Document Sample

Bid-Ask Spreads and Asymmetry of Option Prices. Raisa Beygelman∗ This version: November 30, 2005 ∗ Faculty of Economics and Business Administration, Goethe University, Mertonstr. 17, D-60054 Frank- furt am Main, Germany, e-mail: beygelman@ﬁnance.uni-frankfurt.de Bid-Ask Spreads and Asymmetry of Option Prices. Abstract In theory the prices of traded assets are unique and can be observed on the market. In reality, however, due to market frictions only bid and ask quotes are available to investors and researchers. In the existing option pricing literature one usually assumes that the mid quote is a good proxy for the true value of an option. This assumption follows the intuition that ex ante buyer and seller initiated trades are equally likely. In this paper we use a sample of DAX options and show that this assumption is not justiﬁed empirically for option markets. To estimate the true op- e tion price we use the model of Nord´n (2003) and adjust it for the non-synchronicity of the observed option prices. We show that the location of the true option price relatively to the bid-ask quotes changes over time and can be explained by some market variables. This analysis considerably contributes to the understanding of the price ﬁnding process on option markets. 1 Introduction and Motivation In the existing literature diﬀerent option pricing models have been ﬁtted to the data to test and compare their performances. One of the inputs in these studies are the prices of options traded on the market. In the theory of option prices markets are assumed to be frictionless, so that the equilibrium asset prices are unique and can be observed by investors. Unfortunately, in the reality the assumption of perfect markets does not hold. Investors buy options to higher ask prices and sell them to lower bid prices. The question we investigate in this paper is how to detect the option prices which we would observe if the markets were frictionless. In the following we will refer to those prices as the true or the equilibrium option prices. In the existing literature the problem of unobservable true option prices has by now been solved by just using the mid quotes of option prices assuming to apply the best proxy. This assumption follows the intuition that ex ante buy and sell transactions are equally likely, so bid and ask quotes must then be symmetrically located around the equilibrium prices. However, some arguments against this point of view can also be found in the literature. For example, Chan and Chung [10] argue that mid quotes are biased proxies for the true option prices because due to information asymmetry market makers are more averse to buy than to sell orders. Options exhibit a limited liability property meaning that the maximal 1 loss of an option buyer is just the option premium, whereas the loss of a holder of a short positions is unrestricted. So, to ensure themselves against large losses market makers place ask orders above the optimal values inducing that the true prices are located closer to best bid than to best ask price. Another argument for the asymmetric price distribution can be found in Bossaerts and Hilton [7]. They ﬁnd asymmetries on the foreign exchange markets and explain them with occasional government interventions. Bessembinder [5] argues that asymmetric price distributions on the foreign exchange markets can be due to inventory-control variables. Market makers use asymmetric quotations to reduce or raise their inventory positions according to their needs. Finally, order imbalance can also be an explanation. For example Cao, Hansch, and Wang [8] use order imbalance as a proxy for the true value of an asset. Several issues provide motivation for dealing with the identiﬁcation of the equilibrium option prices. First, by using mid quotes instead of the true prices to calibrate option pricing models, it becomes much more diﬃcult to distinguish between them. Dennis and Mayhew [12] show that in presence of usual bid-ask spreads the power of empirical tests of option pricing models reduces considerably. Second, it is possible to reject the true option pricing model even if it is true. This is due to the fact that non-zero pricing errors can arise and model parameters, which by deﬁnition are assumed to be constant, can become time dependent if we use biased market prices in the analysis. Furthermore, Schoutens, Simons, and Tistaert [22] show that even small errors in the estimated parameter values reduce the quality of the pricing of exotic options considerably. Third, measurement errors in option prices aﬀect the estimations of the implied volatility and the calculation of hedge ratios signiﬁcantly, especially for options far away from the money (see Hentschel [15, 16]). Forth, it is important to use the true option prices in models, which analyze the lead/lag relationship between the option and the underlying prices. Chan, Chung, and Johnson [9] show that using the mid quotes in place of the transaction prices reverses the leading relationship between the assets. This indicates that the results in this area are extremely sensitive to the choice of the proxy for the true prices. Finally, in some models, mid quotes are taken to be the clue for the decision if a transaction is buyer or seller initiated (i.e. see Ball and Chordia [4]). Here transaction prices are compared to the most recent bid and ask quotes. If a transaction occurs above the mid point, it is assumed to be buyer 2 initiated, otherwise it is assumed to be seller initiated. If indeed the true price is located closer to the best bid quote and market participants realize that, a transaction can be buyer initiated even if it takes place below the mid quote (but above the true price). This paper contributes to the existing literature in several ways. First, we check the assumption of independent transactions which supports the idea of using mid quotes as proxies for the true prices. By performing an analysis of transaction directions on the DAX option market we show that this assumption cannot hold for option markets. We draw this conclusion after detecting a signiﬁcant autocorrelation in trade directions and dependencies of the trades on some market variables. Thus, we provide an additional explanation for the option price asymmetry. To our knowledge we are the ﬁrst who perform that kind of analysis for option markets. Second, we show the role of the asymmetry for the calibration of option pricing models. Our experiments indicate that making wrong assumptions about the true option price can lead to large biases when dealing with option pricing models with and without model risk. e Third, we use the model presented by Nord´n [20] to detect the true option prices and show that non-synchronicity at the option markets can lead to biased estimation results. Then, we extend the model by adjusting for non-synchronically observed call and put prices and demonstrate that our model performs well. Fourth, this paper is the ﬁrst to investigate the relationship between the option price asymmetry and the maturity of the option. Finally, we detect that the location of the true option price relative to the bid and ask quotes varies over time exhibiting a mean-reverting process and analyze market variables which inﬂuence these dynamics. Our analysis contributes signiﬁcantly to the understand- ing of the price ﬁnding process on option markets. The rest of the paper is organized as follows: in Section 2 we analyze the variables driving the transaction direction on option markets, Section 3 shows the role of option price e asymmetry for the option pricing, Section 4 presents the model of Nord´n [20], whereas in Section 5 we perform the robustness checks and introduce the model extension. Section 6 deals with empirical results and Section 7 concludes. 3 2 Transactions Directions A trade takes place if a market order is placed at the market. If an order is placed at the current best bid quote, then we observe a sell, otherwise it is a buy. In this section we test if buy and sell orders in the market do arrive independently. According to the existing literature the independence of the transaction direction is an argument for using mid quotes as proxies for the true prices. If, in contrast, there are in contrast some factors driving transaction direction the true prices do not necessarily correspond to mid quotes any more. Thus, we try to provide additional motivation for using a method to estimate the location of the true option prices. By now the analysis of the driving factors of the transaction direction has extensively been performed for stock markets. Among others Bessembinder [5] and Porter [21] ﬁnd systematic patterns in the probability of a trade at the ask or at the bid quote demonstrating positive autocorrelation in the direction of the trades. Aitken, Brown, Izan, Kua, and Walter [1] analyze the inﬂuence of trade volume, bid-ask spreads and other variables on the probability of trading at the ask and show evidence for the inﬂuence of the variables on this probability. Whereas we know a lot about the structure of transactions in stock markets, not much analysis has been performed for options markets in this research area. Kalodera and Schlag [18], by investigating the inﬂuence of stock market activity on the liquidity in option markets, ﬁnd out that the number of call or put purchases and sales can be diﬀerent for rising and falling markets. In general call purchases increase with positive and decrease with negative returns, although the reverse cases could also be found in the data. Drawing upon these results we test the following model: tradet = a + b1 tradet−1 + b2 tradet−2 + b3 tradet−3 +b5 rt + b5 ∆midt + εt , (1) where tradet is the trade direction for a given option at time t with tradet = 1 if the transaction is buyer initiated and tradet = 0 if the transaction is seller initiated. The variables tradet−1 to tradet−3 are the lagged trade directions. So, in the ﬁrst place we intend to capture the autocorrelation structure of trades. Additionaly to that, we deﬁne the underlying return since last transaction as rt = ln(St ) − ln(St−1 ), where St is the underlying price at time t, since it has been shown to have an impact on the trades. 4 Furthermore, we control for changes in mid quotes of the option since last transaction by adding the variable ∆midt = (midt − midt−1 )/midt−1 into the regression model. Changes in options mid quotes between two transactions can occur if some orders are canceled during that time period. We calculate this value in relative terms to make options with diﬀerent moneyness and maturity levels comparable. Finally, a is a constant and εt is the error term. Note, that the time spans between the transactions and thus between the observations of the explanatory variables can diﬀer across the observation. We construct our study this way to answer the question, what changes of the market situation induce investors to perform the next transaction just at the time point we observe it? Further- more, if a transaction occurs, is it a buy or a sell (depending on the market conditions)? Thus, the time span between two transactions is not of interest for us, but only the fact, that a transaction occured. We estimate the model (1) by means of a Logit analysis using the transaction data of the DAX options for January 1998. For every option and every transaction our data contains the trade direction, options bid and ask quotes and the underlying price. The advantage of the data is that we do not need to estimate the trade direction as for example has been done by Chung, Li, and McInish [11], thus avoiding additional noise in the results. We perform the analysis separately for call and put options and also divide the sample in at-the-money (ATM), out-of-the-money (OTM) and in-the-money (ITM) options1 . OTM options are the most frequently traded in our sample since these contracts are often used for hedging and speculation purposes. In Tables 1 and 2 the estimation results for call and put options are presented. Our results show that there is a signiﬁcant autocorrelation in trade directions for all option groups meaning that a buy is more likely to occur given that the last transaction was also a buy. The coeﬃcients of the lagged transactions are in most cases signiﬁcant also, but are of much lower magnitude. Furthermore, we ﬁnd a positive (negative) relationship between the trade direction of a call (put) and the underlying return. Thus, since the value of a call (put) rises (declines) in rising underlying markets, it seems that the market is following a positive feedback strategy. On the other hand there is a signiﬁcant inverse relationship between the trade direction and the changes in mid quotes for both call and put options. 1 ITM call (put) options are those with 0.98 <= X/S < 1.02, where X is the strike price of the option. 5 This points to the contrarian feedback strategy, i.e. that investors buy if options become cheaper. The controversial results indicate that there is no clear strategy followed by the investors. They do rather weight up the changes in the underlying values and the corresponding changes in the option prices and buy if options seem to be relatively cheap given these changes. To clarify this point we extract from our sample the data where a rise (fall) in the underlying price since last transaction accompanies with a decline of a call (put) mid quote2 . Then we count the number of buys and sells for diﬀerent option groups. The results show that independently of the group more than 70% of transactions were buyer initiated ones. We repeat this test for the data where a decline (rise) in the underlying value corresponds to the rise of a call (put) value. As expected, the most of the transactions (also about 70%) were seller initiated. To check the robustness of our estimation we exclude both sub-samples from the original data set and estimate the model (1) once more. We also repeat the experiment for smaller time periods. In all of cases the results were similar to those presented in this section. Our analysis indicates that the assumption of randomly occuring buy and sell transac- tions is not true. Thus, we must reject the hypothesis that mid quotes are equal to the equilibrium option prices. In the following, we investigate the location of the true option prices relative to the bid and ask quotes. 3 Model Risk After providing an explanation why mid quotes do not always correspond to the true op- tion prices we ask ourselves, what drawbacks result from ignoring the issue of asymmetric option price distribution. One of the large ﬁelds of option research is the research on option pricing models. The most prominent model for pricing plain vanilla options is the model of Black and Scholes (Black and Scholes [6]) (BS), which assumes that underlying prices follow a geometric Brownian motion. Several models that allow to consider more risk factors like stochastic volatility or stochastic jumps have been proposed in the literature (see e.g. Bakshi, Cao, and Chen [2] (BCC)). Therefore, the question arises, which one is doing best in terms of pricing 2 See Bakshi, Cao, and Chen [3] for more analysis on that topic. 6 or hedging performance. Dennis and Mayhew [12] take a closer look at the problem and show that given the usual levels of bid-ask spreads, it is impossible to distinguish between the Black Scholes model and the Merton’s jump diﬀusion model (Merton [19]) (SJ) in many cases. Hentschel [15] investigates the eﬀect of observation errors in option prices on the estimation of implied volatilities and concludes that even small errors in option prices have considerable impact on the estimation results. In another paper Hentschel [16] analyzes the eﬀects of observation errors on hedging parameters of the Black-Scholes model and draw a similar conclusion. Thus, previous studies emphasize that ﬁnding the exact location of the true option values e would help improving the performance of option pricing models. Furthermore, Nord´n [20] shows that by taking the location parameters into account, the smile is not as pronounced as in situations where mid-prices are used. Ignoring the problem of asymmetric option price distribution and using mid quotes to calibrate option pricing models can lead to the following drawbacks, even if the true model is known: • Estimated parameter values can be biased • Pricing errors can become non-zero • Parameters which should be constant by deﬁnition become time dependent. To demonstrate this we assume that the true value of a European call option at time t Ct can be expressed as follows: Ct = θBc,t + (1 − θ)Ac,t , (2) where Bc,t and Ac,t are the bid and the ask price of the option3 . The parameter θ is the location parameter. It determines where the true option price is located relatively to the bid-ask quotes. If θ is equal to 1, the true value would coincide with the bid price. Similarly, if it is equal to 0, the true value and the ask price would be equal. 3 For simplicity in this experiment we consider call options only. Using put options alone or additionally to the call options would not change our results. 7 We have a cross section of 31 call options with maturity of 30 days and strike prices ranging from 85 to 115. The volatility of the underlying asset is assumed to be 15%, the risk free rate is 5% and the current underlying price is 100. First, we calculate the true values of the assets. To do this we in the ﬁrst run use the model of Black and Scholes and in the second run the model of Merton. Repeating the experiment for two diﬀerent models allows us to provide deeper insights into the problem. The dynamics of the underlying stock under the risk-neutral measure in the BS model are given by the geometric Brownian motion (GBM) as √ dSt = rSt dt + V St dWt , (3) whereas the dynamics in the SJ model are given by √ dSt = [r − λµj ]St dt + V St dWt + Jt dqt . (4) In (3) and (4) r is the risk free rate, V is the variance of the GBM, λ is the frequency of jumps per year, Jt is the percentage jump size, which is lognormally, identically and independently distributed with unconditional mean µj . The standard deviation of ln[1+Jt ] is σj . dWt are the increments of a Wiener process and qt is a Poisson jump counter with intensity λ. For our experiment we chose the parameter values to be similar to those attained by Bakshi, Cao, and Chen [2]: V = 0.02, λ = 0.6, µj = −0.15, σj = 0.09. Second, we calculate the bid and the ask prices for every asset by ﬁrst setting the θ to a certain value and assuming that the diﬀerence between the both quotes divided by the mid quote (the relative spread) for all options is 5% (high liquidity case) or 15% (low liquidity case). Then, the ask price of a call option at time t is deﬁned as Ac,t = Ct + midc,t · sc,t · θ, (5) Bc,t +Ac,t where Ac,t is the ask price, Ct is the true option price, midc,t = 2 is the mid price and sc,t is the relative spread. After rearranging and simplifying we get the following formula to calculate the ask price of a call option at time t: Ct · (1 + 0.5sc,t ) Ac,t = . (6) 1 + sc,t · (0.5 − θ) 8 The bid price is then given by Bc,t + Ac,t Bc,t = Ac,t − · sc,t 2 1 Ac,t (1 − 2 sc,t) = . (7) 1 + 1 sc,t 2 Bid-ask quotes for put options are calculated accordingly. The diﬀerentiation between the two liquidity cases is important since location parameters report the asymmetry of option prices relative to the spread. In contrast to that, the observation error which is important for calibration of option pricing models, is measured by the diﬀerence between the mid quote and the true option value independently of the spread. Thus, for the same value of θ, larger observation error would occur with larger spreads and vice versa. Finally, the two option price models are calibrated for diﬀerent values of the location parameters θ. Thereby in each case we use the same model as the model we used to calculate the true option prices, thus assuming that the true model is known to the investor. Furthermore, mid quotes instead of the true option prices are used here, since we assume that the true option prices are not observable. To ﬁnd the model parameters best ﬁtting the data, we minimize the sum of squared percentage pricing errors of all options deﬁned as 31 [(midn − Pn (S, r, Kn , τn ; Φ))/midn ]2 , (8) n=1 where midn is the mid price of option n, Pn (S, r, Kn , τn ; Φ) is the model option price and Φ is the parameter vector of one of the models we take into account. To solve the optimization problem we use the simulated annealing algorithm4 . It searches asymptotically over the whole parameter space and changes the direction of the searches randomly. This feature is important, because global minima will then be found almost surely. The main drawback of the method is its long computing time. We put up with this disadvantage to reduce estimation risk to the minimum. Table 3 shows the estimated parameters and Table 4 the average pricing errors in percent for the BS and the SJ model for both the liquid (in brackets) and the illiquid (without brackets) markets. 4 See i.e. Goﬀe, Ferrier, and Rogers [14] 9 Of course, if we set θ = 0.5 (meaning that there is no observation error), for both option price models the estimated parameter values are very close to the true ones and the average errors are equal to zero. This supports our claim that the estimation risk is very small if we use the Simulating Annealing algorithm to estimate model parameters. The more θ deviates from 0.5 the larger are errors in estimated model parameters and the larger are the average pricing errors. In the worst case the errors exceed 5% for BS model and 2% for SJ model in the illiquid market or 1% and 0.2% in liquid markets respectively. Again, in liquid markets the errors are smaller, because of the smaller spreads. It is noticeable that the estimated parameter values vary substantially with changing values of the θ, although we use the true model to estimate them. Thus, if the location parameters are diﬀerent for options with diﬀerent underlying assets, we would observe diﬀerent parameter estimates, even if both underlyings follow the same dynamics. Our results indicate that ignoring the location parameters would lead to improper in- ference about the true option pricing model or to a rejection of the true option pricing model. Using wrong parameter values would also lead to hedging errors as for example shown by Hentschel [16]. Schoutens, Simons, and Tistaert [22] exemplify that even small biases in estimated model parameters can lead to large errors if they are used to price exotic options. Apart from the biases discussed above, ignoring the risk of asymmetric price distribution can result in a problem of diﬀerentiating between two option pricing models. To clarify this point we repeat the experiment described above by employing a model similar to the BCC model5 . In this model the instantaneous variance of the underlying is stochastic. The dynamics of the underlying asset and of the instantaneous variance are given by dSt = [r − λµj ]St dt + Vt St dWts + Jt dqt dVt = [θv − κv Vt ]dt + σv Vt dWtv , (9) where κv , θv /κv , σv denote the speed of adjustment, the long-run mean and the variance of Vt respectively. dWts and dWtv are correlated increments of Wiener processes with 5 The only diﬀerence between the model used here and the BCC model is that we assume the interest rate to be constant. 10 correlation coeﬃcient ρ. The parameter values are set to V = 0.02, λ = 0.2, µj = −0.15, σj = 0.09, κv = 1.1, θv = 0.04, σv = 0.4, ρ = −0.6. As above, for every call option bid and ask prices are calculated according to the chosen location parameter θ and the relative spread, which is set to 15% here. Now we assume that the true option pricing model is not known to the investor and ﬁt the SJ model and the model of Heston (see Heston [17]) (SV) to the mid quotes of the options. In the SV model the risk neutral dynamics are deﬁned as dSt = rSt dt + Vt St dWts dVt = [θv − κv Vt ]dt + σv Vt dWtv . (10) We have chosen these two models, because they show similar pricing performance (see Bakshi, Cao, and Chen [2]). Thus, it is much more diﬃcult to distinguish between them than for example between the SV and the BS model, especially if the observed option prices are noisy. We ﬁt the models by minimizing the sum of squared percentage pricing errors as deﬁned in (8). The results are shown in Table 5. In the ﬁrst column the values of θ are shown. The second and third columns represent the average percentage errors for the SJ and the SV model respectively. In the last column the diﬀerences between the pricing errors of both models are calculated. In cases, in which θ ≥ 0.5 the SJ model performs slightly better providing smaller average pricing errors. In these situations we would prefer the SJ model to price derivatives. But, if θ < 0.5, this relationship reverses. Now, the pricing errors are smaller, if we use the SV model to ﬁt the data. Thus, in case of option prices asymmetry it can be impossible to distinguish between some option pricing models. Of course, our results also depend on the parameter values of the true BCC model, but the idea remains the same. e In the next section we present the model of Nord´n [20] which can be used to calculate the location parameters. 11 4 e The Model of Nord´n The model we use in our paper was ﬁrst presented by Chan and Chung [10] and was e then extended by Nord´n [20]. As mentioned above we assume that the true value of a European call option C at time t can be expressed as Ct = θBc,t + (1 − θ)Ac,t . (11) Taking ﬁrst diﬀerences leads to ∆Ct = θ∆Bc,t + (1 − θ)∆Ac,t , (12) where ∆Ct , ∆Bc,t and ∆Ac,t denote changes in the true value, the bid and the ask price, respectively. After deﬁning ∆SP Dc,t = ∆Ac,t − ∆Bc,t as the change in the bid-ask spread and rearranging equation (12) we get the formula for changes in ask price as being ∆Ac,t = θ∆SP Dc,t + ∆Ct . (13) Analogous to equation (13), the formula for a European put option is ∆Ap,t = γ∆SP Dp,t + ∆Pt , (14) where ∆Pt denotes the change in the true put option values and γ is the location parameter of the put option6 . In equations (13) and (14) the changes in the true option values ∆Ct and ∆Pt are the only e parameters, which cannot be observed on the market. Nord´n [20] solves this problem by using the put-call parity. It says that the following relationship must hold: Ct − Pt = St − Ke−r·T , (15) where St denotes the value of the underlying, r is the risk free rate and T is the time to maturity of the options. Again, expressing the relationship in diﬀerence notation leads to the following relationship: ∆Ct − ∆Pt = ∆St − ∆(Ke−r·T ). (16) 6 Since the locations of the call and put options need not be the same, we use diﬀerent variable names here. 12 After combining the equations (13), (14) and (16) we get the relationship between the changes in ask prices of the call and the put values and other parameters: ∆Ac,t − ∆Ap,t = θ∆SP Dc,t − γ∆SP Dp,t + ∆St − ∆(Ke−r·T ) (17) To estimate θ and γ we can now perform a regression according to equation (17) of the following form: ∆Ac,t − ∆Ap,t = β0 + β1 ∆SP Dc,t − β2 ∆SP Dp,t + β3 ∆St + εt , (18) where εt is the error term. Since ∆(Ke−r·T ) is a constant, it is assumed to be represented through the constant term β0 . Estimation results for regression coeﬃcients β1 and β2 correspond to the estimates of θ and γ respectively. If the assumption that the true option values correspond to mid prices holds, estimates should result in 0.5 for both of them. There are several points concerning the methodology above to mention. First, all input parameters needed to estimate the values of location parameters are observable on the market. Thus, we do not have to choose any proxy. Second, we do not need to know the true dynamics of the underlying and of state variables, i.e. we do not have to know the true option pricing model and thus we are not aﬀected by model risk. Note that the above model is valid for European options only, since we used the corre- sponding put-call parity to derive it. If we wish to estimate the location parameters of American style options, we have to modify our result by adding an early exercise premium as the additional independent variable into the estimation regression (18). Since the early exercise premium is not observable and has to be estimated ﬁrst, which is not an easy exercise, one could drop this variable, by assuming it to be included in the error term ε e as it has been done by Nord´n [20]. e In the following section we test the model of Nord´n [20], show, what problems can arise by implementing it and suggest some extensions to improve its performance. 5 Implementation Issues By implementing the method described above one can face some problems due to the available data. For example, by now we have assumed that the true value of the underlying 13 asset is known. In the reality, this is in general not the case. Like in option markets, in stock markets we cannot observe the true market value of an asset, but instead we observe bid and ask quotes. Although stocks exhibit much smaller spreads than options, the estimated option’s location parameters could be biased, if we just use the mid prices of the underlying asset for the regression. The bias might be more pronounced if the true value of the asset is instead located very close to the bid or to the ask price. Another problem one could face is the problem of non-synchronicity of the observed option prices. For the estimations described above one observation vector consists of the ask price of a call option, the ask price of the corresponding put option, the spreads of these contracts and the underlying price. In illiquid markets however, options are traded very infrequently, so that often call and put prices cannot be observed at the same point in time. The problem of non-synchronicity called attention of many researches. For example, in Dennis [13] the author uses options observed within a 5 minute interval only. Chan and Chung [10] restricts his data to call and put prices with less then 30 minutes of time diﬀerence between the observations. In this section we construct a simulation study to investigate whether observation errors in the underlying price and the non-synchronicity of option prices can lead to biases in estimates of the location parameters. 5.1 Observation Error in the Underlying Prices Besides the true values of options, the true values of the underlying assets are in general not e observable as well. Instead only bid and ask prices can be observed on the market. Nord´n [20] assumes that the true value of the underlying assets corresponds to the mid point of the bid-ask spread. To justify this assumption, he analyzes bid and ask quote changes of 27 stocks, which were listed on the Stockholm stock exchange in 1995. Diﬀerences in the sum of bid and ask changes can be one possible explanation for asymmetric location e of the true prices within the spread (see Chan and Chung [10]). Nord´n [20] ﬁnds that in contrast to options, where variations in the bid are usually lower than those in the ask, the changes for underlying assets were approximately equal. He uses this result as an explanation for taking mid prices of the underlying in his study. 14 Despite this ﬁnding, we check if the incorrect assumption about the true value of the underlying asset does aﬀect the estimation results in (18) by constructing the following simulation study. Analogous to the deﬁnitions for the options we deﬁne the true value of the underlying asset to be located between the respective bid and ask price at time t as follows: St = φBs,t + (1 − φ)As,t, (19) where φ is the location parameter, Bs,t is the bid price and As,t ist the ask price of the asset at time t. Furthermore, we consider a call and a put option each with strike price equal to 100 and time to expiration equal to 30 days. We assume that the current underlying stock price is 100, the risk free interest rate is 5% and the volatility of the underlying process is 15%. First, we simulate a path of the underlying price for the time period of one month with 10 minutes intervals between every observation using the geometric Brownian motion. Second, given the underlying price at every point in time we calculate the BS price of the call and the put option. Third, we determine the bid and ask prices for every observed option price as follows. To be more ﬂexible, we allow the relative spreads at each time point to change within deﬁned bounds. Thus, for each asset we deﬁne a relative spread lower bound (SLB) and a relative spread upper bound (SUB), and then determine the relative spread for each time point by drawing a number from the uniform distribution between SLB and SUB. In this analysis we chose the following values: SLBs = 0.01, SUBs = 0.02, SLBc = SLBp = 0.02, SUBc = SUBp = 0.08 (the subscripts p, c and s distinguish the respective quantities for puts, calls and the underlying stock). Since the results of this study are sensitive to the set up, it is important to chose the parameter values carefully. Thus, we analyze the dynamics of the relative spread of the at-the-money DAX options and detected similar dynamics as assumed here. The spread bound for the underlying asset is chosen to be smaller, since underlying markets are in general more liquid than option markets. In our example we set θ and γ values to 0.2 and 0.7 respectively. Armed with dynamics of the relative spread and the values of the location parameters we are able to calculate the bid and ask quotes for the call and the put option at every time point as shown in (6) and (7). 15 Finally, we perform the regression (18) for diﬀerent values of φ and use the mid quotes of the underlying stock instead of the true value St . For every calibration we compare the θ and the γ estimates with the true ones. By doing so, we can show how large the estimation biases in θ and γ are depending on the observation error in the underlying asset. All other parameters are left unchanged for every regression. Table 6 shows the results. As expected, the best result is achieved for the case where the true value of the stock corresponds to the mid price (φ = 0.5). Here, the estimations of θ and γ are closest to the true values of 0.2 for θ and 0.7 for γ apart from a small estimation error. Also, the adjusted R2 is close to one as one would expect. The further away from the mid point the true value of the underlying is located, the more biased are the estimates for regression parameters. In the worst cases (where φ = 0 and φ = 1) we get an estimate for θ of around 0.15 and for γ of around 0.68. The values of the adjusted R2 go down to approximately 0.3 indicating only a moderate ﬁt of the model. On the one hand, these errors are not that pronounced. On the other hand, however, as has been shown in previous sections, the deviations from the true values of the location parameters of that magnitude can in some cases lead to biases in estimated parameters of option pricing models and to non-zero pricing errors of the models. We conclude that in some cases using the right underlying value for estimation of location e parameters becomes an important issue. Both, Chan and Chung [10] and Nord´n [20] use mid prices for the underlying price, so their results are probably biased. To avoid the problem one can for example use approximative values for the true underlying asset prices as supposed in Cao, Hansch, and Wang [8]. Another possibility is to use the model presented in Bessembinder [5]. Here, the location parameter of the underlying asset can be estimated by using the following simple regression model: ∆Bs,t = α0 + α1 ∆SP Ds,t + ηt , where ∆Bs,t are the changes in the bid price of the underlying, ∆SP Ds,t are the changes in the spreads and ηt is the error term. Estimation results for α1 correspond to the estimated value of φ. To detect the performance of the suggested correction methods more tests are needed. We leave this issue for the further research. 16 5.2 Non-Synchronicity of Option Prices To implement the model described above one needs observation vectors, each containing an ask price of a call option, an ask price of the corresponding put option and an underlying price observed at the same time. However, in illiquid markets options are traded at a lower frequency. A quote is assumed to be valid as long as no new quote arrives. If the price of the underlying however changes in the meanwhile, the true value of an option must also change, because otherwise the put-call parity (which were used to derive the model) would be violated. To account for this liquidity eﬀect, we adjust the put-call parity, derive an adjusted regression model and test whether we are able to eliminate the problem of non-synchronicity. Consider four points in time: t1 , t1 , t2 , and t2 with t1 < t1 < t2 < t2 . Assume that in t1 c p c p c p c p c and in t2 we observe the bid and the ask price of the call option of interest Bc,t1 , Ac,t1 , c c c Bc,t2 and Ac,t2 but no quotes of the corresponding put option. In t1 and t2 we observe c c p p the put values Bp,t1 , Ap,t1 , Bp,t2 and Ap,t2 , but no call quotes are available. We assume p p p p that the underlying asset is traded frequently enough to ﬁnd the true underlying asset price at any point in time. Since stock markets are in general more liquid than option markets, this is a realistic assumption. We now have to build regression vectors at time t1 and at time t2 out of the quotes described above. Since the call and the corresponding c c put prices are observable at diﬀerent points in time, we must assign to the observed call price a ”wrong” put price. To correct for that problem we suppose an adjusted put-call parity. At time t1 it can be written as c Ct1 − (Pt1 + ω(t1 )) = St1 − Ke−r·T , c p c c (20) Pt1 c where St1 is the underlying price observed at time t1 , and ω(t1 ) is the error corresponding c c c to the problem of non-synchronicity. The term Pt1 + ω(t1 ) is the unobserved true price p c Pt1 at time t1 . Similarly, the put-call parity at time t2 can be deﬁned as7 c c c Ct2 − (Pt2 + ω(t2 )) = St2 − Ke−r·T . c p c c (21) Pt2 c 7 Note that the term T in (20) is smaller than the term T in (21) by (t2 − t1 ). Since this diﬀerence is c c negligible small relatively to the magnitude of the values T , we ignore this fact. 17 The error terms ω(t1 ) and ω(t2 ) cannot be observed, but they can be approximated by c c using the Taylor expansion as dPt1 dPt2 ω(t1 ) = c p (St1 − St1 ), ω(t2 ) = c p (St2 − St2 ). (22) dSt1 p c p dSt2 p c p For simplicity, we denote the terms (St1 − St1 ) and (St2 − St2 ) by Std1 and Std2 respectively c p c p in the following. Then, we proceed as in Section 3 by additionally inserting equations (22) in (20) and (21) to get the following regression relation at time t2 : c dPt2 dPt1 ∆Ac,t2 − ∆Ap,t2 = θ∆SP Dp,t2 − γ∆SP Dc,t2 + ∆St2 + p Std2 − p Std1 −∆(Ke−r·T ).(23) c c c c c dSt2 p dSt1 p E Compared to the equation (17), we have the additional term E. It represents the adjust- dPt2 dPt1 ment for option prices observed non-synchronically. The terms dSt2 p and dSt1 p represent the p p deltas of the put option at times t2 and t1 respectively. If we assume that the moneyness p p dPt2 dPt1 of the option does not change too much over time, we can set dSt2 p = p dSt1 . Then, the term p p dPt2 E can be simpliﬁed to p dSt2 (Std2 − Std1 ), leading to the ”new” regression equation (24): p ns ns ns ns ns d ∆Ac,t − ∆Ap,t = β0 + β1 ∆SP Dp,t − β2 ∆SP Dc,t + β3 ∆St + β4 (Std − St−1 ) + εt , (24) ns where β4 represents the estimated delta of the put options as described above. The subscript ”ns” emphasizes that the parameters were estimated after the adjustment for the non-synchronicity. To check whether our approximation method is useful, we conduct an experiment similar to the one described in Section 5.1. As above, we simulate a path of the underlying prices using the GBB with equal time intervals between the observations. However, in contrast to the previous study, we calculate call and put prices (using the BS model) at every second point in time. Additionally, we alternate call and put price calculations, so that at one point in time either a call or a put price is available (but never both of them at the same time point). The larger the time diﬀerence between the time points, the less liquid the market, and the larger is the problem of the non-synchronicity. Again, our underlying price process starts at 100, the volatility of the price process is 15%, the risk free rate is 5% and we consider a call option with strike price 100 and an initial maturity of 30 days. In contrast to the last experiment we do not deﬁne the length of the 18 whole observation period, since this would lead to less data points for larger time spans between the observations and could inﬂuence the results. We instead generate the data for 1000 time points. If an option expires before the simulation is over, we introduce a new option with 30 days to expiration. To avoid additional biases, we set the bid-ask spread of the underlying equal to 0. The bounds of the relative bid-ask spreads of the call and put options are set to the following values: SLBc = SLBp = 0.02, SUBc = SUBp = 0.08. The true values of the location parameters are left unchanged at 0.2 for the θ and 0.7 for the γ. As above, we are able to calculate the bid and the ask quotes using the true option prices, the relative spreads and the values of θ and γ. We perform the simulation for diﬀerent time spans between the observations and apply the regressions (18) and (24) for each of the data sets. Finally, we compare the estimation results of the two models. Table 7 shows the results of the study. The ﬁrst column represents the diﬀerences between the corresponding call and put observations in minutes. The next three columns show the e results obtained by using the model proposed in Nord´n [20], i.e. the regression equation (18). In the last three columns the results of the adjusted regression equation (24) are presented. If we compare the results for θ and for θns , we do not ﬁnd much diﬀerence, although in general the results of the adjusted model are slightly better than those of the unadjusted one. The reason for such a good performance is that by construction always the ”right” call prices are used for estimation. In contrast to that, the put prices are observed with delay, and so we can observe large diﬀerences between the results for the location parameters of the put γ and γ ns . In our scenario 10 minutes of time span between call and put values are suﬃcient to obtain a diﬀerence between the estimated value and the true value of more than 0.3, whereas the adjusted model still performs very well. For the largest time diﬀerence of 60 minutes, the unadjusted model provides us with extremely wrong values. In contrast to that the results of the adjusted model are still reliable. We have checked the results using the SV model instead of the BS model and the results were similar to those presented here. One possible problem of using the approximation in (22) is the changing moneyness of the options, since moneyness is one of the factors, which aﬀects the delta of an option. Putting options with diﬀerent delta values in one regression leads to biases. This problem can be avoided by using only options with similar moneyness 19 in one regression. As will be shown later, the subdivision of the data sample into diﬀerent moneyness categories is one important part of dealing with location parameters anyway, since the location parameters take diﬀerent values for diﬀerent moneyness levels. Thus, we do not have to make additional assumptions to eliminate the problem of diﬀerent deltas. At this point we can conclude that the problem of non-synchronicity of the observations is an important issue by dealing with location parameters. In very illiquid markets one should use the adjusted model to estimate the values of the location parameters, because otherwise in some cases wrong results can be the consequence. Chan and Chung [10] impose the restriction that the call and the corresponding put values have to be observed within a thirty-minute interval. According to our example allowing for this time span could lead to biased results. However, to be able to appraise the magnitude of the error, an extensive analysis of the data structure is needed. In particular one has to relate the number of observations with large time spans to the whole number of the observations. As shown in this subsection one should use the extended model introduced in this section to be sure that errors are minimized. 6 Empirical Results 6.1 Data To investigate the location parameters empirically we collect a time series of best bid and best ask quotes of call and put options on the German blue chip index (DAX) traded in 1998 on the electronic trading platform. The index cobrains the 30 biggest and most actively traded German shares. The quote data for the underlying is obtained from Xetra. The choice of our data set has some advantages compared to other alternatives. First, the DAX is a performance index, where dividends are reinvested. Thus, we do not have to adjust our model for dividend payments. Second, options written on the DAX are all of European style, so we avoid any biases due to an early exercise premium. Third, we avoid the problem of observation errors in the underlying. Finally, these options are very liquid making the liquidity bias relatively small. This also ensures that we have enough data to e make a detailed analysis using options on one underlying only. Nord´n [20], for example, 20 uses options on many underlying assets which can lead to biases. We use one, two, three and sixth month Fibor rates as a proxy for the risk free interest rate. We applied several ﬁlters to the data set. First, we eliminate observations which do not satisfy the put-call parity. Second, observations with a relative spread larger then 50% are deleted from the sample. Finally, options with less than 7 days prior to maturity are eliminated. Observation vectors are constructed by ﬁrst searching for put-call pairs with the smallest time diﬀerence between the corresponding observations. Then, pairs with a time diﬀerence of more than 10 minutes were deleted from the sample (according to the results in Section 3.2., this should provide us with reliable results, if we use the adjusted regression model (24)). 6.2 Location Parameters and Time to Maturity Some previous studies show that the values of the location parameters vary with the moneyness of the options8 . In this subsection, we test if θ and γ also depend on option’s time to maturity. First, we deﬁne four maturity groups: T <= 2 weeks (Group I), 2 weeks <= T <= one month (Group II), one month <= T <= 2 months (Group III), and T >= 2 months (Group IV), where T denotes the maturity of the option. Then we perform the regression model (24) by additionally introducing a dummy variables Di for every of the maturity groups: 4 4 4 ns ns ns ∆Ac,t − ∆Ap,t = Di β0,i + Di β1,i ∆SP Dp,t − Di β2,i ∆SP Dc,t i=1 i=1 i=1 4 4 ns + Di β3,i ∆St + ns d Di β4,i (Std − St−1 ) + εt , (25) i=1 i=1 where D1 is equal to one if maturity of the option belongs to the Group I and zero otherwise, D2 is equal to one if the maturity belongs to the Group II and zero otherwise e and so on. As already pointed out by Nord´n [20] the introduction of dummy variables allows to test the null hypothesis that the values of location parameters for puts and calls are equal. Furthermore, we test whether the location parameters are equal for diﬀerent 8 e see i.e. Nord´n [20] 21 maturity groups. To answer this question we additionally divide our data sample into three moneyness9 categories: 0.9 <= X/St < 0.98, 0.98 <= X/St < 1.02, 1.02 <= X/St < 1.1, and perform the estimation (25) separately for every moneyness category. Figure 1 presents the location parameter as a function of the time to maturity for diﬀerent levels of moneyness. The x-axis shows the maturity groups in months, whereas the value for the last moneyness category is set to 3. The y-axis represents the results for θ (solid line) and γ (dashed line)10 . Consistently with previous results the location parameters of the OTM DAX calls and puts are in general higher than θ and γ of the ITM options and are larger than 0.5. This implies that the true values of these options are on average closer to the bid quotes, and thus the mid quotes are a biased proxy for the true option prices. In fact, OTM options (especially the OTM puts) are sometimes found to be too expensive. Our results seem to support this ﬁnding. Furthermore, we can observe pronounced patterns of θ and γ depending on the time to maturity for ITM and OTM options. While the prices of the short term maturity calls and puts exhibit a considerable asymmetry, the true values of the longer term options are located around the mid quote which consequently seem to be a good proxy for the true prices in this case. For ATM options no signiﬁcant relationship can be found. We could not reject the hypothesis that the θ parameters are diﬀerent for diﬀerent maturity groups at 5% signiﬁcance level. 6.3 Time Series of Location Parameters In the previous subsection we assumed the location parameters to be constant in time. However this assumption is only an approximation to the reality, since empirically we can observe a signiﬁcant variation of the location parameters over time. To clarify this issue we re-estimate the location parameters at intervals of two hours. The results for ATM call options with time to maturity between 2 weeks and 2 months are represented in Figure 2. Obviously, the dynamics of θ oscillate signiﬁcantly. However, the time series is stationary 9 Moneyness at time t is deﬁned as X/St . 10 All of the parameter values are highly signiﬁcant with relatively small standard errors indicating stable results. 22 and exhibits a strong mean reversion. Interestingly, we rarely observe that the location parameter leaves the 0–1–bound. This implies that the true value does not lie within the bid-ask spread. However, those periods are only short-lived and the market participants seem to adjust their quotes immediately. Next, we analyze the relative error that may result if mid quotes are used instead of true midt −Pt prices. We deﬁne the relative observation error of an asset at time t as ψt = midt , where Pt is the true price of the asset at time t. Using our data sample we estimate ψ. Figure 3 represents the median of the estimated ψ for call options as a function of time of day11 . One can observe that the relative observation error is more pronounced immediately after the opening of the exchange and before the closing of the market. The accumulation of the information overnight and the absence of an opportunity to trade may imply a higher price uncertainty in the morning. Since the return distribution overnight is diﬀerent from those over continuous trading intervals, investors may be reluctant to open new positions during the last hours of trading. Thus, liquidity decreases which may result in higher price uncertainty. Given these observations, if option price data is needed at a daily frequency, it seems advisable to collect the time series around noon and not to use opening or closing prices. 6.4 Location Parameters and Market Variables In this subsection we analyze the impact of important market variables on the dynamics of the location parameters. For this purpose we estimate the following regression d d d θ(γ)t = α + β1 θ(γ)t−1 + β2 θ(γ)t−2 + β3 θ(γ)t−3 d d d +β4 rt−1 + β5 vt + β6 T It + εd , h (26) where θ(γ)t is the location parameter of call (put) options calculated at intervals of two hours and θ(γ)t−1 to θ(γ)t−3 are the lagged location parameters. rt−1 is the underlying asset return calculated over the last interval and vt is the volatility of the underlying estimated for the current interval. T It (trade imbalance) is the diﬀerence between the number of buyer and seller initiated trades. The results are represented in Tables 8 and 9 11 The results for put options are similar. 23 for call and put options respectively. The location parameters are positively autocorrelated up to the second lag. This seems to be consistent with the results presented in Figure 2. The return of the underlying has a negative (positive) impact on the location parameters of call (put) options. This seems to be intuitive. Imagine a situation where the price of the underlying is going up. Then, the true value of a call option increases as well. However, if market participants do not adjust the bid-ask spread, the parameter θ decreases. Market participants may be reluctant to adjust the bid-ask spread of the option immediately as the relative tick size of options is usually much higher than those of the underlyings. A symmetric explanation can be derived for put options. If market participants recognize that the change of the underlying price was signiﬁcant they adjust their order submission strategy and thus the bid-ask quotes. Again, imagine a call option. If the underlying price increases signiﬁcantly the true value of the option increases as well and market participants submit more market buy orders. Thus, the trade imbalance increases and the true value returns to the center of the bid-ask spread. This intuition is supported by the sign of the regression coeﬃcient of the trade imbalance. The volatility of the underlying has a positive impact on the location parameter of both call and put options. If volatility shocks occur market participants seem to buy aggressively options by submitting market orders and thus consuming liquidity on the ask side of the order book. As a consequence, the ask price rises and the true value appears to be closer to the bid quote, so that the location parameter increases. 7 Conclusions This paper deals with the question of how to identify the true option prices which cannot be observed due to a number of market frictions. We argue that mid quotes are in general a biased proxy for the true option prices and provide an explanation for that by analyzing a sequence of transactions of call and put options. We detect a signiﬁcant autocorrelation in transaction directions which contradicts the assumption of independently arriving buy and sell orders. An additional motivation for dealing with the topic is provided by showing that consider- able drawbacks can arise if one uses biased option prices when dealing with option pricing 24 models. e To identify the true prices we use the model of Nord´n [20]. However, we show that if call and put prices are observed non-synchronically, it maylead to biased results. To correct for that we extend the model by adjusting it for the case of non-synchronicity using Taylor expansion. We then use the DAX option data to analyze the location parameters for the year 1998. In line with the existing literature the true option prices are on average closer to bid then to ask quotes indicating that mid quotes are indeed biased proxies for the true option prices. By additionally dividing the data sample into time to maturity groups we are able to get new insights into the structure of the location parameters. In the last subsection we show that the location of the true option prices does not remain constant but changes over time exhibiting dynamics similar to a mean-reverting process. The observation error is the highest in the morning and at the evening, thus it seems advisable to collect daily time series of option prices around noon and not to use opening or closing prices. Furthermore, we ﬁnd that the location parameters are autocorrelated and are partly driven by the underlying return and the underlying volatility. Finally, we ﬁnd an evidence that market participants adjust their trading strategies in times of high option price asymmetry. Our results help to understand the price ﬁnding process on option markets. 25 ITM S. Error ATM S. Error OTM S. Error Intercept -0.642 0.116 *** -0.732 0.0527 *** -0.831 0.0472 *** Lag (dt ) 0.650 0.124 *** 1.157 0.0618 *** 1.447 0.0531 *** Lag2 (dt ) 0.387 0.116 *** 0.278 0.0567 *** 0.354 0.0465 *** Lag3 (dt ) 0.433 0.116 *** 0.112 0.0563 ** 0.211 0.0463 *** Underl. return 72.00 12.811 *** 166.00 15.003 *** 207.00 11.273 *** Mid change -4.687 0.579 *** -7.803 0.427 *** -11.736 0.406 *** pseudo R2 0.135 0.169 0.236 Observations 1334 5963 9544 Table 1: Estimated parameters of the model tradet = a + b1 tradet−1 + b2 tradet−2 + b3 tradet−3 + c rt + d ∆midt + εt for call options, where tradet is the trade direction for a given option at time t with tradet = 1 if the transaction is buyer initiated and tradet = 0 if the transaction is seller initiated. The variables tradet−1 to tradet−3 are the lagged trade directions and rt , ∆midt and εt are the underlying asset return, change of the option mid quote since the last transaction and the error variable at time t, respectively. The table presents the results for ITM, ATM und OTM options. ***, ** and * correspond to 1%, 5% and 10% signiﬁcance level, respectively. 26 ITM S. Error ATM S. Error OTM S. Error Intercept -0.552 0.142 *** -0.985 0.0555 *** -1.0688 0.044 *** Lag (dt ) 1.239 0.153 *** 1.293 0.0689 *** 1.435 0.0538 *** Lag2 (dt ) 0.099 0.143 0.248 0.0611 *** 0.377 0.0478 *** Lag3 (dt ) 0.209 0.142 0.231 0.0607 *** 0.316 0.0475 *** Underl. return -119.70 17.750 *** -195.90 15.501 *** -220.50 11.998 *** Mid change -9.608 1.195 *** -12.219 0.614 *** -11.596 0.433 *** pseudo R2 0.206 0.200 0.236 Observations 996 5250 8946 Table 2: Estimated parameters of the model tradet = a + b1 tradet−1 + b2 tradet−2 + b3 tradet−3 + c rt + d ∆midt + εt for put options, where tradet is the trade direction for a given option at time t with tradet = 1 if the transaction is buyer initiated and tradet = 0 if the transaction is seller initiated. The variables tradet−1 to tradet−3 are the lagged trade directions and rt , ∆midt and εt are the underlying asset return, change of the option mid quote since the last transaction and the error variable at time t, respectively. The table presents the results for ITM, ATM und OTM options. ***, ** and * correspond to 1%, 5% and 10% signiﬁcance level, respectively. 27 θ V λ µj σj BS 0 0.0203 (0.0201) 0.3 0.0201 (0.0201) 0.5 0.0200 (0.0200) 0.7 0.0199 (0.0200) 1 0.0197 (0.0199) SJ 0 0.0197 (0.0203) 0.1116 (0.1453) -0.9914 (-0.5759) 1.9534 (0.3412) 0.3 0.0203 (0.0202) 0.1032 (0.3701) -0.8093 (-0.2324) 0.6310 (0.1284) 0.5 0.0200 (0.0200) 0.6007 (0.6024) -0.1499 (-0.1496) 0.0900 (0.0898) 0.7 0.0185 (0.0197) 3.3333 (1.0032) -0.0412 (-0.0979) 0.0385 (0.0663) 1 0.0072 (0.0188) 99.9964 (2.5250) -0.0050 (-0.0627) 0.0125 (0.0430) Table 3: Estimated parameter values of the BS model and the SJ model in case of a liquid (in brackets) and an illiquid (without brackets) market without model risk for diﬀerent values of the location parameter θ. θ Avg. error in % BS Avg. error in % SJ 0 0.0558 (0.0194) 0.0168 (0.0018) 0.3 0.0223 (0.0078) 0.0022 (0.0008) 0.5 0.0000 (0.0000) 0.0000 (0.0000) 0.7 0.0222 (0.0078) 0.0064 (0.0014) 1 0.0553 (0.0194) 0.0258 (0.0052) Table 4: Estimated average percentage pricing errors of the BS model and the SJ model in case of a liquid (in brackets) and an illiquid (without brackets) market without model risk for diﬀerent values of the location parameter θ. 28 θ Avg. error SJ Avg. error SV Diﬀerence 0 0.0475 0.0442 -0.0033 0.3 0.0236 0.0187 -0.0049 0.5 0.0071 0.0093 0.0022 0.7 0.0119 0.0217 0.0098 1 0.0374 0.0477 0.0104 Table 5: Estimated average percentage pricing errors of the SJ and the SV models for diﬀerent values of the location parameter θ. The true option pricing model is the BCC model. In last column the diﬀerences between the errors are calculated. φ θ (0.2) γ (0.7) Adj. R2 0 0.1553 0.6721 0.3405 0.3 0.1652 0.6662 0.4405 0.5 0.2081 0.7005 0.9946 0.7 0.1572 0.7003 0.4198 1 0.1514 0.6880 0.3290 Table 6: Estimated location parameters of call and put options for diﬀerent location parameters of the underlying asset. In parenthesis the true values of θ and γ are presented. Minutes θ (0.2) γ (0.7) Adj. R2 θns (0.2) γ ns (0.7) Adj. R2 0.1 0.2016 0.7044 0.9675 0.2 0.7 1 1 0.2056 0.7263 0.7604 0.2012 0.7023 0.9986 5 0.2012 0.8034 0.4411 0.1938 0.6912 0.9665 10 0.2129 0.9153 0.3678 0.2060 0.6861 0.9609 30 0.1934 1.3744 0.3138 0.2045 0.6493 0.9466 60 0.2305 2.0098 0.3221 0.2458 0.5253 0.9478 e Table 7: Estimation results of the model of Nord´n [20] and the adjusted regression (24) (columns 5 to 7) for diﬀerent time intervals between the call and the corresponding put prices. In parenthesis the true values of θ and γ are presented. 29 θ Std. Error Intercept 0.27520 *** 0.02547 lag (θ) 0.34416 *** 0.03304 lag2 (θ) 0.17234 *** 0.03460 lag3 (θ) -0.00233 0.03310 Lagged underl. return -4.11384 *** 1.32646 Underl. volatility 63.1121 ** 29.19256 Trade imbalance -0.00249 *** 0.000642 adj. R2 0.2336 Observations 932 Table 8: Estimation results of the regression model (26) for call options. ***, ** and * correspond to 1%, 5% and 10% signiﬁcance level, respectively. γ Std. Error Intercept 0.35705 *** 0.02973 lag (θ) 0.28110 *** 0.03149 lag2 (θ) 0.05955 * 0.03226 lag3 (θ) 0.02959 0.03096 Lagged underl. return 6.11364 *** 1.36277 Underl. volatility 136.37045 *** 46.63772 Trade imbalance -0.00478 *** 0.000726 adj. R2 0.1825 Observations 986 Table 9: Estimation results of the regression model (26) for put options. ***, ** and * correspond to 1%, 5% and 10% signiﬁcance level, respectively. 30 0.9 <= X/S < 0.98 0.98 <= X/S < 1.02 1.02 <= X/S < 1.1 1 1 1 theta / gamma theta / gamma theta / gamma 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 Time to Maturity Time to Maturity Time to Maturity Figure 1: Estimations of the location parameter θ (solid line) and γ (dashed line) for diﬀerent maturity and moneyness levels. The results on the left graph were generated using options with 0.9 <= X/S < 0.98, the middle graph using options with 0.98 <= X/S < 1.02, the right graph using options with 1.02 <= X/S < 1.1. 1.5 1 0.5 theta 0 -0.5 -1 Time Figure 2: Location parameter θ for call options with 1.02 <= X/St < 1.1 and time to maturity between 2 weeks and one month for a half year period. Estimation frequency: 120 minutes. 31 0.015 psi 0.005 8 10 12 14 16 18 -0.005 Time of Day Figure 3: Median observation error for call options as a function of time of day. 32 References [1] Aitken, M., P. Brown, H. Y. Izan, A. Kua, and T. Walter, 1995, An Intraday Analysis of the Probability of Trading on the ASX at the Asking Price, Australian Journal of Management 20, 115–154. [2] Bakshi, G., C. Cao, and Z. Chen, 1997, Empirical Performance of Alternative Option Pricing Models, Journal of Finance 52, 2003–2049. [3] Bakshi, G., C. Cao, and Z. Chen, 2000, Do Call Prices and the Underlying Stock Always Move in the Same Direction?, Review of Financial Studies 13, 549–584. [4] Ball, C. A., and T. Chordia, 2001, True Spreads and Equilibrium Prices, Journal of Finance 56, 1801–1835. [5] Bessembinder, H., 1994, Bid-ask spreads in the Interbank Foreign Exchange Markets, Journal of Financial Economics 35, 1655–1689. [6] Black, F., and M. Scholes, 1973, The Pricing of Options and Corporate Liabilities, Journal of Political Economy 81, 637–654. [7] Bossaerts, P., and P. Hilton, 1991, Market Microstructure Eﬀects of Government Inter- vention in the Foreign Exchange Market, The Review of Financial Studies 4, 513–541. [8] Cao, C., O. Hansch, and X. Wang, 2004, The Informational Content of an Open Limit Order Book, Working Paper. [9] Chan, K., P. Y. Chung, and H. Johnson, 1993, Why Option Prices Lag Stock Prices: A Trading-Based Explanation, Journal of Finance 48, 1957–1967. [10] Chan, K., and Y. P. Chung, 1999, Asymmetric Price Distribution and Bid-Ask Quotes in the Stock Options Market, Working Paper. [11] Chung, K. H., M. Li, and T. H. McInish, 2004, Information-Based Trading, Price Impact of Trades, and Trade Autocorrelation, Working Paper. [12] Dennis, P., and S. Mayhew, 2004, Microstructural Biases in Empirical Tests of Option Pricing Models, Working Paper. 33 [13] Dennis, P. J., 2001, Optimal No-Arbitrage Bounds on S&P500 Index Options and the Volatility Smile, Journal of Futures Markets 21, 1091 – 1196. [14] Goﬀe, W. L., G. D. Ferrier, and J. Rogers, 1994, Global Optimisation of Statistical Functions with Simulated Annealing, Journal of Econometrics 60, 65–99. [15] Hentschel, L., 2003, Errors in Implied Volatility Estimation, Journal of Financial and Quantitative Analysis 38. [16] Hentschel, L., 2004, Option Hedging in the Presence of Measurement Errors, Working Paper, Simon School, University of Rochester. [17] Heston, S. L., 1993, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies 6, 327–343. [18] Kalodera, I., and C. Schlag, 2004, An Empirical Analysis of the Relation Between Option Market Liquidity and Stock Market Activity, Working Paper. [19] Merton, R. C., 1976, Option Pricing When Underlying Stock Returns are Discontin- uous, Journal of Financial Economics 3, 125–144. e [20] Nord´n, L., 2003, Asymmetric Option Price Distribution and Bid-Ask Quotes: Conse- quences for Implied Volatility Smiles, Journal of Multinational Financial Management 13, 423–441. [21] Porter, D. C., 1992, The Probability of a Trade at the Ask: An Examination of Interday and Intradey Behavior, Journal of Financial and Quantitative Analysis 27, 209–227. [22] Schoutens, W., E. Simons, and J. Tistaert, 2004, Model Risk for Exotic and Moment Derivatives, Working Paper. 34

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 2 |

posted: | 2/14/2012 |

language: | |

pages: | 35 |

OTHER DOCS BY xiagong0815

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.