The Effect on Volatility of Stock Market After Launching Stock Index Options based on Information Structure of Stock Index Options

Transcription

1 The Effect on Volatility of Stock Market After Launching Stock Index Options based on Information Structure of Stock Index Options 1 Shangmei Zhao, Guiping Sun, 3 Haijun Yang 1,3 School of Economics and Management, Beihang University, Beijing, China, zsleng@sdu.edu.cn, navy@buaa.edu.cn *,Corresponding Author School of Economics and Management, Beihang University, Beijing, China, sgping@163.com Abstract In this paper, the stock market and stock index options market were built through the method of the agent-based computational finance, and the effect on volatility of stock market after launching stock index options was researched from the angle of information structure of stock index options. The results show that the introduction of stock index options affects obviously the stock market. When the information of stock index options for the stock market is little, the volatility of stock market becomes more violent than before the introduction of stock index options, and the stock market liquidity decreases, but with more information of stock index options being introduced, the stock market become more stable and the liquidity increases. Also the size of information weight affects the stability of stock market and asymmetry of stock return volatility. 1. Introduction Keywords: Agent-based, Information Structure, Stock Index Options Along with the quick development of the global financial derivative, stock index options also develop quickly. options are useful tools for investment and avoiding risks of financial market, and they play an important role in the improvement of whole financial derivative market. Now the countries where their own stock index options can trade are America, Germany, France, Korea, India, Hongkong, Taiwan and so on. America is the cradle of stock index options, and Korea is the country which stock index options trade volume is the largest in the world. In Apr, 1 China launched own CSI 3 stock index futures contracts, and stock index futures market develops quickly. According to experience of other countries, stock index futures and stock index options are two important tools for investment and avoiding risks in the financial market, shortly after the stock index futures were launched the stock index options were introduced. Now China is researching the stock index options and will launch them at the right moment. The effect of launching stock index options on stock market is an important problem which attracts the attention of academia, business and regulator and has been researched by many country academics. Danielsen et al. [1] researched the effect on stock market liquidity after launching stock index options in the American stock market and found that due to launching stock index options the market quality is improved and stock market liquidity increases. Sorescu[] researched the effect on American stock market after the introduction of stock index options. The data used in his paper during was split into two sections, the results shows that because the trade of stock index options improves the transfer of disadvantageous information, the return of stock market is positive during , the return is negative during Kumar et al. [3] researched the effect on quality of stock market after the introduction of stock market options trade in American market, they found that stock market options could decrease the spread of stock price, and increase market depth, trade frequency, trade volume and deal size. Xiong et al. [4] took KOSPI index options for instance and researched the effect on volatility of stock market and stock index futures market after launching stock index options, the result show that after the KOSPI index options is introduced, the effect on volatility and asymmetry of KOSPI index and KOSPI index futures is obvious and asymmetry of KOSPI index become bigger. Kang et al. [5] also took KOSPI index options for study sample, used the intra-day data of KOSPI index options, researched that the information which is contained in net buying pressure, Advances in information Sciences and Service Sciences(AISS) Volume5, Number4, Feb 13 doi: /AISS.vol5.issue

2 the result show that net buying pressure can make the liquidity of call option increase and the liquidity of put option decrease. Wei et al. [6] form the market efficiency angle researched the relation among the stock index options, stock index futures and stock index in the Hongkong HSI market, the research shows that HSI options and HSI futures play major role in the discovery of market price, the HSI options improve the arbitrage mechanism, and increase the liquidity of futures market and spot market. The research methods used in the above papers are traditional econometrics methods, the problems are researched from the view of empirical. Along with the continuous discovery of stock market anomalies, because of limits of hypothesis and research method existing in traditional financial economics theory, the solution of these problems faces unprecedented challenges. These existing limits make financial economist innovate in epistemology and methodology. The agent-based method can be used for complexity research of many areas [7, 8]. The agent-based computational finance is developed from the complexity research of finance system, and it breaks away from the limit of analytical thinking and become more and more popular in the researchers of economy and finance. The agent-based computational finance has grown quickly this decade, and this area has attracted many researchers from such diverse areas as computer science, physics and psychology etc. After the agent-based computational finance first appears, many artificial stock market models have been developed and used to research many financial problems. Along with the quick development of agent-based computational finance, model also quickly develops from small strategy model [9] and dynamic learning model [1] to complex agent-based model, the latter have SFI-ASM (Santa Fe Institute Artificial Stock Market) [11, 1], SUMWeb [13] and Genoa artificial market [14] etc. The research of interaction effects between options market and stock market remains small from the agent-based computational finance angle, Sabrina etc. [15] made pioneering research in effect on stock market after launching the options trade based on Genoa artificial market, the research shows that the introduction of stock option can decrease the volatility of and liquidity of stock, the wealth of investors is obviously influenced by using the option investment. In this paper the stock market and stock index options market were built through the method of the agent-based computational finance based on SFI-ASM, here single stock is not considered, but whole stock market is considered. The interaction mechanism between the stock index options and the stock market is built by way of information transfer of the stock index options. The effect of stock index options on volatility of stock market is researched from the angle of information structure of stock index options and the effect of price discovery of stock index option on stock market is also researched.. Model.1. module The model used in this paper consists of two parts: stock index module and stock index options module. SFI-ASM is used to simulate the stock index, the stock index is a method of measuring the value of a section of the stock market. It is computed from the prices of selected stocks. The purpose of this paper is to study the effect of stock index options on stock market, not to study the stock index itself. So in order to meet the need of simplifying the research process, because the stock index can represents the whole stock market, in this paper the stock index is treated as single stock, the investors can trade stock index as if these investors trade stock. The change of stock index represents the change of whole stock market. The price of stock index is stock index itself, the dividend of stock index represents the dividend of whole stock market, and the trade volume of stock index represents the trade volume of whole stock market. In the stock index module there is a riskless asset besides stock index and the both assets can are traded by investors. Riskless asset pay fixed 1% interest in every period. The external information of model is only stock index dividend which is generated by lag-one autoregressive stable stochastic process in every period, the expected value and of dividend do not change with the time. At the beginning, every agent is given a certain wealth, after the model running so as to avoid the agent wealth quick increase and keep the relative balance, every agent needs to pay 1% property tax, the property equals to the sum of riskless asset and risky asset. Every agent has 1 forecast rules to construct the investment strategy set. Every rule is made up of 3 parts which are market condition, forecast parameters and forecast. The market condition 851

3 represents the status of the time series of stock index and stock index dividend in history. When every agent forecasts the stock index and stock index dividend in the next time, the rule with smallest forecast is chosen from their own rules suited for market status, and the best rule is used to forecast the stock index and stock index dividend in the next time. Et ( pt d ) 1 t 1 a( pt dt ) b (1) Where a and b are forecast parameters, pt and d t are stock index and stock index dividend in the current time, respectively. After the agents got the forecast parameters of the stock index and stock index dividend in the next time, they maximize the expected utility value of asset in the next time to get the most investment return. The expected utility function of agent is constant absolute risk aversion. The proper percentage of asset allocation between riskless asset and risky asset is got through the maximization of the expected utility value of asset. The proper percentage is related to stock index in current time. Market maker continuously adjusts the stock index to get balance between demand and supply in stock market, and at last the stock index and stock index trade volume in the current time are got. In the model agents have artificial intelligence, they can periodically adjust investment strategy set to adapt to the external changes. The artificial intelligence is achieved through genetic algorithm. Every once in a while, agents choose rules which are not frequently used or which is big from the rule set, these rules are replaced by new rules which are generated by genetic algorithm. The size of forecast rules remains same, and the execution period of genetic algorithm represents the learning speed of response to external changes. In the model, the expected value of dividend is 1, the of dividend is.74, lag-one autocorrelation coefficient is.95. The number of stock index trade equals to the number of agents, short selling is allowed. The initial asset of every agent is, the execution period of genetic algorithm is 5... options module In the model two options can be traded, European call option and European put option, their underlying assets are stock index. According to the actual situation, the length of option s lifetime equals to 6 days, here one day is defined as model running one cycle. In the model agents can trade stock index options and stock index. The option trade agents are dividend into three trade types in the model; they are respectively random option trader, speculation option trader and hedge option trader. Random option trader represents the noise trader existing in the real market, their investment decisions are irrational and erratic. Hedge option trader is agent who wants to cover stock market risks through holding option contract. In the model they can buy put option or sell call option to lock market risks. Agents adjust continually the holding options to make the stock index position equal to the sum of all option contracts, so the stock risk can be covered. Speculation option trader is entirely opposite to hedge option trader, these agents will choose to hold option contract same with the holding stock index position. So it can eliminate the quantitative restriction of investment, in case of the fixed assets amount, if the asset purchases are very big, the asset price will increase quickly. Speculation option trader will earn additional profit, but they will also face more serious risk in the stock fluctuation. In the model the stock index options market maker is introduced, the marker collects the option demands of all agents and accomplishes the option trade. In our model the maker does not buy and sell options to participate the option trade, it only plays a role of medium. The price of options is decided through matching mechanism, which means that the price and volume of stock index option can be got from an intersection between supply and demand curves of the stock index options. When agents place stock index option trade order, the amount of order is decided through holding stock index position, the price of order is computed through B-S formula, the volatility used in B-S formula is predicted by agents, and different option trade type uses different forecast method of volatility. At the expiring day if the option is in-the-money, it will be exercised by the maker. But if the sellers of the call options or the buyers of the put options has not enough stock index asset to accomplish the option trade, the rest is accomplished through the riskless asset exchange. trade and stock index option trade are not carried out simultaneously, the stock index is firstly traded and then the stock index option is traded. By doing this, the effect of stock index options information 85

4 on stock market can be well studied..3. The introduction of stock index options trade information In our model, the effect of stock market on stock index options is obvious. During the construction of the effect mechanism of stock index options on stock market, because of SFI-ASM defect, this paper will study the effect from another point of view. The information has important value in the capital market, SFI-ASM model is stock market driven by dividend information which plays a decisive role in SFI-ASM. In the capital market the options and futures play an important role in asset price discovery, the option price and the futures price is a good indication of spot market price. In our model the price discovery of option is introduced. In our model the trade information of stock index options include the options price and options trade volume, if the price or volume of stock index call option increase in a sequential two-period, it will be good news for the stock market and increase expectation of the stock index increase in the next time and vice versa. The effect of stock index put option is opposite with stock index call option. In our model the trade information of stock index options is added to the formula 1. E t ( 1 p1, t 1 d1, t1) ( a t )( pt dt ) b Where t 1 is the integrated information of stock index options market by previous time. ()... ) (3) t1 ( 1 t1 t s ts Where is weight coefficient and is positive value, the bigger the value, the greater the effect,,..., after many experiments the value is set to.. 1 s are weight coefficients of different times,... 1, the subscript s is the time length of option information used in model. 1 s, t1.5 t-1,1.5 t1,.5 t1,3. 5 t1,4 (4) Where t-1, 1, t-1,, t-1,3 and t-1,4 represent the price information of call option, the trade volume information of call option, the price information of put option and the trade volume information of put option, respectively. The importances of the above information are same in the agent decision, so the weight coefficients are.5, and the plus-minus signs of different options are opposite. The price or trade volume of stock index options increase in a sequential two-period ( i 1,,3, 4 )is 1, the price or trade volume of stock index options decrease in a sequential t-1,i two-period ( 1,,3, 4 t-1, i 3. The Experiment Results 3.1. The stock index market i )is -1, In other cases, t-1, i ( 1,,3, 4 i )is. An important feature SFI-ASM model is that if all agents are homogeneous, over time, the model will reach rational expectation equilibrium. In this time, the asset price is only a linear function of dividend in the current time and unrelated to all the information of stock market in the past time, the fluctuation trend of asset price is same as fluctuation trend of dividend. All agents can predict accurately the asset price, by holding assets agents can only get the dividend income of asset, the other income from the price fluctuation is very small, the trade volume of asset is low. In SFI-ASM model the default number of agents is 5, when the model is used to simulate the stock index, the more number of agents and the more number of stock index shares are needed. In order to study the effect of the number of agents on stock market in the experiments, the number of agents will be increased from 5 to 5, and the number of stock index shares also is increased from 5 to

5 Table 1. The features of stock market under different number of agents * kurtosis * Magnitude is 1.e-5 The statistics of table 1 are computed with continuous 15 data after the model starts to stabilize. The table shows that as the number of agents increases, stock index decreases, but the other statistics increases except for return kurtosis slight decrease at,5 and 5. When the number of agents is 5, the stock index has small fluctuation, trade volume is very small, the return has not obvious peak and fat tail, the stock market is in the rational expectation equilibrium. With the number of agents increase, in the stock market the status of rational expectation equilibrium is broken, the reason is that the model is nonlinear system, the increase of agents makes the model complexity increase, a small disturbance is amplified many times to lead to stock market volatility increase. The result also shows that although the volatility of stock market increase as the increase of agents, from the point of view of and of stock index and return, the model is still kept at a relative stable state and the model is robust. 1 1 a 8 7 b c 5 d Figure 1. The time series of the stock market (a: stock index b: trade volume c: return d: forecast ) In reality, the real stock market is not in the rational expectation equilibrium, the market volatility is relative big. So as to simulate the sock index, the number of agents is set to in our model, this way, market volatility is relative suitable. The figure 1 is the time series of the stock market, in Fig1.a the above curve is risk neutral value of stock index which equals to dividend divided by riskless interest rate, the below is stock index; Fig1.d is forecast of one agent which is chosen randomly from agents. From the figure the result shows that after 5 th the stock index, trade volume, forecast start to become stable, the extreme values in return time series emerge irregularly, which makes return kurtosis very big. 3.. The information structure of stock index options market The stock index option is introduced into the stock market, and the effect of information structure 854

6 of stock index options market on the stock market will be researched. According to formula 3, in this paper the information structure denotes the length and the weight coefficient of information, the length reflects the quantity of information which is used for decision of agents, the weight reflects the relative importance of information in different time. In the experiments the percentages of different type of stock index options trader are set to 5%, so 5% of all agents do not trade stock index options. Because the primary concern of this paper is information structure, in the next experiments the percentages of different type of stock index options trader keep same. At the beginning, of running the stock market is very unstable as shown in Fig 1.a, so when the stock market starts to become stable the stock index options are introduced, this moment is 8 th. Considering that the agents only maximize the expected utility value of asset in the next time to invest, so the length of information used in the model is not long, the length ranges from 1 to 1, the length maximum is still much smaller than the real market. The length increases at every experiment. Considering the effect of weight coefficient of information on stock market, both cases are researched; one case is equal weight, the other case is unequal weight, which means the weight of new information is higher than the weight of old information, here normal distribution is used to generate the weight. Table. The features of stock market under different length of information (equal weight) Information length * kurtosis S&P E * Magnitude is 1.e-5 In the table the first row is statistics of stock market without stock index options, the last row is statistics of daily return of American S&P5, 319 data are chosen from Jan. 3, to Dec. 3, 11. The table shows that when the information is little, the volatility of stock market is bigger than without stock index options. The trade volume becomes very low, the market liquidity decreases. Even though the return is very big, the peak and fat tail is not obvious. As the more information is used, the stock market become more stable, for example, after the information length is 7, the stock market become more stable than without stock index options. The stock index become smaller, the trade volume become higher, and the market liquidity become much better, the return and kurtosis become smaller and are closer to the return distribution of S&P5. But when the information continuously increases the volatility of stock market starts to increase. From Fig 1.c and Fig. we can see that the introduction of stock index options changes the features of stock market return time series. Before the introduction of stock index options many extreme values in return time series emerge irregularly, volatility cluster in return time series is not obvious. After the introduction of stock index options, irregular extreme values disappear and return kurtosis is very small, but as the information increase, extreme values start to increase and volatility cluster is more obvious, as the information continue to increase, the effect of stock index options on the stock market become weaker, and the difference of return time series between with and without stock index options become smaller. Next, the standard normal distribution is introduced to compute weight coefficient, the formula is as follows. i ( 3 i / s 1 e u / du 3 ( i 1 ) / s 1 e u / du ), i 1,..., s (5) 855

7 Where s is the length of stock index options, the number 3 in the upper limit of integral, P ( Z 3).9987 in the standard normal distribution, almost equals to 1. From table and table 3 we can see that after the normal distribution weight is adopt, when the information is little, the stock market become more unstable than stock market with equal weight, but when information is much, the stock market is more stable. The distribution of return also is closer to S&P5 than equal weight a.3.5. b c.5. d Figure. The time series of the return (a: S&P 5, b: s =1, c: s =5, d: s =7) Table 3. The features of stock market under different length of information (normal distribution weight) Information length * kurtosis * Magnitude is 1.e Asymmetry of stock index options information In the real stock market the return time series has asymmetry in volatility; the effect of negative shocks on return volatility is different from the effect of positive shocks. The research shows that in America developed market, the negative shocks make the stock market volatility bigger than positive shocks, but in China market the effect of positive shocks is bigger than the negative shocks, which attracts the attention of many researchers [16, 17]. In this section, the effect of asymmetry of stock index options information on stock market is researched, to test whether the return time series have volatility asymmetry, if so, we can see whether it accords with American or Chinese conditions. 856

8 So as to construct asymmetry of stock index options information, the size of ( i 1,,3, 4 )is changed in formula 4. Two cases will be discussed, one case is increase of good information value, the other case is increase of bad information value, and specifically, in the first case if t-1,i ( i 1,,3, 4 ) equals to 1, it will be changed to 1.5 or, the other values keep same, in the second case if t- 1,i ( i 1,,3, 4 )equals to -1, it will be changed to -1.5 or -, the other values keep same. In the experiments the normal distribution weight is used and s =15. Table 4. The features of stock market under different asymmetry of information * kurtosis * Magnitude is 1.e-5 From the table 4 we can see that the effects on stock market are very similar to two cases. After the introduction of asymmetry of stock index options information, as the and of trade volume decrease, the and kurtosis of return also decrease and are closer to the real data. Next the EGARCH is used for quantitative analysis of volatility asymmetry, the EGARCH (1, 1) is as follows and t1 ~ N (,1). ln t 1 t 1 t c ln t 1 t 1 t 1 Table 5. The parameters of EGARCH (1, 1) c ( ) /( ) Before the introduction of asymmetry of stock index options information, ( ) /( ) is.934 and the volatility has slight asymmetry, it shows that the introduction of stock index options lead to volatility asymmetry which is not very obvious. After the introduction of asymmetry of stock index options information, ( ) /( ) value decreases much, and with the increase of asymmetry, the volatility asymmetry becomes bigger. The experiments also show that the negative shocks make the stock market volatility smaller than positive shocks, which is closer to China stock market. 4. Conclusions In this paper, the stock market and stock index options market were built through the method of the agent-based computational finance, and the effect on volatility of stock market after launching stock index options was researched from the angle of information structure of stock index options. In SFI-ASM even if all agents are homogeneous, with the increase of agents, the model complexity increases. So the status of rational expectation equilibrium is broken, market volatility increases, the extreme values in return time series emerge irregularly, which makes return kurtosis become very big. The introduction of stock index options affects obviously the stock market. When the information of stock index options for the stock market is little, the volatility of stock market becomes more violent than before the introduction of stock index options, and the stock market liquidity decrease, but with more information of stock index options being introduced, the stock market become more stable and t-1, i 857

9 the liquidity increases. Compared with equal weight of information of stock index options, the normal distribution weight makes stock market more unstable when the information of options is little, but when the information of options is much, it can makes market more stable. The options information can make the extent of peak and fat tail in return time series decrease, so when the information increases the stock market can become more stable and the return distribution become closer to return distribution of real market. After the introduction of asymmetry of stock index options information, the and of trade volume decrease, the and kurtosis of return also decrease and are closer to the real data. The volatility asymmetry becomes bigger. The result shows that the negative shocks make the stock market volatility smaller than positive shocks, which is closer to China stock market. In this paper the information of stock index options are public information and presume that all agents response to options information are same, but this is not exactly the case in the real market. In the future research the response of all agents will be divided into different types, so the effect of stock index options on stock market can be researched in a more comprehensive way. 5. Acknowledgements The research is supported by National Natural Science Foundation of China (No , 79737), and the Humanities and Social Science Research Projects of Ministry of Education of China (No. 9YJA636). 6. References [1] Bartley R. Danielsen, Bonnie F. van Ness, Richard S, Warr. Reassessing the impact of option introductions on market quality: A less restrictive test for event-date effects, Journal of Financial and Quantitative Analysis, vol. 4, no. 4, pp ,7. [] Sorin M. Sorescu, The effect of options on stock prices: 1973 to 1995, Journal of Finance, vol. 55, no.1, pp ,. [3] Raman Kumar, Atulya Sarin, Kuldeep Shastri, The Impact of Options Trading on the Market Quality of the Underlying Security:An Empirical Analysis, Journal of Finance, vol. 53, no., pp ,1998. [4] Xiong Xiong, Zhang Yu, Zhang Wei, Zhang Yong-jie, The effect on volatility of stock market and stock index futures market after launching stock index options: A case of KOSPI index options, Systems Engineering-Theory& Practice, vol. 31, no. 5, pp , 11. [5] Jangkoo Kang, Hyoung-Jin Park, The information content of net a buying pressure: Evidence from the KOSPI index option market, Journal to Financial Markets, vol. 11, no. 1, pp.36-56,8. [6] Wei Jie, Wang Nan, The market efficiency: the relation among stock index options, stock index futures and stock index-evidences from HongKong Index Option Market, Financial Theory & Practice, no. 9, pp.71-77, 1. [7] August F.Y. Chao, Heng-Li Yang, Knowledge sharing effects on the market of used durable goods: Agent-based simulation approach, JCIT: Journal of Convergence Information Technology, vol. 7, no. 7, pp.53-64,1. [8] Du Lei, Wang Wenjun, Zhang Xiankun, An agent-based decision-making model in emergency evacuation management, JCIT: Journal of Convergence Information Technology, vol. 7, no. 1, pp.197-5,1. [9] Kirman A, Money and Financial Markets, Blackwell, Cambridge,1991. [1] Martin Lettau, Explaining the facts with adaptive agents: The case of mutual fund flows, Journal of Economic Dynamics and Control, vol. 1, no. 7, pp , [11] W. Brian Arthur, The Economy as an Evolving Complex System II, Addison-Wesley, Boston, pp.15-44, [1] Blake LeBaron, W.Brian Arthur, Richard Palmer, Time series properties of an artificial stock market, Journal of Economic Dynamics and Control, vol. 3, no. 9-1, pp ,1999. [13] Alessandron N. Cappellini, Gianluigi Ferraris, Waiting Times in Simulated Stock Markets, Advances in Complex Systems, vol. 1, no., pp.195-6,

10 [14] Marco Raberto, Silvano Cincotti, Agent-based simulation of a financial market, Physica A, vol. 99, no. 1-, pp ,1. [15] Sabrina,Ecca, Michele Marchesi, Alessio Setzu, Modeling and simulation of an artificial stock option market, Computational Economics, vol. 3, no. 1-, pp.37-53, 8. [16] Chen Langnan, Huang Jiekun, An Empirical Investigation on Asymmetry of Volatility in China s Stock Market, Journal of Financial Research, vol. 5, pp.67-73,. [17] Rong Lu, Longbing Xu, Asymmetric Effects of Policy Information on China s Stock Markets, China Economic Quarterly, vol. 3, no., pp ,