Working Capital Management of Indianb Tyre Industry

International Research Journal of Finance and Economics ISSN 1450-2887 Issue 46 (2010) © EuroJournals Publishing, Inc. 2010 http://www.eurojournals.com/finance.htm

Working Capital Management in Indian Tyre Industry

Jasmine Kaur Assistant Professor in Guru Arjan Dev Institute of Management & Technology, New Delhi

E-mail: [email protected] Tel: (91) 9811160007; (91)9811669777; (91) (011) 25133012

Abstract

The management of Working Capital is one of the most important and challenging aspect of the overall financial management. Merely more effective and efficient management of working capital can ensure survival of a business enterprise. Working Capital Management is concerned with the problems that arise in attempting to manage the Current Assets, Current Liabilities and the interrelation that exists between them. This is a two-dimensional study which examines the policy and practices of cash management, evaluate the principles, procedures and techniques of Investment Management, Receivable and Payable Management deals with analyzing the trend of working capital management and also to suggest an audit program to facilitate proper working capital management in Indian Tyre Industry. The study covers a production of 8 year viz, 1999-2007. For the purpose of investigation both primary and secondary data is used. The collected data is analyzed by applying research tool which include accounting tools like Analysis, Cash Flow Analysis, Common Size and Trend Analysis. They reveal that there is a stand off between liquidity and profitability and the selected corporate has been achieving a trade off between risk and return. Efficient management of working Capital and its components have a direct effect on the profitability levels of tyre industry. Keywords: Working Capital Management, Cash Management, Inventory Management,

Receivables and Payables Management, Indian Tyre Industry. Introduction Working Capital Management refers to all management decisions and actions that ordinarily influence the size and effectiveness of the working capital. It is concerned with the most effective choice of working capital sources and the determination of appropriate levels of the current assets and their use. It focuses attention to the managing of current assets, current liabilities and the relationships that exist between them. In the present day of rising capital cost and scarce funds, the importance of working capital needs special emphasis. It has been widely accepted that the profitability of a business concern likely depends upon the manner in which its working capital is managed. The inefficient management of working capital not only reduces profitability but ultimately may also lead a concern to financial crises. On the other hand, proper management of working capital leads to a material savings and ensures financial returns at the optimum level even on the minimum level of capital employed. Both excessive and inadequate working capital is harmful for a firm. Excessive working capital leads to un-remunerative use of scarce funds. On the other hand, inadequate working capital usually interrupts the normal operations of a business and impairs profitability. There are many instances of business failure for inadequate working capital e.g. Modi Rubbers. Further, working capital has to play a vital role to

8 International Research Journal of Finance and Economics - Issue 46 (2010)

keep pace with the scientific and technological developments that are taking place in the area of tyre industry. Also, the current financial parameters of tyre industry are much less than the desired level. In this context, an attempt has, therefore, been made to undertake an indepth study on working capital management of Indian Tyre Industry. Objectives of the Study The primary aim of our study is to examine and assess management of working capital of selected companies of Indian Tyre industry i.e. JK, MRF, Apollo, Ceat.

In this broader framework an attempt will be made to meet out the following specific objectives of the study:-

1. To study the components of Working Capital Management in Indian tyre Industry 2. To assess at length, prevalent practices of inventory management, cash management and

receivables management on the profitability and liquidity of firms in the Tyre Industry. 3. To analyse the relative proportion of different sources of finance for working capital of Indian

Tyre industry. Research Design The present study is focused on understanding the impact of efficient working capital management on the profitability and liquidity of the companies. Hence, it’s a case of purposive sampling requiring an in-depth analysis of each selected company. This has led to selection of four companies representing India Tyre Industry, namely MRF Ltd., Apollo Tyres Ltd., J.K. Tyres Ltd. and Ceat Ltd.

The study covers a period of 8 years (1999-2000 – 2006-2007) and the data is collected from primary and secondary sources. The data so collected is analyzed by applying various research tools which include accounting tools like Ratio Analysis, Cash Flow Analysis, Common Size and Trend Analysis. Key Observations and Findings

The following are the major observations and findings of the study w.r.t each component of the working capital of the sample companies: Inventory Management (a)

• Inventory Turnover ratio signifies the amount of sale generated with each unit invested in raw material. The inventory utilization by J.K. and Ceat is quite effective but Apollo and MRF need to take measures to increase to stock turnover. This is possible only by shortening the operating cycle in days taken from the point of purchase of raw material to its conversion to the final sale to the consumers and the money getting back into the organization to be utilized again by the company to purchase raw materials for the next operating cycle.

Exhibit 1:

Year Apollo J.K. Ceat MRF

2000-01 3.82 3.86 3.83 3.23 2001-02 5.30 7.17 4.34 2.94 2002-03 6.12 -- 5.76 3.67 2003-04 5.75 7.54 6.16 3.92 2004-05 5.67 7.62 7 4.25 2005-06 5.29 7 7 4.69 2006-07 5.75 5.5 8.76 5.9

International Research Journal of Finance and Economics - Issue 46 (2010) 9

Comparative Inventory Turnover Ratio

0123456789

10

2000-01 2001-02 2002-03 2003-04 2004-05 2005-06 2006-07

ApolloJ.K.CeatMRF

• The raw material holding period identifies the number of days the raw material stays in the company before being put into the production process. The raw material holding period of Ceat is quite reasonable but Apollo and J.K. Tyres have to reduce the number of holding days.

Exhibit 2: Comparative Raw Material holding days of Apollo, J.K. Ceat and MRF (in days)


2000-01 34 33 -- 22 2001-02 27 61 21 27 2002-03 30 -- 14 25 2003-04 32 57 23 25 2004-05 38 54 26 21 2005-06 38 34 10 25 2006-07 30 30 15 25

Comparative Raw Material holding day

010203040506070

2000-01

2001-02

2002-03

2003-04

2004-05

2005-06

2006-07

ApolloJ.K.CeatMRF

• J.K .tyres has already started taking the initiative to reduce the number of holding days from 61 days to 30 days. This can be done by avoiding stocking up of raw materials.

• The finished goods holding period of Ceat is very less of 12 days as compared to Apollo, J.K. and MRF which is around 18-20 days.

• Reducing the number of holding days whether raw materials or finished goods is important as this will shorten the operating cycle which would increase the working capital turnover indicating its efficient use.


Exhibit 3: Comparative Finished goods holding period of Apollo, J.K. Ceat and MRF (in times)

Year Apollo J.K. Ceat MRF 2000-01 25 28 -- 36 2001-02 19 15 23 37 2002-03 12 -- 20 32 2003-04 15 12 18 30 2004-05 14 10 12 29 2005-06 17 17 11 23 2006-07 18 18 12 20

Comparative finished goods holdiong period

05

10152025303540

2000-01

2001-02

2002-03

2003-04

2004-05

2005-06

2006-07

ApolloJ.K.CeatMRF

• MRF has the highest inventory to current assets ratio of around 50%. This means that a lot of money of MRF is blocked in excess inventory storage which should be reduced.

Receivables Management (b)

• Receivables management indicates management’s efficiency in getting the money back into the organization in the shortest period of time. The debtors turnover ratio reflects the number of times the money received from debtors is rotated in the business cycle in a year. The Debtors Turnover Ratio of Apollo is the best indicating the management’s efficiency in getting the money back from the debtors and again rotating it in the business to generate sales.

Exhibit 4: Comparative Debtors Turnover Ratio of Apollo, J.K. Ceat and MRF (in times)


1999-2000 -- 7.06 -- -- 2000-01 9.25 6.73 6.04 6 2001-02 9.89 8.46 7.2 6.37 2002-03 16.65 - 8 6.74 2003-04 22.69 5.63 7.59 7.52 2004-05 19 5.52 7.75 7.98 2005-06 18 6.63 7.97 8.45 2006-07 20 6 4.68 8.08


Comparative Debtors Turnover Ratio

0

5

10

15

20

25

1999-2000

2000-01

2001-02

2002-03

2003-04

2004-05

2005-06

2006-07

ApolloJ.K.CeatMRF

• J.K., Ceat and MRF should take stringent steps to get the money back into the organization quickly by reducing the average collection period.

Exhibit 5: Comparative Average Collection Period of Apollo, J.K. Ceat and MRF (in days)


1999-2000 -- 51 55 -- 2000-01 39 53 60 60 2001-02 36 43 50 57 2002-03 22 -- 45 53 2003-04 16 64 47 48 2004-05 19 65 47 46 2005-06 20 54 45 43 2006-07 18 60 77 45

Comparative Average Collection Period

0102030405060708090

1999-2000

2000-01 2001-02 2002-03 2003-04 2004-05 2005-06 2006-07

ApolloJ.K.CeatMRF

• The Debtors Turnover Ratio of Apollo is very high which is quite appreciable. This is due to the fact that the average collection period of Apollo is very short of around an average of 24 days as compared to others ranging from 50 to 55 days.

• The average payment period of MRF is very short which is almost equivalent to its average collection period. MRF should increase the payment period unless and until early payment helps in getting heavy cash discounts.


Exhibit 6: Comparative Average Payment period of Apollo, J.K. Ceat and MRF (in days)

Year Apollo J.K. Ceat MRF 1999-2000 -- 148 -- -- 2000-01 131 156 90 77 2001-02 141 96 97 80 2002-03 100 -- 141 67 2003-04 82 129 143 57 2004-05 82 114 147 52 2005-06 78 97 125 51 2006-07 81 90 111 48

Comparative Average Payment Period

020406080

100120140160180

1999-2000

2000-01

2001-02

2002-03

2003-04

2004-05

2005-06

2006-07

Apollo

J.K.

Ceat

MRF

Cash Management (c)

• Cash management throws light on the judicious and efficient use of cash (which is the most liquid asset of an organization). J.K. seems to have the best cash management system since it is the policy of management to invest excess cash into profitable investment avenues.

• Apollo needs to look into its cash management system and bring some changes as there seems to be unnecessary idle cash lying in the business which could otherwise be used more productively.

• Ceat already has started with remedial measures of utilizing excess cash into suitable ventures.

Exhibit 7: Comparative Cash Balance of Apollo, J.K. Ceat, and MRF (crores)

Year Apollo J.K. Ceat MRF 1999-2000 44.83 18.88 71.89 --

2000-01 56.28 55.16 64.18 34.88 2001-02 66.34 39.07 46.42 34.39 2002-03 97.61 -- 66.15 40.26 2003-04 106.35 38.23 38.89 36.72 2004-05 110.43 36.11 31.23 46.02 2005-06 231.36 39.32 39.61 53.32 2006-07 172 23.87 34.92 72.88


Comparative Cash Balance

0

50

100

150

200

250

1999-2000

2000-01

2001-02

2002-03

2003-04

2004-05

2005-06

2006-07

ApolloJ.K.CeatMRF

• The percentage of cash out of total current assets of Apollo is very high ranging from 15% - 20% as compared to others ranging from 2% - 10%. The company needs to utilize the excess cash and bring down the percentage. This will also help the company to increase its profitability.

Key Observations and Summary The major observations of the study are as follows:-

• The selected sample companies have been following either an aggressive or a conservative approach:

APOLLO - Aggressive approach J.K - Aggressive approach CEAT - No definitive approach MRF - Conservative approach

• All the companies have a positive net working capital except in the case of Ceat Ltd. in 2005-2006 and 2006-2007. On an average, the net working capital is largest in MRF followed by Apollo, J.K. and Ceat.

• When quick assets are compared with current liabilities, it is revealed that the former are insufficient to cover current liabilities in case of J.K. Tyres. For Ceat Ltd. there has been a sudden decline in quick ratio in the year 2005-2006 and 2006-2007. MRF and Apollo are in good position to pay off current debts from quick assets.

• If standard current ratio is to be taken as 2:1 then Apollo and MRF have current ratios equal to or more than two. But incase of J.K. and Ceat the current ratio is less than two which reflects a poor liquidity position of these two enterprises.

• There is a stand off between liquidity and profitability position of the tyre companies. These two don’t go hand in hand, as incase of MRF where liquidity levels are very high as compared to the industry standards but profitability levels do not rise upto expectations even though MRF has the largest market share. There is an inverse relationship between the two as analysed from financial reports. Higher the liquidity levels, lower would be the profitability and vice-versa, therefore, tyre companies have to maintain a delicate balance between the two.

• The efficient management of Working Capital and its components have a direct effect on the profitability levels of the tyre companies:-

o With increase in working capital turnover of Apollo and MRF, the net profit ratio has also improved


o An efficient receivables management by Apollo has led to short operating cycle which has led to high debtor turnover ratio and high profitability levels of the company.

High collection period from debtors of J.K., Ceat and MRF is apparent from their low DTR and further low profitability levels.

• Although J.K. has the highest working capital turnover ratio, much above the industry level, it shows no effect on the profitability levels. This may be due to over-trading which the company should look into as early as possible. Also, there is a gradual decline in the liquidity level and the company should be aware of a liquidity crises coming up.

• The tyre companies have on an average half of their total assets in the form of current assets. The average ratio of current assets to total assets is largest for MRF followed by Ceat, Apollo and J.K. Of the total different components of current assets, the share of inventories in total assets, on an average, is largest followed by receivables and cash. Over a period of time, the share of cash has declined except incase of MRF. Since inventories occupy a major share in current assets and its share has increased over a period of time, the tyre industry should pay more attention to management of inventories. The increasing share of inventories indicates that current assets seem to have become less liquid.

• For financing any working capital requirements, the tyre companies generally prefer: o Bank overdraft/Bank cash credit for immediate solution. o Short term loans from Banks generally secured by hypothecation of Inventories and

book debts. o Loans from financial Institutions like IDBI, IFCI, ICICI by hypothecation of immovable

properties. The study of the turnover ratios compiled over a period of 8 years show that there has been an

improvement in utilization of current assets. Conclusion The present study reflects that the proper management does affect positively on the profitability levels of the sample companies. The companies over the years have realized the importance of efficient working capital management and have worked in bringing about a productive change in WCM techniques. The results reveal that there is a standoff between liquidity and profitability and the selected corporate has been achieving a tradeoff between risk and return.


References Books

[1] Anthony, R.N. et.al.(1998). Management Accounting Taxes and cases Illinois. Richard D. Irwin Inc.

[2] Agarwal N.P. and Mangal S.K.(1988). Readings in Financial Management Jaipur: Rupa Publishers.

[3] Bhalla, V.K. (1987). Financial Management, Delhi; Khosla Publishing House. [4] Chaddha R.S. (2002). Inventory Management in India , Mumbai:Allied Publication. [5] Gupta M.C., Profitability Analysis. (1989). Jaipur , Pointer Publishers. [6] Gupta, S.P. (2002). Statistical Methods, New Delhi: Sultan Chand & sons. [7] Horward, L.R., Working Capital: Its Management and control. (1987). London: McDonald and

events Ltd. Reports/Official Publications

[1] Annual reports of selected Companies of the selected period. Periodicals/Journals/Bulletins

[1] The Journal of Industries and Trade Lok Udyog, New Delhi. [2] The Indian Accounting journal and Finance [3] The Accounting Review Websites

[1] www.ICAI.org [2] www.mrftyres.com [3] www.ceatyres.com [4] www.apollotyres.com [5] http://www.jktyres.com/

International Research Journal of Finance and Economics ISSN 1450-2887 Issue 46 (2010) © EuroJournals Publishing, Inc. 2010 http://www.eurojournals.com/finance.htm

Determinants of the Implied Volatility Skew in

LIFFE Equity Options

Ing-Jye Chang Department of Business Administration, National Taipei College of Business

No. 321, Section 1, Jinan Rd, Taipei 100, Taiwan E-mail: [email protected]

Bing-Huei Lin

Department of Finance, National Chung Hsing University No. 500, Kuo Kuang Rd, Taichung 402, Taiwan

E-mail: [email protected]

Abstract

The aim of this study is to examine the firm-specific and market-wide factors explaining the implied volatility skew in LIFFE equity options. Our theoretical model uses Gram-Charlier series expansion to approximate the distribution of the logarithm of stock prices, and incorporates non-normal skewness and kurtosis terms in the adjusted option price; we also estimate the risk-neutral moments by numerically solving the nonlinear least-squares problem. Investigation of the influence of firm-specific variables on the slope of implied volatility of individual stock options indicates that for firms whose stock return is more volatile the slope of the smile is less negative. For firms with higher leverage, of larger size, larger beta, and larger traded volume the smile tends to have a more negative slope. With regard to endogenous variables, the more negative the risk-neutral skewness, the steeper (more negative) the slope of the smile will be, and the greater the risk-neutral kurtosis, the flatter the slope of the smile. For the market-wide variables, greater volatility in the market is associated with a smile of steeper slope; the more negative skewness of the FTSE 100 index options, the steeper the slope of the smile is for individual stock options, and the greater the kurtosis of the FTSE 100 index options, the flatter the slope of the smile for individual stock options. Keywords: Implied Volatility Skew, Gram-Charlier Series Expansion, Risk-Neutral

Skewness, Risk-Neutral Kurtosis, Risk-Neutral Distribution, Leverage Effect JEL Classification Codes: G12, G13

1. Introduction This article investigates the determinants of the implied volatility skew (smile) using the prices of individual stock options traded on the LIFFE (London International Financial Futures and Options Exchange). There are few investigations detailing the characteristics of the implied volatility smile or implied volatility skew available in the literature, and therefore the results of this study should be useful for implementation. While examining the firm-specific and market-wide factors explaining the implied volatility smile, one can detect at the same time the influence of these variables in the pricing of stock options.


Two classification methods, parametric and nonparametric, are used to extract the risk-neutral distribution from option prices. A criterion documented by Jackwerth (2004) is that a fast and stable method should be chosen to extract the risk-neutral distribution, but he indicates that the choice of a particular method does not significantly alter the results. Coutant, Jondeau, and Rockinger (2001) compare the mixture of log-normal densities method, Hermite expansion method, and maximum entropy method of extracting the risk-neutral distribution implied in PIBOR interest rate future options; their results indicate that the implied volatility across methods is of a similar magnitude and the risk-neutral skewness and kurtosis are of differing magnitudes but have similar variation patterns for a given date. They also indicate that it is difficult to select the best method among these methods.

In this article we use Gram-Charlier series expansion to approximate the distribution of the logarithm of stock price, and incorporate non-normal skewness and kurtosis terms in the adjusted option price to expand the Merton (1973) option price formula. Previous articles have used series expansion in modeling the option prices. For example, Jarrow and Rudd (1982) use Edgeworth series expansion to approximate the lognormal distribution of stock price, and Corrado and Su (1996) use Gram-Charlier series expansion of the normal probability density function to model the distribution of the logarithm of stock price to provide skewness and kurtosis adjustment terms for the Black and Scholes option formula. Vahamaa, Watzka, and Aijo (2005) also use the Gram-Charlier expansion model in pricing bond future options, and Beber and Brandt (2006) use the skewness-kurtosis-adjusted Black (1976) formula to examine the effect of regularly-scheduled macroeconomic announcements on the beliefs and preferences of participants in the U.S. Treasury market. Other articles such as that of Pena, Rubio, and Serna (1999) employ Spanish IBEX-35 index options to examine the transaction costs and market condition variables in order to explain the pattern of the implied volatility curve.

The following articles document the improved model for pricing options. Navatte and Villa (2000) provide evidence that the option price model derived using the Gram-Charlier density is more accurate than the Black and Scholes option formula, with less bias in market price. Lim, Martin, and Martin (2005) indicate that misspecifications for pricing options are due to the occurrence of volatility smile and skew; they also show that the use of an options pricing model with higher order moments and time-varying volatility improves performance and corrects the volatility skew. In addition, Tamaki and Taniguchi (2006) provide evidence that option prices are strongly affected by the non-Gaussianity and dependency of stock log returns when incorporated into the model. In this study, we use the slope of the implied volatility curve as a dependent variable, which can present the change of the implied volatility skew better than the skewness. The variable slope of the volatility smile can be observed directly, unlike the option implied risk-neutral skewness, which is used in the alternative approach.

Our results indicate that firms with a larger volatility of stock return have a smile of less negative slope, and firms with higher leverage, of larger size, larger beta, and larger traded volume tend to have a smile of more negative slope. The put-to-call traded volume ratio is not significantly related to the slope of the smile except the medium-term option data. For the endogenous variables, the more negative the risk-neutral skewness, the steeper (more negative) the slope of the smile, and in general, the greater the degree of risk-neutral kurtosis, the flatter the slope of the smile. In the examination of the influence of market-wide variables on the slope of implied volatility of individual stocks, the Gram-Charlier-approximated estimated FTSE 100 index option implied volatility is negatively related to the slope of the smile of individual stocks, meaning that when the stock index return is of larger volatility and is accompanied by larger volatility of individual stocks return, the more negative the skewness of the FTSE 100 index option; in addition, the steeper the slope of the smile of individual stock options, and the greater the kurtosis of the FTSE 100 index option, the flatter the slope of the smile of individual stock options.

The rest of this article is organized as follows. In Section 2, we describe the option data used in this study as the empirical sample and construct the risk-neutral moments implied in option prices. In Section 3, we describe the construction of dependent and independent variables. In Section 4, we conduct the regression and robustness tests. A summary and conclusion are given in Section 5.


2. Data and Construction of Risk-Neutral Moments 2.1. The Data

In this study we use daily settlement prices for 79 individual stock options and the FTSE 100 index option (ESX) traded on the LIFFE as our research samples. The underlying stocks are traded on the London Stock Exchange (LSE); the individual stock options are American-style, while the FTSE 100 index option is European. The primary options data are obtained from the LIFFE database, and the corresponding stock prices and index data are obtained from the DataStream database; the individual stock’s market value, traded volume, and dividend yield are obtained from the DataStream database. The 3-month Treasury-bill rate is used as the proxy for the risk-free interest rate, which is also obtained from the DataStream database. The overall sample period extends from March 13, 1992 through December 31, 2002.

As very short maturity stock option quotes may not be active, options with a remaining time to expiration of less than 9 days were discarded. Although options with longer-term maturity are illiquid compared with short-term options, we decided to incorporate them in this study to increase the degrees of freedom. We group the sample options into three categories in order to examine the analysis across maturities: if an option has a remaining time to expiration of between 9 and 120 days, it is grouped in the short-term option category; if the remaining time to expiration falls between 121 and 240 days, the option is grouped in the medium-term category; and if the remaining time to expiration is over 240 days, the option is grouped in the long-term category. We then use the short-term, medium-term, and long-term data to run the regression models in order to examine the firm-specific and market-wide variables explaining the implied volatility skew (smile). 2.2. Option Pricing with Gram-Charlier Density

As described by Backus, Foresi, Li, and Wu (1997), and Beber and Brandt (2006), the Gram-Charlier series expansion is used to derive the risk-neutral density of the logarithm of the underlying price; we then derive the expanded Merton (1973) option price model as equations (1) and (2), incorporating non-normal skewness and kurtosis terms in the pricing of the European-style FTSE 100 index option.

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formulae of Black and Scholes (1973), adapted so as to consider the dividend payment stock call and put options by Merton (1973), ττ στσθ /])2/()/[ln( 2+−+= ttt rKSd , K is the option strike price, n1γ and n2γ are the skewness and excess kurtosis coefficients of stock index return, respectively, τtr is the continuously compounded τ period interest rate, tθ is annualized dividend yield, )(⋅ϕ is the standard normal density, and )(⋅N is the standard normal cumulative distribution function.

We estimate the three parameters nn 21 ,, γγστ in the expanded Merton (1973) European-style option prices by numerically solving the following nonlinear least-squares (NLLS) object function:


1 2

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where the option prices iKtC ;,τ and

jKtP ;,τ in the brackets of equation (3) are the observed European-style call and put option prices, respectively.

The expanded American call and put option prices for the individual stocks can be defined as the weighted average of the upper and lower bounds of the call and put options, respectively, displayed in equations (4) and (5), which are documented by Melick and Thomas (1997).

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where KeK tt r τθ τ )(' −−= and ττ στσθ /])2/()/[ln( 2'' +−+= ttt rKSd . For the upper bounds of American puts on dividend paying stocks, we use the non-randomized

version of the method of Chen and Yeh (2002) while the interest rate is greater than the dividend yield, and the Melick and Thomas (1997) version otherwise, which are displayed in equations (10) and (12). During the sample period, there is only a short horizontal in which the interest rate is less than the dividend yield for the individual stocks, and the interest rate is only of slightly less magnitude than the dividend yield on average.

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the expanded American put option price is

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PtKt

PSKdNK

dddKedNK

ddKePeP

tt

ttt

ττττ

τττττθ

ττ

ττττθ

ττττθ

ττ

λσσ

σσσϕσγ

σσ

σσϕσγ

σγ

σγ

λ

τ

τ

−−++−−

−++−−−++−+

+−−−++≈

−

−

(11)

When the interest rate is smaller than the dividend yield,


)],0[max(;, ττ +−= ttu

Kt SKEP (12) the expanded American put option price is

),max()1()]}(

)133)(([!4

)](

)2)(([!3

)!4!3

1({

*;,,

4

2323

14231;,,

**;,

KttPt

n

nnnKt

rPtKt

PSKdKN

dddKdKN

ddKPeP t

ττττ

ττττττ

τττττττ

ττ

λσσ

σσσϕσγ

σσ

σσϕσγ

σγ

σγ

λ τ

−−++−−

−++−−−++−+

+−−−++≈

(13)

For the lower bounds of American call and put options we use the version of Melick and Thomas (1997). Equations (14) and (15) represent the maximum of the early exercise value and European-style option prices, as evidenced by Beber and Brandt (2006).

)]),0[max(,][max(;, KSEeKSEC ttr

ttl

Ktt −−= +

−+ τ

τττ

τ (14)

)]),0[max(],[max(;, ττ

τττ

+−

+ −−= ttr

ttl

Kt SKEeSEKP t (15) Here we also use a logistic function to approximate the maximum operator used by Melick and

Thomas (1997).

)](5exp[11],max[itlog

yxyx

−−+= (16)

yyxxyxyx ⋅−+⋅≈ ]),max[itlog1(],max[itlog],max[ (17) We also estimate the five parameters P

tCtnn τττ λλγγσ ,,21 ,,,, in the expanded American-style

option prices by numerically solving the following object function:

∑∑= =

⋅−+⋅−N

i

M

jKtKtKtKt jjii

Pt

Ctnn

PPCC1 1

2**;,;,

2**;,;,

,,,,

})]([)]({[min,,21

ττττλλγγσ τττ

(18)

where the option prices iKtC ;,τ and

jKtP ;,τ in the brackets of equation (18) are the observed American-style call and put option prices, respectively. 2.3. Gram-Charlier Approximated Density

The problem is that using a polynomial expansion to approximate probability density may derive a negative probability. Jondeau and Rockinger (2001) derive a domain in which the skewness and excess kurtosis are bounded on [-1.0493, 1.0493] and [0, 4], respectively, which guarantees positive density to solve the NLLS problem.

In Table 1 we display constrained estimates of the parameters for the short-term, medium-term, and long-term options data. To save space, we only report the results of the twenty largest stocks in the FTSE 100 index; we also report the averages of the 79 stocks. The average implied volatility, skewness, and kurtosis are 0.43, -0.18, and 3.93, respectively, for the short-term individual stock options, 0.36, -0.11, and 3.70 for the medium-term options, and 0.35, -0.08, and 3.53 for the long-term options. The results indicate that the term structure of implied volatility turns into one that is flatter, of less negative skewness, and lower kurtosis for longer maturity individual stock options. The results of the FTSE 100 index option implied volatility are very close, at 0.2023, 0.2049, and 0.2044 for the short-term, medium-term, and long-term options, respectively. In using the Gram-Charlier model to estimate the term structure of implied volatility, the decreasing phenomenon might be modified. Lim, Martin, and Martin (2005) also showed that a generalized Student’s t-option pricing model can be used to correct the volatility skew. In other results, the features of the FTSE 100 index option implied skewness and kurtosis are consistent with individual stock options. The last column of Table 1 displays the average traded volume per day, which for the FTSE 100 index option is 18,410, and the total average traded volume, which is 273 for the 79 individual stock options. The weights of call and put options estimated by equation (18) here we not display in this table. Table 1: Gram-Charlier Model-Estimated Risk-Neutral Moments


Short-term options Medium-term options Long-term options

Ticker GCSTD GCSKW GCKUT GCSTD GCSKW GCKUT GCSTD GCSKW GCKUT Volume 1. AAM 0.38 -0.34 3.91 0.37 -0.26 3.54 0.37 -0.27 3.65 42 2. AZA 0.28 -0.17 4.31 0.28 -0.14 3.85 0.27 -0.11 3.73 719 3. BBL 0.35 -0.19 4.30 0.31 -0.09 3.94 0.29 -0.07 3.63 316 4. BP 0.33 -0.14 3.89 0.26 -0.03 3.49 0.25 0.08 3.29 986 5. TAB 0.42 -0.33 4.06 0.40 -0.33 4.03 0.39 -0.27 3.75 118 6. BSK 0.45 -0.10 3.43 0.36 0.04 3.34 0.35 0.13 3.38 391 7. BTG 0.61 -0.33 3.81 0.38 -0.18 3.39 0.38 -0.29 3.08 1035 8. GNS 0.36 -0.20 4.55 0.25 -0.05 3.92 0.23 0.06 3.54 221 9. GXO 0.36 -0.22 4.21 0.29 -0.09 3.94 0.28 -0.04 3.73 532 10. HAX 0.36 -0.23 3.92 0.35 -0.19 3.73 0.34 -0.15 3.49 143 11. HSB 0.39 -0.18 4.23 0.31 -0.18 4.04 0.29 -0.15 3.66 903 12. TSB 0.39 -0.20 4.23 0.35 -0.18 3.97 0.35 -0.20 3.65 389 13. NGG 0.33 -0.19 4.01 0.30 -0.19 3.69 0.29 -0.20 3.47 129 14. RTZ 0.33 -0.21 4.48 0.28 -0.09 4.01 0.26 0.00 3.69 179 15. RBS 0.39 -0.25 3.61 0.38 -0.23 3.50 0.37 -0.19 3.78 259 16. SHL 0.30 -0.21 4.92 0.24 -0.12 4.38 0.23 -0.08 4.28 520 17. SCB 0.44 -0.12 3.84 0.37 -0.07 3.59 0.37 -0.01 3.49 132 18. TCO 0.34 -0.13 4.18 0.28 -0.06 3.87 0.27 -0.02 3.52 298 19. ULV 0.26 -0.14 4.42 0.24 -0.06 3.95 0.24 -0.06 3.75 154 20. VOD 0.46 -0.14 3.59 0.35 0.04 3.41 0.41 -0.06 3.23 4601 21. ESX 0.20 -0.31 4.16 0.20 -0.27 3.94 0.20 -0.23 3.79 18410 Tot Avg. 0.43 -0.18 3.93 0.36 -0.11 3.70 0.35 -0.08 3.53 273

This table reports the risk-neutral moments as estimated by the Gram-Charlier model implied in the 20 largest stock options and the FTSE 100 index option, displaying the results of the constrained estimate for the categorized sample data. Short-term options are those with a remaining time to expiration of between 9 and 120 days, medium-term between 121 and 240 days, and long-term over 240 days. The last column displays the average traded volume per day. The overall sample period is from March 13, 1992 to December 31, 2002, whereas the sample period for each individual stock option varies, relative to the availability of its price data. The bottom row shows the average of the 79 individual sample stocks.

In Table 2, we show a summary of the implied volatility for all 79 individual stock options of

short-term, medium-term, and long-term maturity. The implied volatilities are average within the natural log of the moneyness and the maturity category. The log-moneyness intervals for OTM puts are [-10%, -5%], [-5%, 0%], [0%, 5%], and [5%, 10%] for OTM calls. As specified above, the average implied volatility of the 79 individual stock options exhibits an asymmetric shape for short-term, medium-term, and long-term options before 1998; however, the implied volatility curve subsequently becomes almost skewed in shape on average, which is consistent with the FTSE 100 index option implied volatility shown in Lin, Chang, and Paxson (2008). The entire sample period from 1992 to 2002 shows an implied volatility curve of asymmetric shape. The changing nature of the implied volatility skew of the LIFFE 79 individual stock options is shown in Figure 1.


Table 2: Average Implied Volatilities for Individual Stock Options across Moneyness and Maturity

Short-term options Medium-term options Long-term options OTM puts OTM calls OTM puts OTM calls OTM puts OTM calls Year -10% to -5% -5% to -0% 0% to 5% 5% to 10% -10% to -5% -5% to -0% 0% to 5% 5% to 10% -10% to -5% -5% to -0% 0% to 5% 5% to 10%1992 26.67 25.75 24.37 25.51 24.67 24.34 22.93 23.36 24.79 24.12 23.01 22.93 1993 25.55 24.99 24.40 24.96 24.66 24.12 23.96 24.08 24.66 24.23 23.96 24.12 1994 25.75 25.71 25.29 25.59 24.69 24.69 24.64 24.62 24.89 24.83 24.62 25.34 1995 22.55 22.16 22.12 22.14 21.97 21.73 21.92 22.22 21.70 21.58 21.80 21.91 1996 24.55 24.89 25.45 26.49 24.71 24.58 25.38 25.68 24.93 25.11 25.47 26.26 1997 28.71 28.21 29.02 29.83 26.43 26.26 26.98 27.34 25.15 26.09 26.04 26.30 1998 34.45 34.29 33.87 34.35 32.30 32.09 32.33 32.42 30.87 31.20 31.14 31.67 1999 36.36 36.38 36.72 36.69 35.83 35.92 36.15 36.09 35.42 35.92 35.85 35.95 2000 53.05 53.36 53.35 53.19 51.20 51.58 51.29 51.48 51.62 52.19 51.81 51.72 2001 40.59 40.48 41.09 40.09 39.70 39.71 40.25 39.43 39.08 39.55 39.32 39.34 2002 41.51 40.50 40.25 39.76 39.18 38.24 38.30 37.87 39.00 36.76 37.74 36.99

1992–2002 31.92 31.39 31.04 31.59 30.58 30.25 30.02 30.30 30.36 29.94 29.67 30.14

Figure 1: The Changing Nature of Average Implied Volatility Skew in 79 Individual Stock Options

Panel A: 79 Individual Stock Options Average Implied Volatility Curve (1992)

Panel B: 79 Individual Stock Options Average Implied Volatility Curve (1995)


Panel C: 79 Individual Stock Options Average Implied Volatility Curve (1998)

Panel D: 79 Individual Stock Options Average Implied Volatility Curve (2002)

3. The Variables 3.1. Dependent Variable

In order to quantify the implied volatility skew using options with a certain time to maturity, we can measure the slope of the implied volatility curve using the following regression model,

iii yySTD εππ ++= )ln()](ln[ 10 Ni ,,2,1= (19) where SKy ii /= is the moneyness of an option with the strike price iK . There are N traded options with a specified time to maturity at a certain time. By regressing the logarithm of implied volatility on the logarithm of moneyness, the regression coefficient 1π , which represents the implied volatility slope as a dependent variable, can be regarded as a measure of the magnitude of the implied volatility skew.

The model represented by equation (19) is estimated daily across our 79 stock samples using a least-squares estimation (LSE) method, and the OTM puts and calls data is used for the short-term, medium-term, and long-term options. 3.2. Independent Variables

Here we outline why these explanatory variables are used and their construction. A. Option Implied Volatility The option implied volatility is estimated using the model of Gram-Charlier as a proxy for the volatility of an individual stock return. The variable is used to examine the hypothesis that higher volatility stocks have a steeper implied volatility curve.


B. Leverage We define the leverage ratio as the sum of the total loan capital and preference capital divided by the sum of the total loan capital, preference capital, and the market value of equity. The total loan capital and preference capital data are also obtained from the DataStream database. We want to test the hypothesis that a firm’s use of higher financial leverage leads to an implied volatility curve of a more negative slope; that is to say, when stock prices decline, leading the market value to decrease and the leverage ratio to rise, the financial risk and the stock volatility increases, and the slope of the implied volatility curve will become more negative, and vice versa. C. Size Size is defined as the logarithm of the market value of equity, the purpose being to test how firm size influences the implied volatility skew and examine what level of firm size leads to a more negative slope of the implied volatility curve. The results of Dennis and Mayhew (2000) show that firm size has a positive relationship with the slope of the implied volatility curve; in addition, Dennis and Mayhew (2002) describe a negative relationship between firm size and risk-neutral skewness in CBOE. D. Beta Bakshi, Kapadia, and Madan (2003) show that when the S&P 100 index options implied risk-neutral skewness becomes more negative, this leads to a more negative risk-neutral skewness implied in individual stock options. These results lead us to expect that the effect might be more pronounced if options are used to hedge the market risk when it is high. Thus, we test the hypothesis that the slope of the implied volatility curve is more negative for firms with more market risk, as measured by beta. The beta for stock i at time t, tiBETA , , is the coefficient that relates the daily returns of stock i to the daily return of the FTSE 100 index from day t-250 to t. In addition to using the traditional method-estimated beta as a measure of market risk, Aijo (2003) also uses the time-varying exponentially-weighted moving average (EWMA) method-estimated beta to measure the market risk (detail in Aijo, 2003), and concludes that better regression results are achieved by using the EWMA model. E. Volume We use the logarithm of daily trading volume (in shares) in the underlying stock as a proxy for the liquidity in order to test the hypothesis that the slope of the implied volatility curve is steeper for larger traded volume stocks. F. Put-to-call Traded Volume Ratio Put-to-call traded volume ratio is defined as the total traded volume of put relative to the total traded volume of call for a certain time to maturity category on a certain day, which is used as a proxy for investor sentiment or trading pressure in order to test whether the implied volatility curve is steeper when accompanied by a higher put options traded volume. We also use the put-to-call open interest ratio to test the robustness of our results. The open interest data were also obtained from the LIFFE database. G. Risk-Neutral Skewness and Kurtosis These are endogenous variables obtained from the option prices and used to test the endogenous relationship between the option prices and their implied moments. We use the endogenous variables as control variables in order to test the relationship between firm-specific variables and the slope. These variables are estimated by the Gram-Charlier model, and we attempt to examine the hypothesis that the more negative the risk-neutral skewness, the steeper the implied volatility slope, and the more fat-tailed the risk-neutral distribution, the flatter the implied volatility slope. Bakshi, Kapadia, and Madan (2003) also clarify the relationship between the volatility smile and risk-neutral skewness and kurtosis.


H. Market-Wide Variables Additional systematic variables are included in the model to investigate the effect of market-wide factors on the implied volatility skew of individual stocks; these variables are stock index option implied volatility, proxy for the volatility of the market, risk-neutral skewness, and risk-neutral kurtosis, estimated using the Gram-Charlier model. We examined whether higher market volatility leads to a steeper implied volatility skew of individual stocks, whether a more negatively skewed risk-neutral distribution of the index option accompanies a steeper implied volatility slope of individual stock options, and whether the more fat-tailed the risk-neutral distribution of the index option, the flatter the implied volatility slope of individual stock options. 4. Empirical Results 4.1. The Determinants of the Volatility Smile

We use the following pooled time series and cross-sectional regression model to investigate the firm-specific and market-wide factors explaining the implied volatility skew (smile),

titittttiti

tititititititi

SLOPEbKUTbSKWbSTDbKUTbSKWbPUTCALLbVOLUMEbBETAbSIZEbLEVERAGEbSTDbbSLOPE

,1,1211109,8,7

,6,5,4,3,2,10,

100100100 ε+++++++

++++++=

−

(20)

where iSLOPE denotes the slope of the implied volatility for firm i, iSTD is the options implied volatility, iLEVERAGE is the leverage ratio, iSIZE is the market value of equity, iBETA is the beta of the stock’s return with the FTSE 100 index, iVOLUME is the stock’s traded volume (in shares),

iPUTCALL is the ratio of put-to-call traded volume, and iSKW and iKUT are the risk-neutral skewness and kurtosis, respectively. tSTD100 is the implied volatility of the FTSE 100 index option, a proxy for the volatility of the market, tSKW100 and tKUT100 are the risk-neutral skewness and kurtosis of the FTSE 100 index option, respectively, and the time lag variable 1, −tiSLOPE proxies for other variables not included in this model1.

In Table 3 we show the results of the pooled regression analysis estimated by the generalized method of moments (GMM). First, the options implied volatility is positively and significantly related to the slope of the implied volatility curve. This means that a firm with a larger volatility tends to have an implied volatility curve of less negative slope, and these results might indicate that the implied volatility is not a continuously-decreasing as shown in Figure 1. This is contrary to the prediction but consistent with the results of Dennis and Mayhew (2002) and Aijo (2003). Second, the coefficients of the variable leverage ratio are significantly negative, meaning that firms with higher financial leverage tend to have a smile of steeper slope, leading to higher volatility in stock price. Previous results regarding this issue were reported by Toft and Prucyk (1997), who document a negative relationship between volatility skew and leverage, and Dennis and Mayhew (2002), who show that the relationship between risk-neutral skewness and leverage is positive. Third, the significantly negative coefficient on SIZE means a smile of more negative slope for large firms. If we define the twenty largest of the 79 individual stocks as large firms and the rest as small firms, for short-term options, the average of the slope of the implied volatility curve is –0.1235 and –0.0676 for large and small firms, respectively, –0.1025 vs. –0.0515 for medium-term options, and –0.0915 vs. –0.0486 for long-term options. Obviously, large firms tend to have a smile of more negative slope.

Fourth, in general, the beta of the firm is negatively related to the slope of the smile, meaning stocks with higher market risk have a steeper implied volatility slope (higher volatility of stock return), which is consistent with the results of Duan and Wei (2009), who indicate that the market risk variable

1 Testing the multicollinearity among independent variables, the variance inflation factor (VIF) values are between 1.00

and 1.71 for short-term options, 1.00 and 2.12 for medium-term options, and 1.00 and 2.57 for long-term options; VIF values are all therefore less than 10, meaning that the multicollinearity problem is not serious.


plays a role in option pricing, and that systematic risk must be incorporated into the option price model in order to capture the impact of systematic risk on option price. In a recent study, Chauveau and Gatfaoui (2002) add systematic risk into the option price model to consider the effect of the volatility of the market factor and the beta on option prices. Our results show that while the market risk is high, the use of options to hedge this risk is more pronounced. Fifth, that the coefficient on traded volume is negative means steeper implied volatility curves for more liquid stocks: the more liquid the stock, the higher the volatility in stock prices, and investors tend to worry about the down side of the risk. Sixth, we predict that a negative relationship exists between put-to-call traded volume ratio and the slope of the implied volatility curve as a measure of market sentiment; however, the results are mixed. Dennis and Mayhew (2000) show that a small options traded volume leads to this result, and use the most active observations to test the robustness, discovering a negative relationship between put-to-call traded volume ratio and the slope of the smile, suggesting that the positive coefficient on this variable was primarily driven by an illiquid traded volume. In this study, we also use the most liquid data and replace the variable by put-to-call open interest ratio in order to test the robustness.

For the endogenous variables, the coefficient on SKW is significantly positive, and that on KUT is the same, with the exception of the short-term options data. The evidence indicates that a more negatively skewed stock exhibits a steeper implied volatility curve, and leptokurtic risk-neutral kurtosis leads to a flatter slope of implied volatility, except for short-term options, which is consistent with the results of Bakshi, Kapadia, and Madan (2003) and Lin, Chang, and Paxson (2008). These variables that vary with time are instrumental in explaining the differential pricing of individual stock options, which displays the expectation of underlying prices by investors. In general, as the volatility of underlying price becomes larger, it is accompanied by more negative skewness and a fat-tailed stock price distribution.

The market-wide variable tSTD100 has a negative relationship with the slope of individual stock options implied volatility curve, indicating that greater volatility of the market is accompanied by a smile of steeper slope. tSKW100 has positive coefficients, and tKUT100 has a positive relationship with the slope of individual stock options implied volatility curve, the results indicating that the more negative (less negative) the skewness, and the less (greater) the risk-neutral kurtosis for the stock index option, the steeper (i.e., more negative) (flatter) the slope of the implied volatility curve for the individual stock options. While the market is of a more left-skewed distribution, the stock price distribution also tends to be left-skewed, and with a higher volatility of stock prices, meaning that when the market is pessimistic, this leads to higher volatility in stock price in the future. Empirical findings from market-wide variables also show that the overall market volatility or investors’ prediction of the stock market index exhibits a similar volatility trend to the stock price of individual companies. These results can increase investors’ awareness of the impact of the market on the share price of individual companies.

Finally, the time lag variable 1, −tiSLOPE proxy for other variables not included in the model has a positive coefficient, indicating that the steeper (flatter) the slope of the implied volatility curve, the steeper (flatter) the next day’s slope; this shows persistence in the slope of the implied volatility curve for individual stock options.

In order to compare the influence of firm-specific and market-wide factors, we drop the market-wide variables from the regression equation, and find that the adjusted r-square for the firm-specific variables is much larger than that for the market-wide variables. This result is consistent with that of Dennis and Mayhew (2002), who find that the firm-specific factors have much more influence on the slope of smile. Table 3: The Determinants of the Slope of Volatility Smile Short-term options Medium-term options Long-term options Variables Coefficient t-stat Coefficient t-stat Coefficient t-stat Intercept 0.1241 10.52** 0.0483 4.36** -0.0010 -0.08 STD 0.0182 3.69** 0.0167 3.50** 0.0770 3.99**


LEVERAGE -0.0000 -2.89** -0.0000 -3.08** -0.0000 -2.42** SIZE -0.0129 -11.95** -0.0058 -4.13** -0.0077 -2.86** BETA -0.0014 -1.13 -0.0003 -0.44 -0.0042 -1.94* VOLUME -0.0048 -7.59** -0.0022 -4.39** -0.0033 -2.56** PUTCALL -0.0000 -0.43 0.0000 2.14** 0.0000 0.10 SKW 0.0674 17.46** 0.0329 5.14** 0.1002 4.20** KUT -0.0034 -4.63** 0.0001 0.17 0.0155 4.20** STD100 -0.0082 -0.96 -0.0149 -1.48 -0.0688 -2.49** SKW100 0.0326 9.54** 0.0074 3.72** 0.0017 0.53 KUT100 0.0090 6.06** 0.0035 3.64** 0.0060 2.58** SLOPE

t-1 0.6174 26.96** 0.7953 17.18** 0.6077 5.31**

Adjusted 2R 0.4774 0.7030 0.4847 * Denotes the test is significant at the 10% level, and ** denotes the test is significant at the 5% level. 4.2. Robustness Tests

We also test the robustness of the results described above, as shown in Table 4. First, we use put-to-call open interest ratio to replace the variable put-to-call traded volume ratio in equation (20). As the coefficient on put-to-call open interest ratio is significantly negative for short-term options, non-significantly negative for medium-term options, and significantly positive for long-term options, this might explain the demand on short-term and medium-term put options in order to defend the downside risk, and investors may hold a larger position on long-term call options. These results reflect that investors feel bearish in the short term and bullish in the long run; therefore, put-to-call open interest ratio can better represent the sentiment of investors than put-to-call traded volume ratio, reflecting whether investors will feel bullish or bearish in the future.

Second, we also use the liquid data to test the robustness of the trading pressure variable, which does not explain the smile of implied volatility. Liquid data are defined as the sum of the call and put options traded volume in the upper quartile of each time to maturity category at a certain time. The coefficient on put-to-call traded volume ratio is non-significantly positive for short-term options, remains the same for medium-term options, and is non-significantly negative for long-term options. In summary, put-to-call traded volume ratio does not represent the sentiment of investors desiring more put options to defend the underlying downside risk. In this instance, we can find no evidence to support the results of Dennis and Mayhew (2000), who report that the positive coefficient on the variable put-to-call traded volume ratio was primarily driven by illiquid traded volume.

For the robustness testing of another category of maturities, we also group the sample options into three categories in order to examine the analysis across maturities. If an option has a remaining time to expiration of between 9 and 90 days, it is grouped in the short-term option category; if the remaining time to expiration falls between 91 and 180 days, the option is grouped in the medium-term category, and if the remaining time to expiration falls between 181 and 270 days, the option is grouped in the long-term category. The results remain the same as for the previous category definitions of option maturity.


Table 4: Robustness Analysis Panel A: Short-term options Open interest Most liquid Variables Coefficient t-stat Coefficient t-stat Intercept 0.1125 11.13** 0.1249 7.00** STD 0.0157 3.62** 0.0430 4.26** LEVERAGE -0.0000 -3.06** -0.0000 -1.23 SIZE -0.0124 -12.50** -0.0134 -8.30** BETA -0.0009 -0.74 -0.0055 -2.86** VOLUME -0.0043 -7.72** -0.0043 -4.29** PUTCALL 0.0000 0.70 PUTCALLOI -0.0010 -2.70** SKW 0.0643 19.29** 0.0737 10.46** KUT -0.0032 -4.94** -0.0037 -3.36** STD100 -0.0120 -1.37 -0.0012 -0.09 SKW100 0.0337 10.11** 0.0391 7.51** KUT100 0.0100 6.88** 0.0087 3.80** SLOPE 1−t 0.6230 30.72** 0.6297 17.95** Adjusted 2R 0.4799 0.5313

** Denotes the test is significant at the 5% level. Panel B: Medium-term options Open interest Most liquid Variables Coefficient t-stat Coefficient t-stat Intercept 0.0402 4.55** 0.0470 4.27** STD 0.0147 3.88** 0.0101 1.22 LEVERAGE -0.0000 -3.19** -0.0000 -3.66** SIZE -0.0052 -4.35** -0.0071 -6.52** BETA 0.0003 0.40 0.0015 0.75 VOLUME -0.0020 -5.02** -0.0011 -1.26 PUTCALL 0.0000 3.32** PUTCALLOI -0.0000 -0.20 SKW 0.0303 5.46** 0.0224 8.05** KUT 0.0004 0.89 -0.0002 -0.20 STD100 -0.0150 -1.66* -0.0061 -0.29 SKW100 0.0076 4.28** 0.0075 2.62** KUT100 0.0036 3.99** 0.0042 3.77** SLOPE 1−t 0.8064 19.80** 0.8216 51.31** Adjusted 2R 0.7137 0.7307

* Denotes the test is significant at the 10% level, and ** denotes the test is significant at the 5% level. Panel C: Long-term options Open interest Most liquid Variables Coefficient t-stat Coefficient t-stat Intercept -0.0085 -0.77 0.0133 0.34 STD 0.0695 3.89** 0.0083 0.27 LEVERAGE -0.0000 -2.02** -0.0000 -3.09** SIZE -0.0071 -2.95** -0.0103 -2.78** BETA -0.0029 -1.57 -0.0005 -0.11 VOLUME -0.0026 -2.41** 0.0031 1.22 PUTCALL -0.0000 -0.68 PUTCALLOI 0.0014 2.06** SKW 0.0906 4.15** 0.0647 4.05** KUT 0.0141 4.14** 0.0113 3.70** STD100 -0.0654 -2.66** -0.0824 -2.15** SKW100 0.0029 0.98 -0.0160 -2.09** KUT100 0.0062 2.89** 0.0025 0.56


SLOPE 1−t 0.6376 5.98** 0.7926 33.65** Adjusted 2R 0.5121 0.3223

**Denotes the test is significant at the 5% level. 5. Conclusion In this article we examine the firm-specific and market-wide factors explaining the implied volatility skew (smile) in LIFFE equity options, and use Gram-Charlier series expansion to approximate the distribution of the logarithm of the stock price, incorporating non-normal skewness and kurtosis terms in the adjusted option price to expand the option price formula of Merton (1973) as a theoretical model and estimate the risk-neutral moments by numerically solving the nonlinear least-squares problem. In this study we first use the Gram-Charlier model to price American equity options, this method providing another theoretical formula with which to model option price. Our empirical results may help to understand the influence of firm-specific and market-wide factors on the slope of the smile in LIFFE equity options.

First, we show that firms with a larger volatility of stock return tend to have an implied volatility curve of less negative slope, and find that firm-specific variables such as leverage ratio, firm size, beta, and traded volume provide useful explanations for the slope of the smile. On average, firms using higher financial leverage, larger firms, higher market risk firms, and larger traded volume firms tend to have an implied volatility curve of more negative slope. The put-to-call traded volume ratio is non-significantly related to the slope of the smile for short-term and long-term options, and significantly positive for medium-term options. In general, the more negative the risk-neutral skewness, the steeper the implied volatility slope, and the more fat-tailed the risk-neutral distribution, the flatter the implied volatility slope.

Second, we also examine how market-wide variables influence the slope of implied volatility of individual stocks. The results indicate that the higher the volatility of the FTSE 100 index, the more negative the slope of the smile of individual stocks, meaning that a higher volatility market is accompanied by greater volatility of stocks. The more negative the skewness of the FTSE 100 index option, the steeper the volatility smile of individual stocks, and the greater the kurtosis of the FTSE 100 index option, the flatter the volatility smile of individual stocks. Persistence of the slope of the smile is also in evidence.

In the analysis of robustness, the results indicate that the variable put-to-call open interest ratio can best represent the sentiment of investors, even though we use the most liquid data to test the robustness; we cannot show that investors wish to attain a larger position in put options to hedge the downside risk. The category of maturity does not on average influence the empirical results.


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