Actual Market Behavior

An Option Pricing Model: Theory, Model & Empirical Test

Abstract

A simple option pricing model based on correlation of underlying stock with actual market behavior as reflected by market index. The simplicity and ease of the model may appeal to the traders, operators and other market participants. Option prices produced by the proposed model are close to the actual price for all range of strike prices. The simplicity and precision of the proposed model are its main advantages over the existing models.

Introduction

Options belong to a class of instruments referred to as ‘Derivatives' because they derive their value from an underlying commodity or financial assets. By definition, “derivative is a contract or an agreement for exchange of payments, whose value derives from the value of an underlying asset or underlying reference rates or indices”. Price of a derivative instrument is contingent on the value of its underlying asset. The underlying commodities and financial assets can range from products like wheat and cotton to precious items like gold, silver, petroleum and financial assets like stocks, bonds and currencies. Options have richer history. Forward contract dated back to Mesopotamian tablets (1750 B.C.). Organized exchanges began trading option on equities in 1973. An Option means a choice. An option in a financial market is created through a financial contract. This financial contract gives a right to its holder to enter into a trade at or before a future specified date. The underlying assets on options include stocks, stock indices, foreign currencies, debt instruments, and commodities and futures contracts. These are called stock options, index options, commodity options and futures options. An option provides a downside protection against risk and also an upside benefit from movements in the underlying asset prices. An option is a contract in which the seller of the contract grants the buyer, the right to purchase from the seller a designated instrument or an asset a specific price which is agreed upon at the time of entering into contract. Option buyer has the right but not an obligation to buy or sell. If the writer gives the buyer of the option the right to purchase from him the underlying assets, it is call option. If the writer gives the buyer of the option the right to sell the underlying asset, the contract is termed as put option. At the time of introducing an option contract, the exchange specifies the period during which the option can be traded or exercised, the period is termed as Expiration period and date at which contract matures is Exercise date. The price at which the underlying asset may be bought or sold is exercise or strike price. Option premium or option price is the amount which the buyer of the option, whether it be a call or put has to pay to the option writer. Intrinsic value of an option is the value of the profits that are likely from the option. The difference between the option premium and intrinsic value is referred as time value. An option whose exercise price is equal to current spot price is said to be at-the-money. A call option is in-the-money when the strike price is below the current spot price of the underlying asset. A put option is in-the-money where the strike price is above the current spot price of the underlying asset. A call option is said to be out-of-the-money when the strike price is above the spot price of the underlying asset. Put option is said to be out-of-the money when the strike price is below the current spot price of the underlying asset. There are two kinds of options—American options and European options. American option can be exercised any time before its expiration date while the European can only be exercised on its expiration date. Trading and pricing of stock options have occupied dominant place in derivative market. Numerous pricing models have been developed, studied and tested. The Black-Scholes model is an option valuation model and it provides a closed form analytical expression for valuation of European style options. It is an option valuation model not a theorem. The model is developed based on assumptions and there are limitations with any such model. When the assumptions of the model are relaxed, discrepancy occurs. According to John C Hull “An option pricing model is no more than a tool used by traders for understanding the volatility environment and for pricing illiquid securities consistently with the market prices of actively traded securities. If traders stopped using Black-Scholes and switched to another plausible model—the prices quoted in the market would not change appreciably”.

Review Of Literature

Mandelbrot(1963) observed that the asset prices returns are highly leptokurtic. Numbers of studies were conducted to test the Black-Scholes model and other pricing models. Latane and Rendleman(1976) observed that out-of-money put options are generally overpriced in the market. Macbeth and Merville(1979) found that implied volatilities tended to be relatively high for in-the-money options and relatively low for the out-of-the-money options. A high implied volatility is indicative of a relatively high option price and low implied volatility is a low option price. Rubinstein(1985) in his study on trades reported on Chicago Board Option Exchange during the period 1976 to 1978 found that for out-of-the-money options, short maturity options had significantly higher implied volatilities than long maturity options. Study of Whaley(1986) showed that, overall the deviation between actual market prices and theoretical prices not significant. The model under prices in-the-money options. Some models assumed that volatility of stock price process is not constant but stochastic. Heston(1993) derived an option pricing formula when the log of underlying price distribution followed a Gamma process. Some option pricing models are based on series expansions of the underlying prices to obtain the model. Corrado and Su (1996) used a Gram-Charlier expansion of the normal distribution of returns. Popova and Ritchken (1998) created bounds on option prices when the underlying asset had the Paretian stable distribution. The study by Raj and Thurston(1998) on an intra day basis found that model under prices both calls and puts. Heston and Nandi (2000) developed a closed form option pricing formula based on a Generalized Autoregressive Conditional Heteroskedastic (GARCH) process and found lower valuation errors. Savickas (2001) developed option pricing formula based on the Weibull distribution. Anurag & Satish(2002), Gururaj and Chug(2002) Varma(2002), Narayana Rao(2003) Schenbagaraman(2003), analyse the volatility and pricing efficiency of options in India. Varma(2003) studies the pricing of volatility in the Indian Index Options market found that volatility is severely misprices and the Indian option market has moved from naïve model to Black Scholes Model. Actual Stock price movement does not follow lognormal distribution. Ait-Sahalia and Duarte(2003) consider call price as a one dimensional function of the strike price alone, by using only options with equal time to maturity and assuming that interest rate and dividend yield are deterministic function of time. Yatchew and Hardle(2006) introduced non parametric least squares estimator. They assume the call price to be a function only depending of strike price. Gatheral(2006) defines profession of models someone who finds equations that fit prices in the market prices with minimal errors. Taleb and Goldstein(2007) show that most professional operators and fund managers use a mental measure of mean deviation as a substitute for variance.

Need For And Objectives Of The Study

For option traders theories should arise from practice. Option price as far as traders are concerned depends on market conditions thereby on stocks and indices. In effect market conditions and related parameters. Option traders normally do not depend on theories. Traders may take decisions based on market conditions, but avoid fragility of theories. Traders specializing in using the put call parity to convert puts into calls or calls into put termed as converters. Dealers who basically operated as market makers can able to operate and hedge most of their risk by hedging option with options or cover their position by off setting.

Objectives Of The Study

To develop a model for pricing of call option to traders, professional and other market participants as reflected by Market Factor Coefficient (). To derive the option pricing formula based on the underlying assumptions. To find the price of varied series of options of different stocks that constitutes NIFTY Index based on the proposed model. To compute the option price based on Black-Scholes Model and compare with proposed model and observed prices and its effectiveness. Operationally price is not valuation. Valuation requires theoretical frame work with its assumptions and the structure of a model. For traders a price means marked to buy or sell an option. Traders are engineers, whether rational or even not interested in any form of probabilistic rationality. Traders produce a price of an option compatible with the instruments in the market, other market parameters, with a holding time that is stochastic. The study seeks to contribute the existing literature in many ways. Study is to examine the market factor, role of trader subsequently by pricing of option

Research Methodology

Assumptions

In developing model there are some underlying assumptions: Value of the option depends on market conditions or market forces. Traders quote their price based on such market conditions. Call Option is function of market factor coefficient () as reflected by Exercise Price, Current Stock Price, time to maturity. Option price depends on current stock price, Exercise price, time to maturity, Market factor coefficient which in turn depends on market movement. Fund required for option transaction by long or short is on borrowed fund, rate of which is considered as rate equivalent to risk free interest rate. Since the trader has already decided to buy or sell option, it is assumed that rate has no impact on option price. As such no adjustment on this factor is required. Transaction cost in buying and selling the option do not be reflected in the price quoted by the trader. Traders normally do not exercise the option. Position is covered by offsetting or reverse transaction i.e., buy or sell. Corporate actions like dividend declaration, bonus issue, right issue, stock split, take over, acquisition, buyout, bankruptcy, window dressing are taken care of by the market thereby reflects in market and stock price reflects such changes or adjustment. It is assumed that no adjustment is required on this count. The option markets are efficient. Market factors reflect in fair value of the option. Price of option depends on market factor as reflected in correlation, current stock price, exercise price, time to maturity. Volatility of the underlying stock has considered in the market factor coefficient. Number of contracts traded by the trader depends of on availability of fund at his disposal. Options are tradable only for some strikes in the region. Traders may take decisions based on market conditions. Traders produce a price of an option compatible with market parameters. For determining the market factor coefficient, historical data is needed in estimation procedure. The current market price of the stock depends on market conditions. Options are tradable only for some strikes in certain range (p,q) around the actual Spot Value St. In a bullish regime actual spot St will be near to q. In bearish regime it is closer to p. In practice the number of tradable options for given expiry could be small.

Main Advantages Of The Proposed Model

(1) Its simple form (2) Ease of the model's implementation (3) Practical application. (4) Adjustment factor and volatility taken care in the market factor coefficient.

Data

The proposed model is to be tested using the actual values in option market. Using market data to test any asset pricing model involves. An asset is incorrectly priced by the model. Asset is incorrectly priced by market. Both the model and the market price the asset incorrectly. The proposed study, covers call options series written on select underlying stocks included in the Nifty index during the period of one year starting from six months prior to the approval of the research proposal. Data for the study is to be collected from Centre for Monitoring Indian Economy (CMIE) PROWESS database and website of the national stock exchange of India, www.nseindia.com. The date, time, contract month, option type, strike price is to be collected from the data source. Trading days is to be considered for analysis and computation and not calendar days. Historical data is used for calculating the market factor coefficient (). Each stock moves variedly depending on market conditions. For calculating the market factor coefficient () values prevailed prior to the day, the option trader marked the price (decided to buy or sell) is to be taken for stock price (kj) and market index (km). Closing price of stock (kj) and the closing price of market index (km) for 20 traded days (n) or any other traded days is to be considered for computation of the market factor coefficient . Exercise or strike price (X) of available series is taken for calculation and St as current stock price. Normal distribution of market factor coefficient from normal distribution table viz, N (). The proposed model is also to be compared with Black-Scholes model. Input parameters are required for estimating theoretical call price of Black-Scholes Model. Time to expiry is annualised by dividing the number of days left for the option to expire by the total number of calendar days. Dividend yield of nifty is to be taken 1.5% p.a Volatility per annum is taken as Volatility per trading day x Trading days is assumed to be 250

Data Analysis :

The easiest way to measure accuracy of the formulae is to compare the calculated values with actual call option prices quoted in the market. The differences between actual and computed values are errors. The formula that produces lowest error can be considered better. In the study, errors are measured using following estimates:

Mean Error (ME)

It can be computed by adding all error values and dividing total error by the number of observations. Where Otp= the theoretical/predicted price of the otpion Oa= Actual price for observation. N= Number of observations. This measure is acceptable when all error data have the same sign (either all are positive or all are negative). A low value of the Mean Error may conceal forecasting inaccuracy due to offsetting effect of large positive and negative forecast errors and make this measure unacceptable.

Percentage Mean Error (PME)

Mean Absolute Error (MAE)

The mean absolute error value is the average absolute error value. The closer this value is to zero, the better is the forecast. MAE is computed using the formula. The neutralization of positive error by negative errors can be avoided in Mean Absolute Error.

Mean Squared Error (MSE)

Mean Squared Error is computed as the average of the squared error values. This measure is very sensitive to large outlier. Commonly used error indicator in statistical fitting procedures. Mean Squared Error is computed as

Root Mean Squared Error (RMSE)

It is square root value of mean squared error and similar to standard deviation.

Thiel's U Statistic

Henri Thiel (1961) developed an inequality coefficient for measuring the degree to which one time series differs from another. Thiel's U statistic is computed as under : Thiels inequality coefficient (Thiel's U) Thiel's U will equal 1 if a forecasting method is found no better than a naive forecast. If Thiel's U is less than 1, it indicates that the method is superior to a naive forecast. A value close to zero indicates a good fit, whereas, value greater than, 1 indicates that the technique is actually worse than using a naive forecast. In comparing the two methods, the method that produces lower U statistic may be considered better than the other. Theil's U statistics is independent of the scale of the variables; it is also constructed in such a way that it necessarily lies between zero and one, with zero indicating a perfect fit. A simple t-test is also to be carried out on the data set to test whether the model predicts correctly on average. The null hypothesis is that the mean pricing bias is zero.

Limitations Of The Study

For computing the market factor coefficient ()., market index data is considered. As there are many indices, the value (),changes from indices to indices. With the changes in the number of trading days, ‘n' viz., 7 days, 14days, 21 days etc., considered for calculating ().value changes depending on trading days.

References

Black F(1975, “Fact and Fantasy in the Use of Options”, Financial Analysts Journal, Vol.31, pp.361-41. Black F and Scholes M S (1973), “The Pricing of Options and Corporate Liabilities”,. Journal of Political Economy, Vol.81, pp.637-654. Corrado, C.J., and Su. T (1996)Skewness and Kurtosis in S&P 500 Index Returns Implied by Option Price,” Journal of Financial Research Vol. 19, pp. 175-192. Heston S (1993), “Invisible Parameters in Option Prices”, Journal of Finance, Vol.48, pp 933-947. Heston S and Nandi S (2000), “A Closed-Form GARCH Option Valuation Model”, Review of Financial Studies, Vol. 13, pp.585-625. Jarrow R and Rudd A(1982), “Approximate Option Valuation for Arbitrary Stochastic Processes”, Journal of Financial Economics, Vol. 10, pp. 347-369. Latane H Rendlemen R J (1976), “Standard Deviations of Stock Price Ratios Implied in Option Prices”, Journal of Finance, Vol. 31, pp-369-381. MacBeth J D and Merville L J (1979), “An Empirical Examination of the Black-Scholes Call Option Pricing Model”, Journal of Finance, Vol.34, pp. 1173-1186. Mandelbrot B B (1963), “The Variation of Certain Speculative Prices,” Journal of Business, Vol.36, pp. 394-419.” Popova I and Ritchken P (1998), “On Bounding Option Prices in Paretian Stable Markets”, Journal of Derivatives, Vol.5, pp.32-43. Raj M and Thurston, D C, (1998), “Transaction Data Examination of the Effectiveness of the Black-Scholes Model for Option Pricing Options on Nikkei Index Futures”, Journal of Financial and Strategic Decisions Vol.11, No.1 Rubinstein M(1995), As Simple as One, Two, Three”, Risk, Vol.8, pp.44-47 “Non-parametric Test of Alternative Option Pricing Models using all Reported trades and Quotes on the 30 Most Active CBOE Option Classes From August 23, 1976 Through August 31, 1973”, Journal of Finance, Vol.40, No.2, 445-480. Savickas. R (2001), “A simple option pricing formula” https://ssrn.com/abstract-265854 Thiel.H (1961), Economic Forecast and Policy, 2nd Revised Edition, North Holland Publishing Co., Amsterdam. Taleb.N. (2007): “Scale-Invariance in Practice: some questions and workable Patches”, working paper. Thorp. E.O. (2007): “ Edward Thorp on Gambling and Trading”, in Hang (2007). Varma, V R (2003), “Mispricing of Volatility in the Indian Index Option Market”, working paper, IIM Ahmedabad. Whaley, R (1986), “Valuation of American future Options: Theory and Empirical Tests”, Journal of Finance, Vol.41, 127-150. REsearch Proposal to Ph. D IN FINANCIAL Management

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