The Arbitrage Opportunities on ETF Options: the Case of Shanghai Stock Exchange 50

UNIVERSITY VAN AMSTERDAM

BUSINESS SCHOOL

Master Thesis

The Arbitrage Opportunities on ETF Options: The case of Shanghai Stock Exchange 50

Supervisor Candidate prof. Liang Zou Mu Xu ID: 11377070

Augest 2017

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.

iii 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 3.3.2 Upper boundary...... 32

iv 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 probability of options may have an inner relation with the remaining time to expiration.

vii 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 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. In these two periods of warrant trading, big pricing bubbles and the speculative

2 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 options

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

3 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 company 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),

3Investopedia: http://www.investopedia.com/terms/e/etf.asp

4 1.4 – ETF options which contains 50 sample stocks with the highest trading volume, the highest liquidity 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.

5 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 0.05CNY for strike price blow 3CNY (included), 0.1CNY for strike price strike 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 0.0001CNY unit Price change limi- The price changing limitation is based on the price change of tation the 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:

6 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

7 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

8 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

9 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 expiration 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 Call Option Volume Put Option Volume SSE50 ETF Total 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.

10 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 Call Option Volume Put Option Volume SSE50 ETF Total 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.

11 12 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 opportunity 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, including 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

13 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 accurate 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

14 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,

15 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 transaction price. They thought that option 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 trading 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 index 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

16 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 using butterfly spread. These studies used similar strategies but different options on different data sample period.

17 18 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 position 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

19 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

20 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 different 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

The total payoﬀ for butterﬂy spread using calls is shown on Figure 3.1.

21 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 equilibrium 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 options in the meantime then invest the total gain 2C2 − C1 − C3 in the monetary

22 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 −C1 - ST − K1 ST − K1 ST − K1 strike price K1 Short two options with 2C2 - - −2(ST − K2) −2(ST − K2) strike price K2 Long the option with −C3 - - - ST − K3 strike price K3 Invest in monetary C1 + C3 − 2C2 I I I I market Sum 0 I I + ST − K1 I + 2K2 − K1 − ST I

Where I is the future cash ﬂow of the investment in monetary market,

r(T −t) I = (2C2 − C1 − C3) × e [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 :

r(T −t) Π1 = (2C2 − C1 − C3) × e > 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:

r(T −t) Π2 = (2C2 − C1 − C3) × e + 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.

23 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:

r(T −t) Π3 = (2C2 − C1 − C3) × e + 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:

r(T −t) Π4 = (2C2 − C1 − C3) × e + 2K2 − K1 − K3 [3.6]

r(T −t) Knowing that 2K2 = K1 + K3, Π4 = (2C2 − C1 − C3) × e > 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.

24 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 equilibrium 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

P1 + P3 − 2P2 ≤ 0.

25 3 – Methodology

The arbitrager could short the overpriced option and long the undervalued options 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 −P1 K1 − ST - - - strike price K1 Short two options with 2P2 −2(K2 − ST ) −2(K2 − ST ) - - strike price K2 Long the option with −P3 K3 − ST K3 − ST K3 − ST - strike price K3 Invest in monetary P1 + P3 − 2P2 I I I I market Sum 0 I I + ST + K3 − 2K2 I + K3 − ST I

Where I is the future cash ﬂow of the investment in monetary market,

r(T −t) I = (2P2 − P1 − P3) × e [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:

r(T −t) Π1 = (2P2 − P1 − P3) × e − 2K2 + K1 + K3 [3.10]

r(T −t) Knowing that 2K2 = K1 + K3, Π1 = (2P2 − P1 − P3) × e > 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.

26 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:

r(T −t) Π2 = (2P2 − P1 − P3) × e + 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:

r(T −t) Π3 = (2P2 − P1 − P3) × e + 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 :

r(T −t) Π4 = (2P2 − P1 − P3) × e > 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:

P1 + P3 − 2P2 + 4tp + 2tpr ≥ 0 [3.14]

27 3 – Methodology

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).

Figure 3.3. Proﬁt for bull spread using calls

28 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

If all the options are fair priced, a box spread should satisfy two equations without considering the transaction costs:

−r(T −t) (C1 − C2) − (P1 − P2) + (K1 − K2)e ≥ 0 [3.15]

29 3 – Methodology

and

−r(T −t) (C2 − C1) − (P2 − P1) + (K2 − K1)e ≥ 0 [3.16]

Taking transaction costs into consideration, the equations change to:

−r(T −t) (C1 − C2) − (P1 − P2) + (K1 − K2)e + 2tc + 2tp + tpr + tcr ≥ 0 [3.17]

and

−r(T −t) (C2 − C1) − (P2 − P1) + (K2 − K1)e + 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.

30 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 −C1 - ST − K1 ST − K1 K1 Short a put option with strike price P1 ST − K1 - - K1 Short a call option with strike price C2 -- - K2 − ST K2 Long a put option with strike price −P2 K2 − ST K2 − ST - K2 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:

r(T −t) I = [(P1 − P2) − (C1 − C2)]e [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:

r(T −t) Π = [(P1 − P2) − (C1 − C2)]e + 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.

31 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 −r(T −t) of the box spread is always (K1 − K2)e . 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 C1 - K1 − ST K1 − ST K1 Long a put option with strike price −P1 K1 − ST - - K1 Long a call option with strike price −C2 - - ST − K2 K2 Short a put option with strike price P2 ST − K2 ST − K2 - K2 Monetary market (P1 − P2) − (C1 − C2) I I I Sum 0 I + K1 − K2 I + K1 − K2 I + K1 − K2

32 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:

r(T −t) I = [(C1 − C2) − (P1 − P2)]e [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:

r(T −t) Π = [(C1 − C2) − (P1 − P2)]e + 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.

33 34 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.

35 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 commission 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 security 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 the trader has no exercise cost

36 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

37 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

38 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

39 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, 2.35CNY, 2.4CNY, 2.45CNY and 2.5CNY.

40 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 correspond 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:

41 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 XD50ETF June XD50ETF June XD50ETF June Contract Name Call 2397A Call 2446A 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

42 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 exercise.

43 5 – Empirical test

Table 5.3. The second arbitrage of butterﬂy spread using calls

Trading Date 30.03.2017 Expiration Date 27.09.2017 50ETF 50ETF 50ETF Contract Name September Call September Call September Call 2350 2400 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 = 2K2

44 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:

−r(T −t) (C1 − C2) − (P1 − P2) ≤ (K2 − K1)e [5.4] −r(T −t) (C2 − C1) − (P2 − P1) ≤ (K1 − K2)e 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 − t are 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.

45 5 – Empirical test

Table 5.5. The arbitrage example of box spread (1)

Expiration Trading Date 05.04.2017 26.04.2017 Date Contract 50ETF April 50ETF April 50ETF April 50ETF April Name Call 2250 Put 2250 Call 2400 Put 2400 Contract Code 10000867.SH 10000868.SH 10000859.SH 10000864.SH Trading Long Short Short Long Strategy Strike Price 2.2500 2.2500 2.4000 2.4000 Closing Price 0.1316 0.001 0.0136 0.029 SHIBOR 4.2525% Arbitrage 0.1460 − (2.4 − 2.25) × e−4.2525%×22/365 = −0.0036 < 0 Condition Trading Cost 7.6 or 8.2 or 7 Margins / 1778.2 2470.2 / Opportunity / 4.564 6.340 / Cost 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 ).

Min(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 ex-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

46 5.2 – Box spread option contract. So the opportunity cost for the put option would be (e4.2525%×22/365− 1) × 1778.2 = 4.564CNY .

According to the assumption of the transaction cost, the trading cost could be one of 7.6, 8.2 or 7CNY. We still choose the highest one to calculate the total cost, which would be 19.1 < 36. So the arbitrage strategy here is still proﬁtable when the transaction cost is taken into account.

Table 5.6 shows another arbitrage example:

Table 5.6. The arbitrage example of box spread (2)

Expiration Trading Date 14.04.2017 24.05.2017 Date Contract 50ETF May 50ETF May 50ETF May 50ETF May Name Call 2250 Put 2250 Call 2300 Put 2300 Contract Code 10000869.SH 10000874.SH 10000870.SH 10000875.SH Trading Long Short Short Long Strategy Strike Price 2.2500 2.2500 2.3000 2.3000 Closing Price 0.1065 0.0038 0.0638 0.0106 SHIBOR 4.2634% Arbitrage 0.0495 − (2.3 − 2.25) × e−4.2634%×41/365 = −0.0003 < 0 Condition Trading Cost 7.6 or 8.2 or 7

In this trading strategy, we could gain 0.0003 × 10000 = 3CNY proﬁt for one trade but, obviously, it could not cover the trading cost. Actually, there are a lot of similar situations in our empirical test results. Only if the transaction cost is low enough, the real arbitrage opportunity is present.

Table 5.7 is the empirical test result sorted by contract types (month):

47 5 – Empirical test

Table 5.7. Empirical test result of box spread

Observation Arbitrage Arbitrage Average Highest Median Month Number Opportunity Probability Profit Profit Profit April 150 94 62.67% 10.9 36.16 8.56 May 100 32 32% 4.4 14.56 3.4 June 1200 294 24.5% 6.99 45.57 5 September 150 23 15.33% 6.71 21.65 5.63 Sum 1600 443 27.69% 7.55 45.57 5.57

As shown above, April contracts seem to have highest probability of arbitrage using box spread. More than half of the observations3 are a “possible” arbitrage opportunity. Furthermore, The average profit and the median profit of all the arbitrage opportunities in April contracts are higher than the 8.2CNY trading cost, which means the trader would have a very high possibility to make profit in the real market. Basically, this result is related to the trading period we choose. The trading days we choose are mostly in April, which means the April option contracts are very close to expiration. That allows a higher probability to do arbitrage. On the opposite hand, the contracts in May and September seem to have high possibility for arbitrage. The May contracts have the lowest average profit and median profit, which means they have the highest possibility to be a “fake” arbitrage opportunity, and the profit a trader makes from the initial trading could not cover the transaction cost. In the meanwhile, there are the fewest possible arbitrage opportunities in September. The percentage is only 15.33%. The highest profit you can make in this period using box spread comes from June contracts which is 45.57CNY. The June contracts have the highest observation number but the average profit and the median profit are quite low as well.

3The observations are not the April Option contracts here. Same as the “observation number” we use in butterﬂy spread, the observation number we use in the above table is the number of the sets of options that could make a box spread, in other words, corresponding with the basic box spread conditions.

48 Chapter 6

Conclusion

6.1 Results

Firstly, this thesis described the background of Shanghai Stock Exchange 50 ETF Option especially the development of option in the Chinese market. As the first option product in the real sense in the Chinese financial market, the importance of this product is quite obvious, and, subsequently, the pricing of this option is very important for the Chinese market. This research also points out the peculiarity of the SSE50 ETF Option, which is an European option while all the other ETF options in the world are American. Based on the fact that the trading of Shanghai Stock Exchange 50 ETF Option is quite active thus far and is becoming more and more dynamic. This research chose two widely used arbitrage strategies from the Law of One Price, the butterfly spread and the box spread, to test the rationality of the SSE50 ETF Option pricing.

We introduced in detail the trading process and arbitrage principles of the two strategies in the article. To use these two arbitrage strategies on the real trading data of Shanghai Stock Exchange 50 ETF Option, we set several rules of transaction cost, margin cost, risk free rate and opportunity cost based on the real trading regulations

49 6 – Conclusion by Shanghai Stock Exchange and China Securities Regulatory Commission to make the empirical test closer to the real market. We also made several assumptions to simplify the calculations. Finally, we got two totally different empirical results from the two arbitrage strategies. When we use butterfly spread, we found only two possible arbitrage opportunities in the whole period in the sample date. After considering the transaction costs, we can make real profit only from one of them, although the profit is as attractive as 225.6CNY. In the empirical test of box spread, we couldn’t find such a big profit by any chance, but the overall arbitrage opportunities are quite considerable. More than 25% of the box spread structure options are possible arbitrage combinations and the average arbitrage profit reaches 7.55CNY when the number of sets is as large as 443. Especially for the “in the month” contracts, more than 60% of the combinations are profitable and the average gain exceeds the trading costs when the opportunity cost is not taken into consideration. We found some very interesting phenomenons according to the empirical test of the Shanghai Stock Exchange 50 ETF Option trading data. Basically, the more complicated the arbitrage strategy we use, the easier we could find the arbitrage opportunity. The box spread arbitrage strategy is obviously more complicated and more difficult to implement than butterfly spread. The number of possible arbitrage structure combinations of box spread in the market are also much bigger than those of butterfly spread (1600 compared to 640). This increases the difficulty of finding the real profitable arbitrage combinations so that the operation complexity increases. It may be a signal that the product pricing is substantially rational and fair in the Chinese financial market or at least in the SSE50 ETF Option market. It increases the threshold of the financial market but in the meanwhile, doesn’t cut off the operability, the profitability and the appeal of the financial market. On the other hand, the difficulty of doing arbitrage does not correspond to the profitability. According to the empirical test, we could gain 225.6CNY for one trade

50 6.2 – Future research using a butterfly spread arbitrage strategy while we could only gain less than 40CNY at most when we use the box spread even if the operation complexity is higher in the box spread arbitrage strategy. This fully shows the uncertainty and the randomness of the financial market especially of the derivatives market. Another interesting phenomenon we can tentatively illustrate is that the arbitrage probability of options has an inner relation with the remaining time to expiration. In the sample, the April contracts have a significantly higher probability to do arbitrage when we use box spread while the sample period are mostly trading days in April. The arbitrage effect of the “in the month options” would be an interesting topic to investigate in further studies. We cannot draw a similar conclusion from butterfly spread because the possible arbitrage combinations are too few.

6.2 Future research

However, the study has some limitations. I didn’t fully take the time value of the currency into consideration, especially regarding the transaction costs. With the purpose of simplifying the calculations, I did several assumptions such as using initial margins instead of the total margin cost and ignoring the maintenance margins, but in the real market, any detailed transaction cost would make an arbitrage operation unprofitable. Furthermore, I only chose 10 consecutive trading days and 66 option contracts in the market to build the analysis sample so the sample size could be not big enough to find the regular patterns. Also, in the sample data, there are some XD options in the June contracts such as XD50ETF June Call 2153A. This means the option is experiencing the dividend exclude. I didn’t add the influence of the dividend excluding day into the analysis, and as we mentioned earlier, the arbitrage effect of the “in the month options” is one of the tentative illustration in the article which needs future researches.

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