party pays a rate that is tied to the rate of inflation. The rate
of inflation is usually based on a well-known inflation index such as the Consumer Price Index (CPI).

A typical Inflation-indexed swap (IIS) has two legs: the index leg and the fixed leg with a series of periods { ,…, }. Party A pays

Party B the inflation rate, and receives a fixed rate. That is, Party A pays and receives at over [ , ],

( )

= − 1
( )
= × × ∆

where N is the notional, K is the fixed rate, ∆ is the fixed side day count fraction of [ , ].

The present value of the i-th floating leg of IIS should be given by

( , )
( ) = × ( , ) × ( ) − 1

( , )
( , )
where ( , ) is the initial CPI at t to T, and ( ) is the convexity adjustment for the ratio ( , ) in the risk-neutral measure of P(t,

). Hence, the present value of IIS is, for t < ,

( , )

( ) = ( ) = ( , ) × ( , ) ( ) − 1

The initial forward index values can be obtained by, say the bootstrapping method, if we have enough many IIS rates. How to
estimate the convexity adjustments c (t)? It is known that the convexity adjustment depends on models. The commonly used

methods in determining convexity adjustments are given in the following sections.

通膨指数互换

通膨指数互换是一种金融衍生品，用于对抗通膨风险。它是一种互换合同，其中一方在名义金额上支付固定利率，而另一
方支付与通膨率相关的利率。通膨率通常基于一个众所周知的通膨指数，例如消费者物价指数(CPI)。

一个典型的通膨指数互换(IIS)有两条腿：指数腿和固定腿，具有一系列的时期{ ,…, }。甲方向乙方支付通膨率，并收到

固定利率。也就是说，甲方在 时支付和收到在[ , ]时间段内的资金。

( )

= − 1
( )
= × × ∆
其中 N 是面值, K 是固定利率, ∆ 是区间的时间长度。

第 i 个浮动区间的现值是

( , )

( ) = × ( , ) × ( ) − 1

( , )
指数腿的现值因此是

( , )
( ) = ( ) = ( , ) × ( ) − 1

( , )

初始指数值可以通过引导法等方法获得，如果我们有足够多的 IIS 利率。如何估计凸性调整 c_i(t)呢？众所周知，凸性调整
取决于模型方法。以下部分给出了确定凸性调整常用的方法。

Put-Call Parity

Put-call parity is a concept in finance that states the relationship between the price of a European call option and a European put
option on the same underlying asset with the same strike price and expiration date. The parity states that the price of the call option
plus the present value of the strike price should be equal to the price of the put option plus the present value of the underlying
asset.

CCFA JOURNAL OF FINANCE March 2023
Page 28 第28页