CHAPTER IV THE VOLATILITY STRUCTURE IMPLIED BY NIFTY INDEX AND SELECTED STOCK OPTIONS

Transcription

1 CHAPTER IV THE VOLATILITY STRUCTURE IMPLIED BY NIFTY INDEX AND SELECTED STOCK OPTIONS 4.1 INTRODUCTION The Smile Effect is a result of an empirical observation of the options implied volatility with same expiration date, across different exercise prices. It typically describes a U-shape from showing higher implied volatility for in-the-money and out-of-the-money options than its corresponding implied volatility figures for at-the-money options. The Black-Scholas (1975) option pricing model is widely used to value options written on a large variety of underlying assets. Despite this widespread acceptance among practitioners and academics, the discrepancies between market and Black-Scholes prices are obvious and systematic. If the market were to price options according to the Black-Scholas model, all options on one stock would have the same implied volatility. However, it is well-known that, at any moment of time implied volatility obtained by the Black-Scholes model varies across time to maturity as well as strike prices. The pattern of implied volatility for different time to maturity is known as the term structure of implied volatility, and the pattern across strike prices is known as the volatility smile or the volatility sneer. A term of volatility structure is used generally to refer to the pattern across both strike prices and time to maturity. ~ Chapter 4 ~ 88

2 Many early studies have documented a U-shape smile pattern for implied volatility in many options markets prior to the 1987 stock market crash. For example, Macbeth and Mervilli (1979) found in-the-money stock option with a short remaining time to expiration tend to have higher implied volatilities than corresponding options with a longer time to expiration. Rubinstein (1985) concluded that systematic deviation from the Black-Scholas model appears to exist, but the pattern of deviations varies over time. Shastri and Wethyavivorn (1987) found that implied volatilities from foreign currency options traded on the Philadelphia Stock Exchange (PHLX) in 1983 and 1984 are a U-shaped function of the exchange rate divided by the strike price. Sheikh (1991) argued that a U-shaped pattern occurred for the S&P 100 options during various sub-periods between 1983 and Fung and Hsieh (1991) discuss informally some empirical smile effect for foreign currency options traded on the Chicago Mercantile Exchange (CME). Heyen (1993) has shown that U- Shaped functions can describe the pattern of implied volatility from nine months transaction prices of European Options Exchange (EOE) stock index options, which are European style options on an index of 25 active stocks on the Amsterdam stock Exchange. Taylor and Xu (1994) fit U-shaped functions to implied volatilities from foreign currency options traded on the PHLX over a longer period from November 1984 to January They also found that the magnitude of smile effect is a decreasing function of time to maturity. Duque and Paxson (1994) provided evidence of smile pattern in equity call options on ~ Chapter 4 ~ 89

3 LIFFE and relatively high implied volatility for in-the-money options. Bakshi, et al. (1997) exhibit a clear U-shaped pattern across moneyness, with the most distinguished smile evident for options near expiration. Brown and Taylor (1997) used Asay Model on the SPI futures option and found that the model tends to overprice call options and under-price put options. Pena, et al. (1999) found smile pattern in Spanish IBEX-35 index. In contrast to the above theoretical and empirical results showing a symmetric pattern for implied volatility against strike price, Rubinstein (1994) develops a new method for inferring risk-neutral probabilities from the simultaneously observed prices of S&P 500 index option for the post-crash of 1987 and found smile pattern in the implied volatilities during pre-crash changed into a sneer in the post-crash period. Dumas, Fleming and whaley (1998) also illustrate that the volatility structure for S&P 500 index options has changed from the symmetric smile pattern to more a sneer since the stock market crash of In other terms, implied call (put) volatilities decrease monotonically as the call (put) option goes deeper out-of-the-money (in-the-money). Brown (1999) extends Dumas, Fleming and Whaley s (1998) results to the SPI 200 futures options on the Sydney Futures Exchange (SFE), covering the period from June 1993 to June In-the-money call (put) options are generally trading at higher (lower) implied volatilities than out-of-the-money call (put) options. Duque and Lopes (2003) empirical support for the notion is ~ Chapter 4 ~ 90

4 that options tend to die smiling in call option quotes on nine heavily traded stock on the London International Financial Futures and Options Exchange (LIFFE) between August 1990 and December However, the changes in the smile pattern are asymmetric. The wry grin found for longer-term options is converted into a reverse grin for options near expiration. For medium term options, the smile is more symmetric. Malin Engstrom (2002) shows U-shaped smile pattern from 27 individual stock options traded in the Stockholm Stock Exchange (StSE) during the period from July 1, 1995 to February 1, Bollen and Whaley (2004) show a sneer pattern for S&P 500 index options and U-shaped smile pattern from 20 individual stock options traded in the Chicago Board Options Exchange (CBOE) over the period from June 1995 to December In contrast, limited research on this subject is available for India due to the nascent nature of its derivatives market. The previous work provides evidence for existence of volatility smile in the Indian options market. Varma (2002) observes mispricing in the Indian index options market and estimates the volatility smile for call & put options and found that is different across option types. Misra. D, et al. (2006) found that deeply in-the- money and deeply out-of-the-money options have higher implied volatility than at the money options, as well as it is higher for far the month option contracts than for near the month option contracts. Malabika, Devanadhen and Srinivasan (2008) found out U-shaped smile pattern for 26 individual stock call options traded in ~ Chapter 4 ~ 91

5 the National Stock Exchange (NSE) covering the period from January to December The implied volatility structure of call option is similar to Duque and Lopes (2003). The Indian studies fail to provide any strong evidence on volatility structure in accordance with time left to maturity, because of small sample size and short study periods, and immaturity of market structure in the initial phase. The study attempts to fill this research gap for the Indian option market literature. The study specifically examines whether asymmetry volatility structure implied by Indian option market is in accordance with time left to maturity. The rest of Chapter 4 is organized as follows. Section 4.2 provides the data and methodology applied in chapter 4. Section 4.3 presents the two dimensional pattern of implied volatility structure against moneyness. Section 4.4 presents three dimensional pattern of implied volatility structure against moneyness and time left to maturity. Section 4.5 provides summary and the conclusion for chapter 4. ~ Chapter 4 ~ 92

6 4.2 DATA AND METHODOLOGY Data Data description This empirical analysis focuses on S&P CNX Nifty option and selected five stock options for the period from January 1, 2002 to June 30, S&P CNX Nifty option is European style and expires on the last Thursday of the expiry month or the preceding trading day, if the last Thursday is a trading holiday. Individual stock options are American style and expire on the last Thursday of the expiry month or before. The corresponding future contract of underlying Nifty index and selected individual stocks were taken during the sample period. The future contract has exactly the same contract specification as the S&P CNX Nifty index and selected five individual stock options. Estimating implied volatilities requires estimates of the time to maturity and risk-free interest rate. Time to maturity is measured as the number of calendar days between the trade date and the expiration date. Mumbai Inter Bank Offer Rate (MIBOR) was used as proxy for risk-free interest rate. 14 days, one month and three month interest rate were taken and converted into continuous rate. One of these three interest rates will be employed depending upon the time to maturity mentioned below: ~ Chapter 4 ~ 93

7 Time to Maturity Interest rate < 30 days 14 days days 1 month > 61 days 3 month Sampling Procedure Nifty option and selected individual stock option constitute 5,94,589 observations for call and put option each. To obtain a relatively accurate measurement of implied volatility, the option must be chosen carefully. Options devoid of informative contents are excluded from the sample based on the following criteria: 1. Option whose time to maturity is lower than 7 days and higher than 90 days. The shorter term options have relatively small time premium, hence the estimation of volatility is extremely sensitive to any possible measurement errors, particularly if the option is not at-the-money. Hentschel (2000) shows other liquidity biases arising due to the rolling over attitude of fund managers. The long term options, on the other hand, are simply not traded. (similar exclusionary criteria are applied among other by Bakshi, et al. (1997) or by Dumas, et al. (1998)). 2. Options which are not traded ~ Chapter 4 ~ 94

8 3. The option price violates the arbitrage condition i.e. the price of the call or put is smaller than Black-Scholas-Merton price in the limit of zero volatility. Options dropped from the sample because they meet at least one of these conditions amount to 1,70,812 for call and 82,949 for put. Only 18.13% of call sample and 13.95% of put sample were taken for the study. Table 4.1 reports the number and percentage of the quotation cleared off the database according to each exclusion criterion. ~ Chapter 4 ~ 95

9 TABLE 4.1 Number of Observations and Percentage of the Contracts Filtered This table presents the number of observations of S&P CNX Nifty index and five selected stock options over the period from January 2002 to June The individual as well as total number of observations and percentage of the quotations cleared off the database according to following criterion. (1) Option whose time to Maturity (TTM) is lower than 7 days and higher than 90 days. (2) Options which are not traded. (3) Option price violates the arbitrage condition. Panel A contains number of observations and percentage of the contracts filtered of the call option sample according to each exclusion criterion. Panel B contains number of observations and percentage of the contracts filtered of the put option sample according to each exclusion criterion. Panel A: Call Option Particular S&P CNX NIFTY INFOSYS ITC RANBAXY RELIANCE SBI Total % Total TTM < 7 days TTM > 90 days No option traded Violation of Arbitrage Final sample Panel B: Put Option Particular S&P CNX NIFTY INFOSYS ITC RANBAXY RELIANCE SBI Total % Total TTM < 7 days TTM > 90 days No option traded Violation of Arbitrage Final sample ~ Chapter 4 ~ 96

10 4.2.2 Methodology Implied volatility is the volatility implied by an option price observed in the market based on an option pricing model. Thus, this study is dependent on option pricing model. In this subsection, the original Black-Scholas model (BS) and its limitations are briefly discussed. Then a dividend-adjusted BS model is presented in detail and used as the option pricing model in this study The Option Pricing Model Black-Scholes (1973) Model The assumptions underlying the original Black-Scholas option pricing model are as follows: 1. The stock price follows a lognormal distribution with a constant volatility. 2. The short selling of securities with full use of proceeds is permitted. 3. There are no transactions cost or taxes. All securities are perfectly divisible. 4. There are no dividends during the life of the derivative. 5. There are no riskless arbitrage opportunities. 6. Securities trading are continuous. 7. The risk-free rate of interest is constant and is the same for all maturities. Based on the assumption above, the Black-Scholas option pricing formulae for call and Put option can be written as ~ Chapter 4 ~ 97

11 (4-1) and (4-2) Respectively, where Where C = the call option price P = the put option price S = the current stock price X = the exercise price of the option T = the time remaining until expiration of the option r = the continuously compounded annualized risk-free rate of interest for the period of T = the annualized volatility of the stock return N ( ) = the cumulative normal density function of ( ) One of the assumptions employed by Black-Scholas (1973) is that the stock pays no dividend (Assumption 4). However, dividends on some stock may be substantial and can have a significant effect on the valuation of option. Therefore a dividend adjustment must be allowed in the option pricing formulae. ~ Chapter 4 ~ 98

12 Merton (1973) Model Merton model generalizes the Black and Scholas (1973) model by relaxing the assumption of no dividend. Merton (1973) allows for a constant continuous dividend yield on the stock and/or stock index. Let denote the volatility implied by an option price and denote the annualized continuously compounded dividend yield during the remaining life of the option, then the Merton (1973) formulae for call and put options can be expressed as (4-3) (4-4) respectively, where Hence, an estimate of the dividend yield is needed. For the purpose of this study, future contracts are utilized, which have exactly the same expiry cycle, expiry date and underlying asset with the option contract. The cost-of-carry model (see, e.g. Hull, 2003) gives ~ Chapter 4 ~ 99

13 (4-5) Combining (2.5) with (2.3) and (2.4) yields and (4-6) (4-7) where, By solving (4-6) and (4-7) numerically, the implied volatility series for call and put options can be constructed. Inverting an option model like that of Black-Scholas is a difficult task, therefore generally implied volatility is extracted from the option prices by using some numerical methods viz., Newton-Rapshson Method; by equating the difference between observed price of the option and the theoretical price of the option to zero ~ Chapter 4 ~ 100

14 and solving for volatility. (See Appendix for Newton-Raphson algorithm implementation in Visual basic) Then two dimensional graphs are plotted to examine the effect of volatility smile or sneer, i.e. how implied volatilities vary across different group of moneyness in the data set. Within the each group moneyness, the average implied volatility is calculated. It is necessary to mention that there is an implicit assumption when taking average in each group, as suggested by Brown (1999), that the trends in implied volatility will influence implied volatilities from options with different strike prices and different maturities equally. Furthermore, to examine how volatility smile or sneer is affected by time to maturity, three dimensional graphs of implied volatility against moneyness and maturity of the option are plotted. This is done by assigning all the options records under investigation into groups according to moneyness and time to maturity, and then taking an average for implied volatilities in each group Moneyness The term Moneyness is defined by various authors in different models. Models prescribed by various authors in literature are given in Table 4.2. ~ Chapter 4 ~ 101

15 TABLE 4.2 Models for Computing Moneyness Authors Moneyness (Mn) formula Gross & Waltners (1995) Malin Engstorm (2001) Natenbery (1994), Dumas (1998) & Tompkin (1999) Jackwerth & Rubinstein (1996) F K log Mn = T X Mn = ln S log Mn = T Mn = X F X F Bollen & Whaley (2002) rt ( S PVD) e Mn = 1 X Brown (1999) F Mn = 1 K S ΣNPV ( Div) Duque & Lopes (2003) Mn = r ( T t) Xe Alessandro Beber (2001) X log F Mn = δ T atm,t X Dumas, Fleming & Whaley (1996) Mn = 1 F ~ Chapter 4 ~ 102

16 Moneyness is defined as the ratio of the exercise price to the future price. The options are divided into five subgroups according to Moneyness. The upper and lower bounds of the moneyness categories are listed in Table 4.3. Options are classified as follows: Deep in-the-money (Deep ITM) A Deep in-the-money (Deep ITM) option is an option that would lead to a high positive cash flow to the holder if it were exercised. An option on the index or individual stock option is said to be Deep in-the-money when Moneyness [Exercise price (X)/Future price (F)] is less than or equal to 0.90 i.e If the moneyness is less than or equal to.90, the call is said to be Deep ITM. In the case of a put, the put is Deep OTM (Deep out-of-the-money) In-the-money (ITM) An In-the-money (ITM) option is an option that would lead to a positive cash flow to the holder if it were exercised. An option on the index or individual stock option is said to be In-the-money when moneyness [Exercise price (X)/Future price (F)] is between 0.90 and 0.98 ( ). If the moneyness is between 0.90 and 0.98, the call is said to be ITM. In the case of a put, the put is OTM (Out-of-themoney). ~ Chapter 4 ~ 103

17 At-the-money (ATM) At-the-money (ATM) option is an option that has zero cash flow at the time of exercising. An option on the index or individual stock option is said to be at-themoney when moneyness [Exercise price (X)/Future price (F)] is between 0.98 and 1.02 ( ). If the moneyness is between 0.98 and 1.02, the call and put is said to be ATM Out-of-the-money (OTM) Out-of-the-money (OTM) option is an option that would lead to a negative cash flow if it was exercised. An option on the individual stock option is said to be out-of-the-money when Moneyness [Exercise price (X)/Future price (F)] is between 1.02 and 1.10 ( ). If the moneyness is between 1.02 and 1.10, the call is said to be OTM. In the case of a put, the put is ITM (In-the-money) Deep out-the-money (Deep OTM) Deep out-the-money (Deep OTM) option is an option that would lead to a high negative cash flow when it was exercised. An option on the individual stock option is said to be Deep out-of-the-money when moneyness [Exercise price (X)/Future price (F)] is greater than If the moneyness is greater than 1.10, the call is said to be Deep OTM. In the case of a put, the put is Deep ITM (Very Deep In-the-money). ~ Chapter 4 ~ 104

18 ~ Chapter 4 ~ 105

19 TABLE 4.3 Moneyness Category Definitions This table displays category number, labels and Corresponding moneyness (Mn) and ranges of option in the sample Category Label Range Deep in-the-money (DITM) call Deep out-of-the-money (DOTM) put In-the-money (DITM) call Out-ofthe-money (DOTM) put At-the-money (ATM) call At-the-money (ATM) put Out-of-the-money (DOTM) call Inthe-money (DITM) put Deep out-of-the-money (DOTM) call Deep in-the-money (DITM) put Mn < Mn < Mn < Mn 1.10 Mn > 1.10 In particular, the 40,952 index call option and 39,242 index put option contracts in the data set are assigned to five groups according to moneyness and nine groups according to the number of days left to expiration, i.e. 45 groups. Time to maturity is measured as the number of calendar days between the trade date and the expiration date. Similarly, the 66,860 stock call option and 43,707 stock put option contracts are placed into five groups according to moneyness and six groups according to the number of days left to expiration, i.e. 30 groups. Since stock options with time to maturity larger than 50 days are not traded as frequently as other stock options with shorter life. These stock options are assigned to one expiration interval. Within each group the average implied volatility is calculated. It ~ Chapter 4 ~ 106

20 is necessary to mention that there is an implicit assumption when taking average in each group, as suggested by Brown (1999), that the trends in implied volatility will influence implied volatilities from options with different strike prices and different maturities equally. 4.3 PATTERN IN IMPLIED VOLATILITY In Table 4.4, the implied volatility of both index call and put options is average over the moneyness groups. For index call options, the result indicates a clear U-shaped smile pattern, with the lowest average implied volatility found for atthe-money options. In the call option sample, the average implied volatility of the category 1 option (DITM calls) is 52.29%, more than 7% higher than the average implied volatility of category 5 options (DOTM calls), 48.76%. For the put option, the average implied volatility of the category 5 option (DITM puts) is 73%, more than 68% higher than the average implied volatility of category 1 options (DOTM puts), 43.54%. The implied volatility of the ATM options generally being lowest and symmetrically increasing with movement in either direction. For the call options, the average implied volatility of the category 1 options (DITM calls) is 141% higher than average implied volatility of ATM options. On the other end of the spectrum, the average implied volatility of category 5 options (DOTM calls) is 125% higher than the average implied volatility of ATM options. For put option, the average implied volatility of the category 1 options ~ Chapter 4 ~ 107

21 (DOTM puts) is 62% higher than the average implied volatility of ATM option (category 3 options). On the other hand, the average implied volatility of category 5 options (DITM puts) is 172% higher than the average implied volatility of ATM option (category 3 options). Index put option exhibits a skew shaped pattern. While comparing to index call option, higher average implied volatility can be viewed in index put option except DITM options. ~ Chapter 4 ~ 108

22 TABLE 4.4 Average Implied Volatility of Index Option against Moneyness This table displays implied volatilities, option premium, contract and number of observations of S&P CNX Nifty index option over the period from January 2002 to June Implied volatilities as well as premium are average into one of 5 groups of moneyness category. Moneyness category is defined as category 1 options for DITM call & DOTM put, category 2 options for ITM call & OTM put, category 3 options for ATM call & put, category 4 options for OTM call & ITM put and category 5 options for DOTM call & DITM put. Category Call Implied Volatility Option premium Contract No. of Observation Put Implied Volatility Option premium Contract No. of Observation ~ Chapter 4 ~ 109

23 CHART 4.1 Graphs of Index Option Average Implied Volatility against Moneyness This figure illustrates the volatility structure implied by the S&P CNX Nifty index option over the period from January 2002 to June The 40,952 call option and 39,242 put option contracts in the data set are assigned to one of 5 groups of moneyness category. Average implied volatility is calculated for each group. Moneyness category is defined as category 1 options for DITM call & DOTM put, category 2 options for ITM call & OTM put, category 3 options for ATM call & put, category 4 options for OTM call & ITM put and category 5 options for DOTM call & DITM put Implied volatility Call Put Moneyness ~ Chapter 4 ~ 110

24 Table 4.5 contains the average implied volatilities of 5 individual stocks in the sample. As the results in the table show, the implied volatilities of stock call and put option exhibit U-shape and skew shape respectively. In stock call option, the average implied volatility of the category 1 options (DITM calls) is 64.54%, more than 15% higher than the average implied volatility of category 5 options (DOTM calls), 56.20%. For the stock put option, the average implied volatility of the category 5 options (DITM puts) is 86.31%, more than 68% higher than the average implied volatility of category 1 options (DOTM puts), 51.53%. For stock call options, the average implied volatility of the category 1 options (DITM calls) is 93% higher than the average implied volatility of ATM options (category 3 options). On the other end, the average implied volatility of category 5 options is 68% higher than the implied volatility of ATM options (category 3 options). In case of stock put options, the average implied volatility of category 1 options (DOTM puts) is 41% higher than the average implied volatility of the ATM options (category 3 options). On the other side, the average implied volatility of category 5 options (DITM puts) is 135% higher than the average implied volatility of ATM options (category 3 options). ~ Chapter 4 ~ 111

25 TABLE 4.5 Average Implied Volatility of Stock Option against Moneyness This table displays implied volatilities, option premium, contract and number of observations of five individual stock options over the period from January 2002 to June Implied volatilities as well as premium are average into one of 5 groups of moneyness category. Moneyness category is defined as category 1 options for DITM call & DOTM put, category 2 options for ITM call & OTM put, category 3 options for ATM call & put, category 4 options for OTM call & ITM put and category 5 options for DOTM call & DITM put. Category Call Implied Volatility Option premium Contract No. of Observation Put Implied Volatility Option premium Contract No. of Observation ~ Chapter 4 ~ 112

26 CHART 4.2 Graphs of Stock Option Average Implied Volatility against Moneyness This figure illustrates the volatility structure implied by the five selected individual stock options over the period from January 2002 to June The 66,860 stock call option and 43,707 stock put option contracts in the data set are assigned to one of 5 groups of moneyness category. Average implied volatility is calculated for each group. Moneyness category is defined as category 1 options for DITM call & DOTM put, category 2 options for ITM call & OTM put, category 3 options for ATM call & put, category 4 options for OTM call & ITM put and category 5 options for DOTM call & DITM put Implied volatility Call Put Moneyness ~ Chapter 4 ~ 113

27 4.4 THREE DIMENSIONAL VIEW OF VOLATILITY STRUCTURE To investigate the volatility structure further, three dimensional graphs of implied volatilities against moneyness and time to maturity of the option are plotted in Chart 4.3 and Chart 4.4 for index call and put options, respectively. This is done by assigning all the options records under investigation to groups according to moneyness and time to maturity, and then taking an average for implied volatilities in each group. Tables 4.6 and 4.7 report the average implied volatility within each group for index call and put options respectively. The average implied volatilities differ for the different time to maturity groups. For index call options, the longer the time left to maturity, the lower the implied volatility tends to be each moneyness group. In addition, the shorter the time left to maturity, the more pronounced the U-shape of the smile appears to be. This pattern is something referred to as the options are dying smiling. When time to maturity is greater than 30 days and less than 80 days, it is expected deep out-of-the-money call options on average trade at higher implied volatilities than deep in-the-money call options. But for short time period, implied volatilities of deep in-the-money are higher than deep out-of-the-money implied volatilities. This finding is similar to Duque & Lopes (2003) that the change in smile shape as time to maturity dies out. This may be seen as in support of the expected wry grin for long term options. But the reverse grin is deeply unexpected for short-term options. In fact, the maturity approach changes the options smile asymmetry, converting a wry grin typical for longer term series into a reverse ~ Chapter 4 ~ 114

28 grin for nearly expiring options, with a more or less symmetric smile in a three dimensional domain. For the index put options, the shorter the time left to maturity, the more pronounced U-shape of the smile appear to be. The deep in-the-money options on average trade are higher implied volatilities than deep out-of-the-money options. Longer the time left to maturity, the lower the implied volatilities tend to be in deep in-the-money put options and more or less constant implied volatilities tend to be in deep out-of-the-money. ~ Chapter 4 ~ 115

29 TABLE 4.6 Group Average of Index Call Option Implied Volatilities with Respect of Moneyness and Time to Maturity The table presents implied volatilities of S&P CNX Nifty index call option over the period from January 2002 to June The 40,952 index call option contracts in the data set are assigned to five groups according to moneyness and nine groups according to the number of days left to expiration, i.e. 45 groups. Average implied volatility is calculated for each group. Time to maturity is measured as the number of calendar days between the trade date and the expiration date. In parentheses the number of observations is given according to the moneyness and time to maturity of call options. Moneyness category is defined as category 1 options for DITM call & DOTM put, category 2 options for ITM call & OTM put, category 3 options for ATM call & put, category 4 options for OTM call & ITM put and category 5 options for DOTM call & DITM put. Moneyness Category Total Time to Maturity (days) Total (230) (323) (301) (146) (148) (50) (10) (5) (4) (1217) (1422) (2255) (2707) (1299) (1030) (608) (294) (180) (89) (9884) (1912) (2871) (3432) (1864) (1526) (861) (515) (461) (262) (13704) (1511) (2010) (2431) (1537) (1229) (584) (482) (402) (175) (10361) (898) (1068) (1268) (879) (669) (417) (299) (175) (113) (5786) (5973) (8527) (10139) (5725) (4602) (2520) (1600) (1223) (643) (40952) ~ Chapter 4 ~ 116

30 CHART D view of Volatility Structure for Index Call Options This figure illustrates a 3-D view of the volatility structure implied by the S&P CNX Nifty index call option over the period from January 2002 to June The 40,952 index call option contracts in the data set are assigned to one of 45 groups by moneyness and time to maturity. Average implied volatility is calculated for each group. Time to maturity is measured as the number of calendar days between the trade date and the expiration date. Moneyness category is defined as category 1 options for DITM call & DOTM put, category 2 options for ITM call & OTM put, category 3 options for ATM call & put, category 4 options for OTM call & ITM put and category 5 options for DOTM call & DITM put. 0.8 Implied volatility Moneyness Time to Maturity ~ Chapter 4 ~ 117

31 TABLE 4.7 Group Average of Index Put Option Implied Volatilities with Respect of Moneyness and Time to Maturity This table presents implied volatilities of S&P CNX Nifty index put option over the period from January 2002 to June2010. The 39,242 put option contracts in the data set are assigned to five groups according to moneyness and nine groups according to the number of days left to expiration, i.e. 45 groups. Average implied volatility is calculated for each group. Time to maturity is measured as the number of calendar days between the trade date and the expiration date. In parentheses the number of observations is given according to the moneyness and time to maturity of call options. Moneyness category is defined as category 1 options for DITM call & DOTM put, category 2 options for ITM call & OTM put, category 3 options for ATM call & put, category 4 options for OTM call & ITM put and category 5 options for DOTM call & DITM put. Moneyness Category Total Time to Maturity (days) Total (802) (1129) (1414) (654) (493) (263) (111) (54) (25) (4945) (2236) (3503) (4198) (1981) (1646) (1064) (610) (433) (235) (15906) (1644) (2438) (2886) (1450) (1125) (598) (344) (294) (165) (10944) (797) (1111) (1241) (776) (85) (247) (150) (97) (54) (5058) (344) (397) (554) (373) (256) (181) (132) (90) (62) (2389) (5823) (8578) (10293) (5234) (4105) (2353) (1347) (968) (541) (39242) ~ Chapter 4 ~ 118

32 CHART D View of Volatility Structure for Index Put Options This figure illustrates a 3-D view of the volatility structure implied by the S&P CNX Nifty index call option over the period from January 2002 to June The 40,952 index call option contracts in the data set are assigned to one of 45 groups by moneyness and time to maturity. Average implied volatility is calculated for each group. Time to maturity is measured as the number of calendar days between the trade date and the expiration date. Moneyness category is defined as category 1 options for DITM call & DOTM put, category 2 options for ITM call & OTM put, category 3 options for ATM call & put, category 4 options for OTM call & ITM put and category 5 options for DOTM call & DITM put Implied volatility Moneyness Time to Maturity ~ Chapter 4 ~ 119

33 In Table 4.8, the implied volatilities for five selected stock call options are averaged over the five groups of moneyness and six groups of time to maturity i.e. 30 groups. Since stock options with time to maturity larger than 50 days are not traded as frequently as other stock options with shorter life. These stock options are assigned into one maturity interval. Within each group the average implied volatility is calculated. The results indicate a clear U-shape smile pattern, with lowest average implied volatility found for the at-the-money in all the groups. When time to maturity is greater than 30 days, it is expected deep out-of-the-money stock call options on average trade at higher implied volatilities than deep in-the-money stock call options. But for short time period, implied volatilities of deep in-the-money are higher than deep out-of-the-money implied volatilities. The volatility term structure of stock call option is similar to that of index call option. That is, the term structure is similar to Duque & Lopes (2003) that the change in smile shape as time to maturity dies out. This may be seen as in support of the expected wry grin for long term options. But the reverse grin is deeply unexpected for short-term options. In fact, the maturity approach changes the options smile asymmetry, converting a wry grin typical for longer term series into a reverse grin for nearly expiring options, with a more or less symmetric smile in a three- dimensional domain (Chart 4.5). For the stock put option, the Table 4.9 and Chart 5.6 show, the shorter the time left to maturity, the more pronounced U-shape of the smile appears to be. The deep in-the-money options on average trade are higher implied volatilities than deep out-of-the-money options. Longer the time left to maturity, the lower the implied ~ Chapter 4 ~ 120

34 volatilities tend to be in deep in-the-money stock put options and more or less constant implied volatilities tend to be in deep out-of-the-money. The volatility term structure of stock put option is similar to that of index put option. That is, the term structure is close to be flat for at-the-money stock put options, and tends to be fluctuant for deep in-the-money option. ~ Chapter 4 ~ 121

35 TABLE 4.8 Group Average of Stock Call Option Implied Volatilities with Respect of Moneyness and Time to Maturity This table presents implied volatilities of five selected stock call option over the period from January 2002 to June The 66,860 stock call option contracts in the data set are assigned into five groups according to moneyness and six groups according to the number of days left to expiration, i.e. 30 groups. Average implied volatility is calculated for each group. Time to maturity is measured as the number of calendar days between the trade date and the expiration date. In parentheses the number of observations is given according to the moneyness and time to maturity of stock call options. Moneyness category is defined as category 1 options for DITM call & DOTM put, category 2 options for ITM call & OTM put, category 3 options for ATM call & put, category 4 options for OTM call & ITM put and category 5 options for DOTM call & DITM put. Moneyness Category Total Time to Maturity (days) Total (536) (682) (626) (113) (39) (8) (2004) (2783) (4542) (5202) (1847) (665) (192) (15231) (2610) (4002) (5139) (2701) (1278) (282) (16012) (3921) (5952) (7475) (3722) (1630) (333) (23033) (2328) (2922) (3244) (1393) (531) (162) (10580) (12178) (18100) (21686) (9776) (4143) (977) (66860) ~ Chapter 4 ~ 122

36 CHART D View of Volatility Structure for Stock Call Options This figure illustrates a 3-D view of the volatility structure implied by the five selected stock call options over the period from January 2002 to June The 66,860 stock call option contracts in the data set are assigned to one of 30 groups by moneyness and time to maturity. Average implied volatility is calculated for each group. Time to maturity is measured as the number of calendar days between the trade date and the expiration date. Moneyness category is defined as category 1 options for DITM call & DOTM put, category 2 options for ITM call & OTM put, category 3 options for ATM call & put, category 4 options for OTM call & ITM put and category 5 options for DOTM call & DITM put Implied volatility Moneyness Time to Maturity ~ Chapter 4 ~ 123

37 TABLE 4.9 Group Average of Stock Put Option Implied Volatilities with Respect of Moneyness and Time to Maturity This table presents implied volatilities of five selected stock put option over the period from January 2002 to June The 43,707 stock put option contracts in the data set are assigned to five groups according to moneyness and six groups according to the number of days left to expiration, i.e. 30 groups. Average implied volatility is calculated for each group. Time to maturity is measured as the number of calendar days between the trade date and the expiration date. In parentheses the number of observations is given according to the moneyness and time to maturity of stock put options. Moneyness category is defined as category 1 options for DITM call & DOTM put, category 2 options for ITM call & OTM put, category 3 options for ATM call & put, category 4 options for OTM call & ITM put and category 5 options for DOTM call & DITM put. Moneyness category Total Time to Maturity (days) Total (1471) (2214) (2538) (667) (152) (38) (7080) (3771) (5931) (7173) (2529) (735) (88) (20227) (2120) (3097) (3588) (1302) (346) (30) (10483) (1331) (1684) (1522) (413) (81) (8) (5039) (277) (292) (223) (68) (13) (5) (878) (8970) (13218) (15044) (4979) (1327) (169) (43707) ~ Chapter 4 ~ 124

38 CHART D View of Volatility Structure for Stock Put Options This figure illustrates a 3-D view of the volatility structure implied by the five selected stock put options over the period from January 2002 to June The 43,707 stock put option contracts in the data set are assigned to one of 30 groups by moneyness and time to maturity. Average implied volatility is calculated for each group. Time to maturity is measured as the number of calendar days between the trade date and the expiration date. Moneyness category is defined as category 1 options for DITM call & DOTM put, category 2 options for ITM call & OTM put, category 3 options for ATM call & put, category 4 options for OTM call & ITM put and category 5 options for DOTM call & DITM put Implied volatility Moneyness Time to Maturity ~ Chapter 4 ~ 125

39 4.5 CONCLUSION In this chapter, the volatility structure implied by S&P CNX Nifty option and selected five individual stock options are illustrated by two dimensional graphs of implied volatility against moneyness and three dimensional graphs of implied volatility against both moneyness and time to maturity. It is found that the U pattern is more pronounced for the call and put options implied volatilities. For both call and put option, deep in-the-money is higher than deep out-of-the-money implied volatilities. In accordance with time left to maturity, the implied volatility of deep inthe-money call options is higher than deep out-of-the-money for shorter time left to maturity. Implied volatility of deep out-of-the money call options is higher than deep in-the-money for longer time left to maturity. In case of put options, the shorter the time left to maturity, deep in-the-money options are higher than deep out-of-the-money. Longer the time left to maturity, the deep in-the-money options implied volatilities tend to decrease monotonically. The term structure of implied volatility is nearly flat of at-the-money call and put options, both more fluctuant for deep in-the-money and deep out-of-the-money options. ~ Chapter 4 ~ 126

40 In general, these results indicate that the Black-Scholes-Merton model applied in this study tends to overprice in-the-money call options and out-of-themoney put options and underprice in-the-money put option and out-of-the-money call options. At-the-money options are often most actively traded and hence they are less likely to be mispriced. ~ Chapter 4 ~ 127