A Kaleidoscopic Study of Pricing Performance of Stochastic Volatility Option Pricing Models: Evidence from Recent Indian Economic Turbulence

Transcription

1 R E S E A R C H includes research articles that focus on the analysis and resolution of managerial and academic issues based on analytical and empirical or case research A Kaleidoscopic Study of Pricing Performance of Stochastic Volatility Option Pricing Models: Evidence from Recent Indian Economic Turbulence Vipul Kumar Singh and Pushkar Pachori Executive Summary A whole host of researchers have modeled volatility as a non-constant stochastic process, based on the principle that volatility follows a stochastic process whose parameters are not directly observable in the market. The objective of this research paper is to empirically investigate the forecasting performance of three most dominant models of this species namely, Hull-White (1988), Heston s (1993), and Heston-Nandi GARCH (2000) option pricing model. These three models have been collaterally compared and contrasted against Black-Scholes and market for pricing S&P CNX Nifty 50 index option of India. The Hull-White model not only warrants a range of stochastic volatility specifications but also incorporates correlation of volatility of asset return and its price changes. The closed form Heston s (1993) model explicitly and elaborately communicates non-lognormal distribution of the assets return, leverage effect, and mean-reverting property of volatility. The model of Heston-Nandi, also in closed form, successfully incorporates variance of asset returns as a range of GARCH process. It strongly permits correlation between returns of the spot asset and variance and also technically accepts multiple lags in the dynamics of the GARCH process. KEY WORDS Option Pricing Black-Scholes Model Hull-White Model Heston Model Heston-Nandi Model GARCH Implied Volatility To decide, determine, and delineate the effectiveness of stochastic models against the Black-Scholes and market, the current paper adopts a structured approach of relative error price, viz., percentage mean error (PME) and mean absolute percentage error (MAPE). The most turbulent period of the Indian economy January 1, 2008 to December 31, 2008 was considered appropriate for testiing the suggested model. It was a testing time for the Indian economy as well as a critical period questioning the sustainability of all financial products/models and challenging their fundamental platform depicted as equity market. How to safeguard investors faith and at least protect their investments if not multiply returns in the face of such financial hardships remained a burning question for all thinkers and experts on the subject. Data pertaining to the specific period of such drastic disturbance was analysed with the help of the proposed models. After rigorous churning of specific data taken across various models, the Heston model was found to outperform and surpass other models. VIKALPA VOLUME 38 NO 2 APRIL - JUNE

2 The epoch-making world famous option pricing formula popularly known as Black-Scholes option pricing model came into existence in the year The formula was first conceptualized by Fischer Black and Myron Scholes and further worked upon by Robert Merton (1973). The framed model was specifically designed to operate upon the category of European call options. The igniting philosophy which was ultimately bound to be a revolutionary phenomenon unearthed the fundamentals of valuation of derivative instruments germinating the seeds of modern finance which finally flourished into the form of financial engineering. The significant contribution of the trio was apparently recognized when the Myron Scholes and Robert Merton were bestowed upon the highest degree of acceptance and appreciation in the form of Nobel Prize in the year The Noble Prize would also have been shared by Fischer Black, but unfortunately he passed away in 1995 and thus was deprived of receiving it posthumously. Prior to the inception of this new concept, the pricing of options was calculated by estimation and speculation basically using a non-mathematical approach. The implemented model was fundamentally based on two important assumptions: (1) Stock/index price follows Geometric Brownian Motion; (2) Return distribution is log-normally distributed with constant volatility. The two basic assumptions of their model namely, constant volatility and asset return log-normal distribution, were severely challenged by the host of contemporary thinkers after the market crash of 1987 (Glosten, Jagannathan, & Runkle, 1993; Derman & Kani, 1994). Singh and Ahmad (2011a) verified the same in the context of the 2008 financial crisis in India. The empirical approach of the constant volatility assumption of Black-Scholes was critically challenged. As per the outcome of the model, the graphical design of implied volatility of different options (on the same underlying asset), when put against time to maturity and strike prices, should result in a neutral facial expression, but in reality, it formed a curved surface opening up or down (Derman & Kani, 1994 (a & b)), widely acknowledged as volatility smile/smirk. Figures 1 and 2 provide evidential existence of volatility smile in the Indian capital market. The downward slope displayed in the Figures is typical for Nifty index options. Backus, Foresi, and Wu (1997) and Singh and Ahmad (2011b) observed that deep-out-of-the-money and deepin-the-money options have higher implied volatilities, as caused by smile. Most of the practitioners have compared the results of their models with the classical Black- Scholes model (Bakshi, Cao, & Chen, 1997; Brockman & Chowdhury, 1997; Corrado & Su, 1998), but we plead that it is more important to observe how models behave with respect to market than that of Black-Scholes. Practitioners have observed that in case of constant volatility, the investors are exposed only to the risk of random movement of asset price process; on the other hand, if the volatility is stochastic (random), investors face the acting risk in addition to the risk of random movement of the volatility process (volatility of volatility). Another distinguished feature of stochastic volatility models is that option values can change without any change in the price of the underlying asset, the reason being that the change in volatility level is strong enough to alter the value of option. The above featured characteristic is an explicit Figure 1: 2D Plot of Volatility Smile: Implied Volatility vs Time to Maturity (April 17, 2008) 62 A KALEIDOSCOPIC STUDY OF PRICING PERFORMANCE OF STOCHASTIC VOLATILITY...

3 Figure 2: 2D Plot of Volatility Smile: Implied Volatility vs Strike Price (October 16, 2008) property of stochastic volatility model which is in no case shared by deterministic volatility models. The reason behind the development of stochastic volatility models was to provide a structured explanation of volatility smile/ smirk and to determine the improvement in the price bias of plain vanilla options. Envisioning the fact, stochastic volatility models can be considered effective and beneficial for option pricing (Hull-White, 1987; 1988; Wiggins, 1987; Nandi, 1996). On having estimated the complex nature of stochastic models, we can presume that the feasible applicability still remains difficult. Considering the complexity of the stochastic models, the practioners were interested in knowing what benefits they would actually reap in pricing of options contracts? Will the stochastic models really fill the gap of theory and practice? Will they improve the pricing bias of the Black-Scholes, significantly? After filtering all the models available in the specified genre of stochastic models, the most updated and latest versions of stochastic volatility (SV) models were shortlisted, namely, Hull-White (1988), Heston s (1993) and Heston-Nandi GARCH (2000) Model. These SV models can further be categorized into two clusters: continuous time and discrete time. An alternative approach to analytically characterize the features of discrete time stochastic model culminates into generalized autoregressive conditional heteroskedasticity (GARCH) model (Bollerslev, 1986; Bollerslev, Chou, & Kroner, 1992). The models described above are of continuous nature and require constant flow of data, but the data this study banked upon is purely of discrete nature where continuity is hampered every now and then making it difficult to filter continuous variables of stochastic models through discrete observations. To furnish parity between the models considered and their applicability with collected discrete data, it was construed that the best possible alternative was to compute parameters directing at market option prices. But truly this approach ultimately proved to be time-absorbing and operationally, an intricate process as it required frequent estimation of fluctuating parameters on every day basis continuing up to the last day of the considered options records, i.e., from first day of January 2008 to the last day of December In order to handle this complex issue, the technique of non-linear least square method (Dumas, Fleming, & Whaley, 1998; Christoffersen & Jacobs, 2004; Rouah & Vainberg, 2007) was deployed and hence, the statistically stable optimal set of obtained parameters was applied in computing the models price. For the concrete findings of this research, the Stochastic-Volatility process was first estimated and then utilized to evaluate option-related parameters. This research work attempts to find the model prices through the direct estimation of optimal set of implied parameters. Previously, researchers (Ball & Roma, 1994; Brockman & Chowdhury, 1997; Shu & Zhang, 2004) had already proved that the pricing bias could not be eliminated completely. But it was further advocated that the stochastic volatility models were consistent with fat tails of the asset return distribution. Thus, to estimate option pricing, the tool of stochastic models became very popular. The authors wished to ensure that the outcome of this research paper should focus upon the applicability of these stochastic models in the Indian scenario operating upon the data of the most turbulent period of the Indian financial market during Besides, the paper also wanted to highlight whether the incorporation of stochastic models could remove or improve the pricing bias of Nifty index options vis-à-vis the market price. It aims to pick out the best model for pricing Nifty index options on the basis of empirical analysis. OVERVIEW OF INDIAN OPTION MARKET India s prominent stock exchange, National Stock Exchange (NSE), was established in the year 1992; however, the actual trading of various derivative instruments commenced in the year 2000 and continued in phases. Soon after the introduction of index future, it became one of the largest stock exchanges of Asia and is now recog- VIKALPA VOLUME 38 NO 2 APRIL - JUNE

4 nized among the top five stock exchanges of the world. NSE, like the other exchanges across the world, uses the benchmarked BS model (despite its shortcomings) for determining the base price of index and stock options. Option series used in this research work is extracted from the database of NSE. The choice of Nifty index options is essentially justified by the fact that they are the most actively traded instrument from the family of derivative instruments. Figure 3 clearly depicts that the trading volume of index option has grown exponentially since its inception, contributing 75 percent of the total turnover of the F&O segment of NSE in DATA DESCRIPTION Data Figure 4 depicts that the year 2008 was the most disturbed phase of Indian capital market and it gave rise to a host of financial experts to estimate the efficiency of various financial models. This study considered this specific period of high turbulence as it provided a range of extreme limits of the Indian capital market and thus the best possible laboratory conditions to test the efficiency/effectiveness of the conceptual models. For the purpose, the following data were collected manually through various sources on a daily basis for the period starting January 1, 2008 to December 31, 2008: For the option contract, index price, strike price, time to maturity, contract month, option type, and traded (closed) prices were obtained from the official website of NSE 1, while risk-free interest rate data, which is equal to the yield of 91 Day T-Bill, were collected from the official website of Reserve Bank of India (RBI) 2. Nifty Financial Characteristics Figures 4, 5, and 6 exhibit Nifty s index movement, returns distribution pattern, and its co-movement with Black-Scholes implied volatility (at-the-money) for the specified period of high turbulence. Figure 7 displays the frequency & QQ plot of the Nifty daily returns encompassing the entire year of 2008 (from June 1, 2001 to December 31, 2008). Figure 7 clearly depicts that contrary to basic assumption of BS, the frequency plot of Nifty index return is not absolutely normal as there are extreme values at both the tails, but the thickness of left tail is more than that of the right tail. This distribution anomaly supports the smile/smirk distribution exhibited in Figures 1 and 2. Portraying Indian capital market, Figures 1, 2, 5, 6, and 7 prove the existence of elements like volatility smile, volatility clustering, and thick tails evidentially, and hence the claim of constant volatility assumption of Black-Scholes model is automatically terminated. Option Categories After having screened the various groups of gathered data and filtering out the useless ones, the remaining data was rigorously analysed and categorized as remaining and important for the purpose of research. It was subsequently put in a matrix comprising three rows and five columns, Figure 3: Business Growth of Index Options Figure 4: Nifty Index Movement Source: Source: A KALEIDOSCOPIC STUDY OF PRICING PERFORMANCE OF STOCHASTIC VOLATILITY...

5 Figure 5: Clustering of Nifty Index Return Figure 6: Leverage Effect Source: Source: Figure 7: Frequency and QQ Plot of the S&P CNX Nifty Index Daily Returns Source: i.e., fifteen different categories of moneyness and maturities depending on the expiration time and ratio of the asset price to the strike price (S/K) as exhibited below. Data Screening Procedure In order to procure the most usable data for conceptual analysis, four exclusionary filters were applied to raw data and the following were ruled out from the research database: 1) Call options having zero trading volume/open interest. 2) Call option prices taken from the market not satisfying the lower boundary (arbitrage) condition VIKALPA VOLUME 38 NO 2 APRIL - JUNE

6 where, S t is the current asset price, K is the strike price, q is the continuous compounded dividend yield of the asset, r is the risk-free interest rate, and C(S t,t) is the call price at time t. 3) Deep-in-the-money (deep-out-of-the-money) options having moneyness ((S/K-1)*100) greater (less) than +15% (-15%), as they are not traded actively on NSE and their price quotes generally do not reflect the true option values. 4) Option strike prices having maturity greater than 90 days and less than 5 days amid liquidity and volatility bias. Having gone through the various filtration steps, the final set of 7,455 call options was collected to testify the formulated concepts and examine the set principles of the existing models in this area. Tables 1 and 2 exhibit the descriptive filtration statistics of Nifty index options for the period are being attempted since 1900 (Bachelier; 1900; Wiener, 1938; Levy, 1948; Itˆo, 1951; Samuelson, 1965; 1973). The French mathematician, Louis Bachelier (1900) saw the first success in ensuring a scientific approach to value options and modeled movement of stock price as conspicuous Brownian motion. Before the coinage of Brownian motion, option-pricing formulas were directly derived by taking the discounted expectations. The model introduced by Black and Scholes (1973) and Merton (1973) was only the modified version of concepts already introduced by Bachelier (1900) and Samuelson (1965). Their model was fundamentally structured on the Geometric Brownian Motion with an assumption that return distribution is log-normally distributed with constant volatility. The classical formula of BS for pricing the European plain vanilla call options on a non-dividend-paying stock is Methodology Model prices were analysed by dividing the option data into fifteen moneyness-maturity groups, depending on the time to maturity and strike price, as defined above. The prices were compared analytically by continuously updating the parameters of the models using cross-sectional option data on daily basis. For estimating the structural parameters, the NLLS method was used an elegant and simple method of optimization and parameter estimation through percentage mean error and mean absolute percentage error (Dumas, Fleming & Whaley, 1998; Christoffersen & Jacobs, 2004; Rouah &Vainberg, 2007). In order to substantiate the concept, the method of analysing the out-of-sample forecasting performance of models was applied. Estimated parameters were then used to figure out the price and volatility forecasting performance of models vis-à-vis Black-Scholes model. To examine the pricing effectiveness of models, relative error was scrutinized critically. For the purpose of this research, only Nifty 50 Index call options were considered as results obtained through call options replicated the results of put options as well. OPTION PRICING MODELS Classical Black-Scholes Option Pricing Model The history of approach to value financial instruments is very primitive; chronologically, the scientific approaches where, C is the price of a call option and S is the underlying asset price (Nifty Index here), K is the option exercise price, t is the time to expiry in years, r is the annual risk free rate of return, N(d) is the standard normal distribution function, σ 2 is the variance of returns of the asset ( index or stock). Similar to call, put option prices can be found with the formula with all usual notations as defined above. BS was successful in deriving an equation consisting of five parameters. The most beautiful specification of the BS is that they succeeded in deriving an equation which can be utilized for pricing any derivative instrument with the desired end term or boundary conditions. Besides that, the greatness of the BS option pricing formula is that except one (volatility), all the four parameters could easily be examined upon by the regular data flow of market. The volatility parameter is latent for any analytical access because of its complex and compound nature, which cannot be testified as the other four parameters could be worked upon. 66 A KALEIDOSCOPIC STUDY OF PRICING PERFORMANCE OF STOCHASTIC VOLATILITY...

7 The Stochastic Hull & White Model (HW) In order to remedy the smile/price bias of classical BS, Hull and White (1987) proposed an extension of BS for option valuation under stochastic volatility. The model was based on the assumption that volatility is uncorrelated with asset price. The HW model is one of the first few models that tackled the problem of volatility stochasticity in option pricing. HW modeled instantaneous variance (v = σ 2 ) as a stochastic process and defined the following stochastic risk neutral relationship between asset and volatility: With usual notations, i.e., φ is the exponential drift rate of asset price (S),η is the instantaneous drift rate of volatility (V), and ξ is the instantaneous standard deviation of. In order to filter continuous variables of model through discrete observations, HW assumed that the parameters of the model remained constant over short intervals instead of time varying amid correlations among time varying parameters, which generally resulted in unstable estimates. Utilizing the second order Taylor series expansion, HW provided an accurate approximation around a constant volatility specification (ξ = 0). Therefore, the HW model, being just an extension of BS incorporating its price bias as a stochastic volatility bias, is defined as: where, the drift of the variance φ and the volatility of the variance ξ do not depend on S. As defined in BS, μ is the exponential drift rate of the asset price (S). HW assumed that the two Wiener processes, dw t 1 and dw t 2, are independent, i.e., the asset price and volatility are uncorrelated. HW priced the call option value as an integral over the BSM formula, defined over the distribution of its mean volatility. Thus, in order to price the European call/put options, the formula of HW cover the distribution of the average variance V of the asset price process over the entire life of the call/put option defined by the stochastic integral The coefficients Q j are further defined as where, C(v) is the BS formula with its usual notations and v is the mean variance of the asset price return, spread over the lifetime of the option, defined as Thus, in order to price European call/put options accurately, HW added this bias to the BS (C BS ) call/put price and obtained the stochastic-volatility adjusted call/put price, defined as: In a related paper, Hull and White (1988) relaxed the zero correlation restriction. This research has considered the second version, i.e., there is correlation (p) between W 1 and W 2. Due to this change, volatility risk premium needs to be calculated. The correlated version of HW was based on the assumption that asset price and its return volatility followed a square root stochastic volatility process defined as: The above equation is the fundamental equation of this research work. In case of constant volatility/variance ( ξ = 0 ), the pricing bias becomes zero, and thus HW formula collapses to the BS formula. Heston Model with Closed-Form Solutions While the HW model only focuses on stochasticity of volatility, the Heston s model (1993) additionally focuses on non-lognormal distribution of the assets return, leverage effect, and mean-reverting property of volatility. Though the HW formula readily lends itself to the estimation of the underlying stochastic process parameters, it is ana- VIKALPA VOLUME 38 NO 2 APRIL - JUNE

8 lytically intractable because of its open form solution whereas Heston s model is not only analytically tractable but also very popular among the practitioners because of the following three features. First, it does not allow negative volatility. Second, similar to HW, it incorporates the correlation of asset returns and volatility. Third, contrary to HW, it has a closed-form pricing formula. Because of the above three specifications, the Heston (1993) Model is one of the most widely used SV models today for pricing index and stock options. Besides that, utilizing correlation (r) between the log-returns and volatility of the asset, the Heston s model also implies a number of different distributions, and thus affects the shape of the tails (Figure 7). Logically, if r>0 (r<0), then volatility will increase as the asset price return increases (decreases). This positive (negative) correlation will spread the right tail (left tail) and squeeze the left tail (right tail) of the distribution and thus create a right (left) fat-tailed distribution. Figure 6 clearly depicts the fact that Nifty index returns and its related volatility are negatively correlated. It is further deduced that the correlation (r) may strongly affect the skewness of the distribution exhibited in Figure 7. The effect of changing the skewness of the distribution impacts not only the shape of normal distribution but also the shape of the implied volatility surface (Figures 1 and 2). It further implies that correlation (r) also affects this. Thus, Heston model can imply a variety of volatility surfaces and thus address the shortcoming of the Black- Scholes-Merton model, viz., constant volatility across differing strike levels. On the side, correlation (r) affects the skewness (tails) of the asset return distribution whereas volatility (s) affects the kurtosis (peak) of the distribution. In cases where volatility is constant (s = 0), it is deterministic further implying that the log-returns will be normally distributed. It has been noticed that increase in s generally increases the kurtosis only and thus creates heavy tails on both sides further impacting the shape of implied volatility. Also, higher s normally makes the skew/smile phenomenon more prominent and makes sense that volatility is more volatile further implying that the market has a greater chance of extreme movements. This justifies the existence of one of the most important financial characteristics observed in almost all financial data volatility clustering i.e., large (small) price variations are more likely to be followed by large (small) price variations. This is depicted in Figure 5. The aforementioned features of this model make it very robust and hence address the shortcomings of the Black-Scholes-Merton model and provide a framework to price index/stock options that is closer to reality. Heston s derived the formula under the assumption that the stock price, S t, and its return variance, v t, follow the stochastic process, defined as where S t and v t are the price and volatility processes, W t 1 and W t 2 are Geometric Brownian motion processes with correlation r, x is volatility of volatility, v t is a square root mean reverting process of asset price with long-run mean è, and rate of reversion k. In order to get statistically stable parameters through set of discrete observations, similar to HW, it is assumed that the parameters of Heston s model, viz., μ, κ, θ,ξ,ρ, are also state and time homogenous over short interval of time. Since the two Wiener process, W t 1 and W t 2, are correlated, the volatility risk premium of the asset needs to be measured. For modeling, volatility risk premium is assumed to be proportional to instantaneous variance, defined as where, χ is a constant. The option pricing formula of Heston s analogy with the Black-Scholes formula, is modeled as with all usual notations of BS. P 1 and P 2 are two probability functions, defined as where, x=ln(s t ) and f j (x,v,t;φ) is the characteristics function, explicitly defined as 68 A KALEIDOSCOPIC STUDY OF PRICING PERFORMANCE OF STOCHASTIC VOLATILITY...

9 Further, for j=1, 2 Though the closed form calibration of the two probability functions is not available, it can be approximated to the extent, integral limits of the probability functions P 1 and P 2 are evaluated exactly. With the advancement in numerical technologies, this is not a difficult task. Although, the formula looks intimidating, it is actually quite easy to evaluate. Closed-Form Heston and Nandi GARCH Model Following the popularity of GARCH model in explaining the non-lognormal distributional and other various financial characteristics such as time varying volatility, volatility clustering, and leverage effects (Bollerslev, 1986; Taylor, 1986; Bollerslev et al., 1992), researchers also attempted to incorporate GARCH effects into option pricing in order to remedy the price bias of models and market (Engel & Mustafa, 1992; Duan, 1995). But none of them have provided a closed-form analytic solution for the option prices and thus the use of such formula remains impracticable. Later, Heston and Nandi (2000) succeeded in providing a formula, which not only incorporated the GARCH financial characteristics in option pricing but also provided its closed form solution. While the Black- Scholes formula is a function of asset price, the formula of HN GARCH is a function of both asset price S(t) and conditional variance h(t + Ä). Since in GARCH, volatility is a function of historical asset price, readily observable from the market thus in contrast to continuous-time stochastic volatility models, it need not be estimated with other procedures. As conditional variance is a function of sequential lagged asset price, the HN GARCH option formula is a function of the current and lagged asset prices. The HN GARCH model is based on the assumption that over time steps of very small lengths Ä, the log-spot asset price S(t) follows a particular GARCH process, defined as where, h(t) is the conditional variance of the log return between t Δ, z(t) is the standard normal disturbance, and r is the continuously compounded interest rate for the time interval Δ. In case the components α i and β i approach zero, the conditional variance converge to constant variance implying that the GARCH diffusion equation will converge to the discrete time Black-Scholes diffusion equation. Similar to the Heston s, the HN GARCH model is also analogous to BS, described as where the probabilities P 1 and P 2 have the form and since the model requires probabilities, P 1 and P 2 to be evaluated exactly. Thus, again similar to Heston s, the HN model is attractive to the extent that call price is obtained in closed form. For the purpose of this research, only first-order GARCH process is considered, i.e., p = q = 1. The HN GARCH model defines the conditional variance as a function of the log spot price, defined as: where, α 1 is the kurtosis of the distribution and γ 1 is the asymmetric influence of shocks (negative/positive news) on asset price. CALIBRATION OF MODELS This section has been devoted to estimating the structural parameters. In order to find the implicit parameters of the model from the market, the study considered the most vibrant technique popularly known as the method of non-linear least square (NLLS). The most significant characteristic of the technique is that it incorporates inform content of both, model and market (Dumas, Fleming, & Whaley, 1998; Christoffersen & Jacobs, 2004). The opti- VIKALPA VOLUME 38 NO 2 APRIL - JUNE

10 mal implied set of parameters (Rouah & Vainberg, 2007) is then used to compute the models price. The set of implied parameters obtained for day (i) was utilized to find the price of the next day, i.e., day (i+1). The process was repeated for the entire sample. Since the BS formula could not be solved for σ in terms of the other parameters, the implied volatilities were extracted numerically utilizing numerical algorithms such as Newton Rapson and Bisection Methods. In order to find out the BS implied volatilities, the objective function f (σ) is defined as linear squared loss function The stochastic volatility process assumed by HW, Heston s, and HN GARCH had parameters ranging from four to six, which needed to be estimated numerically. Similar to BS, the square loss (non-linear) optimize functions for stochastic models is defined as for the first month of 2008, i.e., January and so on. Expecting this calibration process will incorporate information content of both, underlying (Nifty Index) and option market, in the price of Nifty index options. The rest of the parameters of HN GARCH will be evaluated from the non-linear loss function already defined above. EMPIRICAL PERFORMANCE AND RESULTS In order to establish an effective and analytical outcome, the study used a simple and elegant method that encompassed calculation of Error Metrics namely percentage mean error (PME) and mean absolute percentage error (MAPE). To see the performance effectiveness of models, the relative error generated by the models was observed. Tables 3, 5, and 6 report the empirical comparison of model performances based on implied volatility stability and out-of-sample forecasting price performances based on MPE and MAPE. Percentage Mean Error (PME) Mean Absolute Pricing Error (MAPE) where, Ω is a vector of parameters to be determined implicitly by minimizing the error of model and market on daily basis. The above equation will provide us estimates of implied spot variance and values of structural parameters. In addition to other option parameters, the HN GARCH model requires estimation of parameters of GARCH process as well. Therefore, in order to estimate the price of Nifty index options through HN GARCH model, there was the need to first estimate the parameters of the GARCH process which could be further utilized to price Nifty index options along with other parameters of the model. The GARCH process parameters were estimated on a rolling window of S&P CNX Nifty 50 Index spanning over the period of the study, i.e., 2008, starting Jan For the purpose of this research, the size of the rolling window was kept constant, at one year, rolled over the entire sample in steps of twelve months, ending December At the end of each step, a new monthly window was added by deleting the previous window. The first rolling window was expected to provide the forecast where, C i Model is the predicted price of the option and C i Market is the actual price for observation i, and k is the number of observations. To see how well a model performs, the relative error generated by the models was observed. A negative PME means the model under-prices the specific option, whereas a positive PME means the model over-prices the specific option. While a small MAPE means that the model provides a good approximation to the market, a large MAPE indicates a poor approximation to the market. The Implied Volatility Pattern To determine the pricing bias between the BS model and the market, a convenient method is to plot the implied volatility as a function of the exercise price. Previous empirical research work of researchers proved the existence of volatility smile pattern, suggesting that implied volatility tends to vary across exercise prices and time to maturity (Figures 1 and 2), with implied volatility reported higher for DOTM and ITM options and lower for ATM options. The existence of volatility smile shows that the 70 A KALEIDOSCOPIC STUDY OF PRICING PERFORMANCE OF STOCHASTIC VOLATILITY...

11 BS model systematically misprices options across moneyness-maturity. Obviously, a host of researchers felt motivated by the existence of smile pattern. Thus they decided to incorporate stochastic volatility in option pricing models with an object of improving the price performance in terms of more stability in volatility and lower price error. Thus, the study of the implied volatility pattern set the first stage to judge the empirical performance of stochastic option pricing models. Table 3 exhibits the co-movement of implied volatility with strike price and time to maturity. It supports the empirical fact that the implied volatility varies systematically with respect to expiry and maturity, i.e., moneyness and maturity. The Table clearly displays that implied volatility as expected varies from DOTM to DITM options and makes a systematic smile/smirk upward/downward trend when it moves either to DITM and DOTM from ATM call options. Table 3 also depicts that the variation in implied volatility ranging from DOTM to DITM is the highest in case of Heston s followed by HN GARCH, BS, and HW. This reflects the volatility smile-capturing capacity of models. Thus, primarily, it can be claimed that Heston s model may price S&P CNX Nifty index call options better than the classical and other stochastic family models that need to be testified further, empirically. Out-of-Sample Pricing Performance This section analyses the comparative competiveness of family of stochastic models. For this, the study was consolidated with their inter-relations leading towards multilayered correlation juxtaposing the various combinations of moneyness and maturity of stochastic volatility models. Tables 3-6 explicitly state the same. The outcome of cross-sectional, comparative, and analytical empirical study of the Tables will empirically investigate the forecasting/price effectiveness of the stochastic models. Table 4 exhibits the moneyness-maturity statistics of S&P CNX Nifty Index Call Options for the period ranging between January 1, 2008 and December 31, Table 4 validates the fundamentals of call options pricing that the intrinsic value of call option should follow the moneyness (DITM>ITM>ATM>OTM>DOTM) and maturity (Long Term>Medium Term >Short Term) order. In this case too, the moneyness-maturity sequences of Nifty index call options follow the ascending order understood by moneyness and maturity series: DITM>ITM>ATM> OTM>DOTM and Long Term >Medium Term >Short Term. In case of stochastic models, as the primary focus is to find out the model causing undervaluing or overvaluing of Nifty index call options, a systematic decrease in price error going from DOTM to DITM was noticed. Having viewed Table 5, the model s performance sequence based on mean percentage error (MPE) behaviour is jumbled as Black-Scholes It is evident from Table 3 that the BS model over-prices medium- and long-term OTM and DOTM call options and under-prices short-term DOTM, OTM, ITM, and DITM call options. PME is the least for ITM and DITM call options across all moneyness-maturity groups. Our analysis clearly reveals that the BS model prices ATM, ITM, and DITM call options with error less than or equal to 7 percent across all moneyness-maturity groups. The degree of pricing bias increases as one moves from shortterm to long-term options for short- and medium-term call options. The BS model measures short-term ATM, ITM, and DITM options more accurately with a much lower pricing error of 0, -1, and -2 percent compared to the medium-term OTM and DOTM call options. Hull-White The HW model under-prices DOTM, OTM, ITM, and DITM call options with a maturity of less than 30 days and ITM and DITM call options of a maturity less than or equal to 60 days, while it over-prices DOTM options for the maturity of greater than 30 and 60 days. The HW model deeply under-prices DOTM and OTM short-term maturity options to the extent of 27 percent and 6 percent. The HW model prices short- and medium-term ATM, ITM, and DITM options more accurately compared to the short and medium-term options with the degree of pricing bias VIKALPA VOLUME 38 NO 2 APRIL - JUNE

12 being 0-3 percent. For all moneyness-maturity groups, HW model generally produces prices that are very close to BS prices. This may be due to the fact that the HW model is just an extension of the BS model which incorporates stochastic volatility bias. The MAPE of HW model is higher in case of ATM, ITM, and DITM options and lower for OTM and DOTM options for short maturity options and always higher for medium- and long-term options as compared to the Heston s model. Heston s The out-of-sample forecast ability of the Heston s model is not only superior to the classical BS but also to stochastic peers HW and HN GARCH model in 11 out of 15 moneyness-maturity groups. The results show that Heston s model over-prices OTM options and underprices short-term ITM options, but it is less biased compared to BS, HW, and HN GARCH. Table 7 and Figure 8 evidentially prove it (Table 8 is generated randomly). The average percentage pricing error of Heston s model is lower than the BS, HW, and HN GARCH model in 11 out of 15 moneyness-maturity groups while the mean absolute percentage error is lower in 12 out of 15 moneynessmaturity groups. Addition of volatility as a random process strongly manages to improve the pricing accuracy significantly but is still unable to eliminate it completely. Technically, Heston s model is difficult to simulate, yet it is the most popular model, being used extensively by the practitioners, researchers, traders, and investors for pricing index and stock options. HN GARCH HN GARCH model under-prices DOTM call options in all maturity groups. It also under-prices OTM calls of Figure 8: Plot of Relative Price Bias of BS and Stochastic Models short and long-term maturities, while overpricing the long-term ATM, ITM, and DITM options. The model outperforms the benchmark BS model in 7 out of 15 moneyness-maturity categories, but outperforms its counterpart Heston s model in three categories only. This empirical research work clearly reveals that the performance of HN GARCH model is highly dissatisfactory compared to its stochastic counterpart, Heston s. Considering its extremely poor performance relative to others and involving its mathematical, numerical, and computation complexities, it is deduced that this model cannot be used for pricing Nifty Index options. Similar to PME, on the basis of MAPE (Table 6), the price error performance of stochastic volatility models is sequenced as Heston s > HW > BS > HN GARCH, further implying that the efficiency of the Heston s models is superior to others, across moneyness and maturity. While contrary to PME, a systematic increase in price error going from DOTM to DITM options, and short maturity to long maturity options, is noticed. The sequence based on MAPE are arranged as The above analysis strongly suggests that out of the classical BS and stochastic family models, discussed here, Heston s yields values closer to market prices compared to others and outperforms others in most of moneynessmaturity categories during the specific period of turbulence. The Heston model has the smallest out-of-sample valuation errors among all the models and outperforms them in 11 out of 15 categories of moneyness and maturity. An analysis of the empirical research reveals that the Heston model reduces the pricing bias by nearly about 25 percent across moneyness-maturities compared to the other models. Hence, having viewed the data presented in Tables 5 and 6, it can confidently be said that the models in question are either undervaluing or overvaluing Nifty index call options prices. Thus, after critically examining and testing all models, the study could not find 72 A KALEIDOSCOPIC STUDY OF PRICING PERFORMANCE OF STOCHASTIC VOLATILITY...

13 one single model that would best cater to multifarious moneyness-maturity dimensions of option pricing, one which would dominate the rest in all categories of moneyness-maturity. CONCLUSION After testing the concept empirically, the study has reached the conclusion that the stochastic models improve pricing error significantly as compared to classical BS during the waves of recent economic imbalance. The objective of the study was to identify the best suited model which actually worked, outpassing other competitive models in determining the overall outcome and protected a fundamentally estimated benefit; and, Heston s emerged as the dominant model. Though the Heston s outperforms the rest three, it is unable to remove the pricing bias completely. One of the reasons might be of ignorance of factors like random jump, market forces and other various dynamics that exist and enact in the reality of option price volatility. If the suggested points are properly incorporated into the recommended models, the improvement in the pricing bias can be expected. Since the Heston model manifested its potentiality and proved to be a successful model out of available options, it can easily be measured, calculated, and predicted that this model will surely perform better when put in normal, average, and stable conditions. This model fulfills the claim of keeping the investment faith protected if we do not have abnormal conditions in extreme, since the nature of variables and influencing forces can never be pre-imagined or calculated. APPENDIX Table 1: Filter Statistics of Nifty Index Call Options (2008) Total Call Contracts 72,494 Criteria No Trading Volume/Open Interest 56,835 Moneyness > +15% 481 Moneyness < -15% 4,380 Maturity > 90 Days 2,196 Maturity < 5 Days 945 No Arbitrage Relationship 202 Rejected Data 65,039 Rejected Data (%) Remaining Data 7455 Remaining Data (%) Source: Table 2: Nifty Index Call Option Statistics for 2008 (After Filtration) Maturity Moneyness Total/ DOTM OTM ATM ITM DITM Sub Total Short , % 12.6% 22.8% 5.6% 3.0% 54.2% Medium , , % 8.5% 14.6% 3.1% 1.6% 34.4% Long % 3.2% 5.8% 0.3% 0.1% 11.4% Total/Sub Total 1,416 1,805 3, , % 24.2% 43.1% 9.0% 4.7% 100% Source: VIKALPA VOLUME 38 NO 2 APRIL - JUNE

14 Table 3: Descriptive Statistics of Implied Volatility of Black-Scholes, Hull-White, Heston and Heston-Nandi GARCH Option Pricing Models Models Implied Volatility DOTM OTM ATM ITM DITM All Time to Maturity (T 30) BS Average Std Dev HW Average Std Dev Heston s Average Std Dev HN GARCH Average Std Dev No. of Observations Time to Maturity (30<T 60) BS Average Std Dev HW Average Std Dev Heston s Average Std Dev HN GARCH Average Std Dev No. of Observations Time to Maturity (60<T 90) BS Average Std Dev HW Average Std Dev Heston s Average Std Dev HN GARCH Average Std Dev No. of Observations Total BS Average Std Dev HW Average Std Dev Heston s Average Std Dev HN GARCH Average Std Dev No. of Observations 1,416 1,805 3, , A KALEIDOSCOPIC STUDY OF PRICING PERFORMANCE OF STOCHASTIC VOLATILITY...

15 Table 4: Descriptive Statistics of Call Option Price of Black-Scholes, Hull-White, Heston and Heston-Nandi GARCH Option Pricing Models Models Implied Volatility DOTM OTM ATM ITM DITM All Time to Maturity (T 30) BS Average Std Dev HW Average Std Dev Heston s Average Std Dev HN GARCH Average Std Dev No. of Observations Time to Maturity (30<T 60) BS Average Std Dev HW Average Std Dev Heston s Average Std Dev HN GARCH Average Std Dev No. of Observations Time to Maturity (60<T 90) BS Average Std Dev HW Average Std Dev Heston s Average Std Dev HN GARCH Average Std Dev No. of Observations Total BS Average Std Dev HW Average Std Dev Heston s Average Std Dev HN GARCH Average Std Dev No. of Observations 1,416 1,805 3, ,455 VIKALPA VOLUME 38 NO 2 APRIL - JUNE

16 Table 5: Descriptive Statistics of Out-of-Sample Percentage Mean Error (PME) of Black-Scholes, Hull-White, Heston and Heston-Nandi GARCH Option Pricing Models Models PME DOTM OTM ATM ITM DITM All Time to Maturity (T 30) BS Average Std Dev HW Average Std Dev Heston s Average Std Dev HN GARCH Average Std Dev No. of Observations , ,043 Time to Maturity (30<T 60) BS Average Std Dev HW Average Std Dev Heston s Average Std Dev HN GARCH Average Std Dev No. of Observations , ,561 Time to Maturity (60<T 90) BS Average Std Dev HW Average Std Dev Heston s Average Std Dev HN GARCH Average Std Dev No. of Observations Total BS Average Std Dev HW Average Std Dev Heston s Average Std Dev HN GARCH Average Std Dev No. of Observations 1,416 1,805 3, , A KALEIDOSCOPIC STUDY OF PRICING PERFORMANCE OF STOCHASTIC VOLATILITY...

17 Table 6: Descriptive Statistics of Out-of-Sample Mean Absolute Percentage Error (MAPE) of Black-Scholes, Hull- White, Heston and Heston-Nandi GARCH Option Pricing Models Models MAPE DOTM OTM ATM ITM DITM All Time to Maturity (T 30) BS Average Std Dev HW Average Std Dev Heston s Average Std Dev HN GARCH Average Std Dev No. of Observations Time to Maturity (30<T 60) BS Average Std Dev HW Average Std Dev Heston s Average Std Dev HN GARCH Average Std Dev No. of Observations , ,561 Time to Maturity (60<T 90) BS Average Std Dev HW Average Std Dev Heston s Average Std Dev HN GARCH Average Std Dev No. of Observations Total BS Average Std Dev HW Average Std Dev Heston s Average Std Dev HN GARCH Average Std Dev No. of Observations 1,416 1,805 3, ,455 VIKALPA VOLUME 38 NO 2 APRIL - JUNE

18 Table 7: Descriptive Price Bias Statistics of BS and Stochastic Models, Relative to Market (Dated October 1, 2008, Parameters: Index Price: , Expiry: October 29, 2008, Time to Expiry: 28 Days) Moneyness Market Models Price Models Price Bias (%) Price of (Relative to Market) Nifty Index BS HW Heston s HN BS HW Heston s HN Option GARCH GARCH DOTM OTM ATM ITM REFERENCES Amin, K., & Victor Ng (1993). Option valuation with systematic stochastic volatility. Journal of Finance, 48(3), Bachelier, L. (1900). Th e orie de la spe culation,. Annales Scientifiques de l E cole Normale Supe rieure. Series 3, 17, (English translation in: The random character of stock market prices (ed. Paul Cootner), MIT-Press (1964), pp Reprinted Risk Books, London 2000). Backus, D., Foresi, S., & Wu, L. (1997). Accounting for biases in Black- Scholes. CRIF Working Paper Series. New York University. Bakshi, G., Cao, C., & Chen, Z. (1997). Empirical performance of alternative option pricing models. Journal of Finance, 52(5), Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3), Bollerslev, T., Chou, R. Y., & Kroner, K. F. (1992). ARCH modelling in finance: A review of the theory and empirical evidence. Journal of Econometrics, 52(5), Brockman, P., & Chowdhury, M. (1997). Deterministic versus stochastic volatilities: Implications for option pricing models. Applied Financial Economics, 7 (5), Christoffersen, P., & Jacobs, K. (2004). The importance of the loss function in option pricing. Journal of Financial Econometrics, 72(2), Corrado, C., & Su, T. (1998). An empirical test of the Hull- White option pricing model. Journal of Futures Markets, 18(4), Derman, E., & Kani, I. (1994a). The volatility smile and its implied tree. Goldman-Sachs: Quantitative strategies research notes, accessed on June 1, 2007 through pdf Derman, E., & Kani, I. (1994b). Riding on the smile. Risk, 7(2), Duan, J. C. (1995). The GARCH option pricing model. Mathematical Finance, 5(1), Dumas, B., Fleming, J., & Whaley, R. (1998). Implementing volatility functions: Empirical tests. Journal of Finance, 53(6), Engel, R., & Mustafa, C. (1992). Implied ARCH models from 78 A KALEIDOSCOPIC STUDY OF PRICING PERFORMANCE OF STOCHASTIC VOLATILITY...

19 option prices. Journal of Finance, 43, Glosten, L. R., Jagannathan, R., & Runkle, D. E. (1993). On the relation between the expected value and the volatility of the nominal excess return on stocks. Journal of Finance, 48(5), Heston, S. (1993). A closed-form solution for options with stochastic volatility, with applications to bond and currency options. Review of Financial Studies, 6(2), Heston, S., & Nandi, S. (2000). A closed-form GARCH option valuation model. Review of Financial Studies, 13(3), Hull, J. C., & White, A. (1987). The pricing of options on assets with stochastic volatility. Journal of Finance, 42(2), Hull, J. C., & White, A. (1988). An analysis of the bias in option pricing caused by a stochastic volatility. Advances in Futures and Options Research, 3, Greenwich, CT: JAI Press. Itˆo, K. (1951). Multiple Wiener integral. Journal of Mathematical Society, Japan, 3, Levy, P. (1948). Processus stochastiques et mouvement brownien. Gauthier-Villars, Paris. Merton, R. C. (1973). The theory of rational option pricing. Bell Journal of Economics and Management Science, 4 (1), Nandi, S. (1996). Pricing and hedging index options under stochastic volatility: An empirical examination. Federal Reserve Bank of Atlanta Working Paper, Samuelson, Paul A. (1965). Rational theory of warrant pricing. Industrial Management Review, 6(2), Samuelson, Paul A. (1973). Mathematics of speculative price. SIAM Review (Society of Industrial and Applied Mathematics,) 15, Rouah, F. D., & Vainberg, G. (2007). Option pricing models and volatility using Excel-VBA. New Jersey: John Wiley and Sons. Shu, J., & Zhang, J. E. (2004). Pricing S&P 500 index options under stochastic volatility with the indirect inference method. Journal of Derivatives Accounting, 1(2), Singh, V. K., & Ahmad, N. (2011a). Modeling S&P CNX Nifty index volatility with GARCH class volatility models: Empirical evidence from India. Indian Journal of Finance, 5(2), Singh, V. K., & Ahmad, N. (2011b). Forecasting performance of constant elasticity of variance model: Empirical evidence from India. International Journal of Applied Economics and Finance, 5, Taylor, S. (1986). Modelling financial time series. New York: Wiley. Wiener, N. (1938). The homogeneous chaos. American Journal of Mathematics, 60(4), Wiggins, J. B. (1987). Option values under stochastic volatilities: Theory and empirical estimates. Journal of Financial Economics, 19, Vipul Kumar Singh is Assistant Professor of Finance in the Institute of Management Technology, one of the leading business schools of India. He became interested in options in 2005 and since then has published many papers on derivatives. His core area of research is Quantitative Finance and Derivatives. He holds a Ph.D. in Financial Mathematics and Master s degrees in Mathematics, Computer Science, and Finance. He is a member of the International Association of Financial Engineers (IAFE) and the Bachelier Finance Society (BFS). vipul.singh22@gmail.com Pushkar Pachori is currently working with Royal Bank of Scotland (RBS) as Technology Leader Market and International Banking. Prior to joining RBS, he was associated with HSBC, Hong Kong as Quality Specialist, IBM, India as Sr. Consultant, CPA Global, Sydney as Project Leader, and HCL, India as Senior Software Engineer. He holds a Masters degree in Computer Application and has done an Executive MBA in Finance. He has over 12 years of experience in corporate and academics and has deep interest in the computational domain of financial derivatives products. pushkar.pachori@rbs.com VIKALPA VOLUME 38 NO 2 APRIL - JUNE

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