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Moment Estimators: An Affine Jump-Diffusion

Approach

Pakorn Aschakulporn†

Department of Accountancy and Finance Otago Business School, University of Otago

Dunedin 9054, New Zealand beam.aschakulporn@otago.ac.nz

Jin E. Zhang

Department of Accountancy and Finance Otago Business School, University of Otago

Dunedin 9054, New Zealand jin.zhang@otago.ac.nz

First Version: 1 May 2020 This Version: 13 November 2020

Keywords: Risk-neutral moment estimators

JEL Classification Code: G13

Jin E. Zhang has been supported by an establishment grant from the University of Otago and the

National Natural Science Foundation of China grant (Project No. 71771199).

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Bakshi, Kapadia, and Madan (2003) Risk-Neutral

Moment Estimators: An Affine Jump-Diffusion

Approach

Abstract

This is the first study of the errors of the Bakshi, Kapadia, and Madan (2003) risk-neutral moment estimators under the Duffie, Pan, and Singleton (2000) affine jump-diffusion model benchmarked against their true values. This is accomplished using the exact solutions from Zhen and Zhang (2020). To mitigate errors in skewness, interpolating the implied volatility curve with cubic splines and applying constant extrapolation to have a step size of $1 and strikes ranging from half to double the forward price should yield skewness values with errors less than 10−3.

Keywords: Risk-neutral moment estimators

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1

Introduction

This is the first paper to examine the Bakshi, Kapadia, and Madan (2003) (BKM) risk-neutral moment estimators under affine jump-diffusion models with the exact moments known. The BKM estimators have been used by many academics and practitioners to cal-culate risk-neutral moments; this method is also used to calcal-culate the Chicago Board Op-tions Exchange (CBOE) skewness (SKEW) index,1 the forward looking tail-risk or crash

risk indicator of the S&P 500 index. Some papers have analysed the errors of the BKM es-timators, however, they either do not have a clear/exact benchmark or use oversimplified-unrealistic models. This paper analyses the BKM under affine jump-diffusion models with a benchmark which is calculated analytically following Zhen and Zhang (2020).

The CBOE SKEW, the higher-order version of CBOE volatility index (VIX), was cre-ated based on the BKM risk-neutral skewness estimator after the success of the CBOE VIX. The BKM now provides a way of calculating not only skewness but also higher-order risk-neutral moments. The BKM methodology has been used extensively by researchers. Conrad, Dittmar, and Hameed (2020) develop a new method of using equity options data to estimate default probabilities. They find that the correlation is high between default probabilities using their new estimation method and those extracted from CDS spreads. Hollstein, Prokopczuk, and Wese Simen (2020) use high-frequency data to ex-plain asset-pricing anomalies using the conditional capital asset pricing model and find that high-frequency betas are better predictors of future betas compared to those calcu-lated using daily data. Chordia, Lin, and Xiang (2020) document the strong, robust, and positive relationship between the risk-neutral skewness and subsequent stock returns using implied volatility surface data. They also find that the skewness contains incremental in-formation beyond existing option trading signals. Audrino, Huitema, and Ludwig (2019) develop a nonparametric estimation strategy using the BKM estimator. Morellec and Zhdanov (2019) show that product market competition produces negative volatility skew

1

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find that the implied skewness of individual firms tends to be more negative for stocks with larger betas and also during periods of high market volatility. From this list of lit-erature, it is clear that the BKM risk-neutral moment estimator is important and should be studied carefully.

There are few papers which have studied the BKM risk-neutral estimators. However, only one has done so with clear and explicit values of skewness, whereas others have found their benchmark using simulations/approximations. Aschakulporn and Zhang (2019) studied the BKM under Black and Scholes’ (1973) model which was modified to include skewness (and kurtosis) using the truncated Gram-Charlier density. The addition of skew-ness and kurtosis parameters were used as the benchmark as well as the input values. They found that the range of strikes should contain 3/4 to 4/3 of the forward price and the step size should be no larger than 0.1% of the forward price to obtain an error in skewness no larger than 0.001. A major drawback with the truncated Gram-Charlier density that was used is that the valid skewness values do not include typical market skewness values (of the S&P 500). The maximum and minimum skewness of the truncated Gram-Charlier density is ±

q

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they used various measures of correlation to test the effectiveness of interpolating and extrapolating. They find that interpolating and extrapolating help to improve the in-formation content of the estimator. Lee and Yang (2015) study truncation errors using Black and Scholes (1973) and Bates’ (1996) model. Two models are used, Black and Sc-holes (1973) and Bates (1996), the former since it has no skewness or excess kurtosis and the latter since it is a more realistic model. Using these models to generate option prices, the BKM estimator with and without linear extrapolation are compared to test trunca-tion errors. They find that linear extrapolatrunca-tion is able to significantly reduce but not eliminate truncation errors. Aschakulporn and Zhang (2019), Ammann and Feser (2019), Liu and van der Heijden (2016), and Lee and Yang (2015) also use various models to test the BKM estimators; the most advanced of which was Bates’ (1996) stochastic volatil-ity jump (SVJ) model – a combination of Heston’s (1993) stochastic volatilvolatil-ity (SV) with Merton’s (1976) jump. Unlike what Aschakulporn and Zhang (2019) used, the formula for the skewness of BKM is not trivial to find, therefore simulations and relative benchmarks were used. Without a clear expression for the benchmark, more errors are introduced.

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Zhen and Zhang (2020) is neither an approximation nor a simulated value – it is the true skewness. The BKM estimator is derived using Carr and Madan (2001) which requires a continuum of out-of-the-money (OTM) European options over an infinite domain of strike prices. Strike prices in the market are discrete and finite which introduces discretisation errors and truncation errors, respectively. Analysis is also done on the mitigation of these errors – using interpolation (linear, cubic spline, and kernel) and extrapolation (constant and linear) techniques.

The remainder of this paper is organized as follows. Section 2 briefly presents the BKM risk-neutral moment estimators. Section 3 presents the method used to create virtual options and the method used to quantify and analyse the errors and convergence of the BKM method. Section 4 describes the data. Section 5 provides the results, and Section 6 concludes. The appendix gives the details of key derivations.

2

BKM Risk-Neutral Moment Estimators

The BKM risk-neutral estimators can be derived from the normalized nth central moments (viz. standardised moments) equation

nth standardised moment = E Q t h − EtQ[Rτ] ni  EtQ  − EtQ[Rτ] 2n2 (1)

where Rτ ≡ ln ST − ln St is the log return of St from t to T . The key element to the derivation is that BKM makes a small approximation to the mean of the return and defines it as µ: EtQ[Rτ] ≈ µ = erτ − 1 − 1 2E Q t h R2τi− 1 3!E Q t h R3τi− 1 4!E Q t h R4τi (2)

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be included in µ through the exponential function – essentially serving as the convexity adjustment term. The values of EtQ[Rn

τ] for n = 2, 3, and 4 which correspond to volatility, cubic, and quartic contracts, respectively, can be priced through replication. Carr and Madan (2001) show that for any twice differentiable payoff function H (x) ∈C2 and some

constant x0, the payoff can be replicated with stock, bonds, and OTM European options:

H (x) = H (x0) + Hx(x0) (x − x0) + Z x0 0 Hxx(K) max (K − x, 0) dK + Z ∞ x0 Hxx(K) max (x − K, 0) dK (3) Setting H(x) = Rnτ =lnST St n

and applying the Harrison and Pliska (1981) risk-neutral formula, the current value of each contract can be found to be

EtQhe−rτRτni= Z ∞ 0 n K2 " (n − 1)  ln K St n−2 −  ln K St n−1# Q (K) dK (4)

where n specifies the type of power contract and Q (K) corresponds to the OTM European option with strike K. If there exists both put and call at the at-the-money point, then the average of the two is taken. More details are shown in appendix A.

3

Methodology

This section will provide a brief overview of the methodology. To begin with, the chosen model will be presented, from there the sequence in which each subsection is applied, then the errors will be defined.

The model used in this paper is Duffie, Pan, and Singleton’s (2000) stochastic volatility model with contemporaneous jumps in return and volatility (SVCJ). This model can be presented as dSt St = rdt +vtdBtS+ (ex− 1) dNt− λE (ex− 1) dt dvt = κ (θ − vt) dt + σvtdBtv + ydNt− λµydt (5)

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is mean-reverting with a speed of κ, long-term mean of θ, and diffusive volatility of σ. The Brownian motion in the underlying price and variance are denoted as BS

t and Btv, respectively and are correlated with correlation ρ. Nt denotes a Poisson process with intensity λ: P rob(dNt = 1) = λdt and P rob(dNt = 0) = 1 − λdt. When a jump occurs, the marginal distribution of the jump size in variance y is exponential with mean µy. Conditional on the realization of y, the jump size in price x is normally distributed with mean µx+ ρJy and variance σ2x. The correlation between jump sizes is therefore ρJ. The long-term mean is sometimes expressed as θ= θ − λµy

κ (Eraker, Johannes, and Polson, 2003). Forcing µy to zero will reduce SVCJ to SVJ, and setting λ to zero will further reduce the model to SV. From Equation (5) the characteristic function for log returns can be found to be f (φ) = exp (α (φ; τ ) + β (φ; τ ) v) (6) where α (φ; τ ) = riφτ − λτh1 + iφeµx+12σx2 − 1i + κθ − λµy σ2 " (κ − ρσiφ + d) τ − 2 ln 1 − ge 1 − g !# + λ exp  iφµx+ 1 2(iφ) 2 σ2x  1 1 − ρJµyiφ ×   τ 1 − aa (1 − g) d (1 − a) (a − g)ln  1 + (g − a)1 − edτ 1 − g     (7) β (φ; τ ) = κ − ρσiφ + d σ2 " 1 − edτ 1 − gedτ # (8) a = µy 1 − ρJµyiφ κ − ρσiφ + d σ2 (9) d = q

(ρσiφ)2+ σ2iφ (1 − iφ) (10)

g = κ − ρσiφ + d

κ − ρσiφ − d (11)

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Thirdly, using Section 3.3, the exact skewness is calculated to compare with Bakshi and Madan (2000). Finally, using Section 3.4, various interpolation/extrapolation techniques are tested to see if the error can be mitigated. The same can also be done to test the errors of kurtosis.2

As shown in Section 2, the BKM calculation requires the calculation of various power contract values (Equation (A.8)) via integration. This cannot be done directly as there are a limited number of options in the market. A commonly used method to calculate integrals with discrete data is the trapezium rule (viz. trapezoidal integration). Equation (A.8) can be discretised to EtQhe−rτR (t, τ )ni= ∞ X i=1 n K2 i " (n − 1)  ln K i St n−2 −  ln K i St n−1# Q [Ki] ∆Ki (12) where ∆Ki = 1 2        K2− K1, i = 1 Ki+1− Ki−1, 1 < i < m Km− Km−1, i = m (13)

and m is the number of strikes. This slightly differs from CBOE’s discretisation methods. The CBOE method gives equal weights to each strike price interval including the end points, which will slightly overestimate the true value (Aschakulporn and Zhang, 2019). The trapezium rule has been used by many, for example, Chang, Christoffersen, and Jacobs (2013), Chatrath et al. (2016), Conrad, Dittmar, and Ghysels (2013), Dennis and Mayhew (2002), Jiang and Tian (2005), Neumann and Skiadopoulos (2013), Ruan and Zhang (2018), and Stilger, Kostakis, and Poon (2017).

Jiang and Tian (2005), Jiang and Tian (2007), Chang et al. (2012), Liu and van der Heijden (2016), Ammann and Feser (2019), and Aschakulporn and Zhang (2019) study the truncation errors and discretisation errors of various estimators. Following Aschakulporn and Zhang (2019), the truncation and discretisation errors are defined the same way:

2Kurtosis is currently not a focus for practitioners and as skewness is the main interest of this paper,

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1. Truncation errors Z ∞ 0 · · · dK → Z Kmax Kmin · · · dK as K ∈ (0, ∞) → K ∈ [Kmin, Kmax]

The range of strikes are finite; therefore, the range of the integral is truncated to the strikes that are available. This is tested by defining Kmin and Kmax as

[Kmin, Kmax] := h FtT × a, FT t /a i (14)

where a ∈ (0, 1) is the boundary controlling factor. So as a → 0, Kmin → 0 and

Kmax → ∞ and as a → 1, Kmin, Kmax → FtT. For calculations, the maximum and minimum strike prices are rounded to the nearest dollar and a is varied between 0.5 and 0.95 in 0.15 intervals.3 2. Discretisation errors Z Kmax Kmin · · · dK → Kmax X Kmin · · · ∆Ki

To compute integrals numerically, the integrand and region must first be discretised. This can be done using the trapezium rule. The strikes provided in the market is not continuous, but rather, usually in fixed intervals of $1, $5, $25, and $50.4 The step size, ∆K, has been chosen to vary from 1 to 50, in increments of 10.5

The estimation error is defined as

Estimation Error := Estimated Value − True Value (15)

3The boundary controlling factor, a, for S&P 500 options data (from OptionMetrics), based on a

symmetric domain about the forward price, ranges from 0.372 to 0.945 and has an average of 0.797 from January 2010 to June 2019.

4Generally, minimum strike price intervals are as follows: (1) $0.50 where the strike price is less

than $15, (2) $1 where the strike price is less than $200, and (3) $5 where the strike price is greater than $200. (http://www.cboe.com/products/vix-index-volatility/vix-options-and-futures/ vix-options/vix-options-specs)

5S&P 500 options data from OptionMetrics from January 2010 to June 2019 show that, in general,

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3.1

Calibrating the Model

The DPS model is calibrated using the CBOE VIX and SKEW term structures similar to what was done by Zhang et al. (2017), Zhen and Zhang (2020), and Cao, Ruan, and Zhang (2020) (CRZ). The calibration of the Heston (1993) model is shown in Zhang et al. (2017). Much more complicated models are calibrated in Zhen and Zhang (2020) and CRZ. Overall, Zhen and Zhang (2020) present the most comprehensive model. Zhang et al. (2017) and Zhen and Zhang (2020) minimise the root mean squared error (RMSE) whereas CRZ use Markov chain Monte Carlo (MCMC) to calibrate their models using the two indices. A key step in both calibration methods is to find the model-based formulas for the VIX and SKEW. All three papers present clear methods to obtain the VIX and SKEW, however, not all models will have a closed-form solution for the VIX and SKEW. CRZ use finite differences to obtain an approximate closed-form formula for their calibration.

The closed-form formula for the VIX under the DPS model is VIX 100 !2 = 2 τE Q t " Z T t dSt St − d ln St # = 2 τE Q t " Z T t 1 2vudu + (e x− 1 − x) dN u # = κθτ + (vt− θ) (1 − e −κτ) κτ + 2λ   eµx+12σx2 1 − ρJµy − 1 − (µx+ ρJµy)   (16)

and the formula for skewness is shown in Equation (B.6). The relationship between SKEW and skewness is: SKEW = 100 − 10 × skewness.

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function is the sum of squared errors between the actual term structure and model-based theoretical values.6 There are many ways to implement this. A common method is the

two-step iterative procedure used by Christoffersen, Heston, and Jacobs (2009), Luo and Zhang (2012), Zhang et al. (2017), and Zhen and Zhang (2020). This method is modified from Bates (2000) and Huang and Wu (2004). The two-step iterative procedure is, in gen-eral, to initialise the process by setting a set of initial values for the structural parameters then (1) with the given set of structural parameters, optimise to find the latent variables then (2) with the latent variables optimise to find the structural parameters. Steps (1) and (2) are repeated until there is no further improvement. The design of the objective function can be varied in many ways. For simplicity, as calibration is not the focus of this paper, equal weights are assigned. This iterative process is first applied to the VIX to obtain {κ, θ, λ, µx, σx}. Next, calibration is done with SKEW to get {σ, ρ, µv, ρJ}. Finally, with all the structural variables fixed, the latent variable, the instantaneous variance vt, for each day is calibrated using the VIX. The iterative two-step calibration method is potentially faster as fewer parameters need to be calibrated at the same time, however, depending on how the parameters are separated between steps (1) and (2), there may be some oscillations which can slow down convergence. For robustness, all the parameters are simultaneously optimized for potential further improvement.

3.2

Generate Virtual Option Prices

Bakshi and Madan (2000) provide a way of pricing options using characteristic functions and can be written as

ct = StΠ1− Ke−rτΠ2 (17)

6 The objective function:

X

(y −y)b2

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where Πj = 1 2 + 1 π Z ∞ 0 < " e−iφ ln(K/St)× f j(φ) # fj(φ) =    f (φ−i) f (−i), j = 1 f (φ) , j = 2

and f (φ) is the characteristic function of log returns (Equation (6)). To help with com-parisons to Aschakulporn and Zhang (2019), the forward price of the underlying asset is

FtT = $2,000, where the time to maturity τ = T − t = 1/12, and the risk-free rate is set to r = 0.0077%. The time to maturity is set to one month as they tend to be the most liquid. Other parameters are found via model calibration. To analyse the BKM errors caused by truncation and discretisation, option prices are generated with set boundary controlling factors and step sizes.

3.3

Deriving the Exact Skewness

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where T CMH = 3ρσ Z T t A1(u) Et(vu) du − 3 2σ 2Z T t h A21(u) + 2ρ2A2(u) i Et(vu) du +3 4ρσ 3Z T t

[2A1(u) A2(u) + A3(u)] Et(vu) du − 3 8σ 4Z T t A1(u) A3(u) Et(vu) du (19) T CMJ = Z T t  µx3 − 3 2µx2yA1(u) + 3 4µxy2A 2 1(u) − µy3 8 A 3 1(u)  λdu + 3 Z T t  µxyµy2 2 A1(u)  

A1(u) − ρσA2(u) +

1 4A3(u)  λdu (20) V ARH = Z T t  1 − ρσA1(u) + 1 4σ 2A2 1(u)  Et(vu) du (21) V ARJ = Z T t  µx2 − µxyA1(u) + µy2 4 A 2 1(u)  λdu (22) (23) A1(u) = 1 − e−κτκ , τ= T − u (24) A2(u) = 1 − e−κτ− κτe−κτκ2 (25) A3(u) = 1 − e−2κτ− 2κτe−κτκ3 (26) µh(x) = E (h(X)) (27)

Integrating each term will yield Equation (B.6). The skewness can be decomposed into four components: T CMH, V ARH, T CMJ, and V ARJ which correspond to the third central moment (T CM ) and variance (V AR) contribution from Heston (1993) (H) and jumps (J ), respectively. An alternate way to derive skewness is to use moment generating functions or cumulant generating functions, similar to how CRZ derived their estimators.7

As mentioned in CRZ, the method of Zhang et al. (2017) does generate many integral

7The formula for kurtosis can be expressed as

Kurtosis = F CM (V AR)2

where F CM is the fourth central moments which can be calculated with moment generating functions or cumulant generating functions.

F CM = (−i)4 4(ln f (φ)) ∂φ4 φ=0 + 3V AR2

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terms; however, the form of the solution allows for the clear distinction of each compo-nent’s contribution to the skewness. The solution presented in Equation (B.6) has been verified using cumulant generating functions derived from Equation (6) (which is the same method as CRZ but without the finite difference derivatives).

3.4

Interpolation/Extrapolation

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spline interpolated values using piecewise third-order polynomials and kernel regressions uses the superposition of Gaussian curves. A key parameter for kernel regressions is the bandwidth, which can be found using the Nadaraya-Watson kernel regression estimator (Nadaraya (1964) and Watson (1964)) and optimised based on its mean squared error with leave-one-out cross-validation.

4

Data

Data is not required to test BKM directly as arbitrary parameters can be set. However, in order to test the BKM with realistic and relevant parameters, the parameters are obtained by calibrating the models to the S&P 500 index. More specifically, the model parameters are found from the CBOE VIX and SKEW term structures as was done by many such as Zhang et al. (2017), Zhen and Zhang (2020), and CRZ. Currently, only the CBOE SKEW term structure is available on the CBOE website.8 The CBOE VIX term structure used for

calibration is calculated using S&P 500 index options from OptionMetrics data following the CBOE VIX white paper9 without the interpolation step as the term structure is required – not the 30-day VIX. Instead, to obtain the term structure, each maturity’s variance is square rooted and multiplied by 100 so that it has the same dimensions as the VIX. Although the calculated VIX term structure values differ from the one available on the CBOE website,10 when the term structure is used to calculate the CBOE VIX, the

results have a high correlation of 99.68% between the period of 4 January 1996 to 28 June 2019. The only filters applied to calculate the VIX are those specified in the CBOE VIX white paper. Interest rates are downloaded from the U.S. Department of the Treasury.11

Linear interpolation is applied when the required maturity is not available.

8http://www.cboe.com/publish/skewtermstructure/skewtermstructure.csv 9https://www.cboe.com/micro/vix/vixwhite.pdf

10http://www.cboe.com/trading-tools/strategy-planning-tools/term-structure-data 11

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5

Results

Table I show the results of calibrating the SVCJ model using the VIX and SKEW from 1 June 2015 to 30 June 2015. For comparison, the parameters for 2 January 1996 to 31 December 2014 from CRZ are also shown. The forward price and time to maturity are set to $2,000 and 1/12, respectively. The interest rates are the average treasury bill rates over the calibration periods. The instantaneous variance value used is taken as the average of the time series of instant variances. For CRZ the instantaneous variance value has been arbitrarily set to 0.02. As the model used by CRZ has independent jump sizes, ρJ = 0.

[Insert Table I about here.]

The errors of the BKM skewness with parameters calibrated using 1 June 2015 to 30 June 2015 term structure data are shown in Table II. These empirical results are consistent with theory where smaller boundary controlling factors and/or smaller step sizes will reduce errors. With the values shown in Table II, the errors caused by truncation seem to be the greater source of error. Although the design of the boundary controlling factor allows for full control of truncation with a ∈ (0, 1), the step size cannot be tested as exhaustively. The inability to test the step size exhaustively does not detract from the results as the step size of strikes in the market does not extend to infinity and following theory, increasing the step size will only increase errors which is not the direction of this paper.

[Insert Table II about here.]

To obtain an error in skewness of less than 10−3,12 Table II shows that the step size

must be less than $5 (0.25% of the forward price) and the boundary controlling factor must be 0.60 or less. This result is inconsistent with the boundary controlling factor of 3/4 and step size of 1% from Aschakulporn and Zhang (2019). An increase in constraint

12 As the CBOE SKEW is reported to 2 decimal places and SKEW = 100 − 10 × skewness, an error

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conditions is somewhat expected as the mechanism in which skewness is introduced is not a single parameter. This inconsistency regarding the step size is most likely due to the oscillatory nature of its relationship with the error.

The strikes of options are densest near-the-money and strikes far away from the money tend to be sparser. Even though a step size of $4 would be sufficient to obtain an error of less than 10−3 if the boundary controlling factor is 0.60 – over the entire domain the

step size must all be $4. This does not reflect the non-constant step size of strikes in the market. Table III shows the errors when linear, cubic spline, and Gaussian kernel interpolation is applied. These interpolation methods are applied when the boundary controlling factor is 0.50 and interpolation is done to obtain a step size of $1.

[Insert Table III about here.]

In general, all interpolation methods reduce discretisation errors. Cubic spline inter-polation consistently outperforms both linear and Gaussian kernel interinter-polation.

Truncation is, in general, the greater source of error, therefore extrapolation techniques should be applied. Table IV shows the result of applying constant and linear extrapolation so that the boundary controlling factor becomes 0.50 (and the extrapolated region has a step size of $1). Both extrapolation methods, in general, do decrease errors, with the linear extrapolation approach yielding smaller errors especially when the boundary controlling factor is large. However, linear interpolation is not stable and can introduce errors of its own. Constant interpolation is able to consistently reduce errors.

[Insert Table IV about here.]

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lower to satisfy the 10−3 requirement – this is more consistent with Aschakulporn and Zhang (2019).

[Insert Tables V to VII about here.]

Overall, past market conditions show that directly applying the BKM risk-neutral skewness estimator to obtain SKEW without interpolation and extrapolation will unlikely be accurate. Applying a combination of cubic spline interpolation to $1 and constant interpolation to have strike prices from half the forward price to double the forward price (a boundary controlling factor of 0.50) should consistently reduce errors to be less than 10−3.

6

Conclusion

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Appendix

A

BKM Derivation

The standardised skewness is given by Equation (1) when n = 3. Expanding this, the BKM formula for risk-neutral skewness can be found. Similarly, the standardised kurtosis can be found when n = 4. BKM defines µ, V , W , and X as EtQ[Rτ], EtQ[e

−rτ R2τ], EtQ[e−rτR3 τ], and E Q t [e −rτR4 τ], respectively. Skewness = E Q t h (Rτ− µ)3 i n EtQh(Rτ − µ)2 io32 (A.1) = E Q t [R3τ − 3µR2τ + 3µ2Rτ− µ3] n EtQ[R2 τ − 2µRτ + µ2] o32 (A.2) = e W − 3µeV + 2µ3 [erτV − µ2]32 (A.3) Kurtosis = E Q t h (Rτ − µ) 4i n EtQh(Rτ − µ) 2io2 (A.4) = E Q t [R4τ − 4µR3τ + 6µ2R2τ− 4µ3 + µ4] n EtQ[R2 τ− 2µRτ + µ2] o2 (A.5) = e X − 4µeW + 6µ2eV − 3µ4 [erτV − µ2]2 (A.6)

The (annualized) variance σ2 is given by

σ2 = 1 τE Q t h (Rτ − µ) 2i = e V − µ2 τ (A.7)

The volatility (V ), cubic (W ), and quartic (X) contracts are not standard. To find their values, Carr and Madan (2001) is used to replicate the contracts with a continuum of out-of-the-money (OTM) European options. The expected value of each contract is given by EtQhe−rτRnτi= Z ∞ 0 n K2 " (n − 1)  ln K St n−2 −  ln K St n−1# Q (K) dK (A.8)

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B

Exact Skewness

The skewness of the DPS double jump model can be found by building on the skewness of Heston (1993) and Bates (1996) (combination of Heston (1993) and Merton (1976)).

Beginning with the Heston (1993) model,

dSt St = µdt +vtdBtS, (B.1) dvt= κ (θ − vt) dt + σvvtdBtv, (B.2)

the exact skewness, by following Zhang et al. (2017), can be found to be

Skewness = −σv2AκB32 (B.3) where

A = 6e3κτvtσv3− 22e3κτσv3θ + 3e2κτvtσv3+ 15e2κτσ3vθ + 24eκτκ2vtρσv2τ − 12eκτκ2ρσ2 vτ θ − 12e κτκv tσv3τ + 6e κτκσ3 vτ θ + 36e κτκv tρσv2 − 24eκτκρσ2 vθ − 6e κτv tσv3+ 6e κτσ3 vθ − 3vtσv3+ σ 3 vθ − 24e κτκ2v tσv

+ 12eκτκ2σvθ − 48e3κτκ3vtρ + 96e3κτκ3ρθ + 24e3κτκ2vtσv− 60e3κτκ2σvθ + 48e2κτκ3vtρ − 96e2κτκ3ρθ + 48e2κτκ2σvθ − 6e2κτκ2vtσv3τ

2+ 6e2κτκ2σ3

2θ

+ 48e3κτκ2vtρ2σv− 144e3κτκ2ρ2σvθ + 6e3κτκσv3τ θ − 36e

3κτ

κvtρσ2v + 120e3κτκρσ2vθ − 48e2κτκ2vtρ2σv + 144e2κτκ2ρ2σvθ − 6e2κτκvtσ3 + 18e2κτκσ3vτ θ − 48e3κτκ4ρτ θ − 96e2κτκρσ2vθ + 24e3κτκ3σvτ θ + 48e2κτκ4vtρτ − 48e2κτκ4ρτ θ − 48e2κτκ3vtσvτ + 48e2κτκ3σvτ θ − 24e2κτκ4v

2σvτ2+ 24e2κτκ4ρ2σvτ2θ + 48e3κτκ3ρ2σvτ θ + 24e2κτκ3vtρσ2

2 − 24e2κτκ3ρσ2 2θ − 36e3κτκ2ρσ2 vτ θ − 48e 2κτκ3v 2σvτ + 96e2κτκ3ρ2σvτ θ + 48e2κτκ2vtρσ2vτ − 96e 2κτ κ2ρσv2τ θ B = −8e2κτκ2ρσvτ θ + 8e2κτκ3τ θ + 2e2κτκσ2vτ θ + 8e κτ κ2vtρσvτ − 8eκτκ2ρσ vτ θ − 8e2κτκvtρσv+ 16e2κτκρσvθ − 4eκτκvtσv2τ + 4eκτκσv2τ θ + 8e2κτκ2vt − 8e2κτκ2θ + 2e2κτv 2v − 5e 2κτσ2 vθ + 8e κτκv tρσv− 16eκτκρσvθ − 8eκτκ2vt + 8eκτκ2θ + 4eκτσv2θ − 2vtσv2+ σ 2

Including jumps in the stock price (by adding Merton (1976) to Heston (1993)) gives the Bates (1996) model:

dSt St = rdt +vtdBtS+ (e x− 1) dNt− λE (ex− 1) dt dvt = κ (θ − vt) dt + σvtdBtv (B.4)

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be found to be (with the same A and B) Skewness = −σv 1 16κ5e −3κτA + C n 1 3e−2κτB + D o3/2 (B.5) where C = 3µxσ2x+ µ 3 x  λτ D =µ2x+ σ2xλτ

Finally, by adding correlated jumps in volatility to get the DPS model (Equation (5)), the skewness can be found to be

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(29)

Tables

Table I: Model Parameters.

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Table II: Estimation Errors.

This table shows the error of the BKM skewness estimator for various boundary controlling factors, a, and step sizes, ∆K. The options created with the SVCJ model have the following parameters: FT

t = 2,000, r = 0.0077%, τ = 1/12, vt = 0.020267, θ = 0.062058,

κ = 1.1792, σv = 0.70346, ρ = −0.6151, µx = −3.4756 × 10−05, σx = 0.52545, µv = 0.014357, λ = 2.5766 × 10−07, and ρJ = −3.2537 × 10−05. These parameters have been calibrated using the CBOE VIX and SKEW data from 1 June 2015 to 30 June 2015. Except r which was the average treasury bill rate over the same period and both FT

t and

τ were arbitrarily chosen.

The error for each a and ∆K is defined as Error := Estimated Value − True Value. The errors have been scaled by a factor of 1,000. The true skewness is -1.1995.

∆K 1 2 3 4 5 10 20 30 40 50 a Risk-Neutral Skewness ×1,000 0.50 0.05 0.18 -0.12 0.69 1.07 4.25 16.86 -15.73 65.51 100.15 0.60 0.05 0.18 -0.11 0.69 1.07 4.26 16.87 -10.08 65.51 100.16 0.75 3.22 3.35 3.05 3.86 4.24 7.42 20.03 -6.81 -31.66 103.28 0.80 17.81 17.93 17.64 18.44 18.82 21.99 34.55 2.50 82.98 117.47 0.82 33.40 33.52 33.73 34.03 34.41 37.56 50.05 70.46 98.21 22.24 0.84 60.58 60.70 60.42 61.20 61.58 64.69 77.05 51.85 124.68 7.43 0.86 106.17 106.29 106.01 106.78 107.15 110.18 122.28 92.49 168.94 73.38 0.88 179.58 179.70 179.86 180.17 180.49 183.42 194.83 214.06 238.94 205.38 0.90 292.63 292.73 292.34 293.00 293.33 296.05 306.83 287.13 345.88 375.47 0.92 456.93 456.64 456.82 457.02 456.51 458.84 463.31 444.99 498.95 420.98 0.94 676.92 675.53 675.64 672.74 674.48 667.94 674.37 684.94 699.43 473.44 0.95 803.37 800.71 800.55 800.88 803.72 790.53 794.82 689.55 763.18 824.11

Table III: Error Mitigation using Interpolation.

This table shows the result of implementing linear, cubic spline, and Gaussian kernel interpolating to have a step size of ∆K = 1. The boundary controlling factor has been set to a = 0.50. The options created with the SVCJ model have the following parameters:

FtT = 2,000, r = 0.0077%, τ = 1/12, vt = 0.020267, θ = 0.062058, κ = 1.1792, σv = 0.70346, ρ = −0.6151, µx = −3.4756 × 10−05, σx = 0.52545, µv = 0.014357, λ = 2.5766 × 10−07, and ρJ = −3.2537×10−05. These parameters have been calibrated using the CBOE VIX and SKEW data from 1 June 2015 to 30 June 2015. Except r which was the average treasury bill rate over the same period and both FT

t and τ were arbitrarily chosen. The error for each a and ∆K is defined as Error := Estimated Value − True Value. The errors have been scaled by a factor of 1,000. The true skewness is -1.1995.

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Table IV: Error Mitigation using Extrapolation.

This table shows the result of implementing constant and linear extrapolation to have a boundary controlling factor of a = 0.50. The step size has been set to ∆K = 1. The options created with the SVCJ model have the following parameters: FT

t = 2,000,

r = 0.0077%, τ = 1/12, vt = 0.020267, θ = 0.062058, κ = 1.1792, σv = 0.70346,

ρ = −0.6151, µx = −3.4756 × 10−05, σx = 0.52545, µv = 0.014357, λ = 2.5766 × 10−07, and ρJ = −3.2537 × 10−05. These parameters have been calibrated using the CBOE VIX and SKEW data from 1 June 2015 to 30 June 2015. Except r which was the average treasury bill rate over the same period and both FT

t and τ were arbitrarily chosen. The error for each a and ∆K is defined as Error := Estimated Value − True Value. The errors have been scaled by a factor of 1,000. The true skewness is -1.1995.

Risk-Neutral Skewness ×1,000 Extrapolation

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Table V: Estimation Errors.

This table shows the error of the BKM skewness estimator for various boundary controlling factors, a, and step sizes, ∆K. The options created with the SVCJ model have the following parameters: FT

t = 2,000, r = 2.4642%, τ = 1/12, vt = 0.02, θ = 0.037923,

κ = 1.0869, σv = 0.5792, ρ = −0.8343, µx = −0.014, σx = 0.0306, µv = 0.0128, and

λ = 0.68973. These parameters were calibrated by Cao, Ruan, and Zhang (2020) using

data from 2 January 1996 to 31 December 2014. The interest was the average treasury bill rates over the same period. vt, FtT, and τ were arbitrarily set. There is no correlation between jump sizes so ρJ = 0.

The error for each a and ∆K is defined as Error := Estimated Value − True Value. The errors have been scaled by a factor of 1,000. The true skewness is -1.369.

∆K 1 2 3 4 5 10 20 30 40 50 a Risk-Neutral Skewness ×1,000 0.50 0.93 2.18 3.52 4.93 -3.57 0.16 14.92 -22.87 71.79 112.54 0.60 0.94 2.18 -1.48 4.94 -3.57 0.16 14.92 -17.01 71.79 112.54 0.75 3.53 4.78 1.12 7.53 -0.96 2.76 17.52 -14.31 -43.11 115.13 0.80 16.54 17.78 19.10 20.50 12.09 15.81 30.53 -6.76 87.23 127.86 0.82 30.99 32.22 30.83 34.92 26.59 30.29 44.95 68.89 101.43 14.94 0.84 56.76 57.96 54.44 60.62 52.44 56.11 70.65 40.40 126.64 -10.76 0.86 100.83 102.00 103.25 104.59 96.65 100.25 114.55 79.48 169.63 53.92 0.88 173.07 174.18 172.92 176.65 169.09 172.58 186.35 209.04 239.46 195.20 0.90 286.43 287.46 284.42 289.68 282.74 286.03 299.07 274.36 348.57 384.61 0.92 456.85 457.64 458.70 459.61 453.37 456.30 466.55 439.91 511.20 435.52 0.94 701.74 701.91 701.15 702.33 698.22 697.45 706.15 720.41 739.92 555.30 0.95 854.32 853.62 854.17 854.75 852.24 846.98 853.05 776.71 816.13 894.77

Table VI: Error Mitigation using Interpolation.

This table shows the result of implementing linear, cubic spline, and Gaussian kernel interpolating to have a step size of ∆K = 1. The boundary controlling factor has been set to a = 0.50. The options created with the SVCJ model have the following parameters:

FT

t = 2,000, r = 2.4642%, τ = 1/12, vt = 0.02, θ = 0.037923, κ = 1.0869, σv = 0.5792,

ρ = −0.8343, µx = −0.014, σx = 0.0306, µv = 0.0128, and λ = 0.68973. These parameters were calibrated by Cao, Ruan, and Zhang (2020) using data from 2 January 1996 to 31 December 2014. The interest was the average treasury bill rates over the same period. vt,

FtT, and τ were arbitrarily set. There is no correlation between jump sizes so ρJ = 0. The error for each a and ∆K is defined as Error := Estimated Value − True Value. The errors have been scaled by a factor of 1,000. The true skewness is -1.369.

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Table VII: Error Mitigation using Extrapolation.

This table shows the result of implementing constant and linear extrapolation to have a boundary controlling factor of a = 0.50. The step size has been set to ∆K = 1. The options created with the SVCJ model have the following parameters: FT

t = 2,000,

r = 2.4642%, τ = 1/12, vt = 0.02, θ = 0.037923, κ = 1.0869, σv = 0.5792, ρ = −0.8343, µx = −0.014, σx = 0.0306, µv = 0.0128, and λ = 0.68973. These parameters were calibrated by Cao, Ruan, and Zhang (2020) using data from 2 January 1996 to 31 December 2014. The interest was the average treasury bill rates over the same period. vt,

FtT, and τ were arbitrarily set. There is no correlation between jump sizes so ρJ = 0. The error for each a and ∆K is defined as Error := Estimated Value − True Value. The errors have been scaled by a factor of 1,000. The true skewness is -1.369.

Risk-Neutral Skewness ×1,000 Extrapolation

References

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