No-arbitrage implied volatility functions: Empirical evidence from KOSPI 200 index options

Journal of Empirical Finance 21 (2013) 36–53

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Journal of Empirical Finance journal homepage: www.elsevier.com/locate/jempfin

No-arbitrage implied volatility functions: Empirical evidence from KOSPI 200 index options Namhyoung Kim a, Jaewook Lee b,⁎ a b

Department of Industrial and Management Engineering, Pohang University of Science and Technology (POSTECH), San 31 Hyoja Pohang 790-784 South Korea Department of Industrial Engineering, Seoul National University, 599 Gwanak-ro, Gwanak-gu, Seoul 151-744, South Korea

a r t i c l e

i n f o

Article history: Received 28 March 2012 Received in revised form 10 December 2012 Accepted 10 December 2012 Available online 20 December 2012 JEL classification: G13

Keywords: Implied volatility surface Local volatility No-arbitrage constraints Index options Local smoothing

a b s t r a c t Implied and local volatility are very important variables to market practitioners because such variables can be exploited in numerous option models for the pricing and hedging of diverse exotic options. In the present study, we propose a method to implement no-arbitrage constraints in estimating the implied and local volatility surfaces extracted from data on option prices. With the aid of multiple local bandwidths, we increase the functional flexibility of estimators and provide a simple method by which to construct no-arbitrage volatility surfaces, such that the ready-incomputation advantage of the derivatives in local quadratic smoothing is preserved. To show the effectiveness of the arbitrage-free models, we perform a comprehensive empirical study on the performance of the competing models using the KOSPI 200 index options from January 2001 through December 2010. Using experiments, we examine the performance of the models based on three measures: in-sample pricing, out-of-sample pricing, and hedging errors. We find that implied and local volatility modeling under arbitrage-free conditions show better performance in terms of estimation, pricing, and hedging near the out-of-the-money with short maturities. From the range depicted in the findings, we often observe clear differences between the models with and without no-arbitrage conditions imposed. © 2012 Elsevier B.V. All rights reserved.

1. Introduction The modeling of implied and local volatility is a very important and fundamental task to market practitioners because such variables can be exploited in models for the pricing and hedging of other complex derivatives or positions. Moreover, forecasts using option implied and local volatility have shown superior predictive power (Blair et al., 2001; Ederington and Guan, 2002; Fengler, 2005; Kim, 2009; Xu and Taylor, 1995) compared with forecasts using the history of daily asset prices, i.e., the ARCH and GARCH models (Engle, 2001; Klaassen, 2002). Unlike volatility measures based on historical data, implied volatility (IV) reflects market expectations about volatility over the remaining lifetime of the option, and is thus capable of being a predictor of future asset price volatility. IV surface (IVS), estimated from cross-sectional option prices across different strikes over a range of times to maturity, also bears valuable information on the asset price process and its dynamics. IVS results in the generation of smile consistent pricing, as indicated in the local volatility theory, which allows us to recover local volatility surface (LVS) from IVS. In modeling the IVS and LVS, one faces three principal challenges: first, the estimators are required to possess sufficient functional flexibility to optimally fit the shape of the IVS. Second, the estimated IVS should meet no-arbitrage conditions; otherwise, it leads to negative local volatilities or negative transition probabilities. Third, the derivatives of the IVS should be provided in the modeling. Derivative information is essential to recover the LVS from the IVS in a manner that is consistent with the data. Versions of the method have been developed and used for the accurate modeling of IV. Dumas et al. (1998) estimated the volatility function using a number of different structural forms. They used global fits, such as quadratic forms of moneyness and of time to ⁎ Corresponding author. Tel.: +82 2 880 7176. E-mail address: [email protected] (J. Lee). 0927-5398/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jempfin.2012.12.007

N. Kim, J. Lee / Journal of Empirical Finance 21 (2013) 36–53

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maturity to avoid overparameterization. Local fits with more flexible nonparametric smoothing methods have recently been employed for functional flexibility (Benko et al., 2007; Fengler, 2005, 2009). Another advantage of using the local smoothing method is that the derivatives of the regression function are obtained as byproducts of the estimators. However, the usual nonparametric methods do not guarantee that the estimated surface is free of arbitrage. To avoid this problem, Fengler (2009), Kahale (2004) and Roper (2010) proposed approaches to perform preprocessing on raw data and estimate arbitrage-free option prices from call price data using interpolation and the cubic spline smoothing method. Subsequently, they derive the IVS from arbitrage-free option prices. Instead of refining option prices, Benko et al. (2007) proposed the local polynomial smoothing method to estimate arbitrage-free IVS directly from IV. Their algorithm has a drawback in that the optimization problem must be solved under nonlinear constraints. The present study addresses the aforementioned challenges by developing an implementation strategy to impose no-arbitrage conditions in the IVS construction directly from IV data using local smoothing with multiple local bandwidths. We then recover arbitrage-free LVS from IVS using the Dupire formula (Dupire, 1996). One global bandwidth (i.e., the same value of bandwidth applied to all volatility surface grids) is typically employed in implementing the local smoothing estimator. The use of one global bandwidth often yields arbitrage violations, negative transition probabilities and negative local volatilities in obtaining closedform solutions for their derivatives. This condition occurs because the problem of choosing proper values for a small number of parameters is subject to a large number of constraints and results in under-fitting problems. Thus, to overcome such a problem, we increase the number of parameters, using different local bandwidth values for different maturities and moneyness, in order to balance bias and variance, as well as to prevent arbitrage opportunities. To investigate the effectiveness of the arbitrage-free models compared with alternative models, we conduct a comprehensive empirical study on the estimation, prediction, and hedging performance of these competing models using the KOSPI 200 index options from January 2001 through December 2010. In estimating the IVS and LVS for pricing, prediction, and hedging, we compare their results using different types of moneyness and time to maturity. The results are compared with alternative IVS, LVS, and stochastic models because IVS produced by the arbitrage-free models differ from those generated by alternative models around these ranges in numerous option markets. The current study primarily contributes to the literature in two ways. First, our proposed method provides a computationally efficient closed-form solution for an arbitrage-free IVS. It preserves the ready-in-computation advantage of its derivatives for the construction of an arbitrage-free LVS that is consistent with the IVS. Second, our empirical results over KOSPI 200 index options from January 2001 through December 2010 reveal that when no-arbitrage constraints are not imposed, local volatility models do not show statistically better performance compared with the ad hoc Black-Scholes (AH) method. The ad hoc method is also reported in the literature, but the arbitrage-free local volatility model significantly improves pricing performance in both estimation and prediction. With respect to hedging performance, the arbitrage-free AH method shows better performance for the OTM options in short maturities. The paper is organized as follows: We review the no-arbitrage constraints on the implied volatility and its corresponding arbitrage-free local volatility in Section 2. Section 3 details our proposed multiple local bandwidth selection method for the nonparametric smoothing of the arbitrage-free implied and local volatilities. In Section 4, we apply the local volatility option valuation approach to the KOSPI 200 index options for the period January 2001 through December 2010 and compare the empirical performance of the arbitrage-free methods with those of alternative methods. Concluding remarks are given in Section 5. 2. Review of no-arbitrage constraints on implied volatility Given observed market prices, Ctmkt, an IV, denoted by Σ, is defined by BS

mkt

C ðSt ; t; K; T; ΣÞ ¼ C t

ð1Þ

where C BS denotes Black–Scholes price of an option, St is the current price of the underlying asset, t is the current time, K is the strike price, and T is maturity date. The IV for a given price of an option contract can be easily calculated by Black–Scholes option pricing model. The volatility smile phenomenon observed in many real market data results in a strike and maturity dependent implied volatility surfaces. Unlike the case of popular parametric stochastic volatility models that always satisfy the no-arbitrage conditions, implied volatility surfaces require some constraints to be arbitrage-free, given European option prices as follows. (See Durrleman, 2003; Fengler, 2009; Reiner, 2000; Roper, 2010 for the proofs.) [Result 1 (Durrleman, 2003; Fengler, 2009; Reiner, 2000; Roper, 2010):] Given a deterministic risk-free interest rate rt and a ∫T ðr −δ Þds be the forward price, and κ = ln(K/FtT) be the log-forwarddeterministic dividend yield of the asset δt, let F Tt ¼ St e t s s moneyness, and τ = T − t be its time to maturity. The total implied variance (TIV) 1 given by υ(κ,τ) = Σ 2(κ,τ)τ should satisfy the following conditions to preclude arbitrage: (i) (Butterfly spread or Durrleman's condition)


2 κ ∂υ 2 1 1 ∂υ 1 ∂2 υ 1− − þ ≥0 þ 2υ ∂κ 16 4υ ∂κ 2 ∂κ 2 1

ð2Þ

One of the main purposes of using this formulation is to make all the relevant variables dimensionless (or scale-free) and make relevant formulae simple.

38

N. Kim, J. Lee / Journal of Empirical Finance 21 (2013) 36–53

(ii) (Calendar spread) υ(κ,τ) is an increasing function in τ, i.e. ∂υ ≥0: ∂τ

ð3Þ

Given a no-arbitrage total implied variance surface (TIVS), the Dupire formula provides a simple way to pricing and hedging exotic options consistent with the observed volatility smiles as follows (Dupire, 1996; Gatheral, 2006): [Result 2 (Dupire, 1996; Gatheral, 2006):] The local volatility surface (LVS) of the smile consistent local volatility model given by dSt ¼ ðr−δÞdt þ σ ðSt ; t ÞdW t St

ð4Þ

should satisfy the following: 2

σ ðκ; τ Þ ¼  κ 1−

∂υ ∂τ

 ∂υ 2 −ð161 2υ ∂κ

:  2 ∂υ ∂ υ þ 4υ1 Þ ∂κ þ 12 ∂κ

ð5Þ

2

2

3. Estimating IVS and LVS To estimate IVS and LVS, we use nonparametric smoothing methods. A widely used and naive smoothing method is a global polynomial fit that estimates the given data in terms of a linear combination of 2-D polynomials of some lower orders (e.g. quadratic) (Dumas et al., 1998). Polynomials are, however, limited by their global nature-tweaking of the coefficients. It can achieve a functional form in one region but can cause the function to flap about madly in remote regions (Hastie et al., 2009). To overcome this problem, we employ piecewise polynomials that are obtained by dividing the domain into contiguous 2-D box intervals, and estimating the surfaces by a separate 2-D polynomial in each sub-domain. As a result, the local fit in each sub-domain does not influence the behavior of local fits in another domain. In particular, we employ a bivariate local quadratic kernel smoothing since it allows for their derivatives just by computing the coefficients of the estimator. The bivariate local quadratic kernel smoothing has been widely used to estimate volatility surfaces (Benko et al., 2007; Fengler, 2005). 3.1. Local quadratic kernel smoothing To extract the LVS from the TIVS using the Dupire formula (5), the derivatives of the TIVS must be known. Using bivariate local quadratic smoothing methods, we can easily compute the derivatives of the TIV function with respect to the log-forward-moneyness measure and time to maturity. Moreover, the bias problem visible in the sparse region is less evident in local quadratic smoothing compared with other local smoothing methods (Fengler, 2005). Thus, we use bivariate local quadratic smoothing to estimate the TIVS. In the present study, the predictor variable, x ¼ ðκ; τ Þ∈R2 , is a log-forward-moneyness measure and the time to maturity, whereas the response variable υ is the TIV. Given the sample data set (κi,τj,υij), i = 1, …, m, and j = 1,.., n, where υij are totally implied variances observed with log-forwardmoneyness κi, i = 1, …, m, and maturity τj, j = 1, …, n (which can be obtained using interpolation if necessary), a bivariate local quadratic kernel smoothing estimator of υ(κ,τ) can be formulated in terms of the quadratic minimization problem employing kernel weights minβ

n X m n    o2   X υji −β0 −β1 ðκ i −κ Þ−β2 τj −τ −β3 ðκ i −κ Þ2 −β4 ðκ i −κ Þ τj −τ K h κ i −κ; τ j −τ ;

ð6Þ

j¼1 i¼1

where β = (β0, …,β4) ⊤ denotes the vector of coefficients. Note that the squared term of τj − τ is not used because its effect is negligible in our simulations (also reported in the literature (Benko et al., 2007; Fengler, 2005)). In the current work, the local weighting is achieved by bivariate kernel functions Kh with h = (h1,h2), as defined by products of univariate kernels: K h ðκ; τ Þ ¼ K h1 ðκ ÞK h2 ðτ Þ: In our model, the bandwidths hs are used to control the degree of localization or smoothing, and
 1 u ; K hs ðuÞ ¼ K hs hs

ð7Þ

ð8Þ

where K(⋅) denotes a univariate kernel function. The univariate kernel functions commonly employed in nonparametric smoothing are quartic kernel, the Epanechnikov kernel, and the Gaussian kernel. In this paper, we employed the Gaussian kernel with infinite support given by: 1 −u2 =2 K ðuÞ ¼ pffiffiffiffiffiffi e : 2π

ð9Þ

N. Kim, J. Lee / Journal of Empirical Finance 21 (2013) 36–53

39

^ ðκ; τÞ. One distinguished Let the local quadratic estimator for the total IV function υ at (κ,τ) solving Eq. (6) be denoted by υ ^ itself and its derivatives can be easily obtained as follows feature of the resulting local quadratic estimator is that the function υ (Benko et al., 2007; Fengler, 2005). ^ ðκ; τ Þ ^ ðκ; τÞ ¼ β υ 0

ð10Þ

^ ∂υ ^ ðκ; τÞ; ¼β 1 ∂κ

ð11Þ

^ ∂2 υ ^ ðκ; τ Þ; ¼ 2β 3 ∂κ 2

^ ∂υ ^ ðκ; τÞ; ¼β 2 ∂τ ^ ∂2 υ ^ ðκ; τÞ: ¼β 4 ∂κ∂τ

ð12Þ

To estimate the arbitrage-free TIVS from given sample data set (κi,τj,υij), i = 1, …, m, and j = 1, …, n, we impose the no-arbitrage constraints on TIVS in Result 2 to our local quadratic estimator of the form (15). In particular, using the formulas (10), (11), and (12), we can rewrite the butterfly spread and calendar spread constraints in Eqs. (2) and (3) equivalently as, for all i, j, ! !    2        κi  1 1  ^ κ ;τ 2 þ β ^ κ ; τ ≥0; β ^ ^ κ ; τ > 0; 1− κ i ; τj β 1 κ i ; τj − κ i ; τj β þ 1 i j 3 i j 2 i j ^ ^ 16 2β 0 4β 0    ^ κ ; τ bβ ^ κ ; τ ′ ; ∀τ bτ ′ : β 0 i j 0 i j j j

ð13Þ

The third constraint is needed to avoid calendar spread arbitrage, because the second constraint only indicates that the time derivative is positive for each sample data point and does not automatically guarantee the third constraint. Considering these no-arbitrage constraints, we can estimate an arbitrage-free TIVS by solving the constrained nonlinear minimization problem (6) subject to Eq. (13) as in Benko et al. (2007). However, to obtain the solution β(κ,τ), we need to run nonlinear optimization solvers for each (κ,τ), which is computationally intensive in proportion to the running time of the optimization solver. Note that the obtained estimator satisfies no-arbitrage conditions at least on the given initial finite set of points (although not on the entire volatility surface). 3.2. Proposed method To obtain a computationally feasible closed-form solution for arbitrage-free TIVS without losing the ready-in-computation advantage of its derivatives, we propose a two-stage algorithm that estimates arbitrage-free TIVS. The basic idea of our proposed method is based on multiple bandwidths. Specifically, bandwidth h affects the bias and variance of the estimator. The estimated regression function values vary with bandwidth h. In particular, estimating an arbitrage-free volatility surface using one global bandwidth often results in large bias estimation because in practice, large bandwidth values are preferred to avoid arbitrage. To overcome the large bias and variance, we propose a multiple local bandwidth selection method, i.e., using different local bandwidth values for different locations. Multiple local bandwidths are used instead of one global bandwidth for two reasons. First, most option data points are located ATM, usually having lower TIVs due to smile effects. Observations are sparse in the wings of the smile, so different values of the bandwidth are needed. Second, most observations that ruin the no-arbitrage constraints are typically above the estimated volatility function, i.e., whenever there exist option prices violating no-arbitrage conditions in a raw option data set, their corresponding implied volatilities typically have greater values than those of the estimated volatility function. Thus, the estimate will be lowered when using the bigger bandwidths as a result of covering the mass of ATM observations. Setting initial multiple local bandwidths with the same sizes, we will change local bandwidths in the locations where arbitrage occurs. The proposed method is described as follows: κ τ ,hi,j), such that H can take different Let H = (h1,1,h2,1, …,hm,1,h1,2, …,hm − 1,n,hm,n) ⊤ be arbitrarily given, where hi,j = (hi,j bandwidths at different locations. For a given choice of H, the unconstrained optimization problem given by n X m     2   X minβ υji −β0 −β1 ðκ i −κ Þ−β2 τ j −τ −β3 ðκ i −κ Þ2 −β4 ðκ i −κ Þ τj −τ g K hi;j κ i −κ; τ j −τ ; ð14Þ j¼1 i¼1

has the following simple analytic solution:   ^ H ðκ; τÞ ¼ X⊤ W X −1 X⊤ W y: β H H

ð15Þ

where 0

κ 1 −κ κ 2 −κ ⋮ 1 κ m −κ

1 B1 X¼B @⋮

τ1 −τ τ1 −τ ⋮ τn −τ

ðκ 1 −κ Þ2 ðκ 2 −κ Þ2 ⋮ ðκ m −κ Þ2

1 ðκ 1 −κ Þðτ1 −τÞ ðκ 2 −κ Þðτ1 −τÞ C C; A ⋮ ðκ m −κ Þðτn −τ Þ

1 υ11 B υ1 C C y¼B @ ⋮2 A 0

υnm

40

N. Kim, J. Lee / Journal of Empirical Finance 21 (2013) 36–53

0

K h1;1 ðκ 1 −κ; τ1 −τ Þ 0 B 0 K h2;1 ðκ 2 −κ; τ1 −τÞ B WH ¼ B @ ⋮ ⋮ 0 0

1 ⋯ 0 C ⋯ 0 C C: A ⋱ ⋮ ⋯ K hm;n ðκ m −κ; τ n −τ Þ

^ H satisfy the no-arbitrage constraints given by Our goal is then to make β ! !    2  H  κi H  1 1 H ^ κ ;τ 2 ^H κ ; τ β κ i ; τj β − κ ; τ þ 1 i j i j i j ^ ^ 16 4β 0 2β 0 1   ^ H κ ; τ ≥0; ∀i; j þβ 3 i j   ^ H κ ; τ ≥0; ∀i; j β 2 i j

ð17Þ

    ^ H κ ; τ ′ ; ∀i; ∀τ bτ ′ : ^ H κ ; τ ≤β β 0 0 i j i j j j

ð18Þ

1−

ð16Þ

0 0 ,h2,1 , …, To find out a feasible solution for β H, we propose the following procedure: We start from a given initial H 0 = (h1,1 0 0 0 0 0 0 ⊤ ,h ) where h = h = … = h = h . Subsequently, for each pair of (κ ,τ ) in the order of (1,1), (2,1), …, (m,1), hm − 1,n m,n 1,1 2,1 m − 1,n m,n i j 0 such that the constraints in Eqs. (16) to (18) are satisfied by iterating (1,2), …, (m − 1, n), (m,n), we readjust the value of hi,j κ κ τ τ τ κ hi,j ← hi,j ± Δh , hi,j ← hi,j ± Δh with some pre-specified Δh τ, Δh κ until the constraints are satisfied. Given that the proposed process is performed sequentially and returns one proper set of parameter values hi,j for each i, j, we finally obtain one H among numerous candidates. With this choice of H, we obtain the final solution of the form (15).

Algorithm 1. Constrained smoothing (1) Create a log-forward-moneyness by time to maturity grid a set of log-forward-moneyness τ = [τmin : Δτ : τmax], n = number of τ a set of maturities κ = [κmin : Δκ : κmax], m = number of κ 0 0 0 0 ⊤ τ κ ,h2,1 , …,hm (2) Choose initial H 0 = (h1,1 − 1,n,hm,n) > 0, Δh , Δh (3) Estimate the first TIV smile for τmin For i = 1 : n 0 H = hi,1 Repeat ^ H ðκ i ; τ 1 Þ ¼ argmin (14) β If (16) to (18) are satisfied, stop. Otherwise, hκ = hκ + Δhκ End (Repeat) (4) Estimate the entire TIVS For j = 2 : n For i = 1 : m 0 H = hi,j Repeat β H(κi,τj) = argmin (14) Calculate c1, c2. κ κ = hi,j + Δhκ. If (16) is not satisfied, hi,j τ τ κ κ + Δhτ, hi,j = hi,j + Δhκ. Else if (17) is not satisfied, hi,j = hi,j τ κ κ Else if (18) is not satisfied, hi,j = hτ + Δhτ, hi,j = hi,j + Δhκ. Otherwise, stop. End(Repeat)

4. Empirical applications In the current section, we compare the empirical performance of our method with alternative methods in three ways: in-sample estimation performance, out-of-sample prediction performance, and hedging performance.

N. Kim, J. Lee / Journal of Empirical Finance 21 (2013) 36–53

41

4.1. Data For the empirical study, we used market data from KOSPI 200 index options. The KOSPI 200 index, the underlying index of the top 200 stocks traded on the Korea Stock Exchange (KRX), is one of the most actively traded indices in the world. The index base date is January 3, 1990, and the base value is 100. The index base has been listed on option markets and traded on the KRX since July 1997. In 2010, the trading volume reached approximately 350 million contracts, accounting for 65% of the global index option market. KOSPI 200 index options are European-style options where the contract months are the nearest three consecutive months plus one nearest month from the quarterly cycle (March, June, September, and December). The last trading day is the second Thursday of the contract month. Each options contract month has at least five strike prices with an interval of 2.5 points. KOSPI 200 index options are characterized by heavy liquidity in short-term contracts. The sample covered the period from January 3, 2001 through December 29, 2010. For the empirical test, we use the last reported transaction price prior to 2:50 P.M. of each option contract. An option of a particular moneyness and maturity is represented only once in the sample. We estimate volatility surface every Wednesday because this day has the lowest possibility of being a holiday and of being affected by day-of-the-week effects. We use the following trading day if Wednesday is a holiday. Thus, 522 total estimations (one estimation for each week) for each model are obtained. The 91-day certificate of deposit yields is used as the risk-free interest rate due to the low liquidity of the Treasury bill market. The log-forward-moneyness of the KOSPI 200 index option is calculated by   T ð19Þ κ ¼ ln K=F t where FtT = (St − PVD)e rτ. PVD refers to the present value of the dividends on the index portfolio. All the component stocks of the KOSPI 200 index pay annual dividends only at the end of March, June, September, or December, and most of them pay in December. We assume that a year comprises 252 trading days. The sample option data and all the relevant data, including the dividend payments, are obtained from the Koscom Datamall. Actual data for the empirical analysis are selected as follows: First, options with less than 7 or more than 90 days to expiration are excluded from the sample. Options with less than 7 days to expiration may induce biases due to their low time premiums and bid-ask spreads. Because the liquidity of KOSPI 200 index options is concentrated in short-term contracts, longer contract options may yield biases and measurement errors. Second, we use near-the-money and out-of-the-money options for calls and puts. In KOSPI 200 options markets, the trading volumes of deep OTM options are much larger than those of the option with other moneyness. Third, options with a market price lower than 0.02 are excluded to alleviate the effect of price discreteness on option valuation. After applying these exclusionary criteria, a sample of 25,303 total observations remain, with 11,280 (44.56%) call options and 14,033 (55.44%) put options. An overview of the data is given in Table 1. Summary statistics are reported for option prices, IVs, and the number of observations. In Panel A, options are categorized into three levels on the basis of time to maturity (i.e., short maturity, less than 30 days; medium maturity, 30 days to 60 days; and long maturity, over 60 days) and six levels of log-forward-moneyness. The Black–Scholes IV smile is shown to be weak on the average values across the log-forward-moneyness. In Panel B, annual information on option prices and IVs are provided. The average option price over the period is 2.30. Option prices and the IVs change with time. The number of observations shows an increasing trend overall. 4.2. Estimation performance Using the KOSPI 200 index option data described above, we estimate the local volatility functions on the cross section of the options for each Wednesday using the proposed method and other comparative methods. As previously stated, we are interested in the improvement of pricing performance when local estimators are used with no-arbitrage conditions. Thus, we consider the following four models for comparison. The first model is the bivariate local quadratic smoothing method with arbitrage-free conditions (LVNA) applied to generate IV functions and the second one is the general bivariate local quadratic method (LVlocQ), i.e. without arbitrage-free condition. These methods are both local estimators. Hence we also consider the third model using a global quadratic estimator (LVgloQ) suggested in Dumas et al. (1998) for comparison. This estimator switches the following three functions according to the number of different times to maturity (m) in a given date.   2 when m ¼ 1; Model 1 : σ ¼ max 0:01; a0 þ a1 κ þ a2 κ   2 when m ¼ 2; Model 2 : σ ¼ max 0:01; a0 þ a1 κ þ a2 κ þ a3 τ þ a5 κτ   when m≥3: Model 3 : σ ¼ max 0:01; a0 þ a1 κ þ a2 κ 2 þ a3 τ þ a4 τ 2 þ a5 κτ Note that there are several ways to express volatilities in the above models: absolute approach expresses volatilities as a function of strike price K (Dumas et al., 1998; Jackwerrth and Rubinstein, 2001; Li and Pearson, 2007), relative approach as a function of moneyness K/St (Kim and Kim, 2004; Kirgiz, 2001), and forward moneyness approach as a function of forward moneyness κ (Choi and Ok, 2012). We employed the log forward moneyness as above for our experiments to be consistent with our derivations. It is also reported in (Choi and Ok, 2012) that forward moneyness approach performs often better than other approaches.

42

N. Kim, J. Lee / Journal of Empirical Finance 21 (2013) 36–53

Table 1 KOSPI 200 index option data description. This table reports average and standard deviation of option price and IV with the number of contracts for each category and for each year. The IV is obtained by inverting the Black–Scholes formula separately for each option contract. Panel A: KOSPI200 index options by log forward moneyness and by maturities Log-forward moneyness

Maturity b30

Put option κ b −0.06

−0.06 ≤ κ b −0.03

−0.03 ≤ κ b 0

Call option 0 ≤κ b 0.03

0.03 ≤ κ b 0.06

0.06 ≤ κ

All

30–60

>60

All

Mean

Std.

Mean

Std.

Mean

Std.

Price IV Obs. Price IV Obs. Price IV Obs.

0.46 0.36 2955 1.46 0.28 812 2.74 0.27 862

0.75 0.13

1.16 0.33 3348 3.24 0.28 810 4.87 0.27 857

1.40 0.11

2.13 0.31 2719 4.73 0.27 802 6.54 0.27 862

1.96 0.10

Price IV Obs. Price IV Obs. Price IV Obs. Price IV Obs.

2.56 0.24 874 1.12 0.23 888 0.41 0.31 1752 1.09 0.30 8147

1.71 0.10

4.87 0.27 861 2.85 0.23 895 1.13 0.30 2290 2.21 0.29 9078

2.33 0.09

6.54 0.27 866 4.38 0.23 900 2.22 0.28 1919 3.61 0.28 8078

2.80 0.09

1.35 0.10 1.82 0.11

1.19 0.09 0.65 0.11 1.44 0.12

1.98 0.09 2.33 0.09

1.90 0.09 1.30 0.11 2.26 0.11

2.41 0.09 2.81 0.09

2.36 0.08 1.89 0.10 2.90 0.10

Mean

Std.

1.22 0.33 9022 3.14 0.28 2424 4.71 0.27 2593

1.58 0.12 0.00 2.37 0.09 0.00 2.82 0.10 0

4.66 0.24 2620 2.79 0.23 2683 1.27 0.30 5961 2.30 0.29 25,303

2.92 0.09 2.30 0.09 1.56 0.11 2.49 0.11

Panel B: KOSPI200 index options by year Year

Price

2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 All

IV

Observation

Mean

Std.

Mean

Std.

Call

Put

1.33 1.69 1.19 1.31 1.36 1.86 3.02 3.55 2.91 2.24 2.30

1.26 1.59 1.18 1.25 1.38 1.70 3.17 3.36 2.67 2.08 2.49

0.37 0.38 0.30 0.26 0.21 0.21 0.27 0.36 0.34 0.21 0.29

0.08 0.05 0.07 0.06 0.05 0.05 0.09 0.16 0.11 0.05 0.11

745 1017 853 815 698 1022 1392 2273 1297 1168 11,280

776 1111 978 998 1265 1209 2080 1571 2189 1856 14,033

The fourth model for comparison is Heston's stochastic volatility option pricing model (Heston, 1993) that is reported to show better performance in many empirical studies than several alternative parametric option pricing models. The parameters of the third and fourth models are estimated every sample day by minimizing the sum of squared price errors between the reported option prices and their model values (Bakshi et al., 1997; Dumas et al., 1998): min Θ

N h X

model

Cn

i2 mkt ðΘÞ−C n ðΘÞ ;

ð20Þ

n¼1

where Θ is a set of parameters, Cnmodel is the model price, and Cnmkt is the market price for the n-th observation. The estimation results are analyzed based on four measurements. Let εn = Cnmodel − Cnmkt denote the errors. Estimation performance is evaluated by: r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  (1) The root mean square error (RMSE), ∑Nn¼1 ðεn Þ2 =N, measures the volatility of the difference between the model price and the market price. (2) The mean percentage error (MPE), (∑nN= 1εn/Cnmkt)/N, represents the direction of the estimation error. (3) The mean absolute error (MAE), (∑nN= 1|εn|)/N, measures the magnitude of the estimation error. (4) The mean absolute percentage error (MAPE), (∑nN= 1|εn|/Cnmkt)/N, where N is the total number of options in a particular category.

N. Kim, J. Lee / Journal of Empirical Finance 21 (2013) 36–53

43

Table 2 Estimation performance. This table reports estimation errors for KOSPI200 index options of each category with respect to log-forward-moneyness and time to maturity and of each year. SV is Heston's stochastic volatility model. LVgloQ is a local volatility estimate using global quadratic method. LVlocQ is a local volatility estimate using general bivariate local quadratic method. LVNA is an estimate using the bivariate local quadratic smoothing method with arbitrage-free conditions. Each model is estimated every week, and estimation errors are computed using the estimated local volatility function from the current day. We use four r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  N N N model εn/On)/N; (3) MAE, (∑n= − Cnmkt, measurements: (1) RMSE, ∑Nn¼1 ðεn Þ2 =N; (2) MPE, (∑n=1 1|εn|)/N; and (4) MAPE, (∑n=1|εn|/On)/N, where εn = Cn Cnmodel is the model price and Cnmkt is the market price. ΔRMSE is the incremental average RMSE of LVNA over other models. Panel A: Aggregate results Model

SV LVgloQ LVlocQ LVNA

All RMSE

ΔRMSE

t statistic

MPE

MAE

MAPE

0.5360 0.4065 0.2965 0.2843

0.2517 0.1223 0.0122 –

10.3055 14.5578 5.2708 –

−0.0718 0.0658 −0.0076 −0.0048

0.3712 0.2950 0.1988 0.1901

0.1843 0.1808 0.1065 0.1024

Panel B: Estimation errors by categories Log-forward moneyness

Days to expiration b30

30–60

>60

RMSE

MPE

MAE

MAPE

RMSE

MPE

MAE

MAPE

RMSE

MPE

MAE

MAPE

SV κ b −0.06 −0.06 to −0.03 −0.03–0 0–0.03 0.03–0.06 0.06 ≤ κ

0.1207 0.2351 0.3600 0.3173 0.2089 0.1439

−0.0046 −0.0306 −0.0328 0.0594 0.0458 0.1179

0.0788 0.1679 0.2537 0.2183 0.1314 0.0725

0.1075 0.1149 0.1034 0.1061 0.1084 0.1737

0.2771 0.5567 0.7946 0.6143 0.5105 0.2795

−0.0419 −0.1094 −0.1126 −0.0140 −0.0406 −0.0036

0.1765 0.3935 0.5721 0.4221 0.3457 0.1717

0.1303 0.1313 0.1285 0.1007 0.1186 0.1387

0.5504 1.0677 1.4015 1.0839 0.9619 0.6056

−0.0699 −0.1614 −0.1603 −0.0275 −0.0655 −0.0439

0.3554 0.7816 1.0453 0.7784 0.6536 0.3797

0.1688 0.1819 0.1733 0.1401 0.1497 0.1633

LVgloQ κ b −0.06 −0.06 to −0.03 −0.03–0 0–0.03 0.03–0.06 0.06 ≤ κ

0.1888 0.3898 0.5682 0.5982 0.4214 0.2670

0.1272 0.0937 0.0492 0.1412 0.2357 0.4147

0.1038 0.2645 0.3851 0.4272 0.2723 0.1452

0.2648 0.2327 0.1740 0.2151 0.3183 0.4459

0.3549 0.5448 0.6742 0.8485 0.6658 0.5077

0.1818 0.0090 −0.0288 0.0926 0.1079 0.2376

0.2213 0.3881 0.4724 0.6198 0.4676 0.2839

0.2785 0.1410 0.1070 0.1522 0.1953 0.3074

0.4727 0.7204 0.8627 1.0891 0.7505 0.6111

0.0438 −0.0803 −0.0727 0.0890 0.1092 0.1366

0.3052 0.5160 0.6422 0.7432 0.5471 0.3663

0.1998 0.1192 0.1045 0.1340 0.1536 0.1974

LVlocQ κ b −0.06 −0.06 to −0.03 −0.03–0 0–0.03 0.03–0.06 0.06 ≤ κ

0.1257 0.2445 0.3522 0.3186 0.1865 0.1030

−0.0959 −0.0697 −0.0521 0.0489 0.0443 0.0186

0.0666 0.1512 0.2326 0.2094 0.1102 0.0564

0.1190 0.1017 0.0897 0.1053 0.1300 0.1324

0.2156 0.3767 0.5289 0.5464 0.3553 0.2382

−0.0238 −0.0530 −0.0630 0.0671 0.0644 0.0622

0.1147 0.2571 0.3738 0.3741 0.2293 0.1213

0.0999 0.0836 0.0812 0.0955 0.1113 0.1224

0.4361 0.6448 0.8231 0.9261 0.5663 0.4528

−0.0904 −0.0942 −0.0859 0.0847 0.0727 0.0397

0.2548 0.4754 0.6233 0.5998 0.3830 0.2486

0.1225 0.1084 0.1008 0.1099 0.1100 0.1248

LVNA κ b −0.06 −0.06 to −0.03 −0.03–0 0–0.03 0.03–0.06 0.06 ≤ κ

0.1187 0.2297 0.3309 0.3135 0.1874 0.1028

−0.0878 −0.0664 −0.0491 0.0462 0.0507 0.0250

0.0648 0.1443 0.2193 0.2058 0.1102 0.0565

0.1090 0.0965 0.0853 0.1034 0.1394 0.1317

0.1982 0.3554 0.5020 0.5304 0.3498 0.2367

−0.0116 −0.0477 −0.0575 0.0633 0.0617 0.0573

0.1071 0.2397 0.3486 0.3542 0.2201 0.1189

0.0921 0.0787 0.0765 0.0906 0.1066 0.1167

0.4017 0.6111 0.7871 0.8870 0.5635 0.4918

−0.0778 −0.0858 −0.0785 0.0803 0.0704 0.0409

0.2353 0.4405 0.5797 0.5721 0.3701 0.2477

0.1138 0.1017 0.0949 0.1052 0.1067 0.1244

Panel C: Estimation errors by year Year

SV 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

All

Calls

Puts

RMSE

MPE

MAE

MAPE

RMSE

MPE

MAE

MAPE

RMSE

MPE

MAE

MAPE

0.3659 0.3475 0.2991 0.4485 0.5259 0.5800 0.5654 0.6309 0.7616 0.7422

0.0172 0.0072 −0.0337 −0.0593 −0.0950 −0.0898 −0.0280 −0.0283 −0.1040 −0.1078

0.2081 0.1902 0.1687 0.2527 0.2661 0.3273 0.3279 0.3782 0.4496 0.4572

0.1471 0.1078 0.1249 0.1609 0.1529 0.1462 0.1250 0.1201 0.1459 0.1825

0.3332 0.3683 0.2993 0.3992 0.4858 0.5367 0.6731 0.6058 0.7269 0.7447

0.0362 0.0300 0.0082 0.0166 −0.0214 −0.0567 −0.0078 −0.0035 −0.0725 −0.0528

0.1906 0.1990 0.1660 0.2406 0.2687 0.3029 0.4016 0.3518 0.4345 0.4620

0.1313 0.1088 0.1199 0.1523 0.1281 0.1280 0.1264 0.1318 0.1311 0.1760

0.3946 0.3274 0.2989 0.4850 0.5469 0.6142 0.4831 0.6675 0.7824 0.7408

−0.0030 −0.0160 −0.0751 −0.1262 −0.1489 −0.1176 −0.0448 −0.0574 −0.1276 −0.1493

0.2248 0.1822 0.1710 0.2626 0.2646 0.3480 0.2802 0.4184 0.4591 0.4543

0.1637 0.1068 0.1298 0.1684 0.1710 0.1615 0.1237 0.1064 0.1570 0.1874

(continued on next page)

44

N. Kim, J. Lee / Journal of Empirical Finance 21 (2013) 36–53

Table 2 (continued) Panel C: Estimation errors by year Year

All RMSE

Calls MAE

MAPE

RMSE

MPE

MAE

MAPE

RMSE

MPE

MAE

MAPE

0.0974 0.0542 0.0734 0.0659 0.0589 0.0548 0.1472 0.0991 0.0801 0.0871

0.2635 0.3041 0.2074 0.2208 0.2561 0.2543 0.5086 0.4388 0.3899 0.3328

0.2107 0.1876 0.1719 0.1761 0.1754 0.1685 0.2500 0.1782 0.1982 0.1783

0.4794 0.5880 0.5235 0.4073 0.4655 0.3944 0.9439 0.7786 0.7325 0.5453

0.1343 0.0298 0.1355 0.1885 0.1741 0.1078 0.1892 0.1819 0.1277 0.1571

0.3015 0.3456 0.2281 0.2415 0.2920 0.2611 0.6549 0.4697 0.5227 0.4030

0.2363 0.1689 0.1860 0.2167 0.1958 0.1575 0.2489 0.2279 0.2104 0.1876

0.3659 0.4212 0.3044 0.3247 0.4468 0.3776 0.6742 0.5881 0.4897 0.4636

0.0592 0.0786 0.0087 −0.0466 −0.0286 0.0097 0.1159 0.0049 0.0481 0.0374

0.2272 0.2662 0.1895 0.2037 0.2366 0.2490 0.4212 0.3970 0.3175 0.2915

0.1843 0.2064 0.1573 0.1389 0.1599 0.1778 0.2508 0.1217 0.1900 0.1717

0.3022 0.3055 0.3666 0.3242 0.3956 0.3330 0.4927 0.5690 0.4109 0.4000

−0.0092 −0.0078 0.0048 0.0170 0.0049 −0.0114 −0.0166 −0.0250 −0.0097 −0.0012

0.1662 0.1681 0.1340 0.1696 0.1879 0.1954 0.2721 0.3256 0.2514 0.2541

0.1185 0.0952 0.1033 0.1315 0.1294 0.1082 0.0997 0.0972 0.0997 0.1165

0.3065 0.3301 0.4799 0.3396 0.4755 0.3160 0.5393 0.5133 0.4639 0.4096

0.0439 0.0157 0.0613 0.1136 0.1254 0.0639 0.0518 0.0321 0.0588 0.0900

0.1644 0.1873 0.1498 0.1706 0.2322 0.1885 0.2954 0.2659 0.3004 0.2718

0.1156 0.0975 0.1170 0.1517 0.1591 0.1129 0.0992 0.0930 0.1126 0.1219

0.2981 0.2811 0.2261 0.3109 0.3447 0.3454 0.4627 0.6366 0.3789 0.3942

−0.0685 −0.0332 −0.0530 −0.0727 −0.0880 −0.0797 −0.0702 −0.0877 −0.0588 −0.0695

0.1679 0.1506 0.1203 0.1688 0.1638 0.2008 0.2583 0.4066 0.2247 0.2437

0.1217 0.0929 0.0894 0.1127 0.1066 0.1039 0.1001 0.1019 0.0904 0.1124

0.3269 0.3011 0.3606 0.3045 0.3848 0.3267 0.4946 0.5396 0.3721 0.3793

−0.0074 −0.0089 0.0120 0.0139 0.0082 −0.0097 −0.0070 −0.0169 −0.0053 −0.0031

0.1645 0.1640 0.1347 0.1605 0.1799 0.1908 0.2702 0.2972 0.2274 0.2362

0.1156 0.0924 0.1051 0.1244 0.1242 0.1076 0.1006 0.0879 0.0907 0.1078

0.3565 0.3242 0.4686 0.3179 0.4690 0.3097 0.5497 0.5369 0.4235 0.3831

0.0449 0.0118 0.0693 0.1007 0.1237 0.0641 0.0621 0.0311 0.0523 0.0776

0.1621 0.1824 0.1521 0.1626 0.2263 0.1838 0.3060 0.2627 0.2757 0.2477

0.1120 0.0954 0.1213 0.1447 0.1554 0.1124 0.1050 0.0915 0.1039 0.1122

0.2959 0.2783 0.2286 0.2930 0.3302 0.3392 0.4585 0.5432 0.3407 0.3771

−0.0662 −0.0314 −0.0473 −0.0658 −0.0795 −0.0765 −0.0606 −0.0693 −0.0465 −0.0637

0.1667 0.1473 0.1196 0.1588 0.1547 0.1961 0.2488 0.3440 0.2010 0.2295

0.1196 0.0891 0.0883 0.1058 0.1004 0.1033 0.0971 0.0839 0.0813 0.1046

LVgloQ 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

0.4252 0.5077 0.4206 0.3644 0.4535 0.3850 0.7860 0.7040 0.5870 0.4955

LVlocQ 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 LVNA 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

MPE

Puts

In implementing the LVNA algorithm, we use Gaussian kernel and the initial global bandwidth h1 = 0.05 in the log-forwardmoneyness direction and h2 = 0.1 when time to maturity is less than 30 days, 0.15 when time to maturity is less than 60 days, and 0.2 when time to maturity is longer than 60 days in the time to maturity direction. The pre-specified values of Δh τ and Δh κ are 0.01 and 0.001, respectively. The same values of the initial bandwidth are used to estimate LVlocQ, but these values are not changed during the estimation, unlike LVNA. As a result, 204 TIVSs among 522 total estimated TIVS from the general bivariate local quadratic method cannot fulfill the no-arbitrage conditions resulting in negative local volatilities or negative risk neutral transition density (RND). Thus, the proposed algorithm is applied 204 times (39%) in this analysis. Note that the RND of ST = K at time textitT contingent on its value at time t being St, is given by Breeden and Litzenberger (1978) r ðT−t Þ

ϕðK; TjSt ; t Þ ¼ e

∂2 C t ðK; T Þ : ∂K 2

ð21Þ

To price the option values, the MC simulations are performed under local volatility models (LVgloQ, LVlocQ and LVNA) and the fast Fourier transform algorithm under the Heston's model. LVS is recovered from each TIVS, and in the case of occurring negative LVS (i.e., the existence of arbitrage), we give a minimum value of 0.01. The estimation results are provided in Table 2. Aggregate results of each model are in Panel A with t-statistics. Panel B shows the estimation errors of all options by log-forward-moneyness and maturities, and Panel C reports the estimation errors by year with four measurements. With respect to all measures, LVNA shows the best performance, followed by the LVlocQ. The local smoothing method dominates the global estimation in all measurements. From these results, we can say that adding functional flexibility improves the pricing performance of the local volatility model. The reported average RMSE values show that LVNA outperforms other models. The average MAE and MAPE measurements yield the same results. Through the various option categories, the greatest pricing improvement appears to be for OTM call and put options with short and medium maturities. To verify the incremental improvements in going from global to local

N. Kim, J. Lee / Journal of Empirical Finance 21 (2013) 36–53

45

estimation, statistical hypothesis tests, Student's t-tests on the difference between weekly RMSE of each model in particular, are conducted. As a result, LVNA shows a significant improvement over SV, LVgloQ and LVlocQ. The incremental average RMSEs of LVNA over SV, LVgloQ, and LVlocQ are 0.2517, 0.1223, and 0.0122 respectively, and the t-statistics are 10.3055, 14.5578, and 5.2708 respectively. As an illustrative example of the estimation, an estimated smile of 23 days to maturity from August 20, 2008 is presented in Fig. 1. The red solid line is the constrained arbitrage-free smoother, and the blue dotted line is the unconstrained smoother. The left graph shows the observed TIV data and the estimates of TIVS. The estimated TIVS functions can also be used to deduce the RND and LVS, which are given in the middle and right panels, respectively. The estimated TIVs from the constrained and the unconstrained smoothing shown in the left panel do not seem different significantly at first glance. However, relatively large differences exist between the two RNDs and LVSs, particularly at the end of the smile in this case, which can yield differences in pricing and hedging performance under local volatility models.

4.3. Prediction performance The in-sample estimation results in the previous section suggest that performance increasingly improves as we go from SV to LVgloQ to LVlocQ and then to LVNA. As we employ a local volatility model, a local estimator and no-arbitrage conditions, the estimation performance is improved. The estimation performance, however, can be biased because of over-fitting to the in-sample data, which may result in poor predictive performance. Thus, we examine each model's prediction performance with out-of-sample data after estimating the volatility function. To this end, we apply the estimated volatility function to pricing the same options for the next week. Through this process, we can verify the stability of the volatility function over time and the possibility of over-fitting. To establish a benchmark, we use the ad hoc Black–Scholes (AH) method by following (Dumas et al., 1998) for comparison. The AH method prices the option values on t + 7 by applying the Black–Scholes formula using the estimated TIVS on day t. The AH method is not theoretically consistent, but is widely used by market practitioners because of its simplicity and relatively good performance. Given that we estimate TIVS each day in three ways, we call the three corresponding AH methods as global quadratic estimation (AHgloQ), local quadratic estimation (AHlocQ), and arbitrage-free local quadratic estimation (AHNA). AHgloQ fits the same volatility function forms in

−3

4.5

x 10

Total IV smile

RNDs

LV smile

1.3

0.5

1.2 0.45

1.1 1

4

0.4

0.9 0.8

0.35

0.7 3.5

0.3

0.6 0.5

0.25

0.4

3 −0.1 −4

1

−0.05 0 0.05 0.1 log forward moneyness

0.15

x 10

−0.1

−0.05 0 0.05 0.1 log forward moneyness

0.15

0.15

−0.05 0 0.05 0.1 log forward moneyness

0.15

0.15

0.1

0

0.2 −0.1

0.1

0.05

0.05

−1 0

−2 −0.1

−0.05

0

0.05

0.1

0.15

−0.05 −0.1

0 −0.05

0

0.05

0.1

0.15

−0.1

−0.05

0

0.05

0.1

0.15

Fig. 1. Left: smoothed TIV function, constrained (red solid line) and unconstrained (blue dotted line). Middle: corresponding RNDs. Right: corresponding LV function. August 20, 2008. The lower figures present the difference between constrained and unconstrained smoothing.

46

N. Kim, J. Lee / Journal of Empirical Finance 21 (2013) 36–53

Table 3 Prediction performance. This table reports one-week-ahead prediction errors for KOSPI200 index options of each category with respect to log-forward-moneyness and time to maturity and of each year. Performance was evaluated as in Table 2. AHgloQ is an ad hoc method using the IV model with a global fit. AHlocQ is an ad hoc method using an estimate of IV using general bivariate local quadratic method. AHNA is an ad hoc method of an estimate of IV using the bivariate local quadratic smoothing method with arbitrage-free conditions. Each model is estimated every week and prediction errors are computed using estimated local volatility functions from one week ago. Panel A: Aggregate results Model

All RMSE

ΔRMSE

t statistic

MPE

MAE

MAPE

AHglocQ AHlocQ AHNA SV LVgloQ LVlocQ LVNA

0.8052 0.4202 0.4208 0.6109 0.8024 0.4143 0.3996

0.4056 0.0206 0.0212 0.2113 0.4028 0.0147 –

8.6193 4.0233 4.1949 8.8954 8.4903 6.9475 –

0.0730 0.0264 0.0263 −0.037 0.1917 0.0179 0.0136

0.6452 0.3192 0.3198 0.435 0.6321 0.3143 0.3023

0.3131 0.1517 0.1520 0.2351 0.3436 0.1472 0.1412

Panel B: Prediction errors by categories Log-forward moneyness

Days to expiration b30

30–60

>60

RMSE

MPE

MAE

MAPE

RMSE

MPE

MAE

MAPE

RMSE

MPE

MAE

MAPE

AHgloQ κ b −0.06 −0.06 to −0.03 −0.03–0 0–0.03 0.03–0.06 0.06 ≤ κ

0.6365 1.3964 1.5896 2.7326 2.2162 1.3034

0.4652 0.1239 −0.0161 0.1230 0.2840 1.7758

0.2818 0.7774 0.8953 1.2294 1.0017 0.5951

0.5186 0.3518 0.2163 0.2683 0.4359 1.8863

0.9597 2.1568 2.4972 1.9754 1.6089 0.9620

0.3965 0.0458 −0.0441 0.0577 0.1476 0.4229

0.4105 1.0534 1.2886 1.2185 0.9550 0.4816

0.5489 0.3101 0.2635 0.2616 0.3328 0.5600

1.0098 2.0684 2.2371 1.8558 1.4412 1.0643

0.0674 −0.0163 −0.0452 0.1088 0.1417 0.2616

0.4403 0.9957 1.1377 1.1881 0.8983 0.6227

0.2442 0.2194 0.1801 0.2090 0.2347 0.3616

AHlocQ κ b −0.06 −0.06 to −0.03 −0.03–0 0–0.03 0.03–0.06 0.06 ≤ κ

0.2916 0.4912 0.5768 0.5485 0.3788 0.2885

0.1332 −0.0234 −0.0735 0.0286 0.0469 0.1335

0.1786 0.3598 0.4224 0.3948 0.2470 0.1686

0.2828 0.1614 0.1172 0.1185 0.1334 0.2680

0.2777 0.5009 0.6343 0.7887 0.6112 0.5070

0.0583 −0.0333 −0.0584 0.0913 0.0992 0.2424

0.1675 0.3656 0.4767 0.5457 0.3968 0.2650

0.1684 0.1306 0.1097 0.1355 0.1642 0.2954

0.4185 0.7366 0.8771 0.9608 0.7286 0.7389

−0.0349 −0.0774 −0.0843 0.0945 0.0854 0.1479

0.2719 0.5333 0.6708 0.6873 0.4869 0.3885

0.1434 0.1286 0.1132 0.1284 0.1333 0.2211

AHNA κ b −0.06 −0.06 to −0.03 −0.03–0 0–0.03 0.03–0.06 0.06 ≤ κ

0.2915 0.4903 0.5762 0.5484 0.3785 0.2885

0.1333 −0.0232 −0.0733 0.0283 0.0465 0.1335

0.1786 0.3594 0.4221 0.3945 0.2468 0.1686

0.2830 0.1612 0.1170 0.1184 0.1330 0.2680

0.2754 0.5000 0.6371 0.7920 0.6147 0.5056

0.0577 −0.0337 −0.0583 0.0916 0.1000 0.2402

0.1667 0.3652 0.4791 0.5478 0.3999 0.2630

0.1676 0.1304 0.1099 0.1358 0.1649 0.2923

0.4190 0.7441 0.8754 0.9627 0.7340 0.7397

−0.0351 −0.0794 −0.0846 0.0951 0.0871 0.1488

0.2734 0.5398 0.6704 0.6888 0.4925 0.3900

0.1454 0.1300 0.1132 0.1288 0.1347 0.2216

SV κ b −0.06 −0.06 to −0.03 −0.03–0 0–0.03 0.03–0.06 0.06 ≤ κ

0.2957 0.5597 0.6965 0.5753 0.4422 0.2670

0.0892 0.0210 −0.0199 0.0870 0.1481 0.2201

0.1503 0.3313 0.4601 0.3944 0.2719 0.1442

0.3124 0.2261 0.1713 0.1819 0.2874 0.3746

0.4907 0.8420 1.0834 0.9752 0.8013 0.5787

0.0429 −0.0866 −0.1044 0.0036 −0.0174 0.0381

0.2738 0.5780 0.7607 0.6474 0.5424 0.3157

0.2308 0.1773 0.1591 0.1424 0.1831 0.2519

0.7429 1.2679 1.6383 1.4829 1.2139 0.9139

−0.0233 −0.1576 −0.1630 −0.0182 −0.0726 −0.0426

0.4698 0.9237 1.1889 1.0098 0.8675 0.5685

0.2265 0.2040 0.1893 0.1678 0.1975 0.2293

LVgloQ κ b −0.06 −0.06 to −0.03 −0.03–0 0–0.03 0.03–0.06 0.06 ≤ κ

0.5043 1.1640 1.4193 3.4111 2.9293 1.8887

0.6770 0.2981 0.0843 0.2116 0.5962 3.0199

0.3021 0.7741 0.9468 1.6057 1.4171 0.8647

0.7420 0.4439 0.2678 0.3732 0.6981 3.0533

0.8418 1.9660 2.2806 2.2482 1.8896 1.2194

0.4445 0.1158 0.0397 0.1528 0.2682 0.8114

0.3798 0.8999 1.0966 1.3232 1.0683 0.6096

0.5542 0.3032 0.2326 0.2884 0.4196 0.8989

0.9042 1.9378 2.1483 2.1057 1.6921 1.2617

0.1342 0.0056 −0.0222 0.1158 0.1706 0.3331

0.4392 0.9491 1.0736 1.3603 1.0790 0.7320

0.2817 0.2128 0.1710 0.2375 0.2842 0.4406

LVlocQ κ b −0.06 −0.06 to −0.03 −0.03–0 0–0.03 0.03–0.06 0.06 ≤ κ

0.3122 0.4634 0.5570 0.5431 0.4516 0.3443

0.0322 −0.0191 −0.0742 0.0142 0.0127 0.0851

0.1597 0.3345 0.4180 0.3581 0.2800 0.1914

0.1973 0.1497 0.1208 0.1053 0.1357 0.2769

0.2991 0.5119 0.6359 0.7580 0.5792 0.4836

0.0437 −0.0312 −0.0525 0.0898 0.0892 0.1968

0.1752 0.3675 0.4681 0.5268 0.3859 0.2499

0.1701 0.1299 0.1070 0.1341 0.1620 0.2770

0.3809 0.7036 0.8324 0.9796 0.7385 0.7034

−0.0395 −0.0774 −0.0797 0.0945 0.0773 0.1091

0.2532 0.5158 0.6286 0.6943 0.5082 0.3704

0.1338 0.1254 0.1067 0.1302 0.1380 0.2084

N. Kim, J. Lee / Journal of Empirical Finance 21 (2013) 36–53

47

Table 3 (continued) Panel B: Prediction errors by categories Log-forward moneyness

LVNA κ b −0.06 −0.06 to −0.03 −0.03–0 0–0.03 0.03–0.06 0.06 ≤ κ

Days to expiration b30

30–60

>60

RMSE

MPE

MAE

MAPE

RMSE

MPE

MAE

MAPE

RMSE

MPE

MAE

MAPE

0.3002 0.4499 0.5455 0.5440 0.4522 0.3374

0.0105 −0.0467 −0.0844 0.0092 0.0084 0.0968

0.1467 0.3222 0.4148 0.3551 0.2771 0.1839

0.1796 0.1364 0.1214 0.1032 0.1344 0.2844

0.2772 0.4890 0.6207 0.7445 0.5782 0.4942

0.0390 −0.0366 −0.0540 0.0836 0.0808 0.1883

0.1664 0.3561 0.4562 0.5070 0.3719 0.2509

0.1631 0.1263 0.1043 0.1276 0.1526 0.2763

0.3713 0.6782 0.8010 0.9581 0.7412 0.7131

−0.0410 −0.0760 −0.0777 0.0912 0.0765 0.1061

0.2409 0.4891 0.6003 0.6703 0.4919 0.3673

0.1261 0.1197 0.1023 0.1253 0.1335 0.2054

Panel C: Prediction errors by year Year

All RMSE

Calls MAE

MAPE

RMSE

0.0476 0.0286 0.0717 0.2158 0.1332 0.0740 0.2515 0.2532 0.1100 0.1205

0.3424 0.3168 0.2732 0.5430 0.4769 0.5982 1.1435 1.4704 0.8046 0.7509

0.2141 0.1618 0.1988 0.4096 0.3267 0.2691 0.4905 0.4397 0.2664 0.3192

0.5808 0.5154 0.4482 0.9464 1.0671 1.0923 2.4791 2.0201 1.8335 1.5844

0.4424 0.4377 0.3253 0.4283 0.4689 0.4289 0.7793 0.9230 0.4843 0.5442

0.0644 0.0601 0.0495 0.0407 −0.0199 0.0195 0.0077 0.0942 0.0351 0.0178

0.2857 0.2787 0.1991 0.2742 0.2763 0.2976 0.4880 0.5627 0.3354 0.3781

0.1808 0.1518 0.1415 0.1818 0.1540 0.1420 0.1614 0.1863 0.1242 0.1517

AHNA 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

0.4364 0.4373 0.3247 0.4281 0.4689 0.4290 0.7895 0.9210 0.4841 0.5461

0.0637 0.0600 0.0493 0.0407 −0.0199 0.0195 0.0058 0.0943 0.0361 0.0178

0.2851 0.2781 0.1991 0.2741 0.2764 0.2976 0.4968 0.5616 0.3353 0.3793

SV 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

0.4342 0.4060 0.4704 0.5230 0.5548 0.6951 0.9919 1.1636 0.9891 0.8155

0.0484 0.0159 −0.0100 −0.0369 −0.0583 −0.0541 0.0311 −0.0306 −0.0753 −0.0656

LVgloQ 2001 2002 2003 2004

0.6214 0.5688 0.4647 0.8765

0.1849 0.1003 0.1367 0.2505

AHgloQ 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

0.5364 0.4975 0.4549 0.8931 0.8535 1.0544 2.1085 2.8784 1.4489 1.2427

AHlocQ 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

MPE

Puts MPE

MAE

MAPE

RMSE

MPE

MAE

MAPE

0.1643 0.0585 0.0823 0.3045 0.2859 0.2148 0.2277 0.2021 0.2647 0.2519

0.3720 0.3341 0.3019 0.5813 0.6774 0.6641 1.5299 1.2447 1.1011 0.9699

0.2465 0.1570 0.1858 0.4085 0.3820 0.3154 0.4455 0.4420 0.3415 0.4036

0.4922 0.4808 0.4598 0.8507 0.7548 1.0264 1.9087 3.7799 1.2275 1.0240

−0.0805 −0.0014 0.0617 0.1414 0.0389 −0.0464 0.2664 0.3057 0.0191 0.0404

0.3152 0.3013 0.2518 0.5141 0.3993 0.5511 0.9610 1.7927 0.6654 0.6385

0.1785 0.1667 0.2110 0.4105 0.2926 0.2295 0.5188 0.4374 0.2222 0.2678

0.4453 0.4695 0.3983 0.4644 0.5598 0.4586 0.9996 1.0547 0.5787 0.5590

0.1448 0.0987 0.1134 0.1524 0.1070 0.0942 0.0935 0.1779 0.1101 0.0994

0.2953 0.3048 0.2411 0.2996 0.3366 0.3142 0.6700 0.6133 0.4188 0.4021

0.1966 0.1576 0.1647 0.2193 0.1601 0.1452 0.1771 0.2443 0.1379 0.1446

0.4397 0.4069 0.2578 0.3988 0.4285 0.4063 0.6498 0.6928 0.4329 0.5365

−0.0229 0.0210 −0.0144 −0.0565 −0.0976 −0.0442 −0.0482 −0.0036 −0.0084 −0.0349

0.2768 0.2552 0.1678 0.2551 0.2529 0.2857 0.4021 0.4905 0.2963 0.3658

0.1636 0.1459 0.1183 0.1492 0.1503 0.1392 0.1512 0.1185 0.1162 0.1564

0.1810 0.1516 0.1415 0.1818 0.1541 0.1420 0.1624 0.1861 0.1249 0.1522

0.4313 0.4694 0.3966 0.4644 0.5599 0.4590 1.0169 1.0545 0.5779 0.5619

0.1434 0.0990 0.1133 0.1524 0.1071 0.0943 0.0960 0.1779 0.1104 0.1000

0.2927 0.3048 0.2407 0.2996 0.3368 0.3143 0.6864 0.6126 0.4174 0.4036

0.1961 0.1575 0.1645 0.2193 0.1602 0.1453 0.1777 0.2443 0.1375 0.1450

0.4410 0.4062 0.2583 0.3984 0.4285 0.4061 0.6552 0.6869 0.4331 0.5378

−0.0236 0.0205 −0.0146 −0.0565 −0.0977 −0.0442 −0.0526 −0.0033 −0.0069 −0.0352

0.2782 0.2541 0.1680 0.2548 0.2529 0.2856 0.4073 0.4886 0.2967 0.3669

0.1644 0.1456 0.1186 0.1492 0.1504 0.1392 0.1525 0.1182 0.1177 0.1569

0.2684 0.2499 0.2231 0.3195 0.3098 0.4232 0.6072 0.7202 0.6181 0.5248

0.1986 0.1429 0.1617 0.2088 0.1882 0.1990 0.2398 0.2249 0.2036 0.2194

0.4402 0.4124 0.5838 0.4957 0.5559 0.6930 1.2115 1.0400 1.0160 0.8104

0.0752 0.0237 0.0240 0.0369 0.0209 −0.0236 0.0479 −0.0231 −0.0441 −0.0029

0.2647 0.2583 0.2496 0.3208 0.3311 0.4119 0.7778 0.6349 0.6446 0.5656

0.1812 0.1336 0.1691 0.2139 0.1709 0.1776 0.2551 0.2335 0.1947 0.2315

0.4284 0.4002 0.3420 0.5443 0.5542 0.6970 0.8271 1.3429 0.9717 0.8185

0.0204 0.0079 −0.0428 −0.1014 −0.1140 −0.0793 0.0176 −0.0394 −0.0978 −0.1122

0.2720 0.2422 0.2000 0.3184 0.2980 0.4328 0.5015 0.8609 0.6015 0.5009

0.2168 0.1523 0.1545 0.2044 0.2005 0.2166 0.2275 0.2149 0.2101 0.2105

0.4197 0.3860 0.2947 0.5311

0.3056 0.2318 0.2314 0.4071

0.7611 0.6346 0.4853 1.0351

0.2770 0.0541 0.1504 0.3955

0.5298 0.4164 0.3164 0.6486

0.3811 0.2000 0.2172 0.4713

0.4577 0.5022 0.4488 0.7348

0.0889 0.1428 0.1243 0.1268

0.3189 0.3587 0.2785 0.4426

0.2270 0.2609 0.2442 0.3524

(continued on next page)

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N. Kim, J. Lee / Journal of Empirical Finance 21 (2013) 36–53

Table 3 2 (continued) Table Panel C: Prediction errors by years Year

All RMSE

Calls MPE

Puts

MAE

MAPE

RMSE

MPE

MAE

MAPE

RMSE

MPE

MAE

MAPE

LVgloQ 2005 2006 2007 2008 2009 2010

0.8511 0.9888 2.1933 2.9411 1.5608 1.3136

0.2171 0.1129 0.4239 0.3858 0.1929 0.2737

0.4873 0.5373 1.1801 1.5204 0.8913 0.7480

0.3857 0.2731 0.5739 0.5283 0.3480 0.4115

1.1475 1.1614 3.0505 2.3098 2.3421 1.9421

0.3777 0.2695 0.4730 0.4466 0.4174 0.5000

0.7222 0.6763 1.8132 1.4257 1.5039 1.1067

0.4485 0.3498 0.6208 0.6395 0.5168 0.6075

0.7037 0.8436 1.6394 3.6583 1.0023 0.8217

0.1211 −0.0126 0.3954 0.3215 0.0650 0.1392

0.3964 0.4377 0.8811 1.6555 0.6037 0.5639

0.3481 0.2116 0.5466 0.4105 0.2519 0.2949

LVlocQ 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

0.4478 0.4239 0.3173 0.4247 0.4657 0.4602 0.6579 0.9342 0.4993 0.5513

0.0429 0.0320 0.0451 0.0346 −0.0087 0.0004 0.0031 0.0648 0.0267 0.0251

0.2853 0.2725 0.1948 0.2742 0.2567 0.3167 0.4266 0.5600 0.3429 0.3727

0.1744 0.1467 0.1369 0.1815 0.1453 0.1461 0.1405 0.1808 0.1205 0.1497

0.4594 0.4496 0.3771 0.4531 0.6389 0.4186 0.8605 1.0508 0.6106 0.5935

0.1132 0.0512 0.0982 0.1474 0.1214 0.0652 0.0730 0.1492 0.1056 0.1017

0.2932 0.2908 0.2245 0.2932 0.3687 0.2913 0.5997 0.6059 0.4503 0.4154

0.1851 0.1448 0.1548 0.2178 0.1773 0.1324 0.1566 0.2358 0.1452 0.1490

0.4370 0.3993 0.2640 0.4019 0.3778 0.4878 0.5362 0.7365 0.4373 0.5283

−0.0313 0.0130 −0.0059 −0.0644 −0.0880 −0.0542 −0.0417 −0.0318 −0.0188 −0.0242

0.2780 0.2560 0.1726 0.2600 0.2133 0.3350 0.3448 0.4946 0.2925 0.3508

0.1630 0.1487 0.1196 0.1496 0.1257 0.1578 0.1302 0.1178 0.1062 0.1502

LVNA 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

0.4350 0.4148 0.3289 0.4029 0.4602 0.4517 0.6606 0.9289 0.4639 0.5323

0.0359 0.0235 0.0403 0.0264 −0.0073 −0.0048 0.0097 0.0615 0.0223 0.0163

0.2784 0.2616 0.1868 0.2589 0.2494 0.3063 0.4212 0.5538 0.3166 0.3561

0.1676 0.1417 0.1344 0.1713 0.1410 0.1408 0.1388 0.1780 0.1120 0.1415

0.4421 0.4378 0.4069 0.4286 0.6316 0.4103 0.8778 1.0732 0.5784 0.5367

0.1076 0.0428 0.1030 0.1316 0.1197 0.0599 0.0883 0.1428 0.0971 0.0845

0.2833 0.2770 0.2189 0.2807 0.3554 0.2786 0.6059 0.6230 0.4255 0.3672

0.1767 0.1389 0.1574 0.2055 0.1707 0.1279 0.1598 0.2391 0.1391 0.1289

0.4283 0.3928 0.2558 0.3824 0.3733 0.4791 0.5277 0.6711 0.3989 0.5300

−0.0405 0.0041 −0.0197 −0.0647 −0.0846 −0.0596 −0.0409 −0.0308 −0.0210 −0.0279

0.2740 0.2477 0.1629 0.2424 0.2084 0.3261 0.3340 0.4551 0.2654 0.3505

0.1579 0.1446 0.1124 0.1418 0.1230 0.1516 0.1253 0.1086 0.0962 0.1498

estimating LVgloQ to the Black–Scholes IVs. The summary of prediction errors is presented in Table 3 by log-forward-moneyness and maturities and by years, respectively. The four measurements, RMSE, MPE, MAE, and MAPE, are computed in the same way as in the estimation performance analysis. Compared with the results of the estimation, the prediction errors are relatively larger. The LVNA, however, presents the overall best prediction performance among the seven models, giving the similar results as estimation performance was achieved. In Panel B, the prediction errors for all options are provided in categories. The improvement in pricing performance also shows a similar pattern as the estimation results. In terms of average MAE and MAPE, prediction error is large for OTM call and put options with short maturities. The greatest improvements in prediction performance are achieved through OTM put options with short maturities and OTM call options with long maturity. These results demonstrate that our proposed method improves pricing performance without over-fitting by giving more functional flexibility while simultaneously satisfying no-arbitrage conditions. The statistical test supports the fact that LVNA performance is better than that of others. The incremental RMSE improvements in LVNA over AHgloQ, AHlocQ, AHNA, SV, LVgloQ, and LVlocQ are significant with t-statistics of 8.6193, 4.0233, 4.1949, 8.8954, 8.84903, and 6.9475, respectively. Additionally, we analyzed the performance on the nearest option contracts with expiries greater than 6 days only. As mentioned before, KOSPI 200 index options are traded heavily in short-term contracts as did in Choi and Ok (2012). Thus, accurate prediction of the short-term contracts is also very important. To predict an implied volatility smile (not an IVS) of the nearest term, the nearest contracts with greater than 6 days to expiration were chosen. However, on the rollover day, we applied next-to-next strategy, i.e., when the nearest contract's expiry is less than 7 days, next-to-next strategy uses the next-to-nearest contracts on the previous time (Choi and Ok, 2012). For comparison, we considered the following three models. AHgloQ is an ad hoc method using the IV model in Model 1 (i.e. σ = max(0.01, a0 + a1κ + a2κ 2)) as a function of just log-forward moneyness. AHNA is an ad hoc method of an estimate of the IV using the bivariate local quadratic smoothing method with arbitrage-free conditions. For the LV model, we used the LVNA in the same way as the previous part. The prediction results are summarized in Table 4. The result shows that LVNA is slightly better than AHNA and AHgloQ. The incremental RMSE of LVNA is significant over AHgloQ, but not over AHNA with t-statistics 6.4378 and 1.4376, respectively. In summary, local fit models are better than global fit model and that using additional information (not confined to the nearest option contracts) can be helpful in predicting the nearest option contracts well.

N. Kim, J. Lee / Journal of Empirical Finance 21 (2013) 36–53

49

Table 4 Smile prediction performance. This table reports one-week-ahead prediction errors for KOSPI 200 index options of each category with respect to log-forwardmoneyness and of each year. Performance was evaluated as in Table 2. AHgloQ is an ad hoc method using the IV model with a global fit. AHNA is an ad hoc method of an estimate of IV using the bivariate local quadratic smoothing method with arbitrage-free conditions. LVNA is a local volatility model using the bivariate local quadratic smoothing method with arbitrage-free conditions. Each model is estimated every week and prediction errors are computed using estimated local volatility functions from one week ago. Panel A: Aggregate results Model

All RMSE

ΔRMSE

t statistic

MPE

MAE

MAPE

AHgloQ AHNA LVNA

0.3576 0.3286 0.3183

0.0393 0.0103 –

6.4378 1.4376 –

0.0611 0.0375 0.0229

0.2933 0.2634 0.2540

0.1880 0.1668 0.1491

Panel B: Prediction errors by moneyness Log-forward moneyness κ b −0.06 −0.06 to −0.03 −0.03–0 0–0.03 0.03–0.06 0.06 ≤ κ

AHNA

AHgloQ

LVNA

RMSE

MPE

MAE

MAPE

RMSE

MPE

MAE

MAPE

RMSE

MPE

MAE

MAPE

0.3571 0.5181 0.6533 0.6348 0.4771 0.3319

0.2050 −0.0266 −0.0600 0.0688 0.0758 0.4354

0.2379 0.3962 0.5015 0.4738 0.3532 0.2256

0.3008 0.1490 0.1197 0.1270 0.1645 0.4986

0.2549 0.4741 0.6299 0.6043 0.4514 0.3391

0.0967 −0.0409 −0.0720 0.0534 0.0603 0.5062

0.1735 0.3564 0.4687 0.4557 0.3338 0.2229

0.1920 0.1357 0.1129 0.1209 0.1534 0.5615

0.2566 0.4778 0.6068 0.5969 0.4299 0.2661

0.0750 −0.0202 −0.0509 0.0679 0.0669 0.1168

0.1689 0.3647 0.4636 0.4441 0.3289 0.1773

0.1862 0.1379 0.1127 0.1208 0.1521 0.2176

MAE

MAPE

RMSE

MPE

MAE

MAPE

RMSE

MPE

MAE

MAPE

Panel C: Prediction errors by year Year

All RMSE

Calls MPE

Puts

AHgloQ 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

0.3144 0.3392 0.2670 0.3812 0.3380 0.4221 0.5819 0.6236 0.4724 0.5059

0.0616 0.0986 0.0401 0.0805 0.0132 0.0675 0.1365 0.2619 0.1072 0.0564

0.2212 0.2353 0.1857 0.2612 0.2202 0.3106 0.4142 0.4499 0.3524 0.3543

0.1683 0.1651 0.1485 0.2208 0.1534 0.1965 0.2304 0.3778 0.2042 0.1945

0.3602 0.3802 0.2939 0.4013 0.4268 0.3995 0.6053 0.5390 0.5015 0.5300

0.1459 0.1579 0.1159 0.1656 0.0960 0.0900 0.0985 0.4546 0.0903 0.0867

0.2640 0.2759 0.2040 0.2681 0.2609 0.2849 0.4119 0.3986 0.3711 0.3552

0.2047 0.1973 0.1722 0.2351 0.1596 0.1671 0.1666 0.5496 0.1525 0.1693

0.2643 0.2986 0.2430 0.3656 0.2937 0.4392 0.5706 0.7262 0.4573 0.4925

−0.0367 0.0377 −0.0357 0.0072 −0.0404 0.0482 0.1580 0.0541 0.1172 0.0374

0.1810 0.1997 0.1710 0.2561 0.2034 0.3310 0.4153 0.5223 0.3431 0.3538

0.1258 0.1321 0.1248 0.2086 0.1493 0.2220 0.2665 0.1925 0.2346 0.2105

AHNA 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

0.3278 0.3496 0.2624 0.3532 0.3142 0.3500 0.5595 0.5330 0.4089 0.4505

0.0689 0.1012 0.0403 0.0620 −0.0069 0.0361 0.0449 0.3116 0.0356 0.0296

0.2332 0.2402 0.1794 0.2343 0.1957 0.2507 0.3780 0.3821 0.2810 0.3147

0.1843 0.1684 0.1518 0.1926 0.1435 0.1523 0.1756 0.4173 0.1296 0.1688

0.3682 0.4028 0.3058 0.3894 0.3868 0.3643 0.7158 0.4940 0.4231 0.4655

0.1583 0.1600 0.1086 0.1486 0.0806 0.0733 0.0735 0.5833 0.0636 0.0654

0.2724 0.2991 0.2154 0.2588 0.2481 0.2623 0.4979 0.3538 0.3003 0.3335

0.2182 0.2063 0.1834 0.2136 0.1454 0.1455 0.1892 0.6579 0.1190 0.1590

0.2847 0.2952 0.2211 0.3238 0.2788 0.3382 0.4691 0.5836 0.4017 0.4423

−0.0424 0.0408 −0.0315 −0.0144 −0.0651 0.0036 0.0279 0.0079 0.0181 0.0069

0.1965 0.1885 0.1503 0.2163 0.1742 0.2415 0.3221 0.4220 0.2714 0.3047

0.1422 0.1296 0.1187 0.1741 0.1421 0.1582 0.1676 0.1483 0.1361 0.1751

LVNA 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

0.3105 0.3311 0.2455 0.3609 0.3111 0.3771 0.4551 0.4819 0.4221 0.4719

0.0309 0.0503 0.0326 0.0564 0.0058 0.0215 0.0333 0.0303 0.0541 0.0574

0.2114 0.2184 0.1578 0.2407 0.1895 0.2640 0.3111 0.3346 0.3010 0.3259

0.1493 0.1451 0.1338 0.1988 0.1437 0.1566 0.1423 0.1448 0.1413 0.1766

0.3128 0.3394 0.2601 0.3868 0.4259 0.3677 0.5367 0.4344 0.4702 0.4939

0.0733 0.0499 0.0855 0.1484 0.0845 0.0468 0.0663 0.0737 0.0894 0.1032

0.2216 0.2402 0.1690 0.2572 0.2592 0.2549 0.3880 0.3025 0.3417 0.3385

0.1575 0.1458 0.1487 0.2154 0.1609 0.1438 0.1446 0.1601 0.1427 0.1626

0.3084 0.3237 0.2330 0.3405 0.2488 0.3843 0.4116 0.5419 0.3960 0.4596

−0.0170 0.0507 −0.0195 −0.0215 −0.0464 −0.0005 0.0133 −0.0160 0.0326 0.0276

0.2017 0.1992 0.1488 0.2286 0.1608 0.2712 0.2753 0.3797 0.2808 0.3191

0.1401 0.1445 0.1192 0.1848 0.1323 0.1678 0.1409 0.1284 0.1405 0.1857

4.4. Hedging performance To validate the proposed method, the hedging performance of each model is also evaluated. One of the motivations for estimating volatility function is to use it as a method of risk management. Option pricing plays an important role in hedge

50

N. Kim, J. Lee / Journal of Empirical Finance 21 (2013) 36–53

Table 5 Hedging performance. This table reports one-week-ahead hedging errors for KOSPI 200 index options for each year and for each category with respect to mkt model − Ctmkt] − [Ct+Δ − Ctmodel]. log-forward-moneyness and time to maturity. Hedging performance is evaluated by the root mean square of t+Δ = [Ct+Δ Panel A: Aggregate results Model

AHgloQ

AHlocQ

AHNA

SV

LVgloQ

LVlocQ

LVNA

All Call Put

1.1252 1.3721 1.0867

0.7037 0.8799 0.6760

0.7010 0.8820 0.6724

1.2070 1.7266 1.1199

1.3404 1.5931 1.3015

0.9892 1.1421 0.9660

1.2275 1.5293 1.1800

Panel B: Hedging errors by year Year

All

Calls

Puts

AHgloQ 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

0.5504 0.6323 0.4446 0.6371 0.6132 0.9632 1.3468 2.0559 0.9806 1.2638

0.6786 1.0987 0.7480 0.7954 0.8976 1.1661 1.5065 2.1892 1.2832 1.3227

0.5347 0.5630 0.4050 0.6111 0.5757 0.9156 1.3279 2.0257 0.9485 1.2536

AHlocQ 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

0.4938 0.5627 0.3253 0.4106 0.3572 0.4376 0.9805 1.2343 0.5042 0.7615

0.6147 0.9503 0.6085 0.4145 0.4211 0.5837 1.2105 1.4277 0.6194 0.7283

0.4789 0.5064 0.2850 0.4101 0.3499 0.4008 0.9514 1.1880 0.4925 0.7670

AHNA 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

0.4923 0.5605 0.3264 0.4105 0.3572 0.4377 0.9698 1.2317 0.5046 0.7613

0.6083 0.9462 0.6091 0.4145 0.4215 0.5838 1.2274 1.4277 0.6197 0.7257

0.4781 0.5044 0.2863 0.4099 0.3498 0.4009 0.9368 1.1847 0.4928 0.7672

SV 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

0.5289 0.4785 0.4241 0.6371 0.7003 0.7845 1.5589 2.1002 1.1846 1.2896

0.7712 0.8108 0.6224 0.8378 1.3175 1.2244 2.0134 2.4974 1.9283 1.9452

0.4983 0.4319 0.4008 0.6028 0.5887 0.6666 1.5000 2.0132 1.0881 1.1486

LVgloQ 2001 2002 2003 2004 2005 2006 2007 2008

0.6045 0.6478 0.4638 0.5334 0.4275 0.6397 1.4526 3.2573

0.9891 0.9770 0.7656 0.6313 0.5793 0.7889 1.6593 3.2571

0.5468 0.6039 0.4249 0.5179 0.4086 0.6042 1.4277 3.2574

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51

Table 5 (continued) Panel B: Hedging errors by year Year

All

Calls

Puts

LVgloQ 2009 2010

0.9512 1.0338

1.2909 0.9545

0.9144 1.0466

LVlocQ 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

0.5100 0.5554 0.3721 0.4909 0.3826 0.5866 1.2403 2.0520 0.7359 1.0155

0.6758 0.9158 0.6544 0.5863 0.4775 0.6841 1.3575 2.1007 0.7857 0.9537

0.4887 0.5040 0.3339 0.4757 0.3714 0.5643 1.2266 2.0412 0.7312 1.0257

LVNA 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

0.5756 0.5426 0.3633 0.4730 0.3995 0.5770 1.2719 3.0367 0.7363 1.0653

0.6478 0.9631 0.6525 0.5391 0.5006 0.7555 1.4741 3.2911 0.8085 0.9738

0.5672 0.4789 0.3236 0.4628 0.3876 0.5326 1.2473 2.9783 0.7294 1.0801

Panel C: Hedging errors by categories Model

Log-forward moneyness

Days to expiration b30

30–60

>60

AHNA

κ b −0.06 −0.06 to −0.03 −0.03–0 0–0.03 0.03–0.06 0.06 ≤κ κ b −0.06 −0.06 to −0.03 −0.03–0 0–0.03 0.03–0.06 0.06 ≤κ κ b −0.06 −0.06 to −0.03 −0.03–0 0–0.03 0.03–0.06 0.06 ≤κ κ b −0.06 −0.06 to −0.03 −0.03–0 0–0.03 0.03–0.06 0.06 ≤κ κ b −0.06 −0.06 to −0.03 −0.03–0 0–0.03 0.03–0.06 0.06 ≤κ κ b −0.06 −0.06 to −0.03 −0.03–0 0–0.03 0.03–0.06 0.06 ≤κ

1.1976 0.7878 0.7412 1.3075 1.1945 1.1433 0.6221 0.6110 0.6218 0.7688 0.7194 0.6963 0.6185 0.6107 0.6213 0.7688 0.7194 0.6963 0.7110 0.7732 1.1993 0.9868 0.8791 1.3877 2.3976 0.7557 0.8647 2.5503 2.4152 2.7532 1.4955 0.7901 0.7395 1.5232 1.4086 1.7608

0.8088 1.2814 1.5224 1.2759 1.2746 1.1566 0.5213 0.7244 0.8084 0.8059 0.7861 0.7747 0.5187 0.7207 0.8071 0.8053 0.7933 0.7749 0.8283 1.1693 1.4624 1.4708 1.2917 1.6670 0.7968 1.0803 1.2824 1.1461 1.1619 1.7138 0.6874 0.9548 1.1508 0.9187 0.9294 1.2724

1.0709 1.2400 1.2508 1.4837 1.2296 1.4875 0.6797 0.8377 0.8886 0.9754 0.8750 1.1333 0.6805 0.8294 0.8706 0.9801 0.8931 1.1347 1.1938 1.6173 2.0602 2.3949 1.9780 2.4808 1.4609 1.1428 1.2518 1.6562 1.5089 2.2454 0.9418 1.0350 1.1561 1.2275 1.1174 1.6413

AHlocQ

AHNA

SV

LVgloQ

LVlocQ

(continued on next page)

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N. Kim, J. Lee / Journal of Empirical Finance 21 (2013) 36–53

Table 5 (continued) Panel C: Hedging errors by categories Model

Log-forward moneyness

κ b −0.06 −0.06 to −0.03 −0.03–0 0–0.03 0.03–0.06 0.06 ≤ κ

LVNA

Days to expiration b30

30–60

>60

1.9860 0.7960 0.7629 2.2430 2.0974 2.8493

0.6897 0.9387 1.1343 0.9544 0.9596 1.7487

1.4076 1.0577 1.1992 1.7448 1.5342 2.7002

portfolio management. A hedge is an investment position constructed to avoid the risk of future change in underlying asset price. We consider a traditional self-financed single-instrument strategy that is continuously rebalanced through time. The continuous hedge portfolio error is defined as the difference between the change in market option price and the change in model option price as follows (Dumas et al., 1998): h i h i mkt mkt model model − C tþΔ −C t tþΔ ¼ C tþΔ −C t

ð22Þ

where Ctmkt is the market option price on day t and Ctmodel is the model option price on day t. Table 5 shows the RMSE of one-week-ahead hedging, by years in Panel B, and by log-forward-moneyness and maturities in Panel C. Based on RMSE values in Table 5, LVlocQ shows the best performance among the local volatility models, followed by LVNA, and then by LVgloQ. The RMSE of LVlocQ is 0.9892, with 1.2275 and 1.3404 for LVNA and LVgloQ, respectively. Compared with AH models, LVNA does not outperform the Ad-hoc models, except for AHgloQ. Over all sample periods, AHNA has better performance than the other models in terms of overall average RMSE (especially for near ATM and OTM call options), and the other AH methods also show better performance than the LV methods. In this sense, the local volatility model fails to improve hedging performance significantly, and the AH model remains a difficult benchmark to beat. Thus, the fact that the simple models are better for hedging is again proven as in Dumas et al. (1998) and Kim and Kim (2003), although LV methods show better pricing performance. The overall hedging results indicate that AH methods are more stable than local volatility models. The differences among other models, however, are small. 5. Conclusion Estimating volatility is a crucial task in the financial industry. Several methods for estimating the IV function have been proposed over time, including deterministic function methods and local smoothing methods. In the current paper, we focus on developing an algorithm of constrained smoothing for estimating the TIVS where the raw option data includes option prices violating no-arbitrage conditions. Estimating arbitrage-free TIVS is important for the local volatility option pricing model; otherwise, negative values of volatility can occur. The proposed method improves the existing methods in several ways. First, we estimate TIVS directly from the TIV data, not from the price data. Second, the TIV data can be used without having to remove arbitrage violations from the raw data. Third, the nonlinear optimization problem is not required to be solved. Finally, the estimators provide sufficient functional flexibility when using the local smoothing method. The primary purpose of estimating the TIVS is to recover LVS without arbitrage. Using the recovered LVS, we evaluate the option prices under the local volatility model. We apply the local volatility option valuation approach to KOSPI 200 index options for the period January 2001 through December 2010. The empirical test results indicate that the local volatility function recovered from TIVS, considering no-arbitrage constraints, improves the pricing performance in terms of both estimation and prediction. In terms of hedging performance, the AH method is better than the local volatility models. Acknowledgement This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MEST) (No. 2011-0017657). References Bakshi, G.S., Cao, C., Chen, Z.W., 1997. Empirical performance of alternative option pricing models. J. Finance 52, 2003–2049. Benko, M., Fengler, Matthias R., Hrdle, Wolfgang Karl, Kopa, M., 2007. On extracting information implied in options. Comput. Stat. 22, 543–553. Blair, B.J., Poon, S.H., Taylor, S.J., 2001. Forecasting S&P 100 volatility: the incremental information content of implied volatilities and high-frequency index returns. J. Econ. 105, 5–26. Breeden, D., Litzenberger, R., 1978. Prices of state-contingent claims implicit in option prices. J. Bus. 51, 621–651.

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