"Huaxia SSE50 ETF" will deliver dividend and be ex-dividend on December 2, 2019. On that day, 50ETF options will be adjusted accordingly, and the exercise price, contract unit and unit premium will be changed. What impact will these changes have on arbitrage trading? We will explain in this article:

(I) risk free arbitrage of IH futures

In the risk-free arbitrage of IH futures and 50ETF options, investors need to make the spot market value corresponding to long and short portfolio equal, that is, the contract value of stock index futures is equal to the spot market value corresponding to 50ETF options composite futures. IH futures will not be affected by the dividend of 50ETF. According to the first part above, the product of the price of 50ETF after ex dividend and the new contract unit of option will not be changed, and the spot market value corresponding to spot right will not be changed. Therefore, after the dividend of 50ETF fund, IH futures' risk-free arbitrage in the same period does not need to be adjusted.

(II) No risk arbitrage of 50ETF fund in the same period

Considering the difficulty of securities lending, the risk-free arbitrage mode of 50ETF is mainly to buy 50ETF funds and hold synthetic short positions. We use s and K to represent the closing price and option exercise price of 50ETF before dividend, s’ and K’ to represent the corresponding data after dividend, and δ to represent the dividend of each unit fund.

Before dividend:

Spot position: N*S

Option Position: N*(P-C)

After dividend, the position of 50ETF fund becomes:

Spot position: N*S’+N*δ

According to BS formula, after dividend, the premium of short futures per unit option becomes:

P’-C’=K’*EXP(-rT)*N(-d2’)-S’*N(-d1’)+K’* EXP(-rT)*N(d2’)-S’*N(d1’)

For the cumulative distribution of normal distribution: n (x) + n (- x) = 1, so after dividend adjustment: P’ - C’ = k’- s’, which means that after adjustment, each p’ – C’ corresponds to 1 unit -s’, 50ETF fund short position after ex dividend. Therefore, the spot corresponding to 50ETF and option portfolio is as follows:

Spot position match: N*S’+N*δ

Option Position match:-N’*S’

It means that after 50ETF dividend, the number of spot corresponding to the option portfolio is more, and investors can close some option contracts or buy a certain spot long. The specific adjustments are as follows:

(III) Volatility trading: Greek letters and implied volatility changes

The ETF dividend will affect the Greek alphabet in three ways: the underlying price will be lower, the exercise price will be lower and the contract unit will be higher. Here, we omit the specific derivation and give the change results directly:

Delta after adjustment= old Delta * underlying closing price before ex-dividend day/( underlying closing price before ex-dividend day-cash dividend)

Gamma after adjustment= old Gamma * [underlying closing price before ex-dividend day/( underlying closing price before ex-dividend day-cash dividend)]^2

Vega after adjustment=old Vega

Thea after adjustment=old Theta

Basically, after dividend, the absolute values of Delta and Gamma increased, while Vega and Theta remained unchanged.

Due to the adjustment of option contract, the contract unit changes, and there is a difference between the adjusted unit and the standard unit, 10000, so that while the exercise price is almost the same, and there is still a certain difference between the standard contract and the adjusted contracts and it cannot be arbitraged by 1:1 match. Therefore, the implied volatility of the adjusted option may be different from that of the standard contract.

Taking what happened in 2016 as an example, we selected 40 groups of contracts with similar exercise prices and adjusted contracts, and calculated their volume weighted average implied volatility. The results are shown in the figure below. In general, the implied volatility of the adjusted contract is slightly higher, with an average of 0.4 percentage points.

From a more detailed point of view, we can see that in the process of implied volatility changes, the "burr" of adjusted contracts is relatively more.

As a whole, after dividend, the Greek letters of options will change, Delta and Gamma will rise, Vega and Theta will remain unchanged, and the change of Greek letters above the second level will be more complicated; the implied volatility of the adjusted contract may also be different from that of the standard contract. These two factors will increase the analysis difficulty of volatility arbitrage. Combined with the decrease of liquidity of the adjusted contract, it is suggested that investors change positions into standard contracts.

50ETF分红对期权套利策略的影响
“华夏上证50ETF”将于2019年12月2日分红除息，当天50ETF期权将发生相应调整，行权价、合约单位、单位权利金都会改变。这些变化会对套利交易造成什么影响？我们在本文中予以说明：
（一）IH期货同期权无风险套利
在进行IH期货同50ETF期权的无风险套利时，投资者需要使多空组合所对应的现货市值相等，即股指期货的合约价值与50ETF期权合成期货对应的现货市值相等。50ETF基金分红时，IH期货不受任何影响，而根据以上第一部分的内容，50ETF在除息后的价格与期权新合约单位的乘积不变，即期权对应的现货市值不变。因此，50ETF基金分红后，IH期货同期权无风险套利无须调整。
（二）50ETF基金同期权无风险套利
考虑到融券困难，50ETF同期权的无风险套利模式主要为买入50ETF基金并持有合成空头。我们以S与K表示分红前50ETF的收盘价与期权行权价，S’与K’表示分红后的对应数据，δ表示每单位基金的分红。那么
在分红除息前：

在分红除息后，50ETF基金持仓变为：

根据BS公式，在分红后，每单位期权合成期货空头的权利金变为：

对于正态分布的累积分布而言：N(x)+N(-x)=1，所以经过分红调整后：P’-C’=K’-S’，这说明经过调整之后，每一份P’-C’，对应1单位-S’，即除息之后的50ETF基金空头。因此，50ETF、期权组合分别对应的现货如下：

意味着50ETF分红之后，期权组合对应的现货数量更多，投资者可以平仓部分期权合约，或者买入一定的现货多头。具体调整如下表所示：

（三）波动率交易：希腊字母与隐含波动率变化
50ETF分红会从三个方面影响希腊字母：标的价格降低、行权价降低、合约单位上升。在此，我们略去具体推导，直接给出变动结果：
基本上，50ETF分红除息后，Delta、Gamma的绝对值增加，Vega、Theta保持不变。
由于期权合约调整，使合约单位变化，与标准单位10000存在差异，使行权价几乎相同的标准、被调整合约也存在一定差异，不能直接进行1:1买卖套利，因此，被调整后的期权其隐含波动率有可能会同标准合约有差异。
我们以2016年发生的事情为例，挑选了40组行权价接近的标准、被调整合约，分别计算它们的成交量加权平均隐含波动率。结果如下图所示，大体上，被调整合约的隐含波动率略高一些，平均为0.4个百分点。

从更细致的角度来看，可以看到在隐含波动率变化的过程中，被调整合约的“毛刺”相对更多。

整体来说，50ETF分红除息后，期权希腊字母会出现变化，Delta、Gamma上升，Vega、Theta不变，二阶以上希腊字母变化情况则会更加复杂；被调整合约的隐含波动率与标准合约也可能会存在一定的差异。这两个因素会增加波动率套利的分析难度，再结合被调整合约流动性下降的情况，建议投资者换仓为标准合约。