Capturing the Volatility Smile of Options on High-tech Stocks A Combined GARCH-Neural Network Approach

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1 Capturing the Volatility Smile of Options on High-tech Stocks A Combined GARCH-Neural Network Approach Gunter Meissner 1 Noriko Kawano gmeissne@aol.com, el (808) Abstract: A slight modification of the standard GARCH equation results in a good modeling of historical volatility. Using this generated GARCH volatility together with the inputs: spot price divided by strike, time to maturity, and interest rate, a generated Neural Network results in significantly better pricing performance than the Black Scholes model. A single Neural Network for each individual high-tech stock is able to adapt to the market inherent volatility distortion. A single Network for all tested high-tech stocks also results in significantly better pricing performance than the Black-Scholes model. Key words: ARCH, GARCH, Neural Network, Volatility Smile, High-tech Stocks JEL classification: G14 1. Introduction One of the most violated assumptions of the Black-Scholes model is the assumption of constant implied volatility. In trading practice, option traders alter the implied volatility for far out-of-the-money and far in-the-money options. his phenomenon is referred to as the volatility smile. Numerous attempts have been made to solve this volatility bias in an analytically consistent matter. Merton (1976) has proposed a Jump Diffusion Model, where a jump is added to the Geometric Brownian motion. In Merton s model the growth 1 Dr. Gunter Meissner is President of Derivatives Software, and Professor of Finance at the Hawaii Pacific University; Noriko Kawano, MSIS, is currently working as a software engineer at Hawaii Dental Service. he article was presented at the 8 th Asia Pacific Finance Conference, Shanghai, July 000.

2 rate of the underlying asset is - k, where is the expected return, is the rate at which jumps happen, and k is the average jump size. his leads to fatter tails in the log-normal distribution, thus higher implied volatilities for far-outof-the-money calls and puts in the standard Black-Scholes model. his is consistent with the classical volatility smile, which is typical for the currency market and often found in the commodity market: Figure 1: Classical Volatility Smile of Dollar/Yen Options; Data collection December 5 th 1999 to February 5 th 000 Figure : Classical Volatility Smile of Options on Crude Oil; Data collection December 5 th 1999 to February 5 th 000

3 3 Cox and Ross (1976) have suggested a Constant Elasticity of Variance Model, in which the volatility is divided by S. S: asset spot price, 0 1. herefore the volatility is inversely correlated to the stock price. his corresponds with higher implied volatilities for out-of-the-money puts and lower implied volatilities for out-of-the-money calls in the Black-Scholes model. his coincides with the notion that markets drop in a sharp and volatile manner and rises more smoothly and less volatile. he phenomenon of higher implied volatility for out-of-the-money puts (and in-the-money calls due to put-call parity) and lower implied volatility for out-of-the-money calls (and in the-money puts) is often found, especially for short dated options, in the stockmarket and bond market: Figure 3: Right-sided Volatility Grimace for short dated Options and Classical Volatility Smile for longer dated Options on Microsoft; Data collection December 5 th 1999 to February 5 th 000

4 4 Figure 4: Right-sided Volatility Grimace of Options on 30-year -Bonds; Data collection December 5 th 1999 to February 5 th 000 Cox, Ross and Rubinstein (1979) have proposed a Pure Jump Model, in which the asset prices occasionally jump up with a certain probability. his leads to thinner tails for low underlying asset prices and fatter tails for high underlying asset prices compared to the Black-Scholes log-normal distribution. If the Pure Jump Model were reality, the Black-Scholes model would adjust using higher implied volatility for out-of-the-money calls and lower volatility for out-of-the-money puts. his is equivalent to the notion that the market rises in a sharp and volatile manner, and drops in smoother and less volatile manner. Since this is the way the Gold market has behaved in the recent past, it is not surprising that higher volatilities for out-of-the-money calls and lower volatilities for out-of-the-money puts are used for short dated options in the Gold market:

5 5 Figure 5: Left-sided Volatility Grimace for short dated Options on Gold, Right-sided Volatility Grimace for longer dated Options on Gold; Data collection December 5 th 1999 to February 5 th 000 he different implied volatilities used in the financial markets dispute the validity of the Black-Scholes formula. If traders alter the only determinable input parameter of the formula due to the degree of in-the-moneyness, time to maturity, and product, the basic concept of the formula is questionable. his study tries to create an option pricing model, which has a consistent volatility input. he volatility is derived with the help of GARCH volatility modeling. A GARCH (1,1) model gives the best volatility forecasts, in comparison with the tested GARCH (5,5) and GARCH (10,10) models. he GARCH (1,1) model only gives slightly better forecasts than the ARCH (1,1) model. he Neural Networks used in the analysis are fully connected. Four types of neural networks are tested: Multilayer Perceptrons (MLP), Radial Basis Function Networks (RBF), Probabilistic and Generalized Regression Neural Networks (PNN) and (GRNN), and Linear Networks. his study investigates options on 10 high-tech stocks over a period of 9 months. he selection criteria for the stocks were market capitalization and option trading volume. he tested stocks are: Amazon, AOL, Apple, Cisco, Dell, IBM, Intel, Microsoft, Oracle, and Yahoo. After cleaning the data, the data set consisted of 9,386 option quotes.

6 6. he GARCH Volatility Input Over the last two decades, ARCH and GARCH models have become popular in order to forecast volatility in the financial markets. Introduced by Engle in 198, GARCH (General Autoregressive Conditional Heteroskedasticity) models try to express the volatility as a linear function of the past squared errors. In the ARCH model, the volatility is expressed in a discrete stochastic process of the form (1) σ ω t q i 1 α i ε t -1 where : annual volatility > 0 i 0 t : error term, t = z t t, z t = i.i.d., E(z t ) = 0, var(z t ) = 1 he error term is generated by a stochastic process of the general form P t = f (x t-1,) + t where P t : Price of the underlying asset x : Independent variable : Parameter vector In our model, we chose to generate by a simple one-period lagged regression of the form () P t = a + b P t-1 + t For all 10 tested stocks the results of equation () were statistically significant. he average R = 0.94, the p-level = 3.65E-11, and the mean absolute percentage error = 3.00%. An alternative process to the ARCH model of equation (1) is the popular GARCH model, in which a second term is added to the ARCH model:

7 7 (3) σ ω α ε t i t 1 q i 1 j1 p β j σ t j In equation (3), the volatility is expressed with an error term that is at least two-period lagged, since for j=1 the error term of period t- is input in the last term to express σ. [see equation (): for ε t-1 we need P t- ] t In our study, numerous testing was done to determine the best values of p and q in equation (3). While higher orders of p and q as GARCH(5,5) and GARCH(10,10) produced statistically significant R and standard error values, the p-levels were unacceptably high. By far the best results were achieved with GARCH(1,1). hus, equation (3) reduces to (4) σ t ω α 1 ε t 1 β 1 σ t 1 esting equation (4) still gave statistically unsatisfactory results. While p-levels were satisfactorily low, the R levels for the 10 tested stocks were in the range from 0.11 to A modification of equation (4) improved the results dramatically. Instead of using the absolute error t, we input the relative error t /P t into equation (4). Equation (5) σ t ω α 1 (ε t 1 / P t 1 ) β 1 σ t 1 improved the results significantly. he average R for the 10 stocks increased from 0.6 to 0.91, the standard error decreased from 0.07 to 0.0 and the p- levels decreased from 1.14E-07 to 1.13E-9. In order to derive the parameters in equation (5), several non-linear tests such as the Quasi-Newton, Simplex procedure, Hooke-Jeeves pattern moves, Rosenbrock pattern search, and Hessian matrix were conducted. he best results were achieved with the Quasi-Newton method, which evaluates the regression function at different points to estimate first and second order derivatives and then follows a path to the minimum of the loss function. Figure (6) shows the 30-day volatility estimate of IBM using the modified ARCH formula:

8 14-Jan-99 3-Feb-99 3-Feb-99 1-Mar Mar-99 0-Apr-99 7-May-99 6-May Jun-99 -Jul-99 -Jul Aug-99 7-Aug Sep-99 5-Oct-99 -Oct-99 8 q α (ε /P ) t i t -1 t 1 (6) σ ω i % 60.00% 50.00% 40.00% 30.00% 0.00% Historical Volatility ARCH Figure 6: Comparison of historical volatility and ARCH estimated volatility of IBM Using the GARCH equation (5) improves the results of the ARCH equation (6) only insignificantly. For all stocks tested, the average root mean square error RMSE = mean error ME = 1 t 1 absolute error MAE = 1 t 1 (P P ) t t 1 t 1 (P P t t ) reduced from to , the reduced from 1.5 E-0.5 to 8.6 E-17, the mean P t P mean absolute percentage error MAPE = t reduced from to and the 1 t 1 P t P P t t reduced from to he GARCH standard Pearson correlation coefficient R was slightly higher at compared to the ARCH R of he better results of the GARCH equation (5) compared to the ARCH equation (6) are principally unnoticeable when compared graphically:

9 14-Jan-99 3-Feb-99 3-Feb-99 1-Mar Mar-99 0-Apr-99 7-May-99 6-May Jun-99 -Jul-99 -Jul Aug-99 7-Aug Sep-99 5-Oct-99 -Oct % 60.00% 50.00% 40.00% 30.00% 0.00% Hist Vol GARCH Figure 7: Comparison of historical volatility and GARCH estimated volatility of IBM In this study, we decided to use GARCH equation (5) to derive the implied volatility, and input the results into the Neural Network and the Black Scholes model. he volatility was derived for 30-day increments, i.e. for 30- day GARCH, 60-day GARCH, 90-day GARCH etc. Linear interpolation was then done to match the option maturity. 3. he Data Set he data tested consisted of options on ten high-tech stocks, which were monitored over a period of 9 months, from May 1 st 1999 to January 31 st 000. Due to put-call parity, only calls were monitored. Selection criteria for the ten stocks were market capitalization and option trading volume. he ten selected stocks were Amazon, AOL, Apple, Cisco, Dell, IBM, Intel, Microsoft, Oracle, and Yahoo. he original data consisted of 1,37 option quotes. Several criteria were used to clean the original data: he intrinsic value rule C S K e r is violated he option price is lower than 10 basis points he option price has less than 10 days to option maturity he option is very deep in- or out-of-the money S/ K 0.75 or S/ K 1.5 where C : Call price, S : Stock price, K : Strike, : option maturity, r : continuously compounded risk-free interest rate After applying these selection criteria, the number of option quotes in the data set was reduced to 9,386. he data was then divided into three sets: A training set, a verification set, and a test set. he proportions are :1:1 respectively.

10 10 he training set is the set used to train the Network by attempting to minimize an error function. he error function used in this study was the root mean square error (RMSE, formula see above). he verification set is used to check the performance of the trained Network. A deterioration of the verification error during learning indicates overlearning or overfitting, a common problem of Network training. he third subset, the test subset, tests the generalization capability of the trained and verified Network. If the verification and test errors are fairly close, the Network is likely to generalize well. 4. he Black-Scholes Model as a Reference Measure Until today, the 1973 created Black-Scholes model is the dominant model to derive option prices and hedge parameters in the financial markets. herefore, it is a valid reference model to test the performance of our Neural Network models. he original Black-Scholes formula has five inputs, which are the stock price S, the strike K, the option maturity, the interest rate r, and the implied volatility. he call price is derived as where (7) C = S N(d 1 ) - K e -r N(d ) d 1 = S ln ( ) 1 r Ke and d = d 1 - C : Call price S : Stock price K : Strike : option maturity N : cumulative normal distribution. r : continuously compounded risk-free interest rate ln : natural logarithm : annualized implied volatility Since all stocks tested in our model do not pay a dividend, the original formula (7) can serve as the basis formula. As shown by Merton (1973), the model is homogeneous linear in S and K. herefore, dividing by K, the formula is reduced to

11 11 (8) C / K = S / K N(d 1 ) - e -r N(d ) hus, we now have four inputs S / K,, r,, instead of five inputs. his can be considered beneficial for our Network creation, since in most circumstances Neural Networks with less input parameters perform better. Equation (8) is the equation, that is used as a reference model to our Neural Network models. 5. he Neural Networks Models In this study we test four types of Neural Networks and examine their performance against the Black-Scholes model. he four fully connected Network types are Multilayer Perceptrons (MLP), Radial Basis Function Networks (RBF), Probabilistic and Generalized Regression Neural Networks (PNN) and (GRNN), and Linear Networks. he most popular Neural Network, the MLP network can mathematically be expressed as where y NN H h 1 β y NN : Output of the Neural Network (in this study the option price) h : Weighting factor of layer node h j, which reflects the strength h * between h j and the output y NN (see Figure 8) : ransfer function, usually a simple hyperbolic function such as the tangent function. his function standardizes weighted input variables to values between -1 and +1. w ih : Weighting factor of input x i, which reflects the strength between input x i and the a hidden layer node h (see Figure 8) x i : Input variable i, (in our model Spot/Strike S/K, Option maturity, Implied Volatility and Interest rate r) Grahpically, a MLP Neural Network can be expressed as in figure 8: n i 1 w ih x i

12 1 In p u t H id d e n L a y e r O u tp u t a rg e t V a lu e x 1 w 11 x w 1 h 1 1 h y NN y x 3. x n w n3 h 3 3 Figure 8: Sample structure of an MLP Neural Network with various input variables x i, i = 1..n, one hidden layer with three units h j, j =1..3, and one output variable y NN. In the hidden layer or layers, the input variables are activated and transformed. It is a widespread misconception that the structure of the hidden layers is unknown. In most Neural Networks the mathematical algorithms of the hidden layers are clearly expressed and mostly differentiable. In Figure 8 the output variable y NN is the model s option price. his price is compared to the target value y, which is the market option price. he process of learning is implemented with weighting factors. Each weighting factor has a resistance attached to it. Numerous simulations (epochs) of different combinations of weighting factors are run. If the Neural Network output is close to the target value, the weighting factors are strengthened, i.e. the resistance is turned down. If certain weighting factors produce bad results, the weighting factors are weakened, i.e., the resistance is turned up. his recursive process of incrementally adjusting the weighting factors to approach a target value is the backpropagation process. MLP models are the most popular types of Neural Networks models in the financial markets and will result in the best performances for all of the stocks that are tested in this study. Going into detail regarding the other tested Networks would be beyond the scope of this article. herefore, only the major features of the other networks that were tested will be outlined: Radial Basis Function Networks, introduced by Broomhead and Lowe in 1988, consist of an input layer, hidden layers of radial units, and an output

13 13 layer of usually linear units. A single hidden layer is often sufficient to cope with any type of non-linear function. Also, fast traditional linear modeling techniques are applied to the output layer. his reduces the chance of getting stuck at local minima, a problem MLP Networks often suffer. RBF models are therefore faster to train, but are slower in executing than MLP Networks, since they usually require more hidden layer units. For more discussion on RBF s see Broomhead and Lowe (1988), Moody and Darkin (1989), and Haykin (1994). Probabilistic Neural Network (PNN) models are also tested in this study. PNNs usually have three or four layers. Speckt (1990) introduced kernel-based estimation functions into PNNs, estimating the probability density of parameters given by the training data. he probability of various classes can then be calculated to select the most probable one. PNNs usually have simple and fast training functions and are usually slow in executing the network. Generalized Regression Neural Networks (GRNNs) are similar to PNNs but perform regression rather than classification analyses. As the PNNs they are based on Gaussian kernel algorithms performed on each of the training cases. A Unit A derives the regression and a unit B calculates the density. he model function itself is derived directly from the training data herefore, not much training is necessary. However, the number of units in the hidden layer is usually very large. It is recommenced that the number of units is the same as the number of training cases, which is done in this study. Since each stock that is tested has training cases of several hundred prices, the layer units have this magnitude. herefore the PNNs and GRNNs are quick in learning, but rather slow in execution. In comparison to other Network types, PNNs and GRNN Networks perform usually relatively well for a rather heterogeneous data input structure. his will be confirmed in this study. Linear Networks are usually presented by a N x N matrix with a N x 1 bias vector. here are no hidden units, but an output layer with linear units and a linear activation function. o execute the Network, the inputs are multiplied by the matrix s weight, then the bias vector is added. Linear Networks can be valuable in finding simple solutions to a problem that was thought more complex and are a good test to discover overfitting of more complex Network models. A further type of Neural Network are Kohonen Networks, which were developed by euvo Kohonen (198) and modified in the mid 1990 s by Haykin (1994), Fausett (1994) and Patterson (1996). Kohonen Networks have unique features, which differ from the above Networks. hey can be trained unsupervised i.e. without an output variable. he training is conducted by two main parameters, a learning rate and a neighborhood size. Each case is

14 14 presented to the Network and a radial node, which is closest to the training unit, is selected. he neighboring units are then updated to resemble the training unit more closely. One drawback of Kohonen Networks is that they usually only have a nominal, not a metric output value. hus, only a classification not a numerical evaluation is possible. herefore, Kohonen Networks are not well suited to deal with the numerical nature of this study and will not be tested. 6. Analysis Four types of Neural Networks are generated and compared with the Black-Scholes model: Multilayer Perceptrons (MLP), Radial Basis Function Networks (RBF), Probabilistic and Generalized Regression Neural Networks (PNN) and (GRNN), and Linear Networks. All Networks are fully connected. here are four input parameters for each Neural Network, which are the same for the Black-Scholes reference model: Spot price S divided by strike price K, S/K Option maturity in years, Continuously compounded risk-free annual interest rate, r GARCH generated annual implied volatility following equation (5), Neural Networks are generated for each of the 10 selected stocks and one Network for all stocks together. As criteria for judging the quality of a Network the root mean square error RMSE, the correlation coefficient r, and the regression ratio (i.e. the prediction error standard deviation divided by the data standard deviation) were used. hese criteria are applied on the verification set, i.e. the out of sample data. esting was done on a 500MZ, 18MB RAM PC. Generating a Neural Network for individual stocks took about 0 minutes each. Generating a Neural Network for all 10 stocks, i.e. for 9,386 input cases, took a little over 3 hours. 7. Results For each of the individually generated Neural Networks, the two best performing Networks were MLP Networks. A distant second best Network type was the RBF Network. GRNN Networks came in third, Linear Networks did not result in a satisfactory performance. Ranking the 10 best Networks of each stock, giving the best Network10 points and second best Network 9 points etc, the ranking resulted in 8 points for MLP Networks, 139 points for RBF

15 15 Networks, 11 points for GRNN Networks and 8 points for linear Networks. his demonstrates the superiority of the MLP Network for a homogeneous input structure of a single stock. All single stock MLP Neural Networks had a fairly simple structure. Nine Networks had a single hidden layer, whereas Amazon resulted in hidden layers. he number of hidden units in the layers varied from 8 for Oracle to 18 for Dell and Cisco Systems. Figure 9 shows the Network structure of Cisco Systems: In p u t H id d e n L a y e r O u tp u t a rg e t V a lu e w 1,1 h 1 S /K w 1, h 1 r h y NN y w 4,1 8 h 18 Figure 9: Network Structure of Cisco Systems with 4 inputs, S/K,, r and, one hidden layer with 18 units, the Neural Network option price y NN, and the market option price y Comparing the network performance for individual stocks, all best networks had a better performance than the Black-Scholes model. able 1 shows the average results for the 10 stocks: Black-Scholes Model MLP Neural Network RMSE ME MAE MAPE R able 1: Performance comparison between the averages of individual stocks of the Black-Scholes Model and the best performing Neural Network

16 16 As seen in table 1, the MLP networks performed significantly better in each of the performance measures with a smaller RMSE, ME, MAE, and MAPE and a higher R. Figures 10 and 11 show the pricing error of Microsoft using Black- Scholes and the best performing Neural Network: Figure 10: Black-Scholes pricing error of Microsoft stock; C: Market call price; BS: Call price generated by Black-Scholes, K: Strike price Figure 11: Neural Network pricing error of Microsoft stock; NN: Call price generated by the Neural Network Figures 10 and 11 show the better performance of the Neural Network. In Figure 10, the error regression function is upward sloping, verifying the

17 17 Right-sided Volatility Grimace of Figure 3, i.e. the higher implied volatility for out-of-the-money puts and in-the-money calls in trading practice. he Neural Network however, was able to adapt to the volatility grimace, resulting in an almost horizontal regression function. When generating one Neural Network for all stocks, the Network was confronted with a more diverse input data structure. he GARCH implied volatility levels ranged from 31.81% for Cisco Systems to 104.4% for Amazon and the volatility smiles were of different nature. his resulted in a more complex Network structure of the best performing Network, a GRNN Network. he GRNN Network had one hidden layer, the number of hidden units in the layers was, as it is usually the case with GRNN Networks, the same as the number of training cases, in our study 4,693. he execution of the whole Network (i.e. presenting all input cases to the Network, feeding signals through and producing all individual output option prices) takes about 0 seconds. As a comparison, execution of the best performing MLP Network took less than one second. However, executing one single case in the GRNN and deriving a single option price takes less than a second. herefore, using the rather complex GRNN network structure in trading practice is not a problem. he best GRNN Network, as other MLP and RBF Networks, performed significantly better than the Black-Scholes model. able compares the best performing GRNN Network with the Black-Scholes model. Black-Scholes Model GRNN Neural Network RMSE ME MAE MAPE R able : Performance comparison of the Black-Scholes Model and the best Network for all stocks able shows that the GRNN Neural Network outperformed the Black-Scholes Model with a smaller RMSE, ME, MAE, and MAPE and a higher R. Comparing ables 1 and, we can see that due to the more homogeneous input structure, single stock Neural Networks had, except for the MAPE, a better performance than one Network for all stocks.

18 18 Graphically, the performance difference of the best GRNN Network and the Black-Scholes model can be seen in Figures 1 and 13. Figure 1: Black-Scholes pricing error for all stocks; C: Market call price; BS: Call price generated by Black-Scholes, K: Strike price Figure 13: Neural Network pricing error for all stocks stock; C: Market call price; NN: Call price generated by the Neural Network, K: Strike price 7. Conclusion his study has shown that it is possible to produce Neural Networks, which result in a better pricing performance than the Black-Scholes model,

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