Han & Li Hybrid Implied Volatility Pricing DECISION SCIENCES INSTITUTE. Henry Han Fordham University

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1 DECISION SCIENCES INSTITUTE Henry Han Fordham University Maxwell Li Fordham University HYBRID IMPLIED VOLATILITY PRICING ABSTRACT Implied volatility pricing remains an ortant problem in finance with the surge of high frequency trading. However, there is no a widely accepted way in finance community because no systematic performance analytics are available for existing methods. In this study, we propose a novel hybrid lied volatility pricing method and evaluate its performance along with peer methods through a data-driven approach. The proposed method demonstrates its superiority to other methods in terms of convergence order and accuracy. Our method is especially suitable for high frequency trading for its exceptional speed and sureness in convergence. KEYWORDS: Option, Black-Sholes Model, Greeks, Implied Volatility, High Frequency Trading INTRODUCTION With the surge of high frequency trading (HFT), in which a transaction completes in less than a second or even millisecond, lied volatility pricing remains an ortant problem in finance (Carrion, 0, Hagstromer and Lars, 0). Implied volatility measures investor's confidence about the future risk of the underlying asset (e.g. stock) in a financial market, but cannot be observed directly from the historical pricing data. As a key measure of option price, lied volatility provides a prediction or even classification about the future performance of option. It is actually quoted more often than an option s price in trading. Since high frequency trading brings ultra-fast movements in the price of securities, it requires corresponding speedy updates in option quotes for arbitrage or hedge (Foucault et al 06). In other words, it is essential to conduct accurate and fast lied volatility pricing for the sake of profitable trading. Implied volatility is defined as the value that makes the theoretical price of an option under an option pricing model equal to its current market price. That is, it is the numerical solution of the nonlinear equation: F( ) = C( ) C * = 0, in which C ) is an option pricing model with lied volatility as one parameter, and ( * C is the current market price of the underlying asset. Mathematically, ) ( C is a multi-variable function to model option prices, in which each variable is a relevant parameter (e.g. strike price) contributing to option pricing.

2 The Black-Scholes model was the first option pricing model proposed in 97 by modeling option pricing via stochastic partial differential equations (Black and Scholes,97). Merton did excellent extensions to this model (Metron, 97). It has been a classic model widely used in option pricing, especially effective and robust for vanilla options. It has been reported that it can still work correctly even if we drop some assumptions of this model such as constant volatility. There are quite a few option pricing models derived from this model in the previous studies. They include stochastic volatility models (Hull and White, 987; Heston 99.), jump diffusion models, and their variants (Andersen, 000). These polished models perform better than original Black-Scholes model in some aspects, but they require sufficiently accurate parameter estimations, which may not be easily achieved sometimes. As such, we employ the classic Black-Scholes model as the option pricing model in this study for its generalization and robustness. Implied volatility pricing under Black-Scholes model is equivalent to find the root of the nonlinear * function F( ) = C( ) C numerically. Root-finding is a relatively mature mathematical topic with a lot well-developed algorithms. However, only bisection and Newton methods are mentioned mostly in lied volatility pricing literature (Haug 007). The bisection method finds the lied volatility by seeking the root of the function F ) through halving a sequence of search intervals. Newton method approaches the lied volatility along the derivative direction of the function F ) from an initial value, i.e. initial lied volatility approximation. Newton ( method is assumed to outperform the bisection method for its second order convergence (Corrado and Miller 996). However, it is an ad-hoc conclusion and cannot meet the demand of lied volatility pricing in trading for the following reasons. First, it is only a theoretical conclusion based on their convergence order. Newton method s convergence actually relies on its initial lied value by the nature of its local convergence. A bad initial value may trigger the risk of divergence (Wu. 005). Second, many mature root-finding algorithms (e.g. Brent) are not mentioned or least investigated in previous studies (Brenner and Subrahmanyam, 988; Corrado and Miller 996; Haug 007) though they have better convergence speed and accuracy (Boyd, 0). As a result, the performance of the algorithms is almost unknown to finance communities. Third, pricing algorithm evaluations in the previous studies are basically model-driven instead of data-driven. The algorithm evaluations are based on very limited data or even case study results. Such a way usually fails to provide comprehensive evaluation on an algorithm s performance. For example, it may miss the divergent cases of Newton method because of only a limited number of option contracts used in pricing. On the other hand, trading in current finance markets is characterized by a huge amount data in a very short time. It requires lied volatility pricing algorithms to have fast speed and guaranteed convergence in such a data-driven environment. In this study, we categorize lied volatility pricing into Greeks free, Greeks-based, and hybrid pricing methods under the framework of Black-Scholes model. Furthermore, we propose an effective hybrid pricing algorithm and evaluate it in a data-driven approach along with its peers. All option data used in our study is crawled from web and further processed for pricing algorithms under Black-Scholes model. We have found that Newton method has the worst performance in terms of accuracy, but bisection method has the worst performance in terms of convergence speed. However, the (

3 proposed method demonstrates its superiority to other methods in terms of convergence order and accuracy. It is especially suitable for high frequency trading for its exceptional speed and sureness in convergence. Our result can be also easily extended to other refined option pricing models to handle more general situations. PRINCIPLE OF BLACK-SCHOLES MODELS In this section, we briefly review Black-Scholes model that is mainly used to calculate option price and derivative terms (Greeks) in lied volatility pricing. Black Scholes (BS) model Black Scholes model views option price as a differentiable function (f) with five variables: expiration time (t), current underlying asset price (S), (lied) volatility of the underlying (), strike price (K), and risk free interest rate (r). It initially assumes that options are only European options, interest rates are constant and always known, the underlying pays no dividends and its returns are normally distributed and there are no transaction costs. Although the assumptions do not match real market situations completely, it has been addressed in its different extensions. Black Scholes models option price f(s, K, r,, t) as the following partial differential equation (PDE): f f f + rs + S rf = 0 () t S S f / t and f / S are the changes of option price with respect to time and underlying price respectively. f / S is the second order changes of option price with respect to underlying price. Although not technically, Black-Scholes model can be summarized conveniently as one sentence, i.e. all weighted changes with respect to time and underlying price is the product of interest rate and option price. The weights are [, rs, S /] and the changes with respect to time and underlying price are [ f / t, f / S, f / S ]. The analytic solutions exist for European call and put respectively. That is, the prices of call and put with time T before maturity are C = SN ( d ) Ke rt N ( d ) and P = Ke rt N ( d ) SN ( d ) respectively. N(.) is the probability distributive function (CDF) of standard normal distribution. The values of parameters are d = ln ( S / K ) + ( r + / )T, d = d T. T Greeks for lied volatility pricing Greeks are a set of derivatives modeling the sensitivity of option price with respect to its parameters. We mainly focus on two Greeks: Vega and Vomma that are used in lied volatility pricing. They both are identical for put and call options. They are needed in the nd and rd order convergence root finding methods (e.g. Newton and Halley methods) in lied volatility pricing respectively. The convergence order of root finding algorithms will be introduced in the Method section. Vega vega = f / represents the change of option price with respect to volatility. It can be calculated as vega = f / = Sn( d) T, in which n(.) is the probability density function of standard normal distribution (Haug 007). When an option attains its maximum vega, it indicates the best time to profit from moves in lied volatility in trading.

4 Vomma is the second order derivative of option price with respect to volatility, i.e. vomma = f / = Sn( d) T ( dd / ). It measures the change rate of vega as volatility changes. Traders usually bet on changes of lied volatility in trading by observing an option s Vomma. METHODS Implied volatility pricing is essentially a root finding problem under Black Scholes framework. Before categorizing lied volatility pricing algorithms, we introduce the concept of convergence order for the convenience of descriptions. Convergence Order Given the true solution * x obtained from root finding algorithm A for F ( x) = 0, n x is the n th * approximation of x, the convergence order of the root finding algorithm is defined as * x xn+ qa = lim () n * x xn If q A =, the convergence is linear; If < q A <, the convergence is superliner; If q A =,, the convergence is quadratic; If q =,, the convergence is cubic. Implied Volatility Pricing Algorithms Categorizations A We categorize lied volatility pricing algorithms into Greeks-free, Greeks-based, and hybrid methods according to the root finding algorithms they use. The Greeks free methods are those root finding methods that don t require the derivatives of the objective function F ). It includes bisection, brent, muller-bisection, and ridder methods in this study (Kiusalaas 00, Boyd 0). Brent method by nature has an advantage over Bisection for its relative complicate algorithm design. It has a convergence order.84 in best case. Muller-Bisection combines the convergent efficiency of Muller s method and global convergence of the bisection method (Wu, 005). It has a convergence order.84. Ridder method has a concise structure than Brent method and has convergence order.44 (Ridders, 979). Since the derivatives of the objective function are equivalent to Greeks in Black Scholes model, it is called Greeks free methods. These methods have the least demand from option pricing model because they only need option price values and don t require the objective function is differentiable. Almost all of them have a superlinear convergence except bisection has a linear convergence order. The Greeks-based methods require the objective function is differentiable. They actually do need the first and second derivatives of the objective function with respect to lied volatility. In other words, they need Vega and Vomma Greeks respectively in lied volatility calculations. They include Newton, Halley methods, and their variants. They have quadratic and cubic convergence orders, but are locally convergent approaches. The locally convergent means they can find the lied volatility provided the initial value is close enough to it. Thus, Greeks-based methods have a risk of divergence, when a bad initial value is chosen. (

5 The hybrid methods generally combine both methods to optimize speed and accuracy of lied volatility pricing. They share the same convergence orders as the corresponding Greek-based methods. Exactly speaking, it employs Greeks free methods to feed Greek-based methods good initial values point in iteration to optimize their convergence. Although hybrid approaches were well discussed in numerical analysis community, they are least investigated in lied volatility pricing. We introduce three hybrid methods based on Brent method, Newton and Halley method in this study Greeks Free Implied Volatility Pricing Greeks free lied volatility pricing does not need Greeks in lied volatility calculations. The derivatives (Greeks) are technically approximated by option pricing values when it is needed. For example, the derivative of secant method is approximated by as the quotient of finite differences. Greeks free methods are globally convergent approaches that always converge to the true lied volatility value irrespective of any initial lied volatility approximations. We use Brent method as an example to illustrate Greeks free approaches. As a method with a superlinear convergence order (.84) at its best case, Brent s method combines bisection method and inverse quadratic interpolation method (Boyd 0). Its basic idea is to compute a sequence of extra provisional values via inverse quadratic interpolation in Bisection. The extra provisional values are used to update the root bracketing intervals in Bisection. Brent lied volatility pricing method can be described as follows. Algorithm Brent lied volatility pricing Input: Option pricing function F (.), tolerance ε 0 0 Max iteration number: MaxIter Initial search interval: [ x, x ] such that F ( x ) F( x ) < 0 Output: Implied volatility such that F ) < ε (. Initial three points: = x, = x, = x, yi = F( xi ), i =,,. if y y > 0 swap and end if. iter= 4. While y ε and y ε and iter <MaxIter 5. = ( + )/ 6. compute y = F( ) 7. If y y y //do inverse quadratic interpolation 8. y y y y y y = + + ( y y)( y y) ( y y)( y y) ( y y)( y y) () 9. else // linear interpolation (secant method) 0. y ( ) = ( y y) (4). end if. compute y = F( ). if > swap and end if 4. if yy < 0

6 5. =, = 6. else 7. if ( yy < 0) = else = end if 8. end if 9. iter=iter+ 0. end while. y or y Greeks-based Implied Volatility Pricing For the convenience of description, we give Greeks-based lied volatility for European call options as follows. Given an European call option with current underlying asset S, strike K, riskless interest rate r, time to maturity, and its market price C *, lied volatility pricing is equivalent to do root finding for nonlinear equation F( ) = C( S, K, r, T, ) C * = 0, in rt which C S, K, r, T, ) = SN( d ) Ke N( ). ( d The Newton lied volatility pricing is equivalent to the following iteration scheme, F( ( n+ ) = F'( ) The objective function derivative is equivalent to Greeks term: Vega, i.e., F '( ) = C. The equation is further updated as * C( S, K, r, T, ) C ( n ) + = (6) Sn( d) T Similarly, the Halley lied volatility pricing is equivalent to the following iteration scheme, F( ) ( ) ''( ) ( + ) ( ) F F n = n (7) F'( ) F'( ) F'( ) C F' '( ) is actually Vomma: F ''( ) =. Plugging Vomma into equation (4), we have Halley lied volatility pricing: ) (5) F( ) ( ) ( + ) ( ) F n = n dd (8) Sn( d ) T Sn( d) T It is noted that both methods are sensitive to initial points in pricing though with different degrees. If the initial points are close enough to the true lied volatility, they can demonstrate excellent convergence order. Otherwise, they may have the likelihood not to converge in pricing. In other words, Greeks-based methods are locally convergent approaches because convergence relies on the selection of an initial lied volatility approximation. Hybrid Greeks Implied Volatility Pricing

7 As a hybrid approach, it employs Greeks free methods to feed Newton or Halley methods better starting point in iteration to optimize their convergence. The first step is to locate an initial lied volatility approximation via running k feed-in steps of a Greeks free method in a (0) (0) search interval, ] i.e. GreeksFree F,,, k). The second step is to seek the [ ( (0) lied volatility value via a Greeks-based method by using as the starting point. Theoretically, any Greeks free methods can be used in the first step to feed a good starting point to Newton or Halley methods (Proinov et al. 05). We choose Brent method in this study for its relatively fast speed and sureness of convergence. We introduce three hybrid pricing methods in this study. The first two are hybrid Newton and Halley methods by using Brent method to feed initial values. We call them new Newton and new Halley for the convenience. The third is called Halley momentum method that adds a momentum item in new Halley s iteration. The Halley momentum method is obtained by adding a momentum parameterγ that is set as 0.9 in our lementation to the new Halley s iteration method, namely, F( ) ( ) ( + ) ( ) F n = n dd γ (9) Sn( d ) T Sn( d) T It is worthwhile to point out that such a momentum approach is widely employed in complicate gradient based learning for the sake of robust convergence. We are especially interested in knowing this algorithm s accuracy and convergence order because it actually changes the original Halley iteration scheme. It is worthwhile to point out that the new Halley method is the hybrid lied volatility pricing method we specially propose in this study. It combines Brent methods sureness in convergence and the cubic order convergence of Halley method. The reason we introduce three different hybrid pricing algorithms is to demonstrate the advantage of new Halley method by comparing it with the two hybrid peers besides the Greeks free and Greeks-based methods. Although hybrid approaches were well discussed for Newton method in numerical analysis community (Boyd, 0), it is still unknown how many feed-in steps should be used in the Greeks free method that is chosen as Brent method in this work. However, we solve this issue in an empirical way in this study. We have found that the number of feed-in steps k, i.e. number of steps Brent method should run, can be as small as k= while guaranteeing its convergence in this study. It further suggests setting k < E(t) / 4 for a more generic situation, in which E(t) is the time expectations of the Greeks free method (see the time expectation definition in Result section). RESULTS We introduce our datasets before presenting our results. Data Collection and Preprocessing All option data in this study is crawled from nasdaq.com by an option web gleaning software: OptionWebGlean we developed. This website includes almost all stock and transactions information on Nasdaq stock market. The data collection consists of the following steps. First, all option data is collected for all 6685 tickers on nasdaq.com (until Feb 7, 07). The raw option data crawled from the site contains more than 0,000 option contracts.

8 Second, we clean the raw data by removing all contracts with missing data to obtain a cleaned option dataset that has 959 option contracts. Each option contract includes stock price, option price, strike price, expiration date and lied volatility besides other features. Third, European options are selected from the 959 option contracts by employing Brent method for the sake of Black-Scholes based option pricing. This means we use Brent method under Black-Scholes model to calculate the lied volatility for each option contract and compare the result with the true lied volatility. An option the option contract is believed as a European option, if the following two conditions are satisfied. The first is the calculated lied volatility is in the range of the true lied volatility under a preset ( B) tolerance β that is set as 0.0 in this study. ( B) < β (0) ( B) The second is the ratio of the absolute difference and true lied volatility should be less than a threshold α set as ( B) < α () The reason why we employ Brent lied volatility pricing is because of its sureness in convergence and good speed. Theoretically it can be any Greeks free methods. Finally, we have a dataset consisting of 7 European option contracts (076 calls and 95 puts). Figure illustrates the distribution of lied volatility distributions for total 076 call and 95 put options in the dataset. The corresponding percentage change of option prices are colored for each group. It is interesting to see the distribution of lied volatility is not subject to a normal distribution. Instead it skews to the left but with a long right tail. That is, most of lied volatility values are centered in the interval (0.0, 0.5) though there exists values larger than (e.g..). Fig. The distributions of lied volatility values for call and put options on our dataset that consists of 076 call and

9 95 put options. The corresponding percentage change of option prices are colored for each group. Most lied volatility values range from 0. to 0.5. Data-driven Evaluations It is desirable to take a data-driven approach to evaluate different methods performance instead of relying on a few test cases to avoid possible biases. Thus, we evaluate different pricing methods based on all 7 European options collected in this study. On the other hands, all pricing methods are iteration methods and different initial values can even affect a method s performance to some degree. In order to avoid biases from specifically selected few initial values because they may favor some methods, we randomly choose initial points from [0, ] for each pricing method for each option case. That is, there are totally 7 randomly selected initial values for each method. Furthermore, we design the following data-driven evaluation measures to evaluate all methods performance with respect to accuracy and running time. These measures include time expectation, mean error, mean square error (MSE) and efficiency, which aim at evaluating the expected performance of a given method among all option data in our experiment. Time expectation is the average iteration steps spent for a given method to attain its convergence under a tolerance ε that is set as 0 - in our lementation for a set of options, N i.e. E ( t) = t i. For example, if a method has time expectation value 8., it means it takes N i= averagely 8. steps for each option among the total 7 options to converge to its lied volatility value. If the method diverges on an option, then its number of iteration steps can be huge (e.g. the method s maximum iteration number: 0,000). For example, Newton and Halley methods both encounter divergence and their time expectations are and 79.7 respectively. Mean error is the expectation of all absolute errors between the calculated lied volatility and the for all N options me = N N i= () In which, i is the calculated lied volatility for the i th option and is the true lied,i volatility value. The smaller the mean error value, the better accuracy a method. Correspondingly, we can define a similar mean square error (MSE) as mse =, i, i, N i= which has the same functionality as the mean error., i, i N It is worthwhile to point out that the values of mean errors and MSE for a method with divergent cases can be much larger than those convergent methods. For example, Newton method always has an infinite mean error value because it does diverge on a number of options. Efficiency is defined as the following ration for a given a pricing method. e mse ς = () log ( + Ε( t))

10 where Ε (t) is the time expectation of the method. The ς = means the ideal state of pricing, that is, mse is 0 and the pricing method only takes one iteration step averagely to achieve the true lied volatility value. Thus, the method is the most efficient one with 00% efficiency. In other words, the closer the efficiency to, the more efficient the method. On the other hand, it is quite easy to see that a divergent method will always have an infinite efficiency value. Implied Volatility Pricing Method Comparisons We employ total 0 different methods in this study. There are five Greeks free methods, which include Brentq, Brenth, Ridder, Bisection, Muller-bisection. Brentq and Brenth both are Brent lied volatility pricing methods. The former uses the classic inverse quadratic interpolation and the latter employs an alternative hyperbolic interpolation in seeking the final lied volatility for each contract. There are two Greeks-based methods: Newton and Halley and three three hybrid pricing methods: new Newton, new Halley, and Halley momentum methods, in which Brent method is employed to feed the initial values for them. We use existing three months U.S. treasury yield as the risk free rate in Black-Scholes model. We set the convergence tolerance ε =0 for all methods, i.e. will be calculated lied volatility when F ( ) < ε. Greeks-based methods encounter divergence. It is interesting to see that both Newton and Halley methods encounter divergence on our dataset. Exactly speaking, Newton method has diverged on 9 options, and Halley method also has diverged on 60 options. This is mainly because their convergence much relies on the initial values close to the true lied volatility value. Although they have attained decent convergence on most option contracts among the total 7 options, it is hard to say they are robust enough methods because of their large time expectations (465.4 and 79.7), infinite mean errors and efficiency values. It is worthwhile to point out that we did not encounter divergent cases for Greeks-based methods when using an option dataset less than 00 options in our previous study. This suggests the advantage of data-driven evaluations over the traditional model-driven evaluations in which only a limited data are involved in method evaluations. All Greeks free and hybrid methods converge. On the other hand, all the other 8 methods all demonstrate convergence in lied volatility pricing. That is given a random initial value in [0,], each of these methods always achieve their convergence, i.e., the objective function F ( ) < ε = 0. Thus, we only focus on the convergent methods in performance measure comparisons exclude Newton and Halley methods in the comparisons of mean error, time expectation, and efficiency. Figure illustrates that the values of mean error, time expectation and efficiency for eight pricing methods. It demonstrates the advantages of the two hybrid pricing methods over those Greek free methods. For example, new Halley and new Newton attain about 4% and 9% efficiency respectively, but the best Greeks free method (riddle) only achieves.67% efficiency. It is noted that original Greeks-based Newton and Halley methods are excluded for divergence, and the number of feed-in steps is set as k= for all hybrid pricing methods. Especially, the proposed new Halley method has the best performance with respect to efficiency among all the 8 methods. It only takes.9 steps to find the lied volatility value for each option. It is about.7 times faster than the best Greeks free method: ridder,. times faster than new Newton method, and more than 0 times faster than bisection method.

11 Moreover, it is interesting to see that all the seven methods except Halley momentum, achieve the same mean error value. This is because they all converge to a same lied volatility value for each option. It further suggests the correctness of the methods. On the other hand, it suggests that Halley momentum loses some accuracy due to the introduction of momentum term in iteration though it still demonstrates convergence in a slow manner. This result suggests the ortance of a right Greeks-based method in hybrid pricing, because adding a momentum term may change Halley iteration scheme itself. It may demonstrate effectiveness in Gradientbased learning, but not a good choice for lied volatility pricing. Fig. The comparisons of mean error, time expectation (average steps) and efficiency of eight convergent methods. Except Halley momentum method, they all achieve the same mean errors. The proposed new Halley has the obvious advantages over the other methods. How Many Feed-in Steps Needed in Hybrid Pricing? How many feed-in steps needed for Brent method to feed a Greeks-base method in hybrid methods? There are no rigorous results on this question except a suggestion to place the initial point close to the true lied volatility as much as possible.

12 We have found that Brent actually only needs one feed-in step in all hybrid pricing to guarantee convergence for all option data in our study. Instead, more feed-in steps may lower the methods efficiency. Figure compare the time expectation and efficiency of three hybrid pricing methods under five different feed-in steps:,, 5, 8, and 0 respectively. It is interesting to see that all three methods decrease their time expectation and efficiency values with the increase of feed-in steps. For example, the time expectation of new Newton, new Halley and Halley momentum are.99, 4.96 and.79 respectively when the number of feed-in steps k=. However, they become 6.4, 6.8 and 6.4 respectively when k=. This tendency continues with the increase of k values. In fact, their efficiency values also share the same tendency. It is also interesting to see that new Newton and new Halley will show a close performance because more feed-in steps from Brent methods dominate the pricing process. Finally, two methods would have a same level performance though Halley itself has a higher convergence order than Newton. Fig The comparisons of time expectation and efficiency of three Hybrid pricing methods under five different feed-in steps (Brent steps):,, 5, 8, and 0. It is clear that all three methods decrease their time expectation and efficiency values with the increase of feed-in steps. Figure 4 illustrates real iteration scenarios of new Halley for five randomly picked option contracts, in which Brent method provides,,5, and 8 feed-in steps in pricing. It is easy to see that Halley method spends at most steps to converge, i.e. the objective function value < 0 -, when the number of feed-in steps is or. However, Halley method may not need to get involved to pricing for some options when the number of feed-in steps increases to 5 or 8. It suggests only very few feed-in steps are need in hybrid pricing. In fact, we have found that new Newton and new Hybrid methods can still keep its advantage over those Greeks free methods

13 when the number of feed-in steps is set as k < E(t) / 4, where E (t) is the time expectation of the Greek free method used in hybrid methods. Fig 4. The comparison of different feed-in steps for a set of randomly selected 5 options in new Halley pricing. When the number of feed-in steps increases, Halley method may only need one or two steps or not involved in pricing.

14 DISCUSSION AND CONCLUSIONS We propose a novel hybrid lied volatility pricing method: new Halley and evaluate its performance along with its nine peer methods through a data-driven approach. The proposed method achieves its superiority to other methods in terms of convergence order and accuracy. It meets the needs of lied volatility pricing in high frequency trading for its exceptional speed and sureness in convergence. Interestingly, it seems that Greeks-based lied volatility pricing methods are not good choices because their divergence risk. Although Greek free methods have sureness in convergence, they suffer from relatively slow convergence than the proposed new Halley method. In fact, it seems that hybrid lied volatility pricing seems to have advantages than Greeks free and Greek-based methods. However, there is still some room for us to further optimize this method. For example, Brent method can be replaced by other more efficient Greeks free method. This is because Brent method has the second largest time expectation values among all Greeks free methods in our study. Moreover, it is possible to replace Halley method by a fourth order convergence method, () in which Greeks Ultima F ( ) = C / is the corresponding derivate term in iteration (Haug 007). Another contribution of this work is our data driven lied volatility pricing method evaluation. It overcomes the weakness of the traditional model based evaluation by listening to data s voice in evaluating an algorithm s performance by using a large amount of real data. For example, Ridder method is found the best Greeks free method in our study. However, its convergence order is only.44, which is much lower than those of Brent and Mull-bisection methods. This is because algorithm convergence order is calculated by making an assumption there are a large number of iterations in finding the solution, i.e. n. But it does not match the real lied volatility pricing situations because most algorithms only need quite limited number of iterations or few steps to find the solution. The traditional convergence order can t model the situation very well or at least accurately. However, our data-driven evaluation approach can handle it very elegantly. To some degree, it is an approach that is more suitable for lied volatility pricing evaluations for the modern trading market. It is noted that our proposed hybrid method can be generalized to the extensions of Black- Scholes model to handle more complicate cases in lied volatility pricing. For example, an easy and natural extension is to handle European options with dividends by sly using Black- Scholes-Merton model (Haug 007, Merton 97). In our future work, we plan to investigate the extension of our proposed method in different option pricing models and integrate such lied volatility pricing with big data analytics techniques. REFERENCES Andersen, L. and Andreasen, J. (000) Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Option Pricing, Review of Derivatives Research, 4, 6 Black F, Scholes M. (97) The pricing of options and corporate liabilities. Journal of Political Economy, 8 () :

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