# Volatility smiles

### Introduction

There seems to be a continued and growing demand and use of options in today's volatile markets for hedging, speculation and other uses. According to the Options Clearing House (2009), the number of options traded in the US in 2009 was 3.59 billion, 2008 was 3.58 billion, and 2007 was 2.86 billion and 2.03 billion in 2006. At the foundation of the 2007 stock market crash popularly referred to as ‘The credit crunch' is the American property market, multiple factors came together to create what is now being considered “the biggest financial market crash since 1929 - The Great Depression [1]”. Brunnermeier (2009) highlights that about $8 trillion was lost in stock market wealth from October 2007 to October 2008. Easy access to borrowing, excessive repackaging and reselling of credit instruments by banks, subprime mortgages offered to people who could hardly afford them, lax financial regulation of banks by the regulatory authorities, more integration of global financial markets and rising interest rates from 1% to 5.35% in America between 2004 and 2006 all came into play. In July 2007 Bear Stearns had told it investors it was unable to repay money from 2 of its hedge funds as rival banks were unwilling to bail it out. On August 9th 2007 the cost of credit skyrocketed affecting the US and most developed nations financial markets when BNP Paribas stopped investors pulling out cash in $2.2 billion worth of funds as a result of the liquidity market drying up, stating inability to price the assets in the funds as the cause. Northern Rock whose business model was based on borrowing from the money markets asked for help from the bank of England, which led to the highest rush on a bank to the value of £1bllion in a day by its customers on the 14th September 2007[2]. From the start of the crisis till date £ trillion has been pumped into the financial system by World Governments to stem the effects of the crisis. (source).

August 16, 2007 saw a new daily trading volume record with 23,75,650 contracts changing hands and November saw a new monthly record when volume reached 310 million contracts marking the first time options volume surpassed 300 million contracts during a trading month. (OCC Timeline, 2009)

“Ever since investors have been calculating implied volatilities, they have plotted them across strike prices for options with the same time to expiration and on the same underlying stock. These plots are called (implied) volatility “smiles”.” Jackwerth (2004, p5).

Most major financial markets are cyclical in nature with bubbles following bursts, the most recent market crash of 2007 was preceded by a property boom, similar to the dot.com (internet) bubble in 1999 before the crash in 2000 and 2001.

Over the 1928 – 96 period the United States went through the 1929 depression, World War II, the post-war growth, the oil crises of the early 1970s, the advent of the computer revolution, the internet bubble crash of 2001 and the modern service economy. The October 19, 1987 market crash is of interest in as it was a pivotal in the study of volatility, research found that after this time straight implied volatility graphs observed that followed the Black & Scholes model started to curve forming volatility skews, smirks[3] and smiles.

Research into predicting the crash of 1987 using information from volatility smiles was carried out by Gemmill (1996) and Bates (1991) where the options on the FTSE 100 Index and Standard & Poor 500 Index were used respectively as data sources.

### Importance of Topic:

The study will explore if option traders could use differences in the shape of volatility smile distributions of an index can provide information on whether options were able to deduce the market crash from market data and either cover their positions by their by buying puts to hedge exposed positions or short selling to maximise the returns from a likely market crash. The information could also be used as supportive information indicator to predict or confirm big market changes like a market crash or general change in attitude of traders in addition to the VIX index which is used as measure of the level of fear in the markets. The existence of skewness trading is discussed where out of sample skewness forecasts are used to trade in ATM delta neutral strips and straps in addition to straddles, Volatility trading is also discussed[4]. A major aspect of trading is stock market forecasting using technical analysis, there is a possibility with the evidence obtained from the study could be beneficial for trading applications by the Financial Sector by traders, market makers, and institutions.

### Relevance

Numerous research is available in the study of the 1929, 1987 crash in terms of stocks, options[5] and other financial instruments and the factors leading to the crashes. There is a gap in academic literature on options used for insurance or even hedging in the 2007 credit crunch. This could be partly due to the fact that even as at today, January 2010, although it seems the worst of the recession is over, however some banks have no fully shed their risky or bad debts, countries have large deficits and increasing unemployment. It seems the 2007 crash is still not over, nearly 3 years on, so it is understandable why there is not a lot of research done on the crash as the benefit of hindsight, which is usually better seen after recovery is not available yet. This paper however, looks at the period before the beginning of the crash which is unrelated to the magnitude of the crash, length of the crash, after effects and its recovery. It looks at whether options and in particular information from volatility smiles showed that traders were able to predict the 2007 market crash by the changes in the shapes of the smiles. This paper possibly will add to the growing area of financial economics that is looking into the information content of volatility smiles, their practical implications and the ability to shows patterns leading a market crash before. The most recent stock market crash ‘2007' is being investigated in this paper.

### Motivation:

The study aims to carry out a further study on Gemmill's work (1996) on volatility smiles using studying the changing shape of the volatility smile of U.K FTSE 100 index which found out that to measure if option traders anticipated the crash of 1987. There is an acknowledgement by Gemmill (2006) in the conclusion of his paper, that a gap in understanding exists regarding the factors that leads to a smile changing its shape over time, which leads to problems in trying to develop a model for smiles. More recent data from the 2006 and 2007 crash period is applied to this research to find out if similar or conflicting results are achieved. The motivation is to find if the movement that occurs is left skewness of the volatility smiles of the FTSE 100 Index options confirms the prediction of the crash by option traders.

### Literature Review

Studying the information content in volatility smiles of stock indexes is a growing area of financial economics; the focus of this paper is to investigate if this information was used to predict the market crashes. A body of research following different avenues has grown following the 1987 market crash where implied volatility graphs changed shape from a straight line usually found using the Black & Scholes (1973) model, to a volatility smile, smirk or skew.

Benhamou (2000) found options pre the 1987 crash and found that graphs of implied were straight. The Black & Scholes (1973) model is one of the models used for pricing of options, it assumes that the volatility of a stock is constant. Based on this model, which is generally recognised and used in pricing options in finance, the volatility smile which is a graph of the strike price against volatility or is it a graph of volatility against strike price (confirm) is a straight line.

The 1987 market crash signalled the beginning of a trend where it was noticed that the implied volatilities of options[6] differed between strike prices on the same underlying. In 1987, there was a market crash that changed the volatility of options and the volatility smiles curve showed asymmetric U- shape with the implied volatility in the high exercise price option (deep OTM calls ) generally higher than in the near the money option becoming described as a skewed or smile. (Chang, Lin and Paxson, 2008). Many papers[7] have been written about the effect of the crash Rubinstein (1994), Bates (1991), Duque and Lopes (2003), Gemmil (1996), Doran, Peterson and Tarrant (2007) researched into the factors that lead to changes in the volatilities across strike prices. All the papers found implied volatility of prices had skews post 1987 crash[8]

The volatility used in the Black and Scholes formula is not readily accessible, all other variables are easy to access[9]. there are three ways volatility is estimated - historical, implied and realised. The first is the implied volatility that makes the market price of option equal to the model price, and the second is the historical volatility that is estimated from the past time-series of the underlying asset returns and the third realized volatility is the movement in the underlying asset between the date when the option is traded and its expiry. Usually implied volatility is regarded as a superior forecast for future realised volatility because it both is compatible with option pricing theories and has a forward-looking property. There have been, however, extensive arguments regarding the informational efficiency and the degree of bias of the implied volatility parameter.”Kang, Kim and Yoon (2009)

The assumption of Black – Scholes model is that the volatility of the underlying asset is constant; and time to expiration is the only factor that affects the price of the option, which should produce a volatility smile that is a straight line. (Jackwerth, 2004). “Much of the modern financial economics is based on the assumption of a Black – Scholes type economy in which asset returns are lognormally distributed. But even in the early days of finance, Mandelbrot(1963) and others noted that the distribution of daily log returns is leptokurtic; the distribution exhibits fat tails and is too peaked to be normal.” Jackwerth (2004, p 39). Normal distribution as mentioned above has a skewness value of 0 and a kurtosis value of 3. Volatility models in general can be grouped into deterministic volatility models and stochastic or jump volatility models, volatility smiles fall under the later group. Volatility smiles have been widely researched, Bates (1991, 2000), Gemmill (1996), Jackwerth (2004), Bakshi, Cao and Chen (1997), Pan (2002). Most graphs show how options pricing skewness changes across the moneyness – the ratio of strike price to index level. Jackwerth (2004. p6).

After the 1987 crash, option prices began to present non straight lines that were negatively skewed, which show the implied volatility of out of the money options were higher than at the money options (Doran, Peterson , Tarant, 2007, Rubinstein 1985). Rubinstein (1985) found slightly U- Shaped volatility smiles for individual stock options, the violations of the Black & Scholes model were mainly within the bid-ask spreads. Different studies looking at the Black & Scholes implied and risk neutral skews have been done by Alexander and Lazar (2004), Bakshi et al (2003), Bates (1997, 2000), Jackwerth (2000). The findings show[10] that the “index skews are too pronounced, too persistent and have too much leverage to accord to standard portfolio theory and time –series analysis of the conditional densities of the US index returns post the 1987 crash.” Bates (2007) highlights the fact that index options have the tendency to overpredict volatility and jump risk.

Numerous studies on implied volatility smiles on equity indexes and individual stock are been carried out investigating the risk neutral distributions (Jackwerth 2000, Jondeau and Rockinger, 2000). Jackwerth (2004) found “that due to the difficulty of obtaining and working with individual stock volatility smiles, most studies have investigated index smiles”. Another study found the slope of the implied volatility skew of equity options traded on the LIFFE exchange(e.g FTSE 100) is flatter than that on the Chicago Board of Exchange.

### Methodology

### Data

The time period for the data used in this study is between 15 July 2006 and 15 December 2007. These dates were selected as 9 August 2007 is considered as the beginning of the credit crunch, marked by the increase in the cost of credit due to the lack of confidence generated amongst banks triggered by BNP Paribas' statement about market liquidity[11]. Another source [12]reveals that from July to September (Q4) of 2006 which is the period between is when the first signs of major problems were filtering through the system with higher bankruptcies of subprime lenders and massive jumps in interest rates on Collaterized Debt Obligations & Securitization products related to Subprime mortgages.

The implied volatility on the FTSE 100 stock index options with 18 month expiratory options are used to calculate the skewness measure. The skewness measure highlights the change taking place in the volatility smile. It compares the implied volatilities of calls, which have striking prices 2.5% above the forward price with calls, which have striking prices 2.5% below the forward price. Out of the money options have been used to generate implied volatilities for volatility smiles in some papers Gemmill (1996) and the Black and Scholes Model is based on at the money options. Chalamandaris and Tsekrekos (2009), found The out of the money calls and puts are the most traded and will be used more often for hedging, shorting or speculating the market movement. The data values used for this paper are the implied volatilities for options on the FTSE 100 Index and were downloaded from Bloomberg® terminals using the UKX Index GV functions and the dates were backtracked to dates used. Bloomberg's GV function allows you to chart the historical, Implied volatility price/ yields for up to four securities and allows for historical comparison of at the money volatility. It has the ability to compare volatility for different measures of option strikes and terms over time. Getting options prices of the FTSE 100 was not possible as this proved to be a very costly exercise from Euronext – Liffe in the region of hundreds of Euros depending on the number of years of data required even with a 95% student discount. Euronext - Liffe claimed the historical nature of the data is one of the reasons the price is high. In Gemmill (1996) the implied volatility is interpolated from the option price of puts and calls using 5 exercise prices and a 50 – step binomial model which allows for early exercise and which accounts for the actual dividends paid over the life of each option. In this paper the above step has been avoided by getting the implied volatility straight from Bloomberg. The dividends (in Index points) for the FTSE 100 and S& P 500 are not adjusted for in this paper and will be included as part of the indexes. Three month interest rates for the UK and US obtained from Datastream.

### Methodology

The study is create volatility smiles from the option prices of the indexes and compare the smile changes over a time period. According to Gemmill (1996), the skewness measure is designed to capture changes in volatility smile. It compares the implied volatilities of calls, which have striking prices 2.5% above the forward price, with puts, which have striking prices 2.5% below the forward price. The ±2.5% around the forward price was the closest strike point in Bloomberg that was in the range and also in line with Gemmil (1996) which supports the views of Gemmill (1991).

Where

Skew is the skewness measure in %

σ is the implied volatility obtained from Bloomberg

and

(x %) represents an exercise price which is x% above or below the forward price

### Results

The graphs plotted from the implied volatility of the calls and puts show that for the FTSE 100 the levels of implied volatility for the call options were slightly higher in the 2 months before the 9th of August 2007. However from November 2006 to around June 2007 the implied volatility of the puts were slightly higher than calls. This could mean that the 9th of August 2007 used in this study could be too late a date of reference.

Frequency distribution of implied volatilities Graph of FTSE 100

The graph is positively skewed with a longer right tail.The table bello

### Table 1.

Skewness Call

Skewness put

3.92%

4.08%

MIN

13.73

MAX

27.00

AVERAGE

17.14

MODE

16.23

MEDIAN

15.86

### Table of Results

Conclusion

The right skewness of the implied volatility graph for the FTSE 100 over the period preceding the crash shows that traders did not anticipate the crash. The positive value skewness measure means the distribution is positively skewed further confirms that option traders did not anticipate the crash. A negative skewness confirms that optraders anticipated a crash a

Bates (1991), Jackwerth (2004), Bhabra et al (1999) found the option traders did not anticipate the 1987 and 1997crashes, while Schwert (1990) show the traders did not anticipate the crash of 1987 as options traders attention to the increasing volatility did not catch on till Oct 19th – (the crash date) and stayed high for a few more months.

Volatility smiles do not predict the future movement of or crashes of stock indexes.

The high cost for the data led to a shorter period being chosen to examine, however a longer data period of options prices on the FTSE 100 possibly about 2-3 years may have produced a better skew and provided more information. Jha and Kalimipalli (2009) find that “the performance of skweness based trading is significantly improved when conditional skewness model forecasts combined with forward looking option implied volatilities”. However trading costs diminished any significant gains made.

### References

Bates, D. (2007) “The market for crash risk” Journal of Economic Dynamics and Control. Vol 32, Issue 7 pp 2291 – 2321.

Bates, D. (1991) “The Crash of '87: Was It Expected? The Evidence from Option Markets” The Journal of Finance. Vol. 6, No. 3 Accessed at : http://links.jstor.org/sici?sici=0022-1082%28199107%2946%3A3%3C1009%3ATCO%27WI%3E2.0.CO%3B2-1&origin=repec

Bhabra, G. Gonzalez, M. Powell, J. Sup, K. (1999) “Did option traders Predict the Korean Financial Crisis? University of Wollongong Working Paper Series

Benhamou, E. (2000) “Option Pricing with Levy Process” LSE Working Paper. (July 11, 2000). [Online] Available at SSRN: http:ssrn.com/abstract=26565 or doi: 10.2139/ssrn.265685. Accessed (21 December 2009)

Brunermeier, K. (2009)‘Deciphering the Liquidity and Credit Crunch 2007–2008'. Journal of Economic Perspectives. Vol. 23, No. 1, p 77 – 100.

Black, F and Scholes, M. (1973). “ The Pricing of Options and Corporate Liabilities,” Journal of Political Economy.

Chalamandaris, G. Tsekrekos, A. (2009) “Predictable dynamics in implied volatility surfaces from OTC currency options”. Journal of Banking and Finance.16 November 2009 [Online]

Available at: http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6VCY-4XPP11W-2&_user=983321&_coverDate=11/16/2009&_rdoc=1&_fmt=high&_orig=article&_cdi=5967&_sort=v&_docanchor=&view=c&_ct=1977&_acct=C000044920&_version=1&_urlVersion=0&_userid=983321&md5=2fc4bc49bc17c1ef800aaab30d58cd03

(Accessed 21 January 2010)

Can Crashes be Forecasted?

http://www.stock-market-crash.net/forecast-crash.htm. (Accessed 16 January 2010)

Chang, I. Lin, B. Paxson, D. (2008) “Smiling Less at LIFFE” Journal of Futures Markets, Vol. 28, No. 1, pp 57- 81

Gemmill, G. (1996) “Did option traders anticipate the Crash, Evidence from volatility smiles in the U.K with U.S comparisons.” Journal of Futures Markets Vol. 16, No. 8, pp 881- 897.

Gemmill, G. Saflekos A. (2000) “How useful are Implied Distributions? Evidence from Stock Indexes Options,” Journal of Derivatives, Vol. 7, No. 3, pp. 83-91.

Jha, R. and Kalimipalli, M. (2009) “Economic Significance of Conditional Skewness” Journal of Futures Markets, Vol. 00, No. 0 pp 1-29

Kang, B. Kim, T.Yong, S. (2009) ‘Information content of Volatility Spreads'. The Journals of Futures Markets.Vol. 00, No. 0, 1 pp. 1-26.

Kenourgio, D. Paltalidis, N. Samitas, A. (2007) Financial crises and stock market dependence [Online]. Available at:

http://74.125.155.132/scholar?q=cache:L7e2BwC6d6wJ:scholar.google.com/+2007+Market+Crash+stock&hl=en&as_sdt=2000. (Accessed 20/01/10)

Latane, H. Rendleman, R. (1976) ‘Standard deviations of Stock Price Ratios Implied in Option Prices'. The Journal of Finance. Vol.31, No. 2 pp 369-381.

OCC Timeline (2009), Options Clearing House [Online]

http://www.optionsclearing.com/components/docs/about/press/occ_timeline.pdf (Accessed: 19 January 2010).

Pre-budget report : The Shadow of 1937 (2008) Guardian.co.uk, 2 December 2009 [Online] Available at: http://www.guardian.co.uk/commentisfree/2009/dec/02/pre-budget-report-shadow-1937. (Accessed at 20 January 2010).

Rubeinstein, M. (1994). Implied binomial trees, Journal of Finance. Issue 69 p771-818.

Schwert, N. (1990) “Stock Market Volatility and the Crash,” The Review of Financial Studies, Vol. 3. No 1. [Online] Available at : http://www.jstor.org/stable/2961958

(Accessed 16 January 2010).

‘Timeline: Credit crunch to downturn' (2009) BBC Business News, 9 August 2009 [Online]. Available at: http://news.bbc.co.uk/1/hi/7521250.stm#top (Accessed at 14 January 2010)

The Stock crash of 1987 [Online]

http://www.stock-market-crash.net/1987.htm.(Accessed: 16 January 2010)

[1] Pre-budget report: The Shadow of 1937 (2008).

[2] (Timeline: Credit Crunch to Downturn, 2009).

[3] A Smirk is when the 5% OTM put options have a higher implied volatility than ATM put options and the ATM put options has a higher implied volatility than 5% ITM put options. Doran, Peterson and Tarrant (2007).

[4] Jha and Kalimipalli (2009).

[5] options were not traded in 1932

[6] Volatilities implied by options

[7]

[8] Meaning they possess higher implied volatility for OTM puts than ATM & ITM puts.

[9] See Black, F and Scholes, M. (1973). “ The Pricing of Options and Corporate Liabilities,” Journal of Political Economy.

[10] Alexander and Lazar (2004).

[11] According to Timeline: Credit Crunch to Downturn, (2009)

[12] Timeline: The Credit crunch of 2007/2008, (2008)