Bakshi, Cao and Chen (1997) not only provide an empirical study for the performance of alternative option pricing models but also demonstrate the detail of the estimation procedure which is the least square type estimation method. They implement each model by adapting this procedure. Hence, the following study for obtaining the liquidity parameter is based on the spirit of the estimation procedure. We present a test of the performance of the option pricing model under illiquidity for individual stock option prices for the sample period from January 1, 2000 through December 31, 2004.
Our empirical study is based on call option contract's close quote price taken to be the bid-ask mid-point price and the rollover effect is conducted. Moreover, we check the integrity of the quotes and remove unreasonable data in our empirical work. First, we compute the Black-Scholes implied volatility (BS-IV) for each exercise price everyday and then the BS-IV with higher than 100% or lower than 0% are excluded in our dataset. Let
C denote the daily close price of options i at day j . In
i j, addition, we denote C as call option prices from the Frey nonlinear PDE model. The liquidity parameter is estimated numerically with two stages by the following equations25. numerical solution from the Frey nonlinear PDE Modelprices obtain the accurate value of three parameters at the same time and it is time-consuming for getting three parameters. Therefore, we only focus on the estimation of the liquidity parameter
ρ
rather than the other two in our empirical study.( )
N numerical solution from the Frey nonlinear PDE Modelprices where the variable
M presents the number of exercise quote price in each day and
j n presents the number of trading day. The above Equation (22) is a nonlinear least square (henceforth, NLS) estimator because the element C is the numerical solution from the Frey nonlinear PDE model. On the other hand, we can also claim that the above Equation (22) is a type of estimation loss function and therefore the parameterρ
* is a minimum (squared) distance estimator.The following illustrates the detail procedure regarding the estimation of the liquidity parameter. First, we use the moving window method to do the whole estimation procedure. Second, we utilize the pattern search algorithm into the whole estimation procedure. Here the Pattern search algorithm can be employed for finding the minimum of objective function and obtaining the parameter
ρ
which correspond to the local minimum objective function. Third, we determine the number of day N in the moving window and then calculate BS-IV for each exercise price per day. After that we compute the arithmetic mean of BSIV after excluding the unreasonable value of the BS-IV. We use the data of the firstNdays to determine the parameter of the1
N+ -th day. i.e., if we select N=2 for the number of the moving windows which means that we use the first two days’ observations to determine the parameter of the third day. Hence, the third parameter represents the first parameter that we estimated in our empirical study and then we use the same approach for getting the rest of parameters. As a result, we obtain a sequence of the parameter
ρ
: ** * * *
1 2
( N , N , , n)
ρ
=ρ
+ρ
+ "ρ
, (25)where nis the length of date in the sample. We use the pattern search algorithm to determine the optimal parameter in the minimization of the square of the difference between the theoretical price and the observed price. We propose the pattern search algorithm for estimating liquidity parameter of the Frey model. The pattern search algorithm is a popular approach in optimization especially for solving bound constrained nonlinear programs, linear unconstrained problem and some kind of minimization problem26.
However, we should give an initial-guess value, the lower bound and the upper bound of the parameter in this algorithm. Since the pattern search algorithm can not obtain the global minimum of the objective function. Hence, we set a vector of initial-guess which can be applied to find every optimal parameter for every different initial-guess value and then we pick up the most appropriate parameter27 to be the first estimator. After we get the first estimator, the rest of estimation procedures are the same except we impose the former estimator
ρ
S* to be the initial-guess value aswe estimate
ρ
S*+1 in the rest of estimation procedure.The empirical study can be divided into three parts. In the first part, we focus on the analysis of the bear market in short term period and then investigate the fitting ability of the Frey model from January 2002 to December 2002. Furthermore, we compare the pricing error ($MAE) with the BS and the Frey model. In the second part, we select a number of companies to be the sample of our empirical study and choice the period of time from 2000 to 2004. We want to figure out the practicability of the Frey model regarding the Top 20 of the average daily volume (ADV) of the underlying in stock option market and also make a comparison of the Top 20 with
26 If the reader have more interest in the pattern search algorithm, you might search related article or take a look at the MATLAB help file where provide the concise programming code and the condensed introduction.
27 We denote that the most appropriate parameter corresponds to the smallest objective function.
other companies which have worst liquidity than the Top 20. In the last part, we not only verify the suitability of the estimation loss function but also check the validation of the Frey model via a variety of loss function.
4.1 In bear market
The underlying asset IBM is arbitrarily selected by us. We observe the trajectory of the IBM stock price and find the underlying asset having the down trend phenomenon in 2002. According to the market microstructure theory, we argue that the bear market often occur the market illiquidity. Thus, we have more interest in the fitting ability of the Frey model especially when illiquid market happened. Figure 4 displays the trajectory of the underlying stock price of the IBM Company from 2000 to 2004 and only 2002 respectively and Table 5 reports the descriptive statistics of the sample from January 2002 to December 2002.
[Insert Figure 4 here]
[Insert Table 5 here]
The at-the-money (ATM) call options are only used in our empirical study because we consider that the OTM and ITM options are not suitable for the analysis of the pricing error. There are several empirical studies showing that the pricing error will generate more bias result from the volatility smile and skew pattern.
[Insert Table 6 here]
In Table 6, we compute the theoretical price of the BS model and the Frey model.
First, we compare the mean of the ATM option pricing error of the IBM Company in
2002 and showing the numerical result with the different N and UB and given the fixed LB. According to the numerical result of the Table 6, the pricing of the Frey model is smaller than the BS model significantly and the result of pricing error is nothing to do with the upper bound of the parameter. Moreover, we find the number of the moving window does not have great impact on the pricing. Thus, the second part of the empirical study set N=128. If the number of the moving window is given, no matter what the setting of UB, the pricing error of two option pricing model is not change almost surely. In Figure 5, we show that the profile of the liquidity parameter estimated from the Equation (23) & (24) by the pattern search algorithm. Obviously, we observe that the liquidity parameter is quite stable29 so that we obtain accurate and reliable result from the pricing error of two option pricing model. Figure 6 shows the graph of the stock option price and the pricing error respectively.
[Insert Figure 5 here]
[Insert Figure 6 here]
4.2 The performance of the Top 20 & other companies
We select a number of companies which is listed on Chicago Board Options Exchange (CBOE). Moreover, we pick up the Top 20 active stock options in CBOE and also select less liquidity companies for our analysis. We want to figure out the fitting ability of the Frey model with respect to differ underlying stock options with different liquidity state. Table 7 reports the symbol and the name of the sample in the Top 20
28 The more N we set, the more computational time we need. However, the length of the moving window impact on the computational cost significantly but it does not affect the result of the pricing error of two option pricing model in our empirical study. As a result, we denote N=1 in the following analysis.
29 The graph of liquidity parameter seems like unsmooth and volatile result from the scale of the vertical axis. In fact, the liquidity estimator is quite smooth as we readjust the range of the vertical axis.
active stock options and the less liquidity stock options. Table 8 provides the descriptive statistics of 27 stock options from Jan 2000 to Dec 2004 and the sample of the option prices are classified by moneyness and can be divided into three categories, respectively. We might notice that CE and XMSR are excluded in our empirical work due to their sample period are less than the length of the period from Jan 2000 to Dec 2004. However, YAHOO is also excluded in our dataset result from the first day of the implied volatility (IV) is not available and it can not be replaced by the former value of the IV. Hence, we ignore three improper samples for the following study.
[Insert Table 7 here]
[Insert Table 8 here]
The Frey model displays unexpected fitting ability and it can track the asset dynamics for every stock option. Moreover, all of the stock options have pass through the $MAE of the pair t-test except the AMR. Although the pricing error of the Frey model 0.3494 is smaller than the BS model’s 0.4447, the outcome of the pair t-test is not significant result from the option price of the AMR have two jump phenomena happened on March 2000 and January 2001 respectively30.
[Insert Table 9 here]
[Insert Figure 7 here]
If we eliminate the suspicious sample data which have jumps, the pricing error of the Frey model still significantly differs from the BS model in the AMR stock option
30 The jump effect is not the consideration of the Frey model since we can not significantly distinguish the Frey from the BS when jump phenomenon happened. Therefore, we get a very reasonable consequent on the underlying asset of the AMR.
during the period from February 2001 to December 2004. Hence, we conclude that the Frey model exhibits a gorgeous practicability for the stock option and it obtains more precise solution than the BS model especially for illiquid market.
4.3 The loss function
Christoffersen and Jacobs (2004) emphasize the consistency in the choice of the loss function is important. If a theoretical model is implemented using an inappropriate estimation loss function, then the more mean squared error (MSE) we get. In the following analysis, we introduce many loss functions for investigating the accurate estimation of the liquidity parameter when evaluating the Frey model. We compare the value of MSE between the Frey model and the BS model since the loss function can be treated as the criteria of the model selection.
There are many loss functions are employed in literature and practice. First, the traditional loss function is composed of the dollar loss function and the percentage loss function and those can be divided into two categories respectively. Thus, mean squared dollar errors ($MSE), mean absolute dollar errors ($MAE), mean squared percentage errors (%MSE) and mean absolute percentage errors (%MAE) can be defined as where Ci and
C are the model call option prices and the observations respectively.
iSecondly, we introduce the implied volatility loss function; that is, the implied volatility MSE and it also can be defined by
( )
2 given as we calculate the implied volatility MSE.[Insert Table 10 here]
Table 10 reports the results of the loss functions of two models with respect to 24 samples. The traditional loss functions show that the Frey model apparently performs better than the BS model and the IVMSE displays that the Frey model somewhat better than the BS model. Since jump phenomena are found by the trajectory of the implied volatility in the sample of AMR and WDC, the Frey model reduces the fitting ability per se. However, the rest of sample still exhibit a fabulous performance in the Frey model with smaller MSE or MAE and therefore we identify the performance of two theoretical option pricing models. Unquestionably, the Frey model shows that it can capture more the pattern of the market than the BS model by tracking the trajectory of the underlying asset. Furthermore, the estimation method of the liquidity is checked and it can obtain an accurate estimator by NLS method with respect to most of sample. Thus, we not only claim that the choice of the loss function is appropriate but also obtain the reliable results in our empirical work.