Measures of Model Risk in Continuous-time Finance Models
aa r X i v : . [ ec on . E M ] O c t Measures of Model Risk in Continuous-timeFinance Models
Emese Lazar ∗ , Shuyuan Qi † , Radu Tunaru ‡ October 29, 2020
Abstract
Measuring model risk is required by regulators on financial and insurance markets.We separate model risk into parameter estimation risk and model specification risk, andwe propose expected shortfall type model risk measures applied to L´evy jump modelsand affine jump-diffusion models. We investigate the impact of parameter estimationrisk and model specification risk on the models’ ability to capture the joint dynamics ofstock and option prices. We estimate the parameters using Markov chain Monte Carlotechniques, under the risk-neutral probability measure and the real-world probabilitymeasure jointly. We find strong evidence supporting modeling of price jumps.
Keywords : jumps, MCMC, model specification risk, parameter estimation risk, stochasticvolatility.
JEL Classification Codes : C11, C52, C58 ∗ ICMA Centre, Henley Business School, University of Reading, Whiteknights, Reading, RG6 6BA, UnitedKingdom; [email protected]. † ICMA Centre, Henley Business School, University of Reading, Whiteknights, Reading, RG6 6BA, UnitedKingdom; [email protected]. ‡ Corresponding author, University of Sussex Business School, University of Sussex, Brighton, BN1 9SL,UK; [email protected].
Introduction
Model risk is currently considered one of the most overlooked risks faced by financial firms.The Basel Committee on Banking Supervision (2009), Federal Reserve Board of Governors(2011) and European Banking Authority (2012) require banks to measure and report modelrisk as with any other type of risk. The sources of model risk are parameter estimation risk(PER) and model specification risk (MSR). The specification of models includes identifyingand modeling decisive factors that can jointly describe the dynamics of an economic asset.PER denotes the risk of inaccurate estimation of parameters for a given model. The PERor MSR are the two components of the total model risk (TMR). The majority of studies use point-wise estimation methods and consider model risk asmodel mispricing, thus ignoring parameter estimation risk. Kerkhof et al. (2010) is onenotable exception where model risk is separated into PER and MSR, but their approachrelies on gaussian returns and the asymptotic distribution of a function of the parame-ters. PER can be captured via Bayesian estimation methods. Jacquier and Jarrow (2000)study the PER of the Black and Scholes model. Furthermore, Jacquier et al. (2002) applyBayesian estimators for stochastic volatility models and find that the Bayesian approachproduces more robust results by comparison with the moments and likelihood estimators.There are many other advantages of using Markov chain Monte Carlo (MCMC) techniquesin extracting inference on continuous-time models in finance that have been highlightedin a series of works by Eraker (2001), Polson and Stroud (2003), Jacquier et al. (2007),Johannes and Polson (2010) and Yu et al. (2011).In this paper, we propose an expected shortfall (ES) based method to measure modelrisk. This ES-type model risk measurement is potentially superior in capturing modelrisk because it is able to capture model tail risk. We provide an applicable framework toseparate and measure PER and MSR for a very competitive class of option pricing models. In this paper, we use the following abbreviations: PER = parameter estimation risk; MSR = modelspecification risk; TMR= total model risk; ES = expected shortfall; VaR = value-at-risk; AJD = affinejump-diffusion; MCMC = Markov chain Monte Carlo; SV = stochastic volatility; SVJ = stochastic volatilitywith Merton jumps in returns; SVCJ = stochastic volatility with contemporaneous jumps in returns andvolatility; SVVG = stochastic volatility with variance-gamma jumps in returns; SVLS = stochastic volatilitywith log-stable jumps in returns; MJD = Merton jump-diffusion; SND = standard normal distribution; ATM= at-the-money; KS = Kolmogorov-Smirnov; DIC = deviance information criterion; log-BF = log values ofthe Bayes factors; PE = pricing error; APE = absolute pricing error; DM = Diebold and Mariano; CW =Clark and West.
2n addition, we disentangle the model risk for buyers and sellers and we highlight that thetwo parties in options contracts are exposed to model risk asymmetrically.We apply our new methodology to several different option pricing models with respectto their modeling ability to explain S&P 500 spot prices and option prices. The candidatemodels that we investigate have different specifications: the constant volatility model withMerton jumps (Merton, 1976); the pure stochastic volatility controlled by a mean-reversionprocess; as well as the stochastic volatility with affine jump-diffusion (AJD) or L´evy jumps.Consistent with the findings of Yu et al. (2011), we find that the log-stable jumps modelwith stochastic volatility process has the smallest TMR. Moreover, the TMR of AJD modelsis mainly attributed to PER, and their MSR is less than that of L´evy jump models. Theinfinite-activity L´evy jumps capture many small jumps in the index returns that cannotbe captured by AJD models. Therefore, L´evy jump models may face less MSR duringturbulent periods when small jumps are frequent; however, this also restricts their abilityin capturing both physical and risk-neutral dynamics when the market is calm and thereare far fewer jumps. In our paper, we find that the seller of vanilla European call options isexposed to a higher model risk than the buyer of those options when the market is volatile,which is in line with the conclusions of Green and Figlewski (1999).The remainder of the paper is organized as follows. In Section 2, we introduce themodel risk measurement framework; then we revisit all models that are investigated in thepaper in Section 3. Section 4 provides the details of the numerical methods applied in thiswork. An empirical analysis is presented in Section 5 whilst in Section 6 we analyze theforecasting of model risk. The last section summarizes our conclusions.
Most literature in the area of model risk of continuous-time models concern measuringTMR. Routledge and Zin (2009) distinguish between market risk and model risk; theymeasure market risk by risk-neutral pricing and measure model risk by using the worst-case approach. Further examples are the coherent and convex risk measures in Cont (2006).Another innovative work on model risk is Lindstr¨om et al. (2008), where parameter uncer-tainty is taken into account with a revised risk-neutral valuation formula. Furthermore,3etering and Packham (2016) measure the model risk of option pricing models using ahedging portfolio argument. Coqueret and Tavin (2016) show that model selection canlead to a significant effect on the final results since the discrepancy in prices leads to modelrisk. Moreover, Chen and Hong (2011) emphasize the importance of model specification;they develop an omnibus specification test for continuous-time models and study the AJDand L´evy processes specifically.Using a Bayesian approach, one can obtain the estimated posterior distribution of anasset price, denoted as e F t ( H ; M (Θ) , D , K ). In this case H is the option (call or put)conditional on model M with parameter vector Θ at time t , given an observed dataset D covering the historical series of the option prices and underlying asset observations.For clarity we also insist on the notational K for different computational methodologies(including estimation, calibration and pricing). The posterior distribution of option pricesis produced by the uncertainty in the value of parameters Θ weighted by the combinationof prior assumptions on Θ and the likelihood coming out of historical data.We refer to the risk stemming from model specification as MSR, while the risk relatedto parameter estimation is referred to as PER. For a set of observed data D and a set ofconsistent methodologies K , we define PER, MSR and TMR as follows. Definition 1
For option H and model M with the vector of parameters Θ , the parameterestimation risk refers to the uncertainty in the values of parameters Θ obtained via theestimation process K given dataset D . A model can be misspecified for many reasons, for example, because it ignores significantfactors and fails to capture the features of the market fully; and/or its assumptions areunrealistic (e.g., the constant-volatility assumption in the Black-Scholes options pricingmodel). The MSR measures the risk due to the inherent weakness of the model itself toget the correct results.
Definition 2
For option H and model M with the vector of parameters Θ , the modelspecification risk of model M refers to the risk that, based on dataset D and methodologies K , the model is unable to produce the features of H . The TMR is defined as the sum of the two components. Using the information inthe option markets, Jarrow and Kwok (2015) point out that it is challenging to separate4odel specification risk from estimation risk. Given that our focus is on the model risk ofoption pricing models, under the physical and risk-neutral probability measures definingthe financial markets, we first introduce three properties that a valid model risk measurefor option pricing models should have: (1) time variability : model risk is time-varying; (2) symmetry of MSR : the MSR is the same for both long and short positions; (3) asymmetryof PER : the PER can be different for long and short positions. Our main objective is to measure PER and MSR, respectively, and then to comparethe model risk across different pricing models. Johannes and Polson (2010) state that themarginal posterior distribution through the Bayesian estimation characterizes the sampleinformation regarding the objective and risk-neutral parameters and quantifies the parame-ter estimation risk. Chung et al. (2013) take the posterior distribution through the Bayesianapproach as relevant for PER.Let b F represent an estimated price of the target option, the model price adjustedposterior distribution e Λ for long ( L ) and short ( S ) positions is defined as: f Λ t ( H , L ; M (Θ) , D , K ) = e F t ( H ; M (Θ) , D , K ) − b F t ( H ; M (Θ) , D , K ) , f Λ t ( H , S ; M (Θ) , D , K ) = b F t ( H ; M (Θ) , D , K ) − e F t ( H ; M (Θ) , D , K ) . (1)For example, if we consider the expected value of the posterior price distribution withrespect to Θ as the estimated price, then the above formulae can be expressed as: f Λ t ( H , L ; M (Θ) , D , K ) = e F t ( H ; M (Θ) , D , K ) − E Θ h e F t ( H ; M (Θ) , D , K i , f Λ t ( H , S ; M (Θ) , D , K ) = E Θ h e F t ( H ; M (Θ) , D , K ) i − e F t ( H ; M (Θ) , D , K ) . (2)Let V aR
P ERη, t ( H , ~ ; M (Θ) , D , K ) denote the value-at-risk (VaR) at a critical level η ∈ (0 , η quantile of the adjusted posteriordistribution f Λ t ( H , ~ ; M (Θ) , D , K ) computed in (2), where ~ = L for a long position and Theoretically, a short position for an option leads to a larger PER. As the option price ranges from 0to infinity, the left side is bounded while the right side is open, which might lead to a broader right tailof the posterior densities of the estimated price. Options are affected by the asymmetry between buyingand writing, in that the option buyer has liability limited to the amount invested, but the option writeris exposed to the risk of losses that can greatly exceed the initial premium received (Green and Figlewski,1999). e F denotes the estimated price distribution and b F is a point estimate. We consider η to be 5% in this paper. = S for a short position. The ES-type model risk measure for PER, at level η , foroption H , given a model M with parameter vector Θ, dataset D , methodology K , and along(short) position ~ is defined as: ρ P ERη, t ( H , ~ ; M (Θ) , D , K ) = 1 η Z η V aR
P ERx, t ( H , ~ ; M (Θ) , D , K ) dx. (3)The market price of the option at time t is C t ( H ). We can further compute the distri-bution of profit and loss from pricing with the model for both long and short positions byusing the linear functions below:Λ t ( H , L ; M (Θ) , D , K ) = e F t ( H ; M (Θ) , D , K ) − C t ( H ) , Λ t ( H , S ; M (Θ) , D , K ) = C t ( H ) − e F t ( H ; M (Θ) , D , K ) . (4)The VaR of the profit and loss distribution at level η of a long(short) position, de-noted by V aR η, t ( H , ~ ; M (Θ) , D , K ), is computed as the absolute value of the η quantile ofΛ t ( H , ~ ; M (Θ) , D , K ). The ES-type model risk measure of the TMR for option H , given amodel M with parameter vector Θ, dataset D , and methodology K is defined as: ρ T MRη, t ( H ; M (Θ) , D , K ) = R η V aR x, t ( H , L ; M (Θ) , D , K ) dx + R η V aR x, t ( H , S ; M (Θ) , D , K ) dx η . (5)The PER of option H , given a model M with parameter vector Θ, dataset D , andmethodology K is defined as the average of the PER values for long and short positions. ρ P ERη, t ( H ; M (Θ) , D , K ) = ρ P ERη, t ( H , L ; M (Θ) , D , K ) + ρ P ERη, t ( H , S ; M (Θ) , D , K )2 . (6)Then, the MSR of option H , given a model M with parameter vector Θ, dataset D ,and methodology K is measured as the difference between TMR and PER: ρ MSRη, t ( H ; M (Θ) , D , K ) = ρ T MRη, t ( H ; M (Θ) , D , K ) − ρ P ERη, t ( H ; M (Θ) , D , K ) . (7)Because of the symmetry of MSR and asymmetry of PER, the TMR for ~ is the sum6f PER for ~ and MSR, where ~ = L or ~ = S : ρ T MRη, t ( H , ~ ; M (Θ) , D , K ) = ρ P ERη, t ( H , ~ ; M (Θ) , D , K ) + ρ MSRη, t ( H ; M (Θ) , D , K ) . (8)Based on (6), (7) and (8), the following holds: ρ T MRη, t ( H ; M (Θ) , D , K ) = ρ T MRη, t ( H , L ; M (Θ) , D , K ) + ρ T MRη, t ( H , S ; M (Θ) , D , K )2 . (9)It is important to note that ρ T MRη, t ( H , ~ ; M (Θ) , D , K ) = η R η V aR x, t ( H , ~ ; M (Θ) , D , K ) dx ,unless C t ( H ) = E Θ h e F t ( H ; M (Θ) , D , K ) i , and that in this case ρ MSRη, t ( H ; M (Θ) , D , K ) = 0.Moreover, if the model price estimate of H is close enough to the market price, PER isthe main source of model risk. However, MSR is the primary source of model risk if themarket price of H is far from the model price estimate. This is a desirable property of ourproposed model risk measurement. This section describes the set of models that will be compared: the stochastic volatil-ity (SV) model, the stochastic volatility model with Merton jumps in returns (SVJ), thestochastic volatility model with contemporaneous jumps in returns and volatility (SVCJ),the stochastic volatility model with variance-gamma jumps in returns (SVVG) and thestochastic volatility model with log-stable jumps in returns (SVLS). While the model spec-ification is expressed under the physical measure P , it is the return dynamics under risk-neutral measure Q which is required for option pricing. The change of measure between P and Q for these models is also discussed in this section. Let Y t = ln( S t ) denote the logarithm of the asset price. The dynamics, for all models, ofthe continuously compounded return on the asset price under the real-world measure P is7iven by: dY t = µdt + p V t dW Yt ( P ) + dJ Yt ( P ) ,dV t = κ ( θ − V t ) dt + σ V p V t dW Vt ( P ) + dJ Vt ( P ) , (10)where W Yt ( P ) and W Vt ( P ) are standard Brownian motions under P with dW Yt ( P ) dW Vt ( P ) = ρdt , the correlation ρ provides the ability to capture the skewness of the returns’ distri-bution. A negative ρ captures the leverage effect; µ measures the mean return; V t is theinstantaneous variance of returns at time t ; κ represents the speed of mean reversion; θ denotes the long-run mean of the variance process; and σ V is the volatility of volatility.The SVJ and SVCJ are AJD models (Duffie et al., 2000) capturing continuous move-ments of assets with affine diffusions and large discontinuous jumps in asset returns with aPoisson process. The jump processes in AJD models are defined as dJ Yt ( P ) = ξ Y dN Yt and dJ Vt ( P ) = ξ V dN Vt where { N Yt } t ≥ and { N Vt } t ≥ are Poisson processes as in Duffie et al.(2000). The SVCJ model contains simultaneous correlated jumps, where N Yt = N Vt = N t ,in both the return and volatility processes with a constant intensity λ ; the jump size in thevariance process follows an exponential distribution, ξ V ∼ EXP ( µ V ), and the jump size inthe asset log-prices is conditionally normally distributed with ξ Y | ξ V ∼ N ( µ J + ρ J ξ V , σ J ). Thus, the mean of ξ Y is µ J + ρ J µ V and its variance is σ J + ρ J µ V . Moreover, ξ Y is correlatedwith ξ V by ρ J µ V / q σ J + ρ J µ V , the long-run mean-variance of the SVCJ model is θ + µ V λ/κ due to the jump component in the variance process. As explained in Ball and Torous(1985), the return distribution is an infinite mixture of normal distributions under thejump-diffusion models, which leads to an unbounded likelihood function. To circumvent thisissue, we assume that only one jump occurs per trading day. For SVJ, J Vt ( P ) = 0, and theprocess of the jump J Yt ( P ) has the same specification as SVCJ. For SV, J Yt ( P ) = J Vt ( P ) = 0.The AJD model is constructed based on Brownian motions and compound Poissonprocesses, which are just special cases of L´evy processes. AJD models only allow finite-activity jump processes, while the L´evy processes are more flexible, allowing them to achieve This is a standard setting; large movements in equity returns and large shifts in the variance are likelyto occur at the same time (Bardgett et al., 2019). Bates (2000) finds that the model with state-dependent intensities is significantly misspecified whilstAndersen et al. (2002) state that there is no evidence to support the time-varying intensity. EXP denotes the exponential distribution and N denotes the normal distribution. X V Gt ( σ, γ, ν ) = γG νt + σW G νt , (11)where { X V G } is an arithmetic Brownian motion with drift γ and volatility σ ; { G νt } t ≥ denotes the gamma process with unit mean rate and variance rate ν ; and { W t } t ≥ is astandard Brownian motion, which is independent of G νt . Setting J Yt ( P ) = X V Gt ( σ, γ, ν ) and J Vt ( P ) = 0 reduces (10) to SVVG.The SVLS model is an example of infinite-activity and infinite-variation jump model.The log-stable process follows an α -stable distribution ( S α ): X LSt ( α, σ ) − X LSs ( α, σ ) ∼ S α ( β, δ ( t − s ) α , γ ) , t > s, (12)where α ∈ (1 ,
2] is the tail index of the α -stable distribution which determines the shape ofthe stable distribution; β ∈ [ − ,
1] is the skew parameter determining the skewness of thedistribution; δ ≥ − γ ∈ R is the location parameter. We followCarr and Wu (2003) and set β = 0, δ = σ and γ = 0 and then (10) reduces to SVLS, if J Yt ( P ) = X LSt ( α, σ ) and J Vt ( P ) = 0. For Brownian motions, Pan (2002) proposes a standard practice for the change of measure: γ Yt = η s p V t ,γ Vt = − p − ρ (cid:18) ρη s + η v σ V (cid:19) p V t , (13)where γ Yt and γ Vt represent the market prices of risk of Brownian shocks to returns andvariance, respectively. The Brownian motions under Q in the return and variance processes9re: dW Yt ( Q ) = dW Yt ( P ) + γ Yt dt,dW Vt ( Q ) = dW Vt ( P ) + γ Vt dt. (14)For the variance process, following Pan (2002), Bates (2000) and Broadie et al. (2007),we apply the following theoretical restrictions in the change of measure such that bothphysical and risk-neutral probability densities are from the same family: κ P θ P = κ Q θ Q ; ρ P = ρ Q ; and σ P V = σ Q V . Moreover, for jump processes, the following restrictions are imposed: in SVJ, ( λ, µ J , σ J )are able to change between P and Q ; in SVCJ, ( λ, µ J , σ J , ρ J , µ V ) are able to change between P and Q ; in SVVG, γ and σ are able to change between P and Q , while ν remains unchangedunder P and Q ; in SVLS, no parameters of the log-stable process, ( α, β, σ, γ ), are allowedto change between P and Q .In AJD models, all parameters in the jump processes can be different under the physicaland risk-neutral measures. However, this leads to difficulty in econometric identification,as shown in Pan (2002) and Eraker (2004). To bypass this identification issue, they onlyenable µ J to change between measures. We adopt the same methodology here. Finally, thejump parameters under both measures for SVJ, SVCJ, SVVG and SVLS are ( λ, µ P J , σ J , µ Q J ),( λ, µ P J , σ J , µ Q J , µ V , ρ J ), ( ν, γ P , σ P , γ Q , σ Q ) and ( α, σ ), respectively.Under the framework described above, the Radon-Nikodym derivatives of theses pro-cesses can be expressed as: d Q d P (cid:12)(cid:12)(cid:12)(cid:12) t = exp (cid:26) − Z t γ Ys dW Ys ( P ) − Z t γ Vs dW Vs ( P ) − (cid:20)Z t ( γ Ys ) ds + Z t ( γ Vs ) ds (cid:21)(cid:27) exp( U t ) . (15) κ Q = κ P − η v and θ Q = κ P θ P κ Q , we use κ and θ to represent κ P and θ P in this paper. Moreover, forsimplicity, we use ρ to denote ρ P and ρ Q , and use σ V to represent σ P V and σ Q V . U t in (15) is defined in the second part of Sato et al. (1999)’s theorem. As jump processes are restrictedto follow the same processes between measures, the U t ’s of models with different jump specifications areconsidered. In this case, models also differ in terms of Radon-Nikodym derivatives. Additionally, in thispaper we use both e and exp to denote the exponential function. dY t = (cid:18) r t − V t + Φ J ( − i ) (cid:19) dt + p V t dW Yt ( Q ) + dJ Yt ( Q ) ,dV t = [ κ ( θ − V t ) + η v V t ] dt + σ V p V t dW Vt ( Q ) + dJ Vt ( Q ) , (16)where r t is the risk-free rate, Φ J ( − i ) is the jump component, and the expressions of Φ J ( (cid:5) )for different models in this study are documented in Appendix A. Naturally, the drift termof the return process under P can be derived as µ = r t − V t + Φ J ( − i ) + η s V t .When the interest rate is constant, the option price can be deduced from the closed-formsolution to the characteristic function of the log stock price under Q : φ ( t, u ) = exp [ iuY + iu ( r + Φ J ( − i )) t ] exp ( − t Φ J ( u )) exp ( − b ( t, u ) V − c ( t, u )) , (17)where κ M ( u ) = κ − η v − iuσ V ρ ; δ ( u ) = q ( κ M ( u )) + ( iu + u ) σ V ; Y = ln( S ) denoteslog-spot price; V represents the initial variance; b ( t, u ) = ( iu + u )(1 − e − δ ( u ) t )( δ ( u )+ κ M ( u ))+( δ ( u ) − κ M ( u )) e ( − δ ( u ) t ) ;and c ( t, u ) = κθσ V h δ ( u ) − ( δ ( u ) − κ M ( u ))(1 − e − δ ( u ) t )2 δ ( u ) + ( δ ( u ) − κ M ( u )) t i .Taking τ as the time to expiration, one can then price a European call option withstrike K , using the formula below (Yu et al., 2011): F ( Y , V , τ, K ) = E Q [ e − rτ ( S τ − K ) + ] = e − rτ π × Re (cid:18)Z ∞ e − ix ln( K ) φ ( τ, x − i ) − x + ix dx (cid:19) . (18) The models analyzed in this paper are quite complex so parameter estimation may not bestraightforward. One problem is that the latent variables, such as stochastic volatility andsize and arrival rates of jump processes, are difficult to track. L´evy processes themselvesare complex and many of them do not lead to closed-form option pricing formulae. Li et al.(2008) extend the application of the MCMC method to L´evy processes under the real-worldprobability measure ( P ) for spot prices. Then, Yu et al. (2011) further apply the MCMCmethod to L´evy processes under both P and Q probability measures with spot prices andoption prices. Following their work, we summarize the joint dynamics of the daily spot and11he option prices upon discretization as follows: C t +1 − F t +1 = ρ c ( C t − F t ) + σ c ǫ ct ,Y t +1 = Y t + (cid:18) r t − V t + ψ Q J ( − i ) + η s V t (cid:19) ∆ + p V t ∆ ǫ Yt +1 + J Yt +1 ,V t +1 = V t + κ ( θ − V t )∆ + σ V p V t ǫ Vt +1 + J Vt +1 , (19)where ∆ is the one day time interval here; ǫ ct , ǫ Yt +1 and ǫ Vt +1 follow the standard normaldistribution (SND hereafter), ǫ Yt +1 and ǫ Vt +1 are correlated with correlation ρ and are in-dependent from ǫ ct ; C t represents the market option price at time t ; and F t denotes themodel option price at time t given by (18). Following Eraker (2004), we assume that C t +1 − F t +1 ∼ N (cid:0) ρ c ( C t − F t ) , σ c (cid:1) .Let Θ denote the parameter vector of the models. We split the parameters into fourgroups, that is Θ = { (Θ P ) , (Θ Q ) , (Θ risk premia ) , (Θ pricing errors ) } ; the first group contains pa-rameters under P ; the second one includes parameters that are unique under Q ; the risk pre-mia of return and variance are in the third group while the fourth part contains parametersused to describe the option pricing errors of models. The SV model has no parameters thatare unique under the risk-neutral measure, it has Θ = { ( κ, θ, σ v , ρ ) , () , ( η s , η v ) , ( ρ c , σ c ) } ; forSVJ, Θ = { ( κ, θ, σ v , ρ, λ, σ J , µ P J ) , ( µ Q J ) , ( η s , η v ) , ( ρ c , σ c ) } ; for SVCJ, Θ = { ( κ, θ, σ v , ρ, λ, σ J , ρ J ,µ P J , µ V ) , ( µ Q J ) , ( η s , η v ) , ( ρ c , σ c ) } ; for SVVG, Θ = { ( κ, θ, σ v , ρ, ν, γ P , σ P ) , ( γ Q , σ Q ) , ( η s , η v ) , ( ρ c , σ c ) } ;and SVLS has Θ = { ( κ, θ, σ v , ρ, α, σ ) , () , ( η s , η v ) , ( ρ c , σ c ) } .Given the log stock prices Y = { Y t } Tt =0 , the option prices C = { C t } Tt =0 , the variancevariables V = { V t } Tt =0 , the jumps times/sizes J = { J t } Tt =0 , the posterior of parameters andlatent variables can be decomposed into the product of individual conditionals: p (Θ , V, J | Y, C ) ∝ p ( Y, C, V, J,
Θ) = p ( C | Y, V, Θ) p ( Y, V | J, Θ) p ( J | Θ) p (Θ) , (20)where p ( C | Y, V,
Θ) = T − Y t =0 √ πσ c exp (cid:26) − [( C t +1 − F t +1 ) − ρ c ( C t − F t )] σ c (cid:27) . (21)For the SVVG model, a time-changing variable G = { G t } Tt =0 , where G t +1 ∼ G ( ∆ ν , ν ), is G denotes the Gamma distribution. p (Θ , V, J, G | Y, C ) ∝ p ( Y, C, V, J, G,
Θ) = p ( C | Y, V, Θ) p ( Y, V | J, Θ) p ( J | G, Θ) p ( G | Θ) p (Θ) . (22)For the SVLS model, an auxiliary variable series U = { U t } Tt =0 is added: p (Θ , V, J, U | Y, C ) ∝ p ( Y, C, V, J, U,
Θ) = p ( C | Y, V, Θ) p ( Y, V | J, Θ) p ( J, U | Θ) p (Θ) . (23)It is difficult to simulate random draws directly from the joint posterior densities ofthe models shown above; instead, we estimate parameters and latent variables by simulat-ing from complete conditional distributions of each parameter and latent variable with theMCMC method. The complete conditional distributions of AJD models and L´evy processesunder P can be found in several earlier studies, but few investigate the estimation of theparameters by a Bayesian approach under Q . Broadie et al. (2007) simulate posterior dis-tributions of Θ P with derived complete conditional distributions and then calibrate modelswith the estimated parameters under P (represented by c Θ P ) to obtain the values of Θ Q based on the following objective function:Θ Q = arg min T X t =1 O t X n =1 [ IV t ( K n , τ n , S t , r t , V t ) − IV t (Θ Q | c Θ P , K n , τ n , S t , r t , V t )] , (24)here O t is the number of options at time t ; S t , V t and r t denote the spot price, instantvariance and risk-free rate at time t , respectively; K n and τ n represent the strike price andexpiration of the n -th option; and IV is the implied volatility. This is a two-step estimationmethod. A disadvantage of this method is that only the mean values of the posterior of theparameters under P are considered when calibrating the models, overlooking other possiblevalues in the posterior distribution of parameters under P . This two-step estimation methodproduces an interval estimation (under P ) plus a point estimation (under Q ), which isnot ideal for taking parameter estimation risk into consideration. A different approachis presented by Yu et al. (2011) who derive the complete conditional distribution of eachindividual parameter and latent variable under both measures, enabling the simulation ofposterior samples of parameters and latent variables with the MCMC method. For Θ P , theyapply almost the same method with complete conditional distributions as Broadie et al.132007), after which the random draws are accepted/rejected with the Damlen, Wakefildand Walker method (Damlen et al., 1999) based on the likelihood value calculated with(21) whilst the Metropolis-Hasting algorithm is used to estimate Θ Q . In the two-stepmethod of Broadie et al. (2007), Θ P are estimated from spot prices and Θ Q are calibratedwith c Θ P and option data whereas Yu et al. (2011) do these in one step, and parameters areestimated with both spot price and option data. Combining the methods of Yu et al. (2011) and Broadie et al. (2007), we estimate Θ P with only spot prices based on the MCMC methods introduced in Li et al. (2008) while theother parameters are estimated with the method of Yu et al. (2011). Our method has lesscomputational burden than Yu et al. (2011)’s approach and, compared with Broadie et al.(2007), our method estimates risk-neutral parameters and real-world parameters jointly, sothat the PER can be captured fully. More detailed discussions of our MCMC methods aredocumented in Appendix B. The empirical analysis is based on the S&P 500 index spot price and the correspondingS&P 500 index call option prices from January 3, 1996 to December 29, 2017. We followYu et al. (2011) who choose one short-term at-the-money (ATM) call option each day. Theoption is required to have a time-to-expiration between 20 and 50 days and its strike tospot price ratio is closest to 1. We directly use the ATM-forward call options with 30days to expiry as our options data. The ATM-forward option dataset is downloaded inthe Std Option Price file from Option Metrics. This file contains information on ATM-forward options with expiration ranges from 30 to 730 calendar days, and the 30-days-to-expiration call options fully match Yu et al. (2011)’s requirements. The forward price ofunderlying on the expiration date of the option is calculated with the zero-coupon yieldcurve and projected dividends; the strike price of the option equals the forward price; theimplied volatility and premium on these standardized options are calculated daily usinglinear interpolation from the volatility surface, which is computed with a kernel smoothing Candidate points of Θ P are generated based on the posterior distribution with spot prices and only thepoints that also fit option prices are accepted. In addition to the stochastic volatility models, we also use the constant volatility model withMerton jumps proposed by Merton (1976) (MJD) for comparison. The SVCJ model nestsMJD by setting V t +1 = V t = ( σ Q MJD ) , where σ Q MJD is the constant volatility of MJD underthe risk-neutral measure. We employ MCMC to estimate Θ = { (Θ P ) , (Θ Q ) , (Θ risk premia ) , (Θ pricing errors ) } . Table 1 presents the annualized parameter estimates, specifically theestimated posterior mean of the parameters, and their corresponding standard deviation(in parenthesis).The estimated volatility of MJD ( σ Q MJD ) is 0.1149. The estimates of parameters relatedto the variance process of stochastic volatility models are relatively similar. The reversionspeed, κ , ranges from 3.9114 for SVVG to 6.6665 for SVCJ, the faster reversion speedof SVCJ can be explained by the jumps in the variance process, which require a higherreversion speed to drive the variance to the long-run mean level. The estimates of theaverage variance for SV, SVJ, SVVG and SVLS are θ , which are approximately 0.033; thelong-run mean-variance level of SVCJ is θ + µ V λ/κ = 0 . σ V and ρ are also quite close for all models and consistent with existing studies. The estimatedjump intensity, λ , of MJD is 54.1371, indicating about 54 price jumps per annum with verysmall mean jump sizes (-0.0024 for µ P J and almost 0 for µ Q J ). By contrast, jumps in priceare infrequent events in SVJ and SVCJ (both reveal about two jumps per year). The meanjump sizes for both SVJ and SVCJ under the real-world measure ( µ P J ) are negative and themean jump sizes under the risk-neutral measure ( µ Q J ) have larger absolute values. SVJ andSVCJ capture the price movements with infrequent large jumps while L´evy jump models15able 1: Parameter Estimates of Various Models MJD SV SVJ SVCJ SVVG SVLS σ Q MJD κ θ σ V ρ -0.8173 -0.7750 -0.7642 -0.8701 -0.8440(0.0513) (0.0386) (0.0578) (0.0344) (0.0342) η s η v -19.8169 -16.4552 -17.7829 -16.3290 -17.8977(0.9664) (1.2666) (1.2619) (1.0617) (1.1024) ρ c σ c λ µ P J -0.0024 -0.0120 -0.0138(0.0008) (0.0040) (0.0058) µ Q J -0.0003 -0.0872 -0.1150(0.0009) (0.0181) (0.0154) σ J µ V ρ J -0.0023(0.0011) ν γ P -0.1559(0.0573) σ P γ Q -0.3185(0.0863) σ Q α σ are able to capture both infrequent large jumps and frequent small jumps. According to ourresults, the jump distribution under the physical measure for SVVG is negatively skewed, γ P is -0.1559, while the risk-neutral jump distribution is also negatively skewed with evenlarger jump sizes.For a goodness-of-fit comparative analysis, we apply the Kolmogorov-Smirnov (KS) test16able 2: KS and Abadie’s Test Results MJD SV SVJ SVCJ SVVG SVLSPanel A. Return ResidualsKS statistics 0.0419 0.0362 0.0321 0.0291 0.0453 0.0493KS p-values 0.0000 0.0000 0.0000 0.0002 0.0000 0.0000Abadie p-values 0.0002 0.0062 0.0193 0.0384 0.0003 0.0001Panel B. Volatility ResidualsKS statistics 0.0319 0.0378 0.0289 0.0294 0.0226KS p-values 0.0000 0.0000 0.0002 0.0001 0.0068Abadie p-values 0.0183 0.0035 0.0213 0.0299 0.0990NOTE: This table provides the KS statistics and corresponding p-values for return residuals (for all models) and forvolatility residuals (of stochastic volatility models). The p-values calculated with Abadie’s bootstrap method are alsoreported. with the improved adjustment of Abadie (2002) to the return residuals (for all models) andthe volatility residuals (for stochastic volatility models). The results are reported in Table2. According to these, the KS tests reject the null hypothesis that the residuals of allmodels follow the SND; by contrast, Abadie’s test fails to reject the null hypothesis forthe volatility residuals of SVLS. Abadie’s p-value of the SVCJ return residuals is relativelylarge at 0.0384, indicating that the specification of jumps in volatility improves modelingperformance.Moreover, we apply the deviance information criterion (DIC) (Spiegelhalter et al., 2002and Berg et al., 2004) and Bayes factors (Kass and Raftery, 1995 and Chib et al., 2002)to compare models from a Bayesian inference perspective. The DIC values are shownin the last row of Table 1; a smaller DIC value indicates a better fit of the model to theindex returns. MJD has the lowest DIC value because it generates many jumps (over54 jumps per year) to capture the abnormal movements of market prices. For stochasticvolatility models, SV is outperformed by all other stochastic volatility models with jumps.This is consistent with Chernov et al. (2003); the authors state the tradeoffs among variousmodel specifications and find that stochastic volatility and jumps in returns are crucialspecifications for affine models. Except for MJD and SV, the L´evy jump models (SVVGand SVLS) have the smallest DIC values, followed by SVCJ and SVJ. Table 3 reports the logvalues of the Bayes factors (log-BF) of the SVLS, SVVG, SVCJ, SVJ and SV models overthe MJD, SV, SVJ, SVCJ and SVVG models, respectively. A negative log-BF indicates theunderperformance of the column model. Based on the log-BF values, MJD outperforms all For the definition of the DIC and Bayes factors see the Supplementary Appendix.
SVLS SVVG SVCJ SVJ SVMJD -791.75 -722.04 -1008.28 -1062.31 -1152.43SV 360.69 430.39 144.14 90.12SVJ 270.57 340.27 54.03SVCJ 216.54 286.24SVVG -69.71NOTE: Log values of the Bayes factors of the SVLS, SVVG, SVCJ, SVJ and SV models over the MJD, SV, SVJ,SVCJ and SVVG models, respectively.
Figure 1 shows the 5% to 95% quantiles of estimated prices, after burn-in during theestimation process for all models. The MJD model cannot accurately capture the dynamicsof option prices due to the inherent inability of MJD in modeling time-varying varianceprocesses. By contrast, the estimated prices of all stochastic models have a similar trendto the market prices. For stochastic volatility, models with jumps can capture the volatileoption prices well, as the peaks of market prices are almost all in the 5% to 95% quantileof estimated prices of these models. Most of market prices lie in the estimated prices rangeof SVJ due to wide credibility intervals. Although with a narrower range, the estimatedprices of SVCJ also cover the market prices well, except for the apparent mispricing before2010. SVVG has excellent performance during market turmoil; however, it fails to capturethe market dynamics during tranquil periods (the period after 2004 until the financial crisisand the period after 2016); by contrast, SVLS performs well during these tranquil periods.Besides, SV reveals a comparatively better pricing ability in 2017.The model risk and pricing performance of models are reported in Table 4. TMR,PER and MSR are defined in (5), (6) and (7), respectively. According to Panel A, SVhas the largest TMR, followed by SVVG and MJD. However, the dominant model risk ofSVVG is PER while the dominant model risk of SV and MJD is MSR. SVLS has the lowestTMR, followed by SVCJ and SVJ. The TMR of these two AJD models mostly consists ofPER; moreover, SVCJ has lower PER and greater MSR than SVJ. Panel B reports themean values of the percentage model risk, which are calculated as the values of model risk18igure 1:
Call Option Prices under Various Models.This figure presents the 5% to 95% quantiles of estimated prices distribution; the black line is themarket price of options as provided by Option Metrics. The results are based on daily spot priceson the S&P 500 and daily prices of standardized at-the-money-forward call options with 30 days toexpiration between January 3, 1996 and December 29, 2017. Estimated prices are updated withinthe estimation process at the end of each iteration after burn-in. divided by option prices. Panel C reports the PER of long and short positions; this showsthat the mean values of PER of long positions are lower than those of short positions forL´evy jump models and SV, while the PER of a long position for AJD models and MJD isslightly higher than that of a short position. As reported in Panel D, AJD models have thelowest pricing errors (PE, APE and APE(%)); followed by L´evy jump models and MJD.SV reveals the worst pricing performance.In order to analyze the model risk of models under different market periods, we splitthe sample period into seven time windows by detecting abrupt changes in the impliedvolatility of the standardized ATM options based on the method proposed by Killick et al.(2012). Using average values of implied volatility, we identify the following time windows:period 1 ○ : January 3, 1996 to June 4, 1997; period 2 ○ : June 5, 1997 to October 13, 2003;period 3 ○ : October 14, 2003 to July 24, 2007; period 4 ○ : July 25, 2007 to September 23,2008; period 5 ○ : September 24, 2008 to May 1, 2009; period 6 ○ : May 4, 2009 to June 29,2012; and period 7 ○ : July 2, 2012 to December 29, 2017.19able 4: Model Risk and Pricing Performance of Models SVLS SVVG SVCJ SVJ SV MJDPanel A. Model Risk of ModelsTMR 6.0562 8.3272 6.2299 6.3251 8.7559 8.2853PER 2.7469 4.5345 4.6207 4.9616 3.2687 1.4138MSR 3.3092 3.7927 1.6092 1.3635 5.4872 6.8715Panel B. Model Risk of Models (%)TMR(%) 21.81% 35.15% 24.15% 24.71% 32.57% 29.18%PER(%) 10.73% 18.74% 18.80% 20.39% 12.91% 5.77%MSR(%) 11.07% 16.41% 5.34% 4.32% 19.66% 23.41%Panel C. PER of Long and Short PositionsLong 2.5957 4.1590 4.6669 4.9995 3.0368 1.4255Short 2.8982 4.9100 4.5746 4.9237 3.5006 1.4021Panel D. Pricing PerformancePE 4.6671 -3.8331 0.9257 1.0386 8.8167 4.0031APE 5.7338 7.4055 5.1075 4.8393 8.8407 8.2018APE(%) 20.22% 31.47% 19.50% 18.43% 32.79% 28.81%NOTE: Results are based on the daily spot price on the S&P 500 and daily price of standardized at-the-money-forward call options with 30 days to expiry between January 3, 1996 and December 29, 2017. Panel A presents meanvalues of model risk; Panel B shows the mean values of the percentage model risk, calculated as model risk over optionprice; Panel C displays PER of long and short positions; Panel D presents pricing performance, where PE denotespricing error, APE represents absolute pricing error, and APE(%) is the percentage absolute error.
The values of the mean and standard deviation of the implied volatility for the stan-dardized ATM options during these seven time windows are reported in Panel A of Table 5.Period 3 ○ is the most tranquil period, followed by periods 7 ○ and 1 ○ , whose mean impliedvolatility values are all below 0.2. During these three periods the market is stable. Incontrast, the market is turbulent during periods 6 ○ , 4 ○ and 2 ○ . Period 2 ○ is very long andcovers the 1997 Asian financial crisis, the Russian financial crisis that hit on August 17,1998, the Brazilian currency crisis from 1998 to 1999, the dot-com crash from March 11,2000, to October 9, 2002, the 1998-to-2002 Argentine great depression, the 911 (September11, 2001) and the WorldCom accounting scandal in 2002; period 4 ○ involves the subprimecrisis; and period 6 ○ captures the European credit crisis since the end of 2009. Period 5 ○ marks the well-know Financial Crisis of 2008, when the mean implied volatility reaches0.4486 with a standard deviation of 0.0966.Figure 2 illustrates the model risk estimates of our models. The black columns mark thePER of models while the grey columns are the values of the TMR of models; the differencesbetween grey and black columns being the MSR. The MJD has the smallest PER andremarkably high MSR. SV reveals very large MSR during volatile periods 2 ○ , 4 ○ and 6 ○ ,and relatively small MSR during tranquil periods 1 ○ , 3 ○ and 7 ○ . Stochastic volatility20able 5: Statistics on Implied Volatility and the Differences between the PER of Long andShort Positions Period 1 ○ ○ ○ ○ ○ ○ ○ Panel A. The Mean and Standard Deviation of the Implied Volatility of the Standardized ATM OptionsMean 0.1568 0.2265 0.1244 0.2182 0.4486 0.2080 0.1272Std 0.0236 0.0484 0.0207 0.0351 0.0966 0.0557 0.0336Panel B. Mean Differences between the PER of Long and Short PositionsMJD 0.0122** 0.0199** 0.0215** 0.0240** 0.0151** 0.0205** 0.0342**SV -0.2101** -0.3627** -0.4606** -0.3355** -0.3412** -0.4233** -0.7126**SVJ 0.0568** -0.0224* 0.1729** -0.0715** 0.1547** -0.0126 0.2006**SVCJ 0.1331** -0.1416** 0.2406** -0.4108** 0.0108 -0.0074 0.4233**SVVG -0.4528** -0.5240** -0.8259** -0.3697** -0.3366** -0.7433** -1.1700**SVLS -0.1368** -0.3580** -0.1373** -0.7723** -0.3568** -0.3039** -0.2879**NOTE: Panel A reports the mean and standard deviation of the implied volatility of the Standardized ATM optionsin different periods; Panel B reports the mean values of the differences between the PER of long and short positionsfor all models under different time windows. * and ** indicate values significant at 5% and 1% significance levels,respectively. models with jumps have a considerably smaller MSR risk during market turmoil (periods2 ○ , 4 ○ and 6 ○ ) compared with SV. Besides, AJD models and SVLS also have minor MSRin stable periods. By contrast, SVVG reveals large MSR during the tranquil periods 3 ○ and7 ○ . However, during periods 2 ○ , 4 ○ and 6 ○ , characterized by a turbulent market, SVVGhas less MSR compared with other stochastic volatility models. These findings supportadding jumps to stochastic volatility models, especially when the market is volatile. Allmodels reveal substantial model risk during period 5 ○ , the Financial Crisis. The SV model,with no jumps, has the worst performance compared with other stochastic volatility modelswith jumps. It is worth noting that the sizes of the MSR of all models with jumps reveala distinct spike at the end of the research period, while the SV performs very well duringthis period when the S&P 500 index rises considerably in 2017 with few big swings. Panel B of Table 5 reports the means of the differences between the PER of long andshort positions in different periods. For the MJD model, the long position always exhibitsa significantly higher PER than the short position; by contrast, for stochastic volatilitymodels, a short position tends to have a higher PER during turbulent periods 2 ○ , 4 ○ and6 ○ , although the difference is not significant for AJD models in period 6 ○ . A short positionalways reveals a higher PER for L´evy jump models. By contrast, a long position is morelikely to bear higher model risk for AJD models. L´evy jump models and SV reveal larger More results are available in the Supplementary Appendix. The figure of the PER of long and short positions of models can be found in the SupplementaryAppendix.
Model Risk of Models.This figure presents the TMR (grey columns) and PER (black columns) of models. The results arebased on the daily spot price on the S&P 500 and daily price of standardized at-the-money-forwardcall options with 30 days to expiration between January 3, 1996 and December 29, 2017. differences between long and short positions compared with AJD models and MJD.
We examine the performance of models in capturing the risk-neutral dynamics by testingthe null hypothesis that the squared pricing errors of the two models are equal, using theDiebold and Mariano (1995) (DM) tests for non-nested models and the Clark and West(2007) (CW) tests for nested models. Table 6 reports DM and CW statistics for squaredoption pricing errors in the Squared Pricing Error panel. Rejections of the null hypothesisare denoted using stars; for the DM tests, a positive (negative) value indicates that thecolumn model has significantly larger (smaller) squared pricing errors than the correspond-ing row model. According to the DM tests, although the MJD model has excellent abilityto capture index dynamics under the physical measure, it fails to capture the risk-neutraldynamics. The squared pricing errors of MJD are significantly greater than those of SVLS;moreover, SVLS reveals significantly greater squared pricing errors than SVJ. In addition, See the Supplementary Appendix for a presentation of the DM and CW tests.
Squared Pricing Errors MSR(%) MSRPanel A. DM statisticsSVLS SVVG SV SVLS SVVG SV SVLS SVVG SVSVVG -0.9785 -1.6237 -0.7288SVCJ 1.3130 1.5761 3.5859** 2.2955* 5.4824** 1.8497*SVJ 1.8409* 1.6051 4.6174** 2.2979* 4.6850** 1.8859*SV -8.2820** 0.0341 -8.1634** 0.7090 -8.4180** -0.3409MJD -3.5977** -0.4073 -0.9574 -4.7764** -0.3775 -2.1789* -3.8131** -2.0102* -2.5578**Panel B. CW statisticsSVCJ SVJ SVLS SVVG SVCJ SVJ SVLS SVVG SVCJ SVJ SVLS SVVGSVJ 3.7216** 2.7498** 3.3602**SV 0.6632 0.4290 -2.8533 1.7723* -0.7047 -2.0032 -3.2299 2.1046* -3.3965 -3.5998 -4.5702 1.5062MJD -4.1330 -4.1802 -4.4129 -4.3658 -3.6706 -3.3786NOTE: Panel A reports DM statistics for squared pricing errors, MSR(%), and MSR of SVLS, SVVG and SV toother corresponding non-nested models. * and ** indicate values significant at 10% and 2% significance levels for thetwo-sided test, respectively; Panel B reports the CW statistics of SVCJ, SVJ, SVLS and SVVG to other correspondingnested models. * and ** indicate values significant at 5% and 1% significance levels for the one-sided test, respectively. we test the null hypothesis that the MSR(%) and MSR of the two models are equal. MJDhas significantly higher MSR(%) than SVLS and SV; as well as greater MSR compared withSVLS, SVVG and SV. The MSR(%) and MSR of AJD models are significantly less thanthose of L´evy jump models. SV has significantly greater squared pricing errors, MSR(%)and MSR than SVLS. For the CW tests, based on Panel B, SVCJ is rejected comparedwith SVJ in all terms. This indicates that there is no evidence of jumps in the volatilityprocess. Moreover, SVVG is also rejected compared with SV in terms of squared pricingerrors and MSR(%).
In this section, we highlight the necessity of measuring the PER and MSR separately interms of explaining absolute pricing errors. We also consider forecasting model risk usingmarket data.Let ǫ t ( H ; M (Θ) , D , K ) = | b F t ( H ; M (Θ) , D , K ) − C t ( H ) | represent the absolute pricingerror of option H conditional on model M with the parameter vector Θ, observed dataset D , and methodology K at time t . We run the following regression to explore the explanatorypower of the PER and MSR in explaining absolute pricing errors. ǫ t ( H ; M (Θ) , D , K ) = β + β ρ P ERη, t ( H ; M (Θ) , D , K ) + β ρ MSRη, t ( H ; M (Θ) , D , K ) + ε t . (25)23ere, we focus on testing whether β = β , as this would prove that it is not necessary toseparate the PER and MSR from TMR. Taking β − β = α and using (7), (25) can berewritten as: ǫ t ( H ; M (Θ) , D , K ) = β + αρ P ERη, t ( H ; M (Θ) , D , K ) + β ρ T MRη, t ( H ; M (Θ) , D , K ) + ε t . (26)A test of α = 0 in (26) is a test of β = β in (25). The regression results are reportedin Table 7. All α ’s are statistically significant, which supports the necessity of measuringPER and MSR separately. Table 7: Regression Analysis with PER and TMR
C PER TMR Adj. R MJD -0.11** -0.04** 1.01** 1SV -0.10** -0.04** 1.04** 0.98SVJ -0.68** -0.48** 1.25** 0.84SVCJ -0.66** -0.41** 1.23** 0.87SVVG -0.10* -0.44** 1.14** 0.98SVLS -0.38** -0.21** 1.11** 0.97NOTE: The regression results are based on Equation (26). * and ** indicate values significant at 5% and 1%significance levels, respectively.
To understand the structure of model risk, we appeal to a forecasting regression to in-vestigate the association between model risk and several market risk factors. The regressionis based on: ρ MRη, t ( H ; M (Θ) , D , K ) = β + β IV t − + β C P V t − + β DDelta t − + β Gamma t − + β T heta t − + β H L t − + β DSP XV t − + β C P P t − + ε t , (27) ρ MR can be PER, MSR and L S, where L S denotes the difference between the PER of longand short positions. IV is the implied volatility of 30-days-to-expiration ATM-forward calloptions of the S&P 500 Index; C P V is the difference between daily trading volumes of calloptions and put options on the S&P 500 Index over 100,000;
DDelta t = Delta t − Delta t − ;and Delta , Gamma and
T heta are the “greek” sensitivities associated with option prices of30-days-to-expiration ATM-forward call options on the S&P 500 Index, the sizes of
T heta are adjusted by dividing by 100;
H L is the difference between S&P 500 Index High price More results are reported in the Supplementary Appendix.
DSP XV t = SP XV t − SP XV t − , and SP XV is the trading volume ofS&P 500 Index over 100,000,000;
C P P is the difference between 30-days-to-expirationATM-forward call option price and put option price of the S&P 500 Index. Regression results are presented in Table 8. The predictive power of these variables forPER is quite high for all models. The adjusted R of the MJD PER reaches 80%, followedby SVVG (69%), SVJ (65%) and SVLS (61%). IV , C PV , Gamma and
Theta are significantin forecasting the PER of all six models. The PER of all models is negatively related to theprevious day’s IV , C PV , Gamma and
Theta . PER is also inversely linked to
DDelta and
DSPXV . The estimated coefficients of IV , C PV , “greeks” (
DDelta , Gamma and
Theta )and
DSPXV indicate that the implied volatility level, the difference between call and putoptions trades, the sensitivity of the option prices and the number of the underlying tradesare negatively linked to the PER of models.
C PP has a significant effect on the PER ofstochastic volatility models while
H L reveals a significant effect on the PER of SV, SVJ,SVCJ and SVLS.Compared with PER, the MSR is more difficult to predict. The adjusted R is higherfor the MSR of MJD and SVLS (62% and 56% respectively), and lower for SV and SVJat just above 46%; it drops remarkably for SVCJ (38%) and it is only 14% for SVVG. IV , C PV and
DSPXV are significant in predicting MSR of all models. MSR tends to increasewith IV , except for SVVG, whose MSR decreases with IV . Moreover, except for SV, theMSR of all other five models shows a significantly negative relationship with C PV . TheMSR of all models decreases with
DSPXV , indicating that an increase in the number ofunderlying trades decreases the MSR. Option “greeks” have significant power in forecastingthe MSR of MJD.Most of the L S of MJD can be predicted with these factors with an adjusted R valueof 84%, followed by SVVG (32%) and SVLS (17%). The values of adjusted R for L S areonly around 8% for SV and AJD models. IV , Theta and
C PP have significant power inforecasting the L S for all six models. IV , C PV and
DSPXV are negatively linked to theL S of MJD and AJD models and positively linked to the L S of SV, SVVG and SVLS. The Based on the Dickey-Fuller test, the raw series of
Delta and
SP XV are non-stationary, so we computethe differences between consecutive observations of
Delta and
SP XV ; all independent variables in (27) arestationary. We also assess the strength of collinearity among independent variables in (27) with Belsleycollinearity diagnostics, the largest condition index is 11.7834, which does not exceed the tolerance, 30.Thus, there is no evidence of multicollinearity.
C IV C PV DDelta Gamma Theta H L DSPXV C PP Adj. R Panel A. MJDPER 2.42** -5.70** -0.09** -3.14 -83.63** -0.30** 0.05 -0.06** 0.25 0.80MSR -12.67** 51.51** -0.75** -258.87** 298.83** -3.98** 4.01** -0.61** -23.04** 0.62L S 0.60** -2.57** -0.06** -8.36** -39.33** -0.39** 0.24** -0.06** -0.57** 0.84Panel B. SVPER 3.74** -6.65** -0.08** -118.52** -158.29** -1.07** -0.24* -0.32** -1.45** 0.58MSR 0.63 23.97** 0.07** -60.21 -229.39** -1.29** 0.08 -0.72** -3.04 0.46L S -18.80** 67.22** 0.59** -166.43 1088.37** 7.67** 0.70 3.24* 48.65** 0.08Panel C. SVJPER 8.79** -20.28** -0.33** -65.80** -294.09** -0.95** -0.59** -0.37** 2.76** 0.65MSR -5.27** 24.82** -0.26** -9.66 159.58** -0.25* 1.68** -0.28* -7.08** 0.47L S 14.08** -14.29** -0.20 -290.64* -529.40** 3.45** -6.47** -0.61 15.66* 0.06Panel D. SVCJPER 7.37** -16.95** -0.29** -54.65** -249.62** -0.95** 0.73** -0.04 2.96** 0.56MSR -4.36** 23.33** -0.31** -23.31 110.74** -0.41** -0.55 -0.41* -8.22** 0.38L S 22.34** -43.31** -1.13** 35.71 -849.71** 3.31** -19.05** -3.55** -46.89** 0.08Panel E. SVVGPER 8.42** -18.72** -0.32** -33.11* -277.10** -0.63** -0.15 -0.28** 3.19** 0.69MSR 4.11** -12.34** -1.13** -30.67 47.03 -0.02 1.21 -0.62* 4.03 0.14L S -47.53** 100.65** 1.85** 106.19 2527.45** 2.43** -6.83** 1.82* -27.19** 0.32Panel F. SVLSPER 2.52** -6.06** -0.07** -64.86** -108.56** -1.18** 0.31** -0.20** -4.24** 0.61MSR -4.51** 35.09** -0.12** -123.56** 5.41 -0.79** -0.47 -0.67** -4.15* 0.56L S 14.41** 19.79** 0.65** -27.19 -215.56 14.38** -9.22** 2.06 82.10** 0.17NOTE: The regression results are based on Equation (27). * and ** indicate values significant at 5% and 1%significance levels, respectively. coefficients of
Theta for the L S of stochastic volatility models are all positive. Moreover,the L S of SV, SVJ and SVLS increase with
C PP while the L S of MJD, SVCJ and SVVGdecrease with
C PP .Overall, regardless of the model, IV , C PV , Gamma , Theta have statistically significantpower in forecasting PER; IV , C PV and
DSPXV are powerful in predicting MSR; besides, IV , Theta and
C PP can help in predicting L S. IV is the most effective factor in forecastingmodel risk, coming out as significant in all regressions, while C PV and
Theta are also usefulin forecasting model risk.
In this paper, we propose an ES-type model risk measure which is able to estimate PERand MSR of continuous-time pricing models. We then apply this measurement to L´evy26ump models and AJD models to investigate to what extent the MSR and PER affect themodels’ ability to capture the joint dynamics of stock and option prices. Building on theapproaches of Broadie et al. (2007) and Yu et al. (2011), we develop an effective MCMCmethod to jointly estimate parameters and latent variables with both stock and optionprices.We show that the PER is dominant for stochastic volatility models. The introductionof jumps increases PER and decreases MSR. We find that SVLS has the smallest TMR.SVVG has less MSR and PER when the market is turbulent, but its model risk soars whenthe market is calm. By contrast, AJD models and SVLS tend to have slightly higher modelrisk than SVVG under volatile periods while they carry small MSR in tranquil periods.We also find that short positions tend to have a higher model risk when using stochasticvolatility models for call options when the market is turbulent. We further show that AJDmodels have significantly lower MSR than L´evy jump models, but the option pricing errorsof L´evy jump models and AJD models reveal few significant differences.We find that market risk factors can predict more than half of the PER of models. PERdecreases with the previous day’s implied volatility, underlying trading volume and optiongreeks. The MSR of AJD models and L´evy jump models is difficult to predict. The previousday’s implied volatility, the difference between trading volumes of call and put options andthe theta sensitivity of options are the most decisive factors in forecasting model risk.Our results highlight that it is necessary to measure PER and MSR separately, as thetwo components of TMR. A model with small MSR is not necessarily the most desirable ifthe parameters are likely to be very difficult to estimate due to the complexity of the model.Similarly, a model with low PER tends to be simple but it is likely to have large MSR. Thebest models have a tradeoff between PER and MSR. A distinction between PER and MSRcan enhance model risk management. In addition, model risk can influence model parametervalues, and this dependence can be explicitly modeled, as in Bollerslev et al. (2016); infuture research, it would be interesting to explore the dependence of parameter values onPER and MSR separately. Furthermore, non-parametric models can learn from the modelrisk of parametric models, and their pricing performance can be improved as suggested byFan and Mancini (2009). It remains an open question how the pricing performance can beimproved using information on MSR and PER.27 eferences
Abadie, A., 2002. Bootstrap tests for distributional treatment effects in instrumental vari-able models. Journal of the American Statistical Association 97, 284–292.Andersen, T. G., Benzoni, L., Lund, J., 2002. An empirical investigation of continuous-timeequity return models. Journal of Finance 57, 1239–1284.Ball, C. A., Torous, W. N., 1985. On jumps in common stock prices and their impact oncall option pricing. Journal of Finance 40, 155–173.Bardgett, C., Gourier, E., Leippold, M., 2019. Inferring volatility dynamics and risk premiafrom the S&P 500 and VIX markets. Journal of Financial Economics 131, 593–618.Basel Committee on Banking Supervision, 2009. Revisions to the Basel II market riskframework. Bank for International Settlements.Bates, D. S., 2000. Post-’87 crash fears in the S&P 500 futures option market. Journal ofEconometrics 94, 181–238.Berg, A., Meyer, R., Yu, J., 2004. Deviance information criterion for comparing stochasticvolatility models. Journal of Business and Economic Statistics 22, 107–120.Bollerslev, T., Patton, A. J., Quaedvlieg, R., 2016. Exploiting the errors: A simple approachfor improved volatility forecasting. Journal of Econometrics 192, 1–18.Broadie, M., Chernov, M., Johannes, M., 2007. Model specification and risk premia: Evi-dence from futures options. Journal of Finance 62, 1453–1490.Carr, P., Wu, L., 2003. The finite moment log stable process and option pricing. Journal ofFinance 58, 753–777.Chen, B., Hong, Y., 2011. Generalized spectral testing for multivariate continuous-timemodels. Journal of Econometrics 164, 268–293.Chernov, M., Gallant, A. R., Ghysels, E., Tauchen, G., 2003. Alternative models for stockprice dynamics. Journal of Econometrics 116, 225–257.28hib, S., Nardari, F., Shephard, N., 2002. Markov chain Monte Carlo methods for stochasticvolatility models. Journal of Econometrics 108, 281–316.Chung, T.-K., Hui, C.-H., Li, K.-F., 2013. Explaining share price disparity with parameteruncertainty: Evidence from Chinese A-and H-shares. Journal of Banking and Finance37, 1073–1083.Clark, T. E., West, K. D., 2007. Approximately normal tests for equal predictive accuracyin nested models. Journal of Econometrics 138, 291–311.Cont, R., 2006. Model uncertainty and its impact on the pricing of derivative instruments.Mathematical Finance 16, 519–547.Coqueret, G., Tavin, B., 2016. An investigation of model risk in a market with jumps andstochastic volatility. European Journal of Operational Research 253, 648–658.Damlen, P., Wakefield, J., Walker, S., 1999. Gibbs sampling for Bayesian non-conjugate andhierarchical models by using auxiliary variables. Journal of the Royal Statistical Society:Series B (Statistical Methodology) 61, 331–344.Detering, N., Packham, N., 2016. Model risk of contingent claims. Quantitative Finance16, 1357–1374.Diebold, F. X., Mariano, R. S., 1995. Comparing predictive accuracy. Journal of Businessand Economic Statistics 13, 253–263.Duffie, D., Pan, J., Singleton, K., 2000. Transform analysis and asset pricing for affinejump-diffusions. Econometrica 68, 1343–1376.Eraker, B., 2001. MCMC analysis of diffusion models with applications to finance. Journalof Business and Economic Statistics 19, 177–191.Eraker, B., 2004. Do stock prices and volatility jump? Reconciling evidence from spot andoption prices. Journal of Finance 59, 1367–1403.Eraker, B., Johannes, M., Polson, N., 2003. The impact of jumps in volatility and returns.Journal of Finance 58, 1269–1300. 29uropean Banking Authority, 2012. Discussion paper on draft regulatory technical stan-dards on prudent valuation, under Article 100 of the draft Capital Requirements Regu-lation.Fan, J., Mancini, L., 2009. Option pricing with model-guided nonparametric methods.Journal of the American Statistical Association 104, 1351–1372.Federal Reserve Board of Governors, 2011. Supervisory guidance on model risk manage-ment. Federal Reserve SR Letter 11-7 Attachment.Green, T. C., Figlewski, S., 1999. Market risk and model risk for a financial institutionwriting options. Journal of Finance 54, 1465–1499.Jacquier, E., Jarrow, R., 2000. Bayesian analysis of contingent claim model error. Journalof Econometrics 94, 145–180.Jacquier, E., Johannes, M., Polson, N., 2007. MCMC maximum likelihood for latent statemodels. Journal of Econometrics 137, 615–640.Jacquier, E., Polson, N. G., Rossi, P. E., 2002. Bayesian analysis of stochastic volatilitymodels. Journal of Business and Economic Statistics 20, 69–87.Jarrow, R., Kwok, S. S. M., 2015. Specification tests of calibrated option pricing models.Journal of Econometrics 189, 397–414.Johannes, M., Polson, N., 2010. MCMC methods for continuous-time financial economet-rics. In:
Handbook of Financial Econometrics: Applications , Elsevier, pp. 1–72.Kass, R. E., Raftery, A. E., 1995. Bayes factors. Journal of the American Statistical Asso-ciation 90, 773–795.Kerkhof, J., Melenberg, B., Schumacher, H., 2010. Model risk and capital reserves. Journalof Banking and Finance 34, 267–279.Killick, R., Fearnhead, P., Eckley, I. A., 2012. Optimal detection of changepoints with alinear computational cost. Journal of the American Statistical Association 107, 1590–1598. 30i, H., Wells, M. T., Yu, C. L., 2008. A Bayesian analysis of return dynamics with L´evyjumps. Review of Financial Studies 21, 2345–2378.Lindstr¨om, E., Str¨ojby, J., Brod´en, M., Wiktorsson, M., Holst, J., 2008. Sequential calibra-tion of options. Computational Statistics and Data Analysis 52, 2877–2891.Madan, D. B., Carr, P. P., Chang, E. C., 1998. The variance gamma process and optionpricing. Review of Finance 2, 79–105.Merton, R. C., 1976. Option pricing when underlying stock returns are discontinuous.Journal of Financial Economics 3, 125–144.Pan, J., 2002. The jump-risk premia implicit in options: Evidence from an integratedtime-series study. Journal of Financial Economics 63, 3–50.Polson, N., Stroud, J., 2003. Bayesian inference for derivative prices. In:
Bayesian Statistics ,Oxford University Press, pp. 641–650, 7th ed.Routledge, B. R., Zin, S. E., 2009. Model uncertainty and liquidity. Review of EconomicDynamics 12, 543–566.Sato, K.-i., Ken-Iti, S., Katok, A., 1999. L´evy processes and infinitely divisible distributions.Cambridge, UK: Cambridge University Press.Spiegelhalter, D. J., Best, N. G., Carlin, B. P., Van Der Linde, A., 2002. Bayesian measuresof model complexity and fit. Journal of the Royal Statistical Society: Series B (StatisticalMethodology) 64, 583–639.Yu, C. L., Li, H., Wells, M. T., 2011. MCMC estimation of L´evy jump models using stockand option prices. Mathematical Finance 21, 383–422.31 ppendix A The Jump Characteristic Component and Pri-ors for Model Parameters
The jump component Φ J ( u ) in the characteristic function of different models is providedbelow.SVCJ: Φ J ( u ) = λ (cid:18) − exp ( iuµ Q J − σ J u ) − iuµ V ρ J − iuµ V (cid:19) .The expression of Φ J ( u ) for SVJ can be derived by taking µ V = 0 in the equation above.SVVG: Φ J ( u ) = ln (cid:18) − iuγ Q ν + ( σ Q )2 νu (cid:19) ν .SVLS: Φ J ( u ) = ( σ | u | ) α (cid:0) i × sign( u ) × tan (cid:0) πα (cid:1)(cid:1) , where sign( u ) extracts the sign of u . The priors of parameters for all models are detailed here. We choose uninformativeand conditionally conjugate priors whenever possible. Priors for common parameters: κ ∼ N (0 , κ> , θ ∼ N (0 , θ> , ρ ∼ U ( − , σ v ∼ IG (2 . , . η s ∼ N (0 , η v ∼ N (0 , ρ c ∼ N (0 , σ c ∼ IG (2 . , . µ P J ∼ N (0 , µ Q J ∼ N (0 , σ J ∼ IG (10 ,
40) and λ ∼ B (2 , µ V ∼ IG (10 ,
20) and ρ J ∼ N (0 , γ P ∼ N (0 , γ Q ∼ N (0 , ν ∼ IG (10 , σ P ) ∼ IG (2 . , .
1) and ( σ Q ) ∼ IG (2 . , . α ∼ U (1 ,
2) and σ αα − | α ∼ IG (2 . , . N refers to the Normal distribution, IG refers to the Inverse Gamma distribution, U refers toa standard Uniform distribution, and B to a Beta distribution. These priors have similarvalues to those used in Eraker et al. (2003), Li et al. (2008) and Yu et al. (2011). We alsoinvestigate various other priors, and the result does not change significantly. Appendix B MCMC methods for Parameter Estimation
This section introduces the updating algorithms and posterior distributions of model pa-rameters and latent variables.We focus on SVCJ at first, the estimation of SVJ can be derived by setting µ V = 0while taking λ = 0 leads to the estimation of SV. MCMC methods for estimating uniqueparameters and latent variables in SVVG and SVLS are also introduced in this section.Posterior for κ : κ | Y, V, J Y , Θ \{ κ } ∼ N (cid:16) AB , q B (cid:17) κ> , where B = ∆(1 − ρ ) σ v P T − t =0 ( θ − V t ) V t +32; A = σ v (1 − ρ ) P T − t =0 ( θ − V t ) Vt +1 − Vtσv − ρH t +1 V t ; and H t +1 = Y t +1 − Y t − (cid:0) r t − V t + Φ J ( − i ) + η s V t (cid:1) ∆ − J Yt +1 .Posterior for θ : θ | Y, V, J Y , J V , Θ \{ θ } ∼ N (cid:16) AB , q B (cid:17) θ> , where B = κ ∆(1 − ρ ) σ v P T − t =0 1 V t +1; A = κσ v (1 − ρ ) P T − t =0 Dt +1 σv − ρH t +1 V t ; D t +1 = V t +1 + ( κ ∆ − V t − J Vt +1 ; and H t +1 = Y t +1 − Y t − ( r t − V t + Φ J ( − i ) + η s V t )∆ − J Yt +1 .Posterior for σ v : The conditional posteriors of σ v cannot be written as a standard distri-bution; to overcome this difficulty, we apply an independent Metropolis-Hasting algorithm.The conditional posterior of σ v , conditional on Y V , J Y , J V , and Θ \{ σ v } , is proportional to( σ v ) T + c ∗ +22 × π ( σ v ) × exp n − C ∗ σ v o , and π ( σ v ) := exp n − P T − t =0 ( D t +1 − ρσ v ( H t +1 )) (1 − ρ ) σ v V t ∆ o , where c ∗ = 2 .
5, and C ∗ = 0 . σ v ; D t +1 = V t +1 − V t − κ ( θ − V t )∆ − J Vt +1 and H t +1 = Y t +1 − Y t − (cid:0) r t − V t + Φ J ( − i ) + η s V t (cid:1) ∆ − J Yt +1 . Althoughthe posterior distribution is non-standard, it becomes a one-dimensional inverted-Gammadistribution if there is no leverage effect ( ρ = 0). Therefore, we set the proposal den-sity as: σ v | Y, V, J V , Θ \{ σ v } ∼ IG ( A , B ), where A = c ∗ + T ; B = C ∗ + P T − t =0 D t +1 V t ∆ ; and D t +1 = V t +1 − V t − κ ( θ − V t )∆ − J Vt +1 . For a given previous draw σ ( g ) v , we draw σ ( g +1) v fromthe proposal density function, and then accept σ ( g +1) v with probability min (cid:18) π ( σ ( g +1) v ) π ( σ ( g ) v ) , (cid:19) .Posterior for ρ ∝ π ( ρ ) := (1 − ρ ) − T exp {− − ρ ) P T − t =0 ( H t +1 + D t +1 ) + ρ − ρ P T − t =0 H t +1 D t +1 } , where D t +1 = V t +1 − V t − κ ( θ − V t )∆ − J Vt +1 σ v √ V t ∆ and H t +1 = Y t +1 − Y t − ( r t − V t +Φ J ( − i )+ η s V t ) ∆ − J Yt +1 √ V t ∆ .The estimation is updated using following algorithm: (1) Draw ln ρ ( g +1) − ρ ( g +1) from N (cid:16) ln ρ r − ρ r , T − (cid:17) ,where ρ r = Corr ( D , H ), where D = { D t +1 } T − t =0 , H = { H t +1 } T − t =0 , and Corr denotes cor-relation; (2) accept ρ g +1 with min π ( ρ ( g +1) ) π ( ρ ( g ) ) × exp − ( f ( ρ ( g ) ) − f ( ρr ) ) T − ! exp − ( f ( ρ ( g +1) ) − f ( ρr ) ) T − ! , , where f ( ρ ) = ln ρ − ρ .Posterior for µ P J : µ P J | ξ Y , ξ V , Θ \{ µ P J } ∼ N (cid:16) AB , q B (cid:17) , where A = P T − t =0 ( ξ Yt +1 − ρ J ξ Vt +1 ) σ J + c ∗ C ∗ ; B = Tσ J + C ∗ ; c ∗ = 0 and C ∗ = 100 are hyperparameters of the prior of µ P J .Posterior for σ J : σ J | ξ Y , ξ V , Θ \{ σ J } ∼ IG ( A , B ), where A = c ∗ + T ; B = C ∗ + P T − t =0 ( ξ Yt +1 − µ P J − ρ J ξ Vt +1 ) ; c ∗ = 10 and C ∗ = 40 are hyperparameters of the prior of σ J . Posterior for λ : λ | N ∼ B ( A , B ), where A = c ∗ + P T − t =0 N t +1 ; B = C ∗ + T − P T − t =0 N t +1 ; c ∗ = 2 and C ∗ = 40 are hyperparameters of the prior of λ .33osterior for N : N t +1 | Y, V, ξ Y , ξ V , Θ ∼ Bernoulli (cid:16) α α + α (cid:17) . Bernoulli denotes aBernoulli distribution. D t +1 = V t +1 − V t − κ ( θ − V t )∆ − ξ Vt +1 σ v √ V t ∆ ; H t +1 = Y t +1 − Y t − ( r t − V t +Φ J ( − i )+ η s V t ) ∆ − ξ Yt +1 √ V t ∆ ; α = λ exp (cid:16) − − ρ ) (cid:0) H t +1 − ρH t +1 D t +1 (cid:1)(cid:17) ; HH t +1 = Y t +1 − Y t − ( r t − V t +Φ J ( − i )+ η s V t ) ∆ √ V t ∆ ; DD t +1 = V t +1 − V t − κ ( θ − V t )∆ σ v √ V t ∆ ; and α = (1 − λ ) exp (cid:16) − − ρ ) (cid:0) HH t +1 − ρHH t +1 DD t +1 (cid:1)(cid:17) .Posterior for η v ∝ π ( η v ) := Q T − t =0 exp (cid:16) − [( C t +1 − F t +1 ) − ρ c ( C t − F t )] σ c (cid:17) × exp (cid:16) − η v (cid:17) .Posterior for µ Q J ∝ π (cid:16) µ Q J (cid:17) := Q T − t =0 exp (cid:16) − [( C t +1 − F t +1 ) − ρ c ( C t − F t )] σ c (cid:17) × exp (cid:18) − ( µ Q J ) (cid:19) , η v and µ Q J are estimated with the Metropolis-Hasting algorithm.Posterior for η s : η s | Y, V, J Y , J V , Θ \{ η s } ∼ N (cid:16) AB , q B (cid:17) , where B = ∆(1 − ρ ) P T − t =0 V t + 1; A = − ρ ) P T − t =0 (cid:16) H t +1 − ρσ v D t +1 (cid:17) ; D t +1 = V t +1 − V t − κ ( θ − V t )∆ − J Vt +1 ; and H t +1 = Y t +1 − Y t − ( r t − V t + Φ J ( − i ) + η s V t )∆ − J Yt +1 .Posterior for µ V : µ V | ξ V ∼ IG ( A , B ), where A = c ∗ +2 T ; B = C ∗ +2 P T − t =0 ξ Vt +1 ; c ∗ = 10and C ∗ = 20 are hyperparameters of the prior of µ V .Posterior for ρ J : ρ J | ξ Y , ξ V , Θ \{ ρ J } ∼ N (cid:16) AB , q B (cid:17) , where A = P T − t =0 ( ξ Yt +1 − µ P J ) ξ Vt +1 σ J + c ∗ C ∗ ; B = P T − t =0 ( ξ Vt +1 ) σ J + C ∗ ; c ∗ = 0 and C ∗ = 4 are hyperparameters of the prior of ρ J .Posterior for ρ c : ρ c | C, F, Θ \{ ρ c } ∼ N (cid:16) AB , q B (cid:17) , where A = P T − t =0 H t H t +1 σ c + c ∗ C ∗ ; B = P T − t =0 H t σ c + C ∗ ; H t +1 = C t +1 − F t +1 ; c ∗ = 0 and C ∗ = 1 are hyperparameters of the prior of ρ c . Posterior for σ c : σ c | C, F, Θ \{ σ c } ∼ IG ( A , B ), where A = c ∗ + T ; B = C ∗ + P T − t =0 ( H t +1 − ρ c H t ) ; H t +1 = C t +1 − F t +1 ; c ∗ = 2 . C ∗ = 0 . σ c .Posterior for ξ Y : ξ Yt +1 | Y, V, ξ V , N t +1 = 1 , Θ ∼ N (cid:16) AB , q B (cid:17) , where A = H t +1 − ρσv D t +1 (1 − ρ ) V t ∆ + µ P J + ρ J D t +1 σ J ; B = − ρ ) V t ∆ + σ J ; D t +1 = V t +1 − V t − κ ( θ − V t )∆ − ξ Vt +1 ; and H t +1 = Y t +1 − Y t − ( r t − V t + Φ J ( − i ) + η s V t )∆.Posterior for ξ V : ξ Vt +1 | Y, V, ξ Y , N t +1 = 1 , Θ ∼ N (cid:16) AB , q B (cid:17) ξ Vt +1 > , where A = D t +1 − ρσ v H t +1 (1 − ρ ) σ v V t ∆ + ρ J ( H t +1 − µ P J ) σ J − µ V ; B = − ρ ) σ v V t ∆ + ρ J σ J ; D t +1 = V t +1 − V t − κ ( θ − V t )∆; and H t +1 = Y t +1 − Y t − ( r t − V t + Φ J ( − i ) + η s V t )∆ − ξ Yt +1 .Posterior for V ∝ π ( V t +1 ) := V t +1 × exp n − − ρD t +1 H t +1 + D t +1 − ρ ) − D t +2 − ρD t +2 H t +2 + H t +2 − ρ ) (cid:17) ,for 0 < t +1 < T . D t +1 = V t +1 − V t − κ ( θ − V t )∆ − J Vt +1 σ v √ V t ∆ ; and H t +1 = Y t +1 − Y t − ( r t − V t +Φ J ( − i )+ η s V t ) ∆ − J Yt +1 √ V t ∆ . V and V T are estimated in a similar way. This is a nonstandard distribution, and weapply the random-walk Metropolis-Hastings algorithm with the Student’s t distribution,the average standard deviations is 0.25, the degrees of freedom is 6.34elow we describe the algorithm to estimate parameters and latent variables unique toSVVG.Posterior for γ P : γ P | G, J Y , Θ \{ γ P } ∼ N (cid:16) AB , q B (cid:17) , where A = σ P ) J Yt +1 + c ∗ C ∗ ; B = σ P ) G t +1 1 C ∗ ; c ∗ = 0 and C ∗ = 1 are hyperparameters of the prior of γ P .Posterior for σ P : ( σ P ) | G, J Y , Θ \{ σ P } ∼ IG ( A , B ), where A = c ∗ + T ; B = C ∗ + P T − t =0 ( J Yt +1 − γ P G t +1 ) G t +1 ; c ∗ = 2 . C ∗ = 0 . (cid:0) σ P (cid:1) .Posterior for ν ∝ π ( ν ) := (cid:18) ν ∆ ν Γ( ∆ ν ) (cid:19) T (cid:16)Q T − t =0 G t +1 (cid:17) ∆ ν exp n − ν (cid:16)P T − t =0 G t +1 + C ∗ (cid:17)o (cid:0) ν (cid:1) c ∗ +1 , c ∗ = 10 and C ∗ = 20 are hyperparameters of the prior of ν . Γ denotes the Gamma function.Posterior for γ Q ∝ π ( γ Q ) := Q T − t =0 exp (cid:16) − [( C t +1 − F t +1 ) − ρ c ( C t − F t )] σ c (cid:17) × exp (cid:18) − ( γ Q ) (cid:19) .Posterior for σ Q ∝ π ( σ Q | σ Q >
0) := Q T − t =0 exp (cid:16) − [( C t +1 − F t +1 ) − ρ c ( C t − F t )] σ c (cid:17) × exp (cid:18) − ( σ Q ) (cid:19) , ν , γ Q and σ Q are estimated with the Metropolis-Hasting algorithm.Posterior for J Y : J Yt +1 | Y, V, G, Θ ∼ N (cid:16) AB , q B (cid:17) , where A = − ρ ) V t ∆ (cid:16) H t +1 − ρD t +1 σ v (cid:17) + γ P ( σ P ) ; B = − ρ ) V t ∆ + σ P ) G t +1 ; D t +1 = V t +1 − V t − κ ( θ − V t )∆; and H t +1 = Y t +1 − Y t − ( r t − V t + Φ J ( − i ) + η s V t )∆.Posterior for G ∝ π ( G t +1 ) := G ∆ ν − t +1 exp (cid:26) − J t σ P ) G t +1 − G t +1 (cid:18) ( γ P ) σ P ) + ν (cid:19)(cid:27) . We sam-ple G t +1 as G t +1 | J Y , Θ ∼ GIG (cid:16) ν − , ( γ P ) ( σ P ) + ν , ( J Yt +1 ) ( σ P ) (cid:17) directly, where GIG denotes ageneralized inverse Gaussian distribution.The posterior distribution of the parameters and latent variables unique to SVLS iscalculated as follows.Posterior for α ∝ π ( α ) := (cid:16) αα − (cid:17) T exp ( − P T − t =0 (cid:12)(cid:12)(cid:12)(cid:12) J Yt +1 σ ∆ α t α ( U t +1 ) (cid:12)(cid:12)(cid:12)(cid:12) αα − ) × Q T − t =0 (cid:12)(cid:12)(cid:12)(cid:12) J Yt +1 σ ∆ α t α ( U t +1 ) (cid:12)(cid:12)(cid:12)(cid:12) αα − × h ( σ ) αα − i c ∗ +1 × exp n − ( σ ) αα − C ∗ o × . ≤ α ≤ , where c ∗ = 2 . C ∗ = 0 . σ αα − , and t α ( U t +1 ) = sin h παU t +1 + (2 − α ) π i cos [ πU t +1 ] ! × cos [ πU t +1 ] cos h π ( α − U t +1 + (2 − α ) π i ! α − α .The posterior distribution of α is nonstandard, we simulate the posterior based on theMetropolis-Hastings method and take a linearly transformed Beta distribution as the pro-posal density: (1) draw ̟ from B ( A , B ), where A = α ( g ) − . . (5 ln( T ) −
2) + 1 and B =5 ln( T ) − A . Set α ( g +1) = 0 . ̟ + 1 .
01; (2) calculate A ∗ = α ( g +1) − . . (5 ln( T ) −
2) + 1 and B ∗ = 5 ln( T ) − A ∗ ; (3) define f ( α | a, b ) = Γ( a + b )Γ( a )Γ( b ) (cid:0) α − . . (cid:1) α − (cid:0) − α . (cid:1) b − . Accept α ( g +1) withprobability min (cid:16) π ( α ( g +1) ) π ( α ( g ) ) × f ( α ( g ) |A ∗ , B ∗ ) f ( α ( g +1) |A , B ) , (cid:17) .Posterior for σ : σ αα − | J Y , U, Θ ∼ IG ( A , B ), where A = c ∗ + T ; B = C ∗ + P T − t =0 (cid:12)(cid:12)(cid:12)(cid:12) J Yt +1 ∆ α t α ( U t +1 ) (cid:12)(cid:12)(cid:12)(cid:12) ;35 ∗ = 2 . C ∗ = 0 . σ αα − .Posterior for J Y ∝ π ( J Yt +1 ): π ( J Yt +1 ) := exp n − J t +1 − ρ ) V t ∆ h J t +1 − (cid:16) H t +1 − ρσ v D t +1 (cid:17)io × exp ( − (cid:12)(cid:12)(cid:12)(cid:12) J t +1 σ ∆ α t α ( U t +1 ) (cid:12)(cid:12)(cid:12)(cid:12) αα − ) | J t +1 | α − ; D t +1 = V t +1 − V t − κ ( θ − V t )∆; and H t +1 = Y t +1 − Y t − (cid:0) r t − V t + Φ J ( − i ) + η s V t (cid:1) ∆.Posterior for U ∝ π ( U t +1 ) := f ( U t +1 ) × h J t +1 ∈ ( −∞ , ∩ U t +1 ∈ ( − , α − α ) + J t +1 ∈ (0 , ∞ ) ∩ U t +1 ∈ ( α − α , ) i ,where f ( U t +1 ) = exp ( − (cid:12)(cid:12)(cid:12)(cid:12) J t +1 σ ∆ α t α ( U t +1 ) (cid:12)(cid:12)(cid:12)(cid:12) αα − + 1 ) (cid:12)(cid:12)(cid:12)(cid:12) J t +1 σ ∆ α t α ( U t +1 ) (cid:12)(cid:12)(cid:12)(cid:12) αα − . We update U ( g +1) t +1 withthe following steps: (1) if J t +1 <
0, draw U ( g +1) t +1 from U ( − , α − α ); if J t +1 >
0, draw U ( g +1) t +1 from U ( α − α , ); (2) draw u from U (0 , U ( g +1) t +1 if u < f (cid:16) U ( g +1) t +1 (cid:17)(cid:17)