Page 29 - CCFA Journal - Tenth Issue

加中金融 数学建模 Math Modeling
加中金融

The idea behind put-call parity is that if an investor were to sell a call option and buy a put option with the same strike price and
expiration date, they would have created a synthetic position that is equivalent to owning the underlying asset. As a result, the price
of the two options should be related in such a way that the sum of their prices is equal to the cost of owning the underlying asset.

Suppose that over [ ， ], we have the PV of call and put options of CPI against a strike K,

( )
( , , ) = ( , ) − 1 − ∆

( )
( )

( , , ) = ( , ) ∆ − + 1

( )
From the put-call parity,
( , )

( , , ) − ( , , ) = ( , ) ( ) − ∆ − 1

( , )

So we have the i-th convexity adjustment as

( , ) ( , , ) − ( , , )
( ) = + ∆ + 1

( , ) ( , )

In the market, we can observe caps and floors with different strikes. By the put-call parity method, we can get all convexity
adjustments if we have enough many caps and floors. Then an interpolation will be used for convexity adjustments for the missing
terms.

期权的抵消平衡

欧式期权的抵消平衡是金融学中的一个概念，它阐述了同一基础资产上具有相同行权价格和到期日的欧洲看涨期权和欧洲
看跌期权的价格关系。平衡状态表明，看涨期权的价格加上行权价格的现值应等于看跌期权的价格加上基础资产的现值。

期权的抵消平衡背后的思想是，如果投资者卖出一个看涨期权并买入具有相同行权价格和到期日的看跌期权，他们将创造
一个等价于拥有基础资产的合成头寸。因此，这两个期权的价格应该以这样一种方式相关，使它们价格的总和等于拥有基
础资产的成本。

设区间[ ， ], CPI 的关于行权价 K 的看涨和看跌期权 价格分别为

( )

( , , ) = ( , ) − 1 − ∆

( )
( )

( , , ) = ( , ) ∆ − + 1

( )
期权的抵消平衡给出了
( , )
( , , ) − ( , , ) = ( , ) ( ) − ∆ − 1

( , )

所以凸性修正为

( , ) ( , , ) − ( , , )

( ) = + ∆ + 1

( , ) ( , )

在市场中，我们可以观察到不同行权价的上限和下限。通过放空对冲平价法，如果有足够多的上限和下限，我们就可以获
得所有的凸性调整。然后，使用插值法进行缺失项的凸性调整。

CCFA JOURNAL OF FINANCE March 2023
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