UNIVERSITY VAN AMSTERDAM
BUSINESS SCHOOLMSc Finance
Master Thesis
The Arbitrage Opportunities on
ETF Options:
The case of Shanghai Stock
Exchange 50
Supervisor
prof. Liang Zou
Candidate
Mu Xu
This document is written by Mu Xu who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Contents
1 Introduction 1
1.1 Background . . . 1
1.2 Warrant . . . 2
1.3 Differences between share warrants and options . . . 3
1.4 ETF options . . . 4
1.4.1 Exchange Traded Fund . . . 4
1.4.2 Contract content . . . 5
1.5 An overview of the 50ETF Option market performance . . . 10
1.5.1 The performance of the first trading day . . . 10
1.5.2 The performance of the recent trading day . . . 10
2 Literature Review 13 3 Methodology 19 3.1 Generalization . . . 19
3.2 Butterfly spread . . . 21
3.2.1 Butterfly spread using calls . . . 21
3.2.2 Butterfly spread using puts . . . 25
3.3 Box spread . . . 28
3.3.1 Lower boundary . . . 30
4 Data and descriptive statistics 35
4.1 Data selection . . . 35
4.2 Assumption of transaction cost . . . 36
4.3 Risk free rate and opportunity cost . . . 38
4.4 Descriptive statistics . . . 39 5 Empirical test 41 5.1 Butterfly spread . . . 41 5.2 Box spread . . . 45 6 Conclusion 49 6.1 Results . . . 49 6.2 Future research . . . 51
Summary
This article briefly introduces the first option product Shanghai Stock Exchange 50
Option in the Chinese market, which is also an unusual European ETF option. This
article uses butterfly spread and box spread to test the pricing efficiency of SSE50 ETF Option. The result shows that the market is substantially rational and fair. From the empirical test, we can see that the more complicated the arbitrage strategy we use, the easier we could find the arbitrage opportunities. The arbitrage proba-bility of options may have an inner relation with the remaining time to expiration.
Chapter 1
Introduction
1.1
Background
On 9th February 2015, the first option product in the Chinese financial market went public and started trading in Shanghai Stock Exchange, which is called Shanghai Stock Exchange 50 ETF Option. As the first option product in the Chinese market, this new ETF option faces plenty of uncertainties about pricing, trading volume or trader restriction. Both the possible bubble in pricing and a speculative attitude of investors might reduce the lifetime of this product.
ETF options have been widely used by investors all over the world to do arbitrage or hedging especially in the American market and the European market for many years. As early as 26th April 1973, The Chicago Board of Trade established the Chicago Board Options Exchange. The first exchange to list standardized, exchange-traded stock options began its first day of trading in a celebration of the 125th birthday of the Chicago Board of Trade (Markham[1]). After a development of more than 40 years, CBOE offers options on over 2,200 companies, 22 stock indices, and 140 ETFs. Among the four basic tools in financial derivative markets (forwards, futures, options and swaps), options have become the most active derivative product
1 – Introduction
because of their convenience and benefits in short-selling, hedging and leverage. They are also very important in risk management and price seeking.
Compared to the widely use of options in foreigner markets, the 50 ETF Option appeared quite late in the Shanghai Stock Exchange. We can clearly foresee that the Shanghai Stock Exchange 50 ETF Option would increase the market liquidity of the underlying asset, improve the structure of the ETF investors, enrich the investment strategies and also provide a new risk-free arbitrage opportunity.
1.2
Warrant
There used to be a product very similar to options in the Chinese market called share warrant. In June 1992, the first allotment warrant went public in China but the trading of the share warrant was stopped by China Securities Regulatory Commission in June 1996.
Also, in August 2005, to promote the reform of the shareholder structure of listed companies in China, the call warrant and the put warrant were introduced into the Chinese financial market. The shareholder structure reform connects to an endemism in China which is called split of stock shares or equity division. Chinese listed companies have a particular shareholder structure. A part of the shares, called tradable shares (most of which are public), could be traded in the market, while the other part, called untradable shares (most of which are state-owned shares and legal person shares), could not be traded in the market. At the beginning of the Chinese stock exchange, 70% of the shares could not be traded. This complicated shareholder structure fell behind the development of the financial market, so the China Securities Regulatory Commission tried to reform the shareholder structure to eliminate the differences between tradable shares and untradable shares, and the share warrant is one of the reform tools they used.
1.3 – Differences between share warrants and options
marketing environment became the biggest problem of the market. After the last warrant expired in 2008, there was no new warrant announced in the market any-more. In the 9 years warrant trading history, the trading volume has been as much as 2.73 trillion CNY, which is the third place in the international warrant market. Although both the share warrant and option are contracts that define the rights and obligations, these two financial derivatives have significant differences.
1.3
Differences between share warrants and
op-tions
On Investopedia, a warrant is defined as a derivative that confers the right, but not the obligation, to buy or sell a security, normally an equity, at a certain price before expiration1. An option is defined as a financial derivative that represents a contract
sold by one party (the option writer) to another party (the option holder). The contract offers the buyer the right, but not the obligation, to buy (call) or sell (put) a security or other financial asset at an agreed-upon price (the strike price) during a certain period of time or on a specific date (exercise date)2. According to the
definition, both of the two derivatives endow the investors with the right to trade the security on an agreed price on or before expiration. However, the differences are quite clear. Options are standard contracts issued by a stock exchange while warrants are non-standard contracts issued by listed companies or investment banks. Furthermore, option investors could long options or short options if short selling is not banned while warrant investors could only long warrants. Regarding the impact of exercising, options will only lead to the transfer of the underlying asset among different traders, but when traders exercise the warrant, the issuer should
1Investopedia: http://www.investopedia.com/terms/w/warrant.asp 2Investopedia: http://www.investopedia.com/terms/o/option.asp
1 – Introduction
issue additional shares as they promised on contracts, so the total equity increases. Basically, the pricing of warrants, which should be based on the price change of the underlying asset, deviates more easily from the real value.
1.4
ETF options
1.4.1
Exchange Traded Fund
Before we introduce the definition of an ETF option, we should better explain the underlying asset, an ETF. ETF is an acronym that means Exchange Traded Fund. It is a marketable security that tracks an index, a commodity, bonds, or a basket of assets3. As one of the most successful equity market innovations of the past two
decades, these products first appeared in 1993 in the form of Standard & Poor’s Depository Receipts (Ackert and Tian[2], 1998). Basically, an ETF is an investment fund traded on stock exchanges, similar to stocks. An ETF holds assets such as stocks, commodities, or bonds, and trades close to its net asset value over the course of the trading day. Most ETFs track an index, such as a stock index or bond index (Ackert and Tian[3], 2001). The proportion of ETF in derivative markets is growing so fast because of its risk diversification, investment diversity, instantaneous arbitrage opportunities and different available risk-preferred types.
The underlying asset of Shanghai Stock Exchange 50 ETF Option is the first Exchange Traded Fund in the Chinese stock market, Shanghai Stock Exchange 50 ETF (code: 510050), issued on 30th December 2014 and put into market on 23rd February 2015. The fund management company is China Asset Management Com-pany Limited with 0.5% fund management fees, and the fund custodian comCom-pany is Industrial and Commercial Bank of China with 0.1% fund custodian fee. The price of this fund follows Shanghai Stock Exchange 50 index (SSE 50, code: 000016.SH),
1.4 – ETF options
which contains 50 sample stocks with the highest trading volume, the highest liquid-ity and the most representativeness including banking, insurance, energy, real estate firms in Shanghai Stock Exchange by “float-adjusted” capitalization. The sample stocks will be adjusted twice a year. The Fund Net Growth Rate has reached 246% after its execution.
1.4.2
Contract content
Some of the basic contents of the Shanghai Stock Exchange 50 ETF Option are shown in Table 1.1.
1 – Introduction
Table 1.1. Shanghai Stock Exchange 50 ETF Option Contract
Underlying Asset Shanghai Stock Exchange 50 ETF (SSE 50ETF) Contract Type Call option and put option
Contract unit 10,000
Expiration month The current month, the next month and the following two season month
Strike price 5 different price in total (1 price at the money, 2 prices in the money, 2 prices out of the money)
The interval of the
strike price 0.05CNY for strike price blow 3CNY (included), 0.1CNY forstrike price between 3 and 5CNY (included), 0.25CNY for strike price between 5 and 10CNY (included), 0.5CNY for strike price between 10 and 20CNY (included), 1CNY for strike price between 20 and 50CNY (included), 2.5CNY for strike price between 50 and 100CNY (included), 5CNY for strike price above 100CNY
Exercise type European option Delivery method Physical delivery
Expiration date The fourth Wednesday of the expiration month Exercise date 9:15-9:25, 9:30-11:30, 13:00-15:30 on expiration date Settlement date The day after expiration date
Trading time 9:15-9:25, 9:30-11:30 (9:15-9:25 is the opening call auction time)
13:00-15:00 (14:57-15:00 is the closing call auction time) Delegate type Limit Order,
Minimum quote
unit 0.0001CNY
Price change
limi-tation The price changing limitation is based on the price change ofthe underlying asset (SSE 50ETF). The highest daily decrease of call and put options prices is 10% of the last closing price of the underlying asset, but different for the increase.
Trading curb During the continuous auction, when the trading price of the option is 50% higher or lower than the closest comparable price, and in the meantime, the absolute value of the price changing is higher than 5 minimum quote unit, the trading will enter into a 3 minutes call auction.
There are some contents that need further explanations: (1)Expiration month:
1.4 – ETF options
For example, the first batch of option started in March 2005, and the expiration months are March, April, June and September in the same year.
From the table we can see that Shanghai Stock Exchange 50 ETF Option contains 2 kinds of position, call option and put option, 4 kinds of expiration month, 5 different strike prices, which means 40 kinds of contracts in the same time in total. An individual investor could long the call options and put options at the same time, but after the market closes, the trading system would offset these two positions automatically until the investor is on one single position.
(2)Exercise type:
This is a very special and interesting point. In general, most of the ETF options are American options, which means that traders can exercise the option at any time during its life until its maturity date. However, Shanghai Stock Exchange 50 ETF Option doesn’t allow traders to exercise early.
(3)Delivery method:
Basically, most of the derivatives on stock indices and Eurodollars are settled in cash, but most of the derivatives on stocks or ETFs are settled by physical delivery. The physical delivery in this context means that if the ETF option contract is not closed out before maturity, the trader on the short position should deliver the ETFs as the strike price to the trader on the long position.
(4)Price change limitation:
There is always a 10% daily price change limitation in the Chinese stock market. However, this rule in the option market is quite different from the stock market. The most important difference is that in the stock market, the percentage of highest daily increase and decrease of the stock price is based on the closing price of the stock itself in the previous trading day, while in the SSE50 ETF Option market, this price change percentage of the option price is based on the closing price of the underlying asset in the previous trading day and sometimes the strike price of the option as well. The calculation formula of the price change limitation is also different
1 – Introduction
from the other foreigner options. For example, the limitation of Standard & Poor’s 500 index option is 5%. The highest daily decrease of call and put options is 10% of the previous closing price of the underlying asset, Shanghai Stock Exchange 50 ETF. For the maximum increase of option, it has complicated formulas as Formula [1.1] and [1.2]:
Maximum Increase of Calls = max{previous closing price of underlying asset × 0.5%,
min[(2 × previous closing price of underlying asset
−strike price), previous closing price of underlying asset]
times10%}
[1.1]
Maximum Increase of Puts = max{strike price × 0.5%, min[(2 × strike price −previous closing price of underlying asset),
previous closing price of underlying asset] × 10%} [1.2] From the formulas we can see that the highest daily increase is not a fixed percentage number and it is also different among different contract types. The highest increase for all contracts is 10%. The increase of out-of-the-money options is minimum, and the lowest increase limitation could be as small as 0.5%.
(5)Trading curb:
It also called Circuit Breaker. It is a financial regulatory instrument that is in place to prevent stock market crashes from occurring. Since their inception, circuit breakers have been modified to prevent both speculative gains and dramatic losses within a small time frame. When triggered, circuit breakers either stop trading for a small amount of time or close trading early in order to allow accurate information to flow among market makers and for institutional traders to assess their positions
1.4 – ETF options
and make rational decisions. In this case, the 10% price change of the underlying asset would lead to more than 50% price change of the option. When the trading curb occurs, the market makers will play an important role in price adjusting to reduce the influence from abnormal price fluctuation.
(6)Margins:
In options trading, a margin is a collateral that the traders on short position has to deposit with traders on long position to cover some or all of the credit risk. Differently from futures trading, only the investors on short position have to pay the margins because the right and the duty of long and short position are not equal. From the table we can estimate that the margin of Shanghai Stock Exchange 50 ETF Option is around 7%, which means the leverage is more than tenfold. For some out-of-the-money options, the leverage could be as large as twentyfold.
(7)Market maker:
The main function of a market maker is to facilitate options trading and to keep trading stabilization. The Shanghai Stock Exchange stipulates that option trading should follow mixed trading rules, which means a mixture of auction trading and competitive market makers. The market maker quotes both bid and ask prices when requested and the price follows “price priority, time priority” rule. This trading system could reduce bid ask spread, improve trading possibility, and increase market liquidity.
(8)Investor level-to-level administration:
The high risk in option trading asks for the professionalism of the investors. So investor level-to-level administration is necessary for keeping market fairness especially in a country with a population this big. Firstly, high access to the market makes sure that only the investors with some experience and enough capital could take part in the trading. Secondly, based on the professionalism level, different investors have different transaction authority. The Shanghai Stock Exchange divides investors into 3 levels, and we assume we have already had all the authority in
1 – Introduction
our arbitrage opportunity analysis. In fact, the access to the different transaction authority is also one of the initial transaction cost.
1.5
An overview of the 50ETF Option market
performance
1.5.1
The performance of the first trading day
The first trading day of Shanghai Stock Exchange 50 ETF Option is 9th January 2015. The first batch of contracts contain 40 types including 4 kinds of expira-tion months (March, April, June and September) and 5 different strike price. The performance of the first trading day is shown on Table 1.2.
Table 1.2. Trading Information of 50ETF on 09.02.2015
Asset Name Trading Volume in
Total Call Option Volume Put Option Volume SSE50 ETF
Asset Code 18,843 11,320 7,523
510050
The total trading volume is less than 20,000 contracts, and the call option number is slight higher than the put option number. It shows that the first trading day is not active. The reason may lie on the inexperience of the investors to this new financial product.
1.5.2
The performance of the recent trading day
To compare changes of the performance of Shanghai Stock Exchange 50 ETF Option market after 2 years, we collect the trading data on the same date as the first trading day in 2017. The trading volume on 9th January 2017 is shown on Table 1.3.
1.5 – An overview of the 50ETF Option market performance
Table 1.3. Trading Information of 50ETF on 09.02.2017
Asset Name Trading Volume in
Total Call Option Volume Put Option Volume SSE50 ETF
Asset Code 73,260 40,581 32,679
510050
The total trading volume has increased to more than 70,000 contracts, much bigger than that on 9th January 2015. It shows that the Shanghai Stock Exchange 50 ETF Option market has become more active now.
Chapter 2
Literature Review
The ETF option is a special type of option based on the Exchange Traded Fund. Based on the highly maturity of the development of ETF, the ETF option is the perfect explanation of “the derivative of derivative” to fulfil the increasing demand of investors. The study of ETF options is an extension of the study of ETFs. Deville[4](2008) introduced the history and related research information about the Exchange Traded Fund systematicly in his previous study. He especially mentioned that, in the period that the ETF was first introduced, “With the introduction of index futures contracts, program trading became more popular. As such, the oppor-tunity to develop a suitable instrument allowing index components to be negotiated in a single trade became increasingly interesting.” He also summarized the investors in the ETF market into several categories: besides the index providers that develop and provide licences for existing or new indices, ETF players are the stock exchanges, sponsors and trustees, ETF authorized participants, market makers and investors in the secondary market.
Except for the history, the researches about this indexing investment tool, in-cluding the influence to the market, the arbitrage opportunities using ETF and the arbitrage strategies, have become a hot field in derivative study as well. As one of the most important index related product, most of the arbitrage and pricing strategies
2 – Literature Review
for index futures also work on ETFs. According to former studies, Gould[5](1988) added the influence of the interest rates on borrowings and loans into the model of the arbitrage of index futures and spots, and optimized the no arbitrage interval of index futures and spots. He found that a “fair range” of price is more accu-rate than a “fair price” to describe the relationship between a futures contract and its underlying stock index when transaction costs are considered; moreover, prices outside the fair range window will bring arbitrage opportunities. Białkowski and Jakubowski[6](2008) used futures index WIG20 from WSE and deduced the upper and lower bound of the no-arbitrage interval and the arbitrage model.
Nowadays, the studies about the the pricing efficiency of ETFs are more and more detailed. A lot of adjustments on the data collection and model parameters has been made among different researches. Switzer, Varson, Zghidi, et al.[7](2000) studied the relationship between the trading funds in SPDRs and corresponding index futures using intraday data each hours, and pointed out that there was significant positive pricing error in ETF futures. They also showed that ETF improved the pricing efficiency of futures. Furthermore, the strategy of using intraday data improves the feasibility of collecting trading data and will also be used in my empirical analysis. Engle and Sarkar[8](2002) are the first who used high frequency data of each trading in ETF secondary market to study the discounted and premium price of ETF, and they found that the tracking error is quite small using American domestic index as underlying asset while the tracking error of ETF that was issued in the USA but using foreigner index as underlying asset is much bigger. Some other people focused on the field of trading or transactions to draw more conclusions about ETFs. Ivanov[9](2016) documented that ETFs have lower proportion of adverse selection in the bid-ask spread relative to stocks, which means that the order processing cost component is higher in ETFs. Also, he found that ETFs with more quotes have lower adverse selection; whereas ETFs with higher average bid price, higher expense ratio and trust structuring of the ETF have higher adverse selection component of
2 – Literature Review
the bid-ask spread.
As “the derivative of the derivative”, ETF options have most of the advantages as an ETF itself and also provide more flexible leverage, more convenient trading process and more diversified investment strategies. Since it was introduced to the market, the fair pricing of this financial product is very important for keeping the market fair. Basically two approaches were wildly used to test the pricing efficiency. The first approach is using pricing models. Actually, there are not many English researches using this approach. Collin-Dufresne, Goldstein, and Yang[10](2010) in-vestigated a structural model of market and firm-level dynamics in order to jointly price long-dated S&P 500 options and tranche spreads on the five-year CDX index to explain the puzzle reported by Coval, Jurek, and Stafford[11](2009). The model matches the time series of tranche spreads well, both before and during the financial crisis. There are several Chinese researches based on the pricing model on Chinese option products. Chen[12](2016) used B-S model and Monte Carlo method to price the SSE 50 ETF Option. He adopts the modified B-S option pricing model with dividend payout ratio in the BSM model. He compared the theoretical price with the actual price by regression analysis and index analysis. The empirical study showed that the pricing efficiency of B-S pricing model is better than the Monte Carlo pricing model. The put option is more easier to price than the call option. Furthermore, based on the analytical solution of the BSM differential equation and numerical methods of option pricing, Song[13](2015) found that BSM formula and the binary tree model are more suitable for the calculation and forecast of SSE 50 ETF option pricing. The GARCH model improves the accuracy of BSM pricing formula and Monte Carlo method.
The other approach is straightforward arbitrage trading strategies such as box spread, butterfly spread and put-call parity. Bharadwaj and Wiggins[14](2001) used box spread and put-call parity to test the S&P 500 Index LEAPS market. They found that while LEAPS puts are overpriced relative to calls about 80% of the time,
2 – Literature Review
the discrepancy does not happen often enough to produce a reliable arbitrage profit after transaction costs. They gave the reason why it is better to use straightforward arbitrage trading strategies than making a pricing model to estimate the transac-tion price. They thought that optransac-tion pricing models are based on arbitrage, but setting up a delta-neutral position and rebalancing it continuously over an options lifetime is often a difficult and costly trading strategy. Pricing violations, therefore, may not indicate significant market inefficiency, particularly given uncertainty over the volatility parameter. By contrast, put-call parity is an option price relationship that, if violated, generates a straightforward arbitrage trade that does not require rebalancing or informations about volatility. We will also use this explanation of the weakness of model pricing in the thesis methodology. Actually, the approach of no-arbitrage rule is not a new topic. As long ago as in 1989, Ronn and Ronn[15](1989) particularly introduced the box spread arbitrage conditions in the option market. They tested the Chicago Board Options Exchange (CBOE) data across several trad-ing days over an eight-year period and discussed the box spread boundary separately with or without transaction cost. The results indicated that arbitrage conditions in lending opportunities appear to exist only for low transaction-cost agents who can implement options trading expeditiously. Even under these conditions, the magnitude of these arbitrage profits may not be economically meaningful on most trading days. Ackert and Tian[2](1998) used the boundary conditions, call and put parity and box spread to test the pricing efficiency of Toronto 35 stocks in-dex option. They[3](2001) also proved that there are few arbitrage opportunities in S&P500 stocks index option considering trading cost and short selling bans when using box spread, convexity arbitrage, and call and put spread. Capelle-Blancard and Chaudhury[16](2001) used the boundary conditions, call and put parity, box spread, convexity and call and put spread with the data during Jan 2nd 1997 and Dec 30th 1999, and found that even if the market is quite efficiency, the institutional arbitrager with low trading costs could also make profit by doing arbitrage. Davis
2 – Literature Review
and Hobson[17](2007) showed that there are arbitrage opportunities in the market when the initial cost of the portfolio is negative but with positive returns on expir-ing date usexpir-ing butterfly spread. These studies used similar strategies but different options on different data sample period.
Chapter 3
Methodology
3.1
Generalization
To set up an arbitrage in the option market, the options trader would long an un-derpriced position and sell the equivalent overpriced position. As a type of options, ETF option follows the similar arbitrage method as normal options. If puts are overpriced relative to calls, the arbitrager would sell a naked put and offset its po-sition by buying a synthetic call. Similarly, if calls are overpriced relative to puts, the arbitrager would sell a naked call and offset its position by buying a synthetic put. In general, there have been a lot of arbitrage strategies already. No matter what kind of arbitrage strategies the investors use, they are taking advantage of the failure of the no-arbitrage rule.
Arbitrage pricing theory shows that an asset’s returns can be predicted using the relationship between that asset and many common risk factors. Based on the CAPM model, Arbitrage pricing theory thinks that arbitrage is the determinant of the market equilibrium. When the market is in equilibrium, there are no arbitrage opportunities in the market and this equilibrium price of the asset is the rational price. In an equilibrium market, there is no such portfolio with an initial cost equal
3 – Methodology
to zero but in any possible state, the future payoff of this portfolio is positive. Similarly, there is no such portfolio with a positive initial gain but in any possible state, no cost will be paid in expiration.
In this case, our target is to estimate if the Shanghai Stock Exchange 50 ETF Option market is an equilibrium market. If not, we will show some arbitrage cases. However, in the real market, it’s much harder to do arbitrage compared to theory, due to the existence of transaction costs, opportunity cost and other client fees. So in our methodology, firstly we check whether the options are fair priced using different models without taking other transaction costs into consideration. If not, then we put the real transaction costs in the model to see if the arbitrage opportunity still exists. Also we will test how much the costs should be in order to prevent the arbitrage opportunities in the market.
Instead of using a capital pricing model such as the BSM model to compare the real trading price and the theoretical price, in this research, we will use some arbitrage models with different combined options structures, so we do not have to estimate all the parameters for the pricing model or make a lot of assumptions. Actually, in the strategy of model estimation, models are always based on some rigorous assumptions that may be totally different from the reality of the complicated market and some parameters such as volatility in the BSM model is very hard to obtain.
The basic principle we use for empirical test is the Law of One Price. The Law of One Price is an economic concept which posits that “a good must sell for the same price in all locations” (Rashid[18], 2007). The law of one price constitutes the basis of the theory of purchasing power parity, which states that the exchange rate between two currencies equals the ratio of the currencies’ respective purchasing power. In an efficient financial market, if the payoff and risk of two assets are exactly the same, these two assets should have the same price. If the prices are different, the arbitrage opportunity exists and the investors, especially the informed traders, will respond to
3.2 – Butterfly spread
this unbalance by shorting the overvalued asset and longing the undervalued asset until the market turns back to equilibrium.
The butterfly spread and the box spread are two arbitrage strategies we will use for the empirical test.
3.2
Butterfly spread
A butterfly spread is a neutral option strategy combining bull and bear spreads. Butterfly spreads use four option contracts with the same expiration but three dif-ferent strike prices to create a portfolio. The trader sells two option contracts at the middle strike price and buys one option contract at a lower strike price and one option contract at a higher strike price. A butterfly spread can be constructed by four call options or four put options.
Symbols defined:
Ki: Strike price of the options, where K1 < K2 < K3 and K1+ K3 = 2K2
Ci: European call option price with strike price Ki
Pi: European put option price with strike price Ki
t: Trading date T: Expiration date
ST: The price of Shanghai Stock Exchange 50 ETF (spot price) on expiration date
tp, tc: The transaction cost for put option and call option
tpr, tcr: The opportunity cost for the margins of the position on put and call option
r: Risk free rate
3.2.1
Butterfly spread using calls
3 – Methodology
Figure 3.1. Profit for butterfly spread using calls
The trading strategy of butterfly spread using calls is longing one call option with strike price K1, longing one call option with strike price K3 and shorting two call options with strike price K2.
Butterfly spread using calls should satisfy the following equation in an equilib-rium market without considering the transaction cost:
C1+ C3 −2C2 >0 [3.1]
Knowing that K1 < K2 < K3, the price of the call option with a higher strike price should be lower than that of the call option with a lower strike price, so we have C1 > C2 > C3.
In general, the butterfly spread arbitrage strategy is used when the options prices in the market do not satisfy equation[3.1]. When the call option with strike price in the middle is overvalued compared to the call options with higher strike price and lower strike price, which means C1+ C3−2C2 ≤0.
The arbitrager could short the overpriced option and long the undervalued op-tions in the meantime then invest the total gain 2C2 − C1 − C3 in the monetary
3.2 – Butterfly spread
market. The cash flow of this arbitrage operation facing different spot price is shown in Table 3.1.
Table 3.1. Trading strategy of butterfly spread using calls
Date Trading date t Expiration date T
Strategy ST≤ K1 K1< ST≤ K2 K2< ST≤ K3 ST > K3 Long the option with strike price K1 −C1 - ST− K1 ST− K1 ST− K1 Short two options with strike price K2 2C2 - - −2(ST− K2) −2(ST− K2) Long the option with strike price K3 −C3 - - - ST− K3 Invest in monetary market C1+ C3− 2C2 I I I I Sum 0 I I + ST− K1 I + 2K2− K1− ST I
Where I is the future cash flow of the investment in monetary market,
I = (2C2− C1− C3) × er(T −t) [3.2] The arbitrage strategy has no initial cost. On the trading day, the arbitrager longs two call options and shorts two call options.
Depending on different spot price, there are four situations:
1. When ST ≤ K1, all the four options won’t be exercised. The profit comes from the investment in monetary market :
Π1 = (2C2− C1− C3) × er(T −t)>0 [3.3] On the expiration day, the arbitrager doesn’t need to do anything.
2. When K1 < ST ≤ K2, only the option with strike price K1 will be exercised. So the final cash flow is:
Π2 = (2C2− C1− C3) × er(T −t)+ ST − K1 >0 [3.4] On the expiration day, the arbitrager buys the SSE50 ETF as price K1, then sells it in the ETF market.
3 – Methodology
3. When K2 < ST ≤ K3, the arbitrager could exercise the option with strike price
K1 and the short position of the option with strike price K2 will be exercised by other investors. The final cash flow is:
Π3 = (2C2 − C1− C3) × er(T −t)+ 2K2− ST − K1 [3.5] Knowing that 2K2− ST − K1 = K3− ST >0, Π3 >0.
On the expiration day, the arbitrager buys one contract of SSE50 ETF as price
K1 and sells two contracts of SSE50 ETFs as price K2. So the arbitrager needs to buy one contract of SSE 50 ETFs in the ETF market.
4. When ST > K3, all the four options will be exercised. The future cash flow is: Π4 = (2C2− C1− C3) × er(T −t)+ 2K2− K1− K3 [3.6] Knowing that 2K2 = K1+ K3, Π4 = (2C2− C1− C3) × er(T −t)>0.
On the expiration day, the arbitrager buys one contract of SSE50 ETF as price
K1, buys one contract of SSE50 ETF as price K3 and sells two contracts of SSE 50 ETF as price K2 without trading in ETF market.
Therefore, the initial cost of the portfolio is zero but in any possible state, the future payoff of this portfolio is positive. The risk free arbitrage opportunity exists. Taking the transaction costs into consideration, the no-arbitrage condition changes to:
C1+ C3−2C2+ 4tc+ 2tcr ≥0 [3.7] So if the arbitrage opportunity exists, the mispricing should be high enough that the present value of the future cash flow is higher than the sum of transaction cost and the opportunity cost for the margins.
Hence, our purpose is to find three call options with the same expiration date and the butterfly spread strike price conditions K1 < K2 < K3, K1 + K3 = 2K2. The price of the options should also correspond to C1+ C3−2C2 ≤0.
3.2 – Butterfly spread
3.2.2
Butterfly spread using puts
The total payoff for butterfly spread using puts is shown on Figure 3.2.
Figure 3.2. Profit for butterfly spread using puts
The trading strategy of a butterfly spread using puts is shorting one put option with strike price K1, shorting one put option with strike price K3 and longing two put options with strike price K2.
A butterfly spread using puts should satisfy the following equation in an equi-librium market without considering the transaction costs:
P1+ P3−2P2 >0 [3.8]
Knowing that K1 < K2 < K3, differently from butterfly spread using calls, the price of the put option with a higher strike price should be higher than that of the put option with a lower strike price, so we have P1 < P2 < P3.
Similar to the butterfly spread using calls, the butterfly spread arbitrage strategy is used when the put option with strike price in the middle is overvalued compared to the put options with higher strike price and lower strike price, which means
3 – Methodology
The arbitrager could short the overpriced option and long the undervalued op-tions in the meantime then invest the total gain 2P2 − P1 − P3 in the monetary market. The cash flow of this arbitrage operation facing different spot price is shown in Table 3.2.
Table 3.2. Trading strategy of butterfly spread using puts
Date Trading date t Expiration date T
Strategy ST ≤ K1 K1< ST≤ K2 K2< ST≤ K3 ST> K3 Long the option with strike price K1 −P1 K1− ST - - -Short two options with strike price K2 2P2 −2(K2− ST) −2(K2− ST) - -Long the option with strike price K3 −P3 K3− ST K3− ST K3− ST -Invest in monetary market P1+ P3− 2P2 I I I I Sum 0 I I + ST+ K3− 2K2 I + K3− ST I
Where I is the future cash flow of the investment in monetary market,
I = (2P2− P1− P3) × er(T −t) [3.9] The arbitrage strategy has no initial cost. On the trading day, the arbitrager longs two put options and shorts two put options.
Depending on different spot price, there are four situations:
1. When ST ≤ K1, all the four options will be exercised. The future cash flow is:
Π1 = (2P2− P1− P3) × er(T −t)−2K2+ K1+ K3 [3.10]
Knowing that 2K2 = K1+ K3, Π1 = (2P2− P1− P3) × er(T −t)>0.
On the expiration day, the arbitrager sells one contract of SSE50 ETF as price
K1, sells one contract of SSE50 ETF as price K3 and buys two contracts of SSE50 ETF as price K2 without trading in ETF market.
3.2 – Butterfly spread
2. When K1 < ST ≤ K2, the arbitrager could exercise the option with strike price
K3 and the short position of the option with strike price K2 will be exercised by other investors. So the final cash flow is:
Π2 = (2P2− P1− P3) × er(T −t)+ ST + K3−2K2 [3.11]
Knowing that ST + K3−2K2 = ST − K1 >0, Π2 >0.
On the expiration day, the arbitrager sells one contract of SSE50 ETF as price
K3, then buys two contracts of SSE50 ETF as price K2. So the arbitrager needs to sell one contract of SSE50 ETF in the ETF market.
3. When K2 < ST ≤ K3, only the option with strike price K3 will be exercised. The final cash flow is:
Π3 = (2P2− P1− P3) × er(T −t)+ K3− ST >0 [3.12] On the expiration day, the arbitrager buys one contract of SSE50 ETF in the ETF market, then sells one contract of SSE50 ETF as price K3.
4. When ST > K3, all the four options won’t be exercised. The profit comes from the investment in monetary market :
Π4 = (2P2− P1− P3) × er(T −t)>0 [3.13] On the expiration day, the arbitrager does not need to do anything.
Therefore, the initial cost of the portfolio is zero but, in any possible state, the future payoff of this portfolio is positive. The risk free arbitrage opportunity exists. Taking the transaction costs into consideration, the no-arbitrage condition changes to:
3 – Methodology
So if the arbitrage opportunity exists, the mispricing should be high enough that the present value of the future cash flow is higher than the sum of transaction cost and the opportunity cost for the margins.
Hence, our purpose is to find three put options with the same expiration date and the butterfly spread strike price conditions K1 < K2 < K3, K1 + K3 = 2K2. The prices of the options should also correspond to P1+ P3−2P2 ≥0.
3.3
Box spread
A box spread is a combination of a bull call spread and a bear put spread. A bull call spread consists of a long and a short positions on two call options with different strike prices but same expiration date, while a bear put spread consists of a long and a short positions on two put options with different strike prices but same expiration date (see Figure 3.3 and 3.4).
3.3 – Box spread
Figure 3.4. Profit for bear spread using puts
The expiration date and strike prices of the bull call spread and bear put spread in a box spread are the same. In the transaction, the trader longs a call option and shorts a put option with lower strike price K1, and shorts a call option and longs a put option with higher strike price K2.
Symbols defined:
Ki: Strike price of the options, where K1 < K2
Ci: European call option price with strike price Ki
Pi: European put option price with strike price Ki
t: Trading date T: Expiration date
ST: The price of Shanghai Stock Exchange 50 ETF (spot price) on expiration date
tp, tc: The transaction cost for put option and call option
tpr, tcr: The opportunity cost for the margins of the position on put and call option
r: Risk free rate
If all the options are fair priced, a box spread should satisfy two equations without considering the transaction costs:
3 – Methodology
and
(C2− C1) − (P2− P1) + (K2− K1)e−r(T −t) ≥0 [3.16]
Taking transaction costs into consideration, the equations change to:
(C1− C2) − (P1− P2) + (K1− K2)e−r(T −t)+ 2tc+ 2tp+ tpr+ tcr ≥0 [3.17]
and
(C2− C1) − (P2− P1) + (K2− K1)e−r(T −t)+ 2tc+ 2tp+ tpr+ tcr ≥0 [3.18]
If either one of the equations is not satisfied, the arbitrage opportunity exists. Our purpose is to find two call options and two put options in the SSE50 ETF option market that correspond with the box spread conditions we described before.
3.3.1
Lower boundary
Considering the investment strategy of a long position on a bullish vertical call spread and a long position on a bearish vertical put spread, the initial cost will be (C1 − C2) − (P1 − P2). A negative value means an initial gain. This balance could be financed in the monetary market by borrowing or investing. For a box spread with all European options, the value of the spread is the present value of the difference between the strike prices, which means the payoff of the box spread is always independent of the terminal stock price and the value at maturity always equals to K2 − K1. So we have the no-arbitrage lower boundary of this strategy as equation [3.15].
When the lower boundary doesn’t hold, the cash flow of this investment strategy is shown in Table 3.3.
3.3 – Box spread
Table 3.3. Trading strategy of box spread (lower boundary)
Date Trading date t Expiration Date T
Strategy ST ≤ K1 K1< ST ≤ K2 ST > K2
Long a call option with strike price
K1
−C1 - ST− K1 ST − K1
Short a put option with strike price
K1
P1 ST − K1 -
-Short a call option with strike price
K2
C2 - - - K2− ST
Long a put option with strike price
K2
−P2 K2− ST K2− ST
-Monetary market (C1− C2) − (P1− P2) I I I
Sum 0 I + K2− K1 I + K2− K1 I + K2− K1
Where I is the future cash flow in monetary market, from the borrowing of the initial cost or investing of the initial gain:
I = [(P1− P2) − (C1− C2)]er(T −t) [3.19] The arbitrage strategy has no initial cost. On the trading day, the arbitrager longs a bullish vertical call spread and a bearish vertical put spread.
Depending on different spot prices, three situations are possible:
1. When ST ≤ K1, both the put options will be exercised and neither the call options will be exercised. The cash flow at maturity is:
Π = [(P1− P2) − (C1− C2)]er(T −t)+ K2− K1 [3.20] From the conversion of equation [3.15], we have Π > 0.
2. When K1 < ST ≤ K2, only the two long positions will be exercised. The cashflow at maturity is the same as equation [3.20], which is still positive.
3 – Methodology
3. When ST > K2, both the call options will be exercised and neither the put options will be exercised. The cash flow at maturity is the same as equation [3.20], which is still positive.
Hence, the initial cost of the portfolio is zero but in any possible state, the future payoff of this portfolio is positive. At expiration date, the arbitrager should always buy one contract of SSE50 ETF as price K1, sell it as price K2 in the option market and also balance the financing in the monetary market.
3.3.2
Upper boundary
Similar to the lower boundary, equation [3.16] is the upper boundary of the box spread strategy, considering the investment strategy of a short position on a bullish vertical call spread and a short position on a bearish vertical put spread. The initial cost will be (C2− C1)−(P2− P1). A negative value means an initial gain. The value of the box spread is always (K1− K2)e−r(T −t).
When the upper boundary doesn’t hold, the cash flow of this investment strategy is shown in Table 3.4.
Table 3.4. Trading strategy of box spread (upper boundary)
Date Trading date t Expiration Date T
Strategy ST ≤ K1 K1< ST ≤ K2 ST > K2
Short a call option with strike price
K1
C1 - K1− ST K1− ST
Long a put option with strike price
K1
−P1 K1− ST -
-Long a call option with strike price
K2
−C2 - - ST − K2
Short a put option with strike price
K2
P2 ST − K2 ST− K2
-Monetary market (P1− P2) − (C1− C2) I I I
3.3 – Box spread
Where I is the future cash flow in monetary market, from the borrowing of the initial cost or investing of the initial gain:
I = [(C1 − C2) − (P1− P2)]er(T −t) [3.21] The arbitrage strategy has no initial cost. On the trading day, the arbitrager shorts a bullish vertical call spread and a bearish vertical put spread.
Depending on different spot prices, three situations are possible, but similarly to what we have discussed about lower boundary, in each state, the portfolio has the same cashflow at maturity, which is:
Π = [(C1− C2) − (P1− P2)]er(T −t)+ K1− K2 [3.22]
Also, at the expiration date, the arbitrager should always buy one contract of SSE50 ETF as price K2, sell it as price K1 in the option market and also balance the financing in the monetary market.
According to this strategy, the initial cost of the portfolio is zero but in any possible state, the future payoff of this portfolio is positive.
Chapter 4
Data and descriptive statistics
4.1
Data selection
Using the Choice database1, we select the intraday data of the in-trading Shanghai
Stock Exchange 50 ETF option prices including the April expiration contracts, May expiration contracts, June expiration contracts and September expiration contracts including call options and put options during 30th March 2017 and 14th April 2017. This sample period contains 10 trading days of 68 types of contracts (exclude the 1st, 2nd, 8th, 9th April 2017 because of weekends, exclude 3rd, 4th April 2017 because of national holidays). The data includes the expiration date, strike price, closing price, settlement price and trading volume per day. We also exclude the data with 0 trading volume or 0 price (in fact, the sample shows that 2 options are excluded, so the number of contracts changes to 66).
1Choice database is a trading software for Chinese stocks and derivatives. It also provides
authority financial trading data. Because of the particularity of Chinese financial market, there is no such well-known database as Chicago Board Options Exchange (CBOE) database. So we choose to use open market data from a recognized database.
4 – Data and descriptive statistics
4.2
Assumption of transaction cost
The transaction cost of Shanghai Stock Exchange 50 ETF option trading expe-rienced an adjustment on 1st November 2016. The brokerage of Shanghai Stock Exchange decrease from 2CNY to 1.3CNY for each option contract. The exercise cost is 0.6CNY for each option contract2. The settlement cost by China
Securi-ties Depository and Clearing Company Limited is 0.3CNY for each option contract. During the transaction, the security company will also charge investors the com-mission charge for each transaction. This amount of money is different for different security companies. In general, only the long position will be charged by the secu-rity company while the transaction of a short position is for free. According to the commission charge rules of some most famous security companies, the transaction costs in the option market are shown in Table 4.1.
Table 4.1. Transaction costs in the option market
Exercise Not exercise
Long position 2.5 1.9
Short position 1.6 1.6
All the Shanghai Stock Exchange 50 ETF options should be exercised by physical delivery. So we need to long the SSE 50ETFs in the ETF market for a short call or a long put in the SSE 50ETF option market, and short the SSE 50ETFs in the ETF market for a long call or a short put in the SSE 50ETF option market. So the transaction costs for the trading in SSE 50 ETF secondary market should be taken into consideration as well. In practice, one option contract stands for 10,000 ETF contracts. Similarly to trading in the stock market, the stamp tax is not asked but 0.3% commission charge is asked from the security company. So we assume the
2For a short position, the trader only has the obligation but not the right on the contract, so
4.2 – Assumption of transaction cost
transaction cost in the ETF market is 5CNY per ETF option contract of ETF. Hence, the total transaction costs for each arbitrage strategy are:
1. Butterfly spread:
(1) Butterfly spread using calls:
• Π1(ST ≤ K1): No positions will be exercised. So the total cost is 2 × 1.9 + 2 × 1.6 = 7CNY ;
• Π2(K1 < ST ≤ K2): Only the long position with strike price K1 will be exercised and the arbitrager should sell one contract of ETF in the ETF market. So the total cost is 2.5 + 1.9 + 2 × 1.6 + 5 = 12.6CNY ; • Π3(K2 < ST ≤ K3): The arbitrager should exercise his long position with strike price K1 in the option market and also buy one contract of ETF in the ETF market. So the total cost is 2.5 + 1.9 + 2 × 1.6 + 5 = 12.6CNY ;
• Π4(ST > K3): The arbitrager should exercise both the long position in the option market but doesn’t have to trade in the ETF market. So the total cost is 2 × 2.5 + 2 × 1.6 = 8.2CNY .
(2) Butterfly spread using puts:
• Π1(ST ≤ K1): The arbitrager should exercise both the long position in the option market but doesn’t have to trade in the ETF market. So the total cost is 2 × 2.5 + 2 × 1.6 = 8.2CNY .
• Π2(K1 < ST ≤ K2): The arbitrager should exercise his long position with strike price K3 in the option market and also sell one contract of ETF in the ETF market. So the total cost is 2.5 + 1.9 + 2 × 1.6 + 5 = 12.6CNY ;
• Π3(K2 < ST ≤ K3): Only the long position with strike price K3 will be exercised and the arbitrager should buy one contract of ETF in the
4 – Data and descriptive statistics
ETF market. So the total cost is 2.5 + 1.9 + 2 × 1.6 + 5 = 12.6CNY ; • Π4(ST > K3): No positions will be exercised. So the total cost is
2 × 1.9 + 2 × 1.6 = 7CNY ;
2. Box spread:
For box spread, the situations of lower boundary and upper boundary are quite similar, and whatever the situations is, the arbitrager would only trade in the option market, which means no transaction costs in the ETF market will be charged.
• When ST ≤ K1, one of the long positions and one of the short positions will be exercised, but the arbitrager should only pay for the long position exercising cost. So the total cost is 2.5 + 1.9 + 2 × 1.6 = 7.6CNY ; • When K1 < ST ≤ K2, there is a little bit difference for lower and upper
boundary.
1) Lower boundary
Both the long positions will be exercised while the short positions will not be exercised. So the total cost is 2 × 2.5 + 2 × 1.6 = 8.2CNY ; 2) Upper boundary
On the contrary, the long positions will not be exercised. The total cost is 2 × 1.9 + 2 × 1.6 = 7CNY .
• When ST > K2, one of the long positions and one of the short positions will be exercised. So the total cost is 2.5 + 1.9 + 2 × 1.6 = 7.6CNY ;
4.3
Risk free rate and opportunity cost
The risk free rate is used to calculate the future value of the investment in monetary market and the opportunity cost for the margins of the positions on put and call
4.4 – Descriptive statistics
options. Considering the current situation of Chinese financial market, we select the Shanghai Interbank Offered Rate (SHIBOR) as the risk free rate. SHIBOR is a changing rate that is also different for different durations. Considering the location of the monetary market, SHIBOR should be the most accurate to measure the opportunity cost. When we calculate the opportunity cost of the margins, the margin is a market-to-market value which is changing along the time, so we use the initial margin as a constant margin number to simplify the calculation. However, the initial margins are different for different option contracts. The margin mainly depends on the price of the contract so the opportunity costs of the arbitrage strategies are different in each case. We will show the specific ones in the empirical test part. Based on the margin cost and the risk free rate, the formula of the opportunity cost is:
Opportunity cost = (e(SHIBOR rate×t)−1) × Margin cost [4.1]
4.4
Descriptive statistics
An overview of the sample data is shown in table 4.2:
Table 4.2. Descriptive statistics of SSE 50ETF options sample
Expiration Month
Observation Number June September April May
Call Options 160 60 60 50
Put Options 160 60 60 50
Sum 320 120 120 100
The sample period is 30th March 2017 to 14th April 2017, including 10 trading days. In each trading day, there are 66 different option contracts with different expiration dates and strike prices. The observation number is 660 in total. Each call option has a corresponding put option with same strike price and expiration
4 – Data and descriptive statistics
date. For a set of contracts with same expiration date, mostly, the interval of the strike price is 0.05CNY3.
The expiration date for April option is 26th April 2017. The interval of the strike prices is from 2.25CNY to 2.5CNY with 6 × 2 types of contracts.
The expiration date for May option is 24th May 2017. The interval of the strike prices is from 2.25CNY to 2.45CNY with 5 × 2 types of contracts.
June option has a highest proportion in the sample with expiration date on 28th June 2017. The interval of the strike prices is from 2.153CNY to 2.55CNY with 16 × 2 types of contracts, but differently than other months options, the interval of the strike price is not a fixed number.
The expiration date for September option is 27th September 2017. The interval of the strike prices is from 2.25CNY to 2.5CNY with 6 × 2 types of contracts.
3For example, the April call option has 6 different contracts with strike prices 2.25CNY, 2.3CNY,
Chapter 5
Empirical test
5.1
Butterfly spread
By using the result of the methodology analysis, we should find three call options with the same expiration date that correspond to the butterfly spread using calls strike price conditions. The conditions are:
K1 < K2 < K3, K1+ K3 = 2K2, C1 + C3−2C2 ≤0 [5.1] Or we should find three put options with the same expiration date that corre-spond to the butterfly spread using puts strike price conditions. The conditions are:
K1 < K2 < K3, K1+ K3 = 2K2, P1+ P3−2P2 ≤0 [5.2] In practice, for March, May, September contracts, it’s very easy to find 3 options that correspond to K1 < K2 < K3, K1+ K3 = 2K2 regularly, because the interval of the strike prices is always 0.05CNY; but this rule doesn’t exists in June contracts. There is no fixed interval on the strike prices if we sort them in a line. Particularly, there are 16 sets of strike prices that meet the requirement. We list them as following Table 5.1:
5 – Empirical test
Table 5.1. Strike prices of June contracts
K3/K2/K1 2.55/2.5/2.45 2.5/2.45/2.4 2.495/2.446/2.397 2.495/2.397/2.299 2.45/2.4/2.35 2.446/2.397/2.348 2.446/2.348/2.25(2 contracts) 2.4/2.35/2.3 2.397/2.348/2.299 2.35/2.3/2.25(2 contracts) 2.348/2.299/2.25(2 contracts) 2.3/2.25(2 contracts)/2.2
Firstly, we don’t take transaction costs into consideration to find any possible arbitrage opportunities. After one empirical test, based on the butterfly spread arbitrage strategy using calls, we find that only two of them exist in the sample. The first opportunity of butterfly spread using calls is shown in Table 5.2:
Table 5.2. The first arbitrage of butterfly spread using calls
Trading Date 12.04.2017 Expiration Date 28.06.2017
Contract Name XD50ETF June
Call 2397A XD50ETF June Call 2446A XD50ETF June Call 2459A Contract Code 10000739.SH 10000747.SH 10000765.SH Strike Price 2.397 2.446 2.495 Closing Price 0.0307 0.0171 0.0090
Trading Strategy Long Short Long
Arbitrage Condition 2 × 0.171 − (0.307 + 0.0090) = 0.026 > 0 Trading Cost 7 or 12.6 or 8.2 Margins / 2380.6 / SHIBOR 4.2653% Opportunity Cost / 21.798 / Total Cost 34.398
According to the table above, we can long the call option with strike price 2.397 and the call option with strike price 2.495, short two call options with strike price 2.446 to do arbitrage. Without considering the transaction costs, we can gain 0.026× 10000 = 260CNY for one trade.
Taking transaction costs into consideration, we need to pay the margins first. According to the assumptions we made in the methodology part, we only take the
5.1 – Butterfly spread
initial margin as the total margins, and we only have to pay the margin for the short positions on the call options1. In this example, based on the equation we have from
the contract content, the margins are:
(0.0207+Max(12%×2.3780−Max(2.446−2.3780,0),7%×2.3780)×10000 = 2380.6 [5.3]
Where 0.0207 is the ex-settlement price of the option contract, 2.3780 is the ex-closing price of the underlying asset (SSE 50 ETF), and 2.446 is the strike price of the option contract.
To calculate the opportunity cost, we need the appropriate SHIBOR rate as the risk free rate. Considering the trading date and the expiration date, we choose to use the 3 months SHIBOR rate on 12.04.2017, which is 4.2653%. So the opportunity cost will be (e4.2653%×78/365−1) × 2380.6 = 21.798CNY .
Actually, the trading cost part is quite tricky. Depending on the spot price at the expiration date, the trading cost will be different. But we could not know in which situation we would be when we try to do arbitrage on the trading day. So using the highest trading cost on the list will be a secure way to calculate the total cost. Therefore, the total cost will be 12.6 + 21.798 = 34.398CNY , which is still far less than the total gain we will have.
The second opportunity is shown in Table 5.3:
1For the long positions on the call, the trader only has the right but not the obligation to
5 – Empirical test
Table 5.3. The second arbitrage of butterfly spread using calls
Trading Date 30.03.2017 Expiration Date 27.09.2017
Contract Name 50ETF September Call 2350 50ETF September Call 2400 50ETF September Call 2450 Contract Code 10000845.SH 10000846.SH 10000847.SH Strike Price 2.35 2.440 2.45 Closing Price 0.0816 0.0639 0.0455
Trading Strategy Long Short Long
Arbitrage Condition 2 × 0.0639 − (0.0816 + 0.0455) = 0.0007 > 0
Trading Cost 7 or 12.6 or 8.2
As shown above, on 30.03.2017, we have another possible arbitrage opportunity on the September contracts. The gain within the trading would be 0.0007×10000 = 7CNY , but if we take transaction cost into consideration, the trading cost would be at least 7 CNY, not mentioning the opportunity cost of the margins. Thus, the arbitrage opportunity doesn’t exist until the total transaction cost is less than 7 CNY.
Then we used the same procedure on the put options, but unfortunately, we could not find any arbitrage opportunities.
Table 5.4 is the summary of the empirical test for butterfly spread arbitrage strategy.
Table 5.4. Arbitrage opportunity using butterfly spread
Contract Type Observation Number Arbitrage Opportunity Arbitrage Probability
Call Option 320 2 0.625%
Put Option 320 0 0%
Sum 640 2 0.3125%
The total observation number2 is 640 (320 each), but we only find 2 arbitrage
2The observation number is not the number of the options in the sample. It’s the number of the
sets of options that correspond to the basic butterfly spread conditions: K1< K2< K3, K1+K3=
5.2 – Box spread
opportunities in the sample, and one of them is actually costly, which shows that it’s very hard to do arbitrage in the real ETF market using butterfly spread, but on the another hand, as soon as we find one opportunity as in the example of the June call option on 12.04.2017, we could gain at least as much as 221.29 CNY for one trade, which is quite considerable compared to the initial cost.
According to the two arbitrage opportunities we found, one is the June contract with a 0.049CNY interval among the strike price, and the other one is September contract with a 0.05CNY interval. This result might suggest that the arbitrage opportunity is easier to appear in longer term contracts and within the shorter strike price interval contracts.
5.2
Box spread
When we use a box spread as the arbitrage strategy, we should find 2 sets of call and put options with same expiration date, 2 options for each set, and in each set, the call option and the put option should have the same strike price. The arbitrage condition is one of the two equations:
(C1− C2) − (P1− P2) ≤ (K2− K1)e−r(T −t) (C2− C1) − (P2− P1) ≤ (K1− K2)e−r(T −t)
[5.4]
Basically, the left hand of the equation represents the initial cost of the box spread, and the right hand represents the present value of the spread. Thus, we set the right hand of the equation positive, then choose one of the equations to use, but it’s still a quite complicated procedure to find all the possible arbitrage opportunities in the sample because the risk free rate r and the remaining time from expiration
T − tare changing among different trading dates and different contract types.
As in the procedure for the butterfly spread, we ignore the transaction costs in the first step to find all the possible arbitrage opportunities. The empirical results are quite surprising. We show two examples in Table 5.5 and 5.6.
5 – Empirical test
Table 5.5. The arbitrage example of box spread (1) Trading Date 05.04.2017 Expiration
Date 26.04.2017 Contract Name 50ETF April Call 2250 50ETF April Put 2250 50ETF April Call 2400 50ETF April Put 2400 Contract Code 10000867.SH 10000868.SH 10000859.SH 10000864.SH Trading
Strategy Long Short Short Long
Strike Price 2.2500 2.2500 2.4000 2.4000 Closing Price 0.1316 0.001 0.0136 0.029 SHIBOR 4.2525% Arbitrage Condition 0.1460 − (2.4 − 2.25) × e −4.2525%×22/365= −0.0036 < 0 Trading Cost 7.6 or 8.2 or 7 Margins / 1778.2 2470.2 / Opportunity Cost / 4.564 6.340 / Total Cost 19.1
In this example, we long the call option with strike price 2.25, short the put option with strike price 2.25, short the call option with strike price 2.4 and long the put option with strike price 2.4 to do arbitrage. The initial cost is (0.1316−0.0136)− (0.001 − 0.029) = 0.1460. We choose the one month SHIBOR rate on 05.04.2017 as the risk free rate, which is 4.2525%. So we can gain 0.0036 × 10000 = 36CNY for one trade. Actually, this is not a big number but it’s already the highest profit that we could gain from April contracts in this period.
Now we take transaction costs into consideration. We have already shown the calculation of the margins of call option earlier, now we show the calculation of the margins of the put option (50ETF April Put 2250 ).
M in(0.0011+Max(12%×2.3560−Max(2.3560−2.25,0),7%×2.25),2.25)×10000 = 1778.2
[5.5] Where 0.0011 is the settlement price of the option contract, 2.3560 is the ex-closing price of the underlying asset (SSE 50 ETF), and 2.25 is the strike price of the
