Option Pricing Under a Discrete-Time Markov Switching Stochastic Volatility with Co-Jump Model
aa r X i v : . [ q -f i n . P R ] J un Option Pricing Under a Discrete-Time MarkovSwitching Stochastic Volatility with Co-Jump Model
Michael C. Fu
Smith School of Business & Institute for Systems Research, University of Maryland, College Park, MD 20742, [email protected]
Bingqing Li
School of Finance, Nankai University, 300350 Tianjin, China, [email protected]
Rongwen Wu
A Financial Company in the US, rongwen [email protected]
Tianqi Zhang
School of Finance, Nankai University, 300350 Tianjin, China, [email protected]
We consider option pricing using a discrete-time Markov switching stochastic volatility with co-jumpmodel, which can model volatility clustering and varying mean-reversion speeds of volatility. For pricingEuropean options, we develop a computationally efficient method for obtaining the probability distributionof average integrated variance (AIV), which is key to option pricing under stochastic-volatility-type models.Building upon the efficiency of the European option pricing approach, we are able to price an American-styleoption, by converting its pricing into the pricing of a portfolio of European options. Our work also providesconstructive guidance for analyzing derivatives based on variance, e.g., the variance swap. Numerical resultsindicate our methods can be implemented very efficiently and accurately.
Key words : option pricing; stochastic volatility; co-jump; Markov switching; average integrated variance
1. Introduction
The stochastic volatility with co-jump (SVCJ) model introduced by Eraker et al. (2003) is com-monly used to model the price dynamics for an underlying financial asset, because it is able tocapture leptokurtic, skewness, and volatility clustering observed in real-world data. The SVCJmodel simultaneously considers stochastic volatility, jumps in return, and jumps in volatility, gen-eralizing the jump-diffusion model in Merton (1976), the stochastic volatility model in Heston(1993), and the stochastic volatility/jump-diffusion model in Bates (1996), which are all specialcases of SVCJ. Empirical evidence supporting the presence and importance of stochastic volatilitywith jumps in both return and volatility is documented in Eraker et al. (2003), and surveys ofcritical developments in the SVCJ model can be found in Eraker (2004), Broadie et al. (2007),Johannes et al. (2009), Collin-Dufresne et al. (2012), Bandi and Ren`o (2016), Du and Luo (2019)and references therein. SVCJ models primarily price options using Fourier transform methods. ption Pricing under MS-SVCJ Model However, Li et al. (2008) and Kou et al. (2017) point out that the SVCJ model cannot sufficientlycapture volatility clustering, especially for scenarios with persistently high levels of volatility suchas the period from Feb 28 to May 15, 2020, since the CIR process used in the SVCJ model assumesrapid mean reversion of volatility. Grasselli (2017) observes that the volatility process in the SVCJmodel spends much time close to zero, so once an extreme value in volatility is reached, thepreference for moving back to zero volatility leads to an excessively rapid mean-reversion speed.Eraker (2004) also empirically observed slower mean-reversion speed improving performance oftheir model in the out-of-sample period. Here we also refer to the SVCJ model with CIR processas the classical SVCJ model.To address some of the shortcomings of the classical SVCJ model, we propose a discrete-timeMarkov switching stochastic volatility with co-jump (MS-SVCJ) model where the volatility iscomposed of a Markov switching (MS) process and a jump process. Naik (1993), Timmermann(2000), and Guo (2001) pointed out that MS processes can model the persistence of volatility.For example, Naik (1993) stressed that, “by choosing the parameters appropriately, we can modeldifferent levels of persistence of the volatility process in the high and low states.” Moreover,Timmermann (2000) stated that “Markov switching models can generate a wide range of coefficientsof skewness, kurtosis and serial correlation even when based on a very small number of underlyingstates.” Similar arguments are made in related literature, e.g., Pan (2002), Alizadeh et al. (2002),Eraker (2004), Bakshi et al. (2006), Adrian and Rosenberg (2008), Christoffersen et al. (2009),Christoffersen et al. (2010), Chourdakis and Dotsis (2011).In addition, since the MS process can distributionally approximate any diffusion stochastic pro-cess, we can approximate the CIR process by adjusting the parameters of the MS process. Detailedapproximation implementations can be found in Lo and Skindilias (2014), Cai et al. (2015) andCui et al. (2017). The proposed model can distributionally approximate the classic SVCJ model,so it is more robust and flexible than the classical SVCJ model. In short, the proposed modelovercomes some limitations of the classical SVCJ model while retaining its advantages. ption Pricing under MS-SVCJ Model For pricing European options under the proposed model, we consider the method based onaverage integrated variance (AIV) developed by Hull and White (1987), in which the option priceunder the stochastic volatility model is expressed as the expectation of the Black-Scholes formulawith variance replaced by AIV. Although this provides a formal solution for the option price, theprobability distribution of AIV is generally difficult to obtain, which makes the practical applicationof this method challenging. For our MS-SVCJ model, we face the same challenge.We consider a discrete-time MS volatility process with finite state space, so there are a finitenumber of sample paths of volatility, which means the value space of AIV is also finite. Thus,theoretically we can find the probability distribution of AIV by enumerating all the sample pathsof volatility, but such enumeration is generally not computationally feasible, so we propose therecursive recombination (RR) algorithm to efficiently compute the probability distribution of AIV.Then we derive a pricing formula that leads to an analytical solution for European options underour proposed model. Numerical experiments demonstrate the effectiveness of the RR algorithm.Our work extends the existing literature on European option pricing in several ways. First,without the jumps in volatility, the proposed model becomes the MS-SVJ model, so our analyticalsolution for European options applies to the MS-SVJ model with general jump size distribution.Second, removing jumps from both the volatility and asset price, the proposed model reducesto the MS-SV model, so using our analytical solution for the MS-SV model avoids having tosolve a set of intractable ordinary differential equations when pricing options, as in Naik (1993),Guo (2001), and Fuh et al. (2012), or using numerical inverse Fourier transform methods, as inBuffington and Elliott (2002), Elliott et al. (2006), and Liu et al. (2006).We also consider the pricing of American-style derivatives (see Broadie and Detemple (1996)),applying the approach proposed by Laprise et al. (2006) by leveraging our analytical solutionsfor pricing European options. In the approach, the pricing of an American-style option can beconverted into the pricing of a basket of European options. We compare our results with the leastsquares Monte Carlo simulation approach by Longstaff and Schwartz (2001). For more discussion ption Pricing under MS-SVCJ Model of the application of Monte Carlo simulation approaches to American-style derivatives, refer toBroadie and Glasserman (1997a,b) or Fu et al. (2001). Our numerical examples further illustratethe effectiveness of our approach.In sum, our work contributes to the option pricing research literature as follows: • We provide an analytical pricing formula for European options under a discrete-time MS-SVCJmodel that is more robust and flexible than the classical SVCJ model and can explain volatilityclustering for high levels of volatility. • We develop an efficient algorithm to obtain the probability distribution for AIV, which canalso be applied to other related volatility derivatives, e.g., the variance swap. • Our analytical solution for European option prices applies to several well-known models inthe literature, including the MS-SVJ model with general jump size distribution. For the MS-SVmodel, our approach also has computational advantages over existing option pricing methods, byeliminating having to numerically solve a set of ordinary differential equations. • We price American-style options by leveraging the efficiency of our analytical pricing solutionfor European options.The remainder of the paper is organized as follows. In Section 2, we construct the MS-SVCJmodel and analyze the AIV probability measure. Section 3 develops the option valuation underthe MS-SVCJ model. In Section 4, we present the RR algorithm and analyze its computationalcomplexity. Numerical results for pricing both European options and American-style options arepresented in Section 5. Section 6 concludes and discusses future research.
2. Theoretical Framework
In this section, we build the Markov switching stochastic volatility with co-jump (MS-SVCJ) modelfor the underlying asset and define the AIV probability measure.
Under the MS-SVCJ model, the underlying asset price S t is assumed to follow a jump-diffusionprocess, and the asset volatility is also stochastic. Specifically, the dynamics are specified by the ption Pricing under MS-SVCJ Model following two equations under the risk-neutral probability measure Q : dS t S t − = ( r − λζ ) dt + ˆ σ t dB t + ( J t − dN t ˆ σ t = σ t + N t X i =1 f ( J i , t, t i ) (1)where { B t } is standard Brownian motion; { N t } is a Poisson jump process with intensity λ ; theproportional jumps { J t } are independent and identically distributed (i.i.d.) with ζ ≡ E ( J t − { σ t } follows a discrete-time Markov switching (MS) process; f ( · ) which states the impact of jumpsin asset price on variance is the proportional and exponentially attenuating (PEA) process; { B t } , { N t } , { J t } and { σ t } are mutually independent. We regard the risk-free interest rate r as constant.The main distinctive feature of the proposed model is that the variance process { ˆ σ t } is composedof two components: the first explains the exogenous dynamics of variance, e.g., due to changes inthe economy and company announcements, and the second explains the endogenous movement invariance due to jumps in the asset, similar to Duffie et al. (2000). The proposed model assumesthe first part follows an MS process and the second part follows the jump process related to jumpin asset price. In what follows, we provide detailed specifications of the two processes.We assume { σ t } follows a discrete-time Markov switching (MS) process with finite state space { u , u , · · · , u m } and constant time step τ , and one-step transition probability matrix P = [ p ij ] m × m ,i.e., p ij = p ( σ ( k +1) τ = u j | σ kτ = u i , σ ( k − τ , · · · · · · ) = p ( σ ( k +1) τ = u j | σ kτ = u i ) . The MS process hasbeen shown to model reasonably well most of the stylized facts of volatility, volatility clusteringand mean-reversion. See Naik (1993), Ryd´en et al. (1998), Duan et al. (2002), Aingworth et al.(2006), Rossi and Gallo (2006). Furthermore, compared with affine models for volatility, e.g. thesquare-root model, the MS process can better capture the clustering in different volatility levelsand the varying mean-reversion speeds of volatility.The second part N t P i =1 f ( J i , t, t i ) takes into account the sudden movement in variance causedby jumps in the asset price. Here, we model f ( J i , t, t i ) as the product of two components, onerepresenting the instantaneous shock size of variance due to the jump in asset return, and the ption Pricing under MS-SVCJ Model other representing the dynamics of this shock over time. This dependence structure of jumps inasset price and volatility was proposed theoretically by Barndorff-Nielsen and Shephard (2001),Kl¨uppelberg et al. (2004). Todorov (2011) empirically identified the effectiveness of this structure.Specifically, we give the following description of f ( J i , t, t i ). Definition 1.
The { f ( J i , t, t i ) } is a proportional and exponentially attenuating (PEA) process if f ( J i , t, t i ) = f ( J i ) f ( t, t i ) for the ith jump J i in asset price at t i , where f ( J i ) = b ln ( J i ) ,f ( t, t i ) = e − β ( t − t i ) , t i < t ≤ t i + ∆; 0 , otherwise . In the above representation, the term f ( J i ) is the shock size of variance caused by the jumpin asset price, and the shock size is proportional to the square of log-jump in asset price withproportional coefficient b , which is consistent with the definition of variance expressed as the averageof the quadratic function of the decentralized logarithm of the return. As a memory function,the term f ( t, t i ) indicates that the shock of the jump tails off exponentially with attenuatingfactor β and duration ∆, which is analogous to the CARMA kernel in Brockwell (2001). FollowingJ.P.Morgan/Reuters (1996), we assume that the duration ∆ is finite and fixed. Hull and White (1987) show that the option price under the stochastic volatility model can becomputed as the expectation of the Black-Scholes formula with variance replaced by average inte-grated variance (AIV). We derive a similar result in Section 3.1 when the diffusive innovation tothe asset price process is independent of volatility. Under the proposed model, AIV during theinterval [0 , T ] can be expressed as V = 1 T Z T ˆ σ t dt = 1 T Z T σ t dt + 1 T Z T X i f ( J i , t, t i ) dt. (2)Thus, AIV can be expressed as a sum of two terms, one due to the MS process and the other due tothe PEA process. Since AIV due to the MS process plays a key role, we first provide the following ption Pricing under MS-SVCJ Model L − L − L u m u m − u u u LLLLL LLLLL LLLLL LLLLL LLLLL LLLLL
Figure 1
Sample Path ω : σ = u , σ = u , σ = u m − , . . . , σ L − = u , σ L − = u , σ L = u description. We assume that τ is chosen such that L = T /τ is integer-valued, so that L is the totalnumber of time steps. Since the information up to current time is available, we also assume theinitial state of { σ t } is known. The MS process { σ t } generates sample path ω during [0 , T ]. Since { σ t } is a piecewise constant process, we can represent the sample path ω as the following tupleform: ω = (cid:2) σ , σ τ , σ τ , . . . , σ ( L − τ , σ Lτ (cid:3) , where σ is fixed.For notational simplicity, we write σ kτ as σ k . In the following, the MS process { σ t } will berewritten as { σ k } to highlight its discretization, and henceforth, the above sample path ω will beexpressed as ω = [ σ , σ , σ , . . . , σ L − , σ L ] , with weight | ω | and probability p ( ω ) given by | ω | = σ + σ + σ + . . . + σ L − + σ L − L ,p ( ω ) = p σ σ · p σ σ · · · p σ L − σ L − · p σ L − σ L . Here, we denote the set of all sample paths ω as Ω, the sample path space for { σ k } .Figure 1 illustrates an example of sample path ω = [ u , u , ..., u m − , . . . , u , u , u ].AIV due to the MS process, which is the first term defined in Equation (2), is given by V = 1 L L X k =1 σ k − , which is a random variable with probability distribution derived from | ω | and p ( ω ) with value spaceΨ = { v : v = | ω | , ω ∈ Ω } , and corresponding probability p V ( v ) := p ( V = v ) = P ω ∈ Ω: | ω | = v p ( ω ) , v ∈ Ψ . Clearly, | Ψ | ≤ | Ω | , and in general, the number of possible values of V is far less than the totalnumber of sample paths of { σ k } , and we have the following proposition (see Appendix A for proof). Proposition 1. | Ψ | ≤ (cid:0) L + m − m − (cid:1) . ption Pricing under MS-SVCJ Model
3. Option Valuation
In this section, we price a European call option under the proposed model described by Equation(1). A European put option can also be priced by the same method. First, we provide the followingformal solution for the European call option price (see Appendix B for proof).
Lemma 1.
Under the MS-SVCJ model, the price of a European call option with strike price K ,maturity T , and initial price S can be written as C = + ∞ X n =0 p ( N T = n ) X ω ∈ Ω p ( ω ) E J n ( e − rT E ( S T ( J n , ω ) − K ) + ) (3) where N T is the number of jumps in the asset price up to T ; J n := ( J , · · · , J n ) is an n -dimensionalrandom vector of n jump sizes; Ω is the sample path space of { σ k } ; S T ( J n , ω ) is the asset price atthe maturity T given J n and ω . Before deriving a tractable solution, we need the probability distribution of S T ( J n , ω ). Usingthe lemma in Hull and White (1987), we can derive the following (see Appendix C details): W ( J n , ω ) := ln S T ( J n , ω ) S n Q i =1 J i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( J n , ω ) ∼ N (cid:18)(cid:18) r − λζ − V ( J n , ω )2 (cid:19) T, V ( J n , ω ) T (cid:19) , (4)where n Q i =1 J i is the cumulative effect of n jumps in the asset price with Q i =1 J i = 1, ζ ≡ E ( J i − V ( J n , ω ) is the realization of V given ω and J n , V ( J n , ω ) = 1 T Z T ˆ σ t ( J n , ω ) dt = V + 1 T Z T n X i =1 f ( J i , t, t i ) dt = | ω | + 1 T n X i =1 f ( J i ) Z T f ( t, t i ) dt. We now focus our attention on the quantity V ( J n , ω ). For simplicity and technical convenience,we assume that all jumps during the interval [ T − ∆ , T ] occur at the beginning of the interval attime T − ∆; the impact of this jump time assumption on the option price is negligible, as discussedin Appendix I and illustrated numerically in Section 5.1. Hence, V ( J n , ω ) = | ω | + ˆ b n X i =1 ln ( J i ) , ˆ b = b (1 − e − β ∆ ) T β . (5) ption Pricing under MS-SVCJ Model From Lemma 1, combined with Equations (4) and (5), we have: e − rT E ( S T ( J N T , ω ) − K ) + = e − rT E ( S e W ( J NT ,ω ) · e NT P i =1 ln( J i ) − K ) + = BS ( S e − λζT + NT P i =1 ln( J i ) , | ω | + ˆ b N T X i =1 ln ( J i ) , r, T, K ) (6)where BS ( S, σ , r, T, K ) is the classical Black-Scholes formula as a function of initial stock price S , volatility σ , risk-free rate r , maturity T , and strike price K . Substituting Equation (6) intoEquation (3) yields the price of a European call option (see Appendix D for proof). Theorem 1.
Under the MS-SVCJ model, the price of a European call option with strike price K ,maturity T , and initial underlying asset price S is given by C = + ∞ X n =0 p ( N T = n ) X v ∈ Ψ p V ( v ) C n ( v ) , where C n ( v ) = E Ξ n ( BS ( S e − λζT + X n , v + ˆ bY n , r, T, K )) , Ξ n := ( X n , Y n ) := ( n P i =1 ln( J i ) , n P i =1 ln ( J i )) . Ξ n in Theorem 1 is a bivariate random variable, so the n -dimensional integral E J n ( · ) in Lemma1 has been replaced by a double integral E Ξ n ( · ) in Theorem 1. In many special cases, e.g., whenthe jump distribution is lognormal, the probability distribution of Ξ n can be expressed explicitly,so that C n ( Z ) can be easily computed (see Appendix E for proof). Proposition 2.
For ln( J i ) i.i.d. ∼ N ( µ, ε ) , ζ = E ( J i −
1) = e µ + ε − , the probability density functionof ( X n , Y n ) := ( n P i =1 ln( J i ) , n P i =1 ln ( J i )) is given by g ( x, y ) = √ π √ nε e − ( x − nµ )22 nε ( n − ) n − (cid:18) y − x n ε (cid:19) n − − e − y − x n ε /ε , y ≥ x n , n ≥ √ πε e − ( x − µ )22 ε , y = x , n = 10 , otherwise. For lognormally distributed jump sizes, Proposition 2 provides an analytical expression for theoption price, different from the traditional solution using numerical Fourier transform inversion.Moreover, under a general jump size distribution, the following corollary gives the option pricefor the MS-SVJ model, a special case of MS-SVCJ model (see Appendix F for proof). ption Pricing under MS-SVCJ Model Corollary 1.
Under the Markov switching stochastic volatility jump-diffusion (MS-SVJ) model,which can be recovered by setting f ( · , · , · ) = 0 in the MS-SVCJ model, and the jump size J t followsa general distribution, the price of a European call option is given by C = P v ∈ Ψ p V ( v ) C jd ( v ) , where C jd ( · ) is the European call option price under the jump-diffusion model. Thus, given the probability distribution { p V ( · ) } , the option price only depends on C jd ( · ). Whenln( J t ) follows a normal distribution, a mixed-exponential distribution, or a general discrete dis-tribution, C jd ( · ) can be obtained via various approaches, such as Merton (1976), Kou (2002),Kou and Wang (2004), Cai and Kou (2011), or Fu et al. (2017). Hence, under the MS-SVJ modelwith the above jump size distribution, we also provide an analytical solution for the option price.Moreover, we can provide an analytical option price under the MS-SV model, which overcomesthe drawbacks of Naik (1993), Duan et al. (2002), Aingworth et al. (2006). Corollary 2.
Under the Markov switching stochastic volatility (MS-SV) model, which can berecovered by setting f ( · , · , · ) = 0 and the Poisson intensity λ = 0 in the MS-SVCJ model, the priceof a European call option is given by C = P v ∈ Ψ p V ( v ) BS ( S , v, r, T, K ) . Proof.
When f ( · , · , · ) = 0 and the Poisson intensity λ = 0, we have ˆ b = 0, p ( N T = 0) = 1, p ( N T = n ) = 0 , n ≥
1, and C ( Z ) = BS ( S , Z, r, T, K ). (cid:3) This result is similar to Hull and White (1987), who provided a formal solution but did notspecify the distribution of AIV needed for computing the option price.
By leveraging our analytical solution for European option, we can provide an efficient approxima-tion to the price of an American-style option using the approach proposed by Laprise et al. (2006).By converting the price of an American-style option to the price of a portfolio of European options,Laprise et al. (2006) designed algorithms to provide an upper bound and a lower bound for theprice of an American-style option, where early-exercise opportunities were restricted to discretepoints 0 = t < t < · · · < t N = T . ption Pricing under MS-SVCJ Model In order to apply the algorithms, for every interval [ t i , t i +1 ], one only needs to compute two criticalvariables: V t i ( S i , K i , t i +1 − t i ) and ∂∂S i V t i ( S i , K i , t i +1 − t i ), where V t i ( S i , K i , t i +1 − t i ) is the Europeancall option value with initial asset price S i , strike price K i and maturity t i +1 − t i . For the assetprice following geometric Brownian motion and the Merton jump-diffusion model, Laprise et al.(2006) provided a tractable expression for V t i ( S i , K i , t i +1 − t i ).When the volatility σ t follows the MS process, at any given time t i , volatility is a random variablewith value space { u , u , · · · , u m } and corresponding probability π t i ( . ). Hence we have, V t i ( S i , K i , t i +1 − t i ) = X σ ∈{ u , ··· ,u m } π t i ( σ ) C ( S i , K i , t i +1 − t i , σ ) ,∂∂S i V t i ( S i , K i , t i +1 − t i ) = X σ ∈{ u , ··· ,u m } π t i ( σ ) ∂∂S i C ( S i , K i , t i +1 − t i , σ ) , (7)where C ( S i , K i , t i +1 − t i , σ ) is the European call option with initial asset price S i , strike price K i ,maturity t i +1 − t i and initial status σ for the MS process in [ t i , t i +1 ].Theorem 1, Corollary 1, and Corollary 2 provide detailed solution for C ( S i , K i , t i +1 − t i , σ ) in (7)for MS-SVCJ, MS-SVJ, and MS-SV models, respectively. As an example, the detailed expressionsunder the MS-SVCJ model can be written as follows by substituting Theorem 1 into (7): V t i ( S i , K i , t i +1 − t i ) = X σ ∈{ u , ··· ,u m } π t i ( σ ) + ∞ X n =0 p ( N t i +1 − t i = n ) X v ∈ Ψ( σ ) p V ( v ) C n ( S i , K i , t i +1 − t i , v ) ,∂∂S i V t i ( S i , K i , t i +1 − t i ) = X σ ∈{ u , ··· ,u m } π t i ( σ ) + ∞ X n =0 p ( N t i +1 − t i = n ) X v ∈ Ψ( σ ) p V ( v ) ∂∂S i C n ( S i , K i , t i +1 − t i , v ) , (8)where Ψ( σ ) highlights that AIV is dependent on the initial status σ for the MS process, and C n ( S, K, T, v ) = E Ξ n ( BS ( Se − λζT + X n , v + ˆ bY n , r, T, K )).For the MS-SV model, we have V t i ( S i , K i , t i +1 − t i ) = X σ ∈{ u , ··· ,u m } X v ∈ Ψ( σ ) π t i ( σ ) p V ( v ) BS ( S i , v, r, t i +1 − t i , K i ) ,∂∂S i V t i ( S i , K i , t i +1 − t i ) = X σ ∈{ u , ··· ,u m } X v ∈ Ψ( σ ) π t i ( σ ) p V ( v ) ∂∂S i BS ( S i , v, r, t i +1 − t i , K i ) . (9)Up to now, we have provided an analytical solution for option price under the proposed modelwith the lognormal jump size distribution. Practical application requires calculating the probabilitydistribution of V efficiently, which is addressed in the next section. ption Pricing under MS-SVCJ Model
4. An Efficient Algorithm for MS-process AIV
After observing that complete enumeration (CE) based on the definition of V is intractable due tothe enormous computation time and consumed memory, we develop an efficient algorithm for theprobability distribution { p V ( · ) } called the recursive recombination (RR) algorithm. Based on the definition of V in Section 2.2, a complete enumeration (CE) algorithm for Ψ and { p V ( · ) } would traverse all sample paths ω and generate ( | ω | , p ( ω )); then collect distinct values | ω | to get Ψ and sum probabilities p ( ω ) with | ω | = v ∈ Ψ to get p V ( v ). However, for the MS processwith m states, the complexity of CE is clearly O ( m L ), exponential in the number of time steps.Hence, to efficiently derive Ψ and { p V ( · ) } , we propose the RR algorithm. Recall from Section 2 .
2, for the MS process { σ k } , the initial state σ is fixed. Here, we define asubsample path ω l = [ σ , σ , σ , . . . , σ l − , σ l − , σ l ] of { σ k } up to step l , as the first l + 1 elements ofa sample path ω , with weight | ω l | and corresponding probability p ( ω l ): | ω l | = σ + σ + σ + . . . + σ l − + σ l − l ,p ( ω l ) = p σ σ p σ σ ....p σ l − σ l − p σ l − σ l . In addition, we also denote the set of all subsample paths ω l as Ω l , which is the subsample pathspace for { σ k } up to step l .For a subsample path ω l , we extract three fundamental features: the weight | ω l | , the length l and the last element σ l , which generate a triple [ | ω l | , l, σ l ]. Thus, [ | ω l | , l, σ l ] is a random variablewith value space Ψ l = { [ x, l, σ l ] : x = | ω l | , σ l is the last element of ω l , ω l ∈ Ω l } , and correspondingprobability p ([ x, l, σ l ]) = P ω l ∈ Ω l : | ω l | = x,ω l ends with σ l p ( ω l ) , [ x, l, σ l ] ∈ Ψ l .Now we can relate [ | ω L | , L, σ L ] to the random variable V :Ψ = { v : [ v, L, σ L ] ∈ Ψ L } ,p V ( v ) = X [ v,L,σ L ] ∈ Ψ L p ([ v, L, σ L ]) . (10) ption Pricing under MS-SVCJ Model .
4] [0 . , . . , .
2] [0 . , . , . . , . , . . , . , . . , . , .
2] [0 . , . , . , . . , . , . , . . , . , . , . . , . , . , . . , . , . , . . , . , . , . . , . , . , . . , . , . , . Figure 2 Ω l , Subsample Path Space of ω l Up to Step L = 3 . , , . . , , .
2] [0 . , , . . , , . . , , . . , , .
2] [0 . , , . . , , . . , , . . , , . . , , . . , , . Figure 3 Ψ l , Value Space of [ | ω l | , l, σ l ] Up to Step L = 3 We provide an example with state set { . , . } , number of time steps L = 3, and initial state σ = 0 .
4. In Figure 2, all the subsample paths with the same l constitute Ω l . Taking l = 3 as anexample, eight subsample paths generate Ω = { [0 . , . , . , . , · · · , [0 . , . , . , . } . Based onΩ l in Figure 2, Figure 3 presents the corresponding value space Ψ l . As an example with l = 3,the different subsample paths [0 . , . , . , .
4] and [0 . , . , . , .
4] are distinct elements in Ω ,but since they have the same weight . +0 . +0 . = . +0 . +0 . = 0 .
12, the same length and lastelement, then they correspond to the same element [0 . , , .
4] in Ψ . Finally, based on Equation(10), we have Ψ = { . , . , . } , which is consistent with the definition of V . ption Pricing under MS-SVCJ Model intermediary set without recombination . , , . . , , . . , , . . , , .
2] [0 . , , . . , , . . , , . . , , . . , , . . , , . . , , . . , , .
2] [0 . , , . . , , . . , , . . , , . . , , . . , , . Figure 4
Recursion Recombination Relationship From Ψ to Ψ Now we present a recursive algorithm to obtain Ψ L and p ([ x, L, σ L ]). Since the initial state σ isfixed, for l = 1, we have Ψ = { [ σ , , σ ] : σ ∈ { u , · · · , u m }} and p ([ σ , , σ ]) = p σ σ .The main recursive step is based on the following proposition, where ω l +1 can be generated from ω l by taking a step forward. Proposition 3.
The value space and the probability of [ | ω l | , l, σ l ] follow the recursive relationship: Ψ l +1 = { [ z, l + 1 , σ l +1 ] : z = x · l + σ l l + 1 , [ x, l, σ l ] ∈ Ψ l , σ l +1 ∈ { u , · · · , u m }} ,p ([ z, l + 1 , σ l +1 ]) = X [ x,l,σ l ] ∈ Ψ l : x =( z · ( l +1) − σ l ) /l p ([ x, l, σ l ]) p σ l σ l +1 . Continuing with the example above, Figure 4 illustrates recursion and recombination from fromΨ to Ψ . We start from Ψ , then take a step forward to generate an intermediary set withoutrecombination. For example, [0 . , , . ∈ Ψ and [0 . , , . ∈ Ψ taking a step forward generate { [0 . , , . , [0 . , , . } and { [0 . , , . , [0 . , , . } , respectively. After recombining the sameelement in the intermediary set, for example [0 . , , .
4] and [0 . , , . .Table 1 provides the RR algorithm for the value space Ψ and the probability distribution { p V ( · ) } .In terms of computational complexity, we have the following result (see Appendix G for proof): Proposition 4.
The total number of distinct triples [ x, l, σ l ] from step to step L is bounded by m (cid:0) L − mm (cid:1) ; hence the complexity of the RR algorithm is O ( L m ) . ption Pricing under MS-SVCJ Model Table 1
RR Algorithm: Obtaining the Value Space Ψ and the Probability Distribution { p V ( · ) } Input:
State set { u , · · · , u m } , transition probabilities p ij , i, j ∈ { , · · · , m } , initial state σ , number of time steps L . Initialization:
Set Ψ = { [ σ , , σ ] : σ ∈ { u , · · · , u m }} , p ([ σ , , σ ]) = p σ σ . Recursion:
For l = 1 to L − l +1 = { [ z, l + 1 , σ l +1 ] : z = x · l + σ l l +1 , [ x, l, σ l ] ∈ Ψ l , σ l +1 ∈ { u , · · · , u m }} , p ([ z, l + 1 , σ l +1 ]) = P [ x,l,σ l ] ∈ Ψ l : x =( z · ( l +1) − σ l ) /l p ([ x, l, σ l ]) p σ l σ l +1 . Output:
Ψ = { x : [ x, L, σ L ] ∈ Ψ L } , p V ( v ) = P [ x,L,σ L ] ∈ Ψ L : x = v p ( x, L, σ L ).Since L ≫ m in settings of practical interest, the RR algorithm should be far superior to CE interms of computation, which is confirmed in the next section.
5. Numerical Experiments
In this section, we first price the European call option under the proposed model. We conductMonte Carlo simulation to assess the impact of the jump time assumption on the option price.After that, we discuss the result for the Bermudan call option pricing to show the effectiveness ofour approach. Then we compare the proposed RR algorithm with CE, and the results highlightthe computational superiority of the RR algorithm. Lastly, we provide an application example offitting to real-market data. All numerical results are obtained using MATLAB (see Appendix Hfor MATLAB code) on a 2.40 GHz Intel Xeon E5-2680, 128 GB RAM computer.
Before applying our pricing method under the proposed model, we conduct a Monte Carlo simu-lation to assess the impact of the assumption on jump time on the option price. The parametersfor the proposed model are presented in Table 2. Specifically, we follow Todorov (2011) to assignthe values of duration ∆, attenuating factor β , proportional coefficient b (see Appendix J). ption Pricing under MS-SVCJ Model Table 2
Parameter Values for MS-SVCJ Model
Parameter Value Parameter Value Parameter ValueMaturity T = 0 .
25 Jump variance ε = 0 .
005 Initial state σ = 0 . K = 55 Max † N max = 10 State space σ k ∈ { . , . , . , . } Risk-free rate r = 0 .
05 Time step τ = 0 . /
30 Transitionprobabilitymatrix P = .
70 0 .
15 0 .
10 0 . .
03 0 .
90 0 .
06 0 . .
05 0 .
05 0 .
85 0 . .
03 0 .
07 0 .
10 0 . Asset price S = 50 Duration ∆ = 0 . λ = 3 Attenuating factor β = 250Jump mean µ = − .
025 Proportional coefficient b = 2 † max N max such that P ( N > N max ) < ǫ ; for ǫ = 5 . × − with λ and T values, N max = 10. Table 3
Option Valuation Comparison
MS-SVCJ MC Simulation N =600 750 900 1200 1500Option Price 0.9696 0.9680 0.9683 0.9684 0.9687 0.9689(Std Err) (.0063) (.0044) (.0050) (.0066) (.0054)Computation Time (seconds) 30 16575 20213 25014 28411 41034Using an Euler approximation with N equal subintervals, Monte Carlo simulation generates theasset price at maturity. For the case in Table 2, the option prices using Theorem 1, denoted byMS-SVCJ, and Monte Carlo simulation, denoted by MC, are summarized in Table 3. For the MCcolumn, option price, standard error, and computation time are based on 10 sets of 100 ,
000 paths.MC simulation is very time-consuming, with an example of N = 1500 taking around 11 hours.As mentioned in Section 3.1 and Appendix I, the assumption on jump time will increase thevolatility slightly, hence increase the option price slightly. On the other hand, the MC valuesmonotonically increase with N , so the option price without the assumption lies in the interval[0 . , . .
07% = | . − . | . and indicating thatthe impact of the assumption on the option price is negligible. Moreover, the option price withthe assumption stays within the 95% confidence interval of all the MC values, providing furthersupport for the reasonableness of the assumption. ption Pricing under MS-SVCJ Model For an American-style option, we apply the secant and tangent algorithms of Laprise et al. (2006)to establish bounds for its price. Under the MS-SV model, the value V t i ( S i , K i , t i +1 − t i ) is aweighted sum of BS ( S i , v, r, t i +1 − t i , K i ) over the possible AIVs. We use n interpolation points forthe asset price, where the approximation accuracy can be improved by increasing the number ofinterpolation points. As a comparison, we also implement the least squares Monte Carlo simulationapproach of Longstaff and Schwartz (2001) for American option pricing, denoted by LSM, basedon 10 sets of independent runs with 100,000 paths for each run.We illustrate our approach with the following example from Laprise et al. (2006): a three-yearBermudan call option with strike price K = 100, exercisable every 0 . r = 0 .
05, dividend rate δ = 0 .
04, initial state σ = 0 . σ t ∈ { . , . , . , . } , time step τ = 0 . /
30 with the transition matrix P shownin Table 2. We calculate V t i ( S i , K i , t i +1 − t i ) and ∂∂S i V t i ( S i , K i , t i +1 − t i ) using Equation (9).The results in Table 4 show that the upper and lower bounds derived from the secant andtangent algorithms, respectively, converge quickly with the number of interpolating points. Forexample, with 100 interpolating points, the upper and lower bounds are within a penny of thetrue price. For n = 200, option prices with different initial asset prices via the secant and tangentalgorithms all stay with the 95% confidence interval of the corresponding LSM values. In addition,the computation time for LSM is orders of magnitude higher than the time for the secant andtangent approximation approaches.Next we apply our approach to price the same call option under the MS-SVCJ model, which isvery similar to MS-SV model with additional jumps and co-jumps. The model parameters are thesame as in Table 2, and V t i ( S i , K i , t i +1 − t i ) and ∂∂S i V t i ( S i , K i , t i +1 − t i ) are calculated via Equation(8). The results for n = 20, 50 and 100 are illustrated in Table 5. ption Pricing under MS-SVCJ Model Table 4
Bermudan Call Option Pricing Under MS-SV Model
Option Price Computation TimeAlgorithm n S =60 90 100 110 140 (seconds)Tangent 50 1.294 9.846 14.867 20.848 43.204 0.41100 1.302 9.861 14.883 20.861 43.212 0.98200 1.305 9.864 14.886 20.864 43.213 2.97Secant 200 1.307 9.868 14.890 20.867 43.215 1.64100 1.311 9.875 14.897 20.873 43.219 0.5550 1.328 9.904 14.925 20.899 43.234 0.23LSM 1.306 9.866 14.888 20.860 43.210 415(Std Err) (.016) (.019) (.043) (.020) (.074) Table 5
Bermudan Call Option Pricing Under MS-SVCJ Model
Option Price Computation TimeAlgorithm n S =60 90 100 110 140 (seconds)Tangent 20 1.970 11.624 16.815 22.845 44.864 6850 2.040 11.723 16.911 22.932 44.924 390100 2.050 11.737 16.924 22.945 44.933 1504Secant 100 2.060 11.752 16.938 22.957 44.941 89550 2.080 11.780 16.965 22.982 44.958 23020 2.223 11.977 17.154 23.157 45.078 39LSM 2.066 11.773 16.957 22.972 44.903 23934(Std Err) (.028) (.082) (.082) (.079) (.095) We compare the computation time for CE and the RR algorithm as a function of L and m , withthe results provided in Tables 6 and 7, respectively. The results in Table 6 illustrate that for CE,the computation time increases exponentially with respect to the number of time steps L , and theconsumed memory is quickly exhausted, which limits the application of CE in practice. Comparing ption Pricing under MS-SVCJ Model Table 6
Computation Time for CE (seconds, ‘*’ indicates out of memory) m L
15 16 17 18 19 20 25 302 0.007 0.01 0.04 0.04 0.07 0.13 4.5 1513 1.3 4.1 12 38 119 365 * *4 92 * * * * * * *5 * * * * * * * *6 * * * * * * * *
Table 7
Computation Time for RR Algorithm (seconds) m L
20 25 30 35 40 45 502 0.005 0.006 0.007 0.008 0.011 0.013 0.0143 0.009 0.013 0.019 0.024 0.033 0.040 0.0504 0.05 0.11 0.21 0.39 0.66 1.05 1.585 0.27 0.8 1.9 4.0 7.3 12 196 1.2 4.2 10 22 38 60 87Table 6 with Table 7, the improvement using the RR algorithm is significant. For example, for m = 5, L = 40 or m = 6, L = 30, the RR algorithm finishes within about 10 seconds.To illustrate the computational complexity of both algorithms, we graph the computation timeas a function of the number of time steps L for both algorithms. Figure 5 shows representativeplots of the performance of CE with two states and the RR algorithm with five states; additionalresults are provided in Appendix K. The results support the theoretical computational complexityof O ( m L ) for CE and O ( L m ) for the RR algorithm. ption Pricing under MS-SVCJ Model
15 20 25 30-6-4-20246 (a) CE with Two States (b) RR Algorithm with Five States
Figure 5
Computation Time as a Function of the Number of Time Steps L (Log Scales) We calibrate our model to market option prices after estimating the MS and PEA model parametersfor the underlying asset prices using real data from Yahoo Finance consisting of the daily closingstock prices of IBM from January 1, 2005 to June 30, 2019. We estimate the jump process using thebox-plot method. Specifically, we assume values of the daily log-return r at = ln( S t + a /S t ) outside therange ( Q − k f R f , Q + k f R f ) constitute jumps, where a = 1 /
252 is the sampling interval lengthfor the daily price, Q is the lower quartile, Q is the upper quartile, the interquartile range R f isdefined as Q − Q , and k f is a constant. After assigning k f , we can estimate jump intensity, meanof jump size, and variance of jump size for the jump process.We then estimate the MS process using maximum likelihood estimation (MLE) and the PEAprocess using the generalized method of moments (GMM). Following the box-plot method, wedecompose the full sample into two subsamples, the diffusion subsample within the range ( Q − k f R f , Q + k f R f ) and the jump subsample outside the range ( Q − k f R f , Q + k f R f ). Based onthe diffusion subsample, we estimate the MS process by the method provided in Perlin (2015);see Appendix L for the details. Based on the jump subsample, we estimate the PEA process byGMM provided in Todorov (2011); see Appendix M for the details. The parameters for the PEAare duration ∆ = 0 .
02; attenuating factor β = 550; proportional coefficient b = 4 .
45. For the MS ption Pricing under MS-SVCJ Model process, the state space is σ k ∈ { . , . , . , . } , i.e, four states with correspondingtransition probability matrix P = . . . . . . . . . . . . . . . . . Next we discuss the model calibration procedure. The proposed model consists of three parts:the PEA process captures the correlation for model drivers, while the MS process and jump processcapture the volatility of underlying asset. For illustrative purposes, in this example we fix theparameters for the PEA and MS processes estimated from historical data, and solve for the optimalparameters for the jump process by calibrating to the option market prices.Specifically, we calibrate our model to IBM call options on July 1, 2019 with maturity T = 1 . r = 2 .
36% is determined by US Dollar LIBOR rates using two maturities,1 month and 2 months, by linear interpolation to match option maturity. Similar to Cai and Kou(2011), we minimize the objective function N P i =1 ( ˜ C i ( π ) − C i ) /C i over the set of varying parameters π = ( λ, µ, σ ) of the jump process, where ˜ C i ( π ) and C i represent the calibrated price and themarket price for the i th option, respectively. To solve the optimization problem, a random searchalgorithm gave the final optimal solution: λ = 4 . µ = − . σ = 0 .
6. Conclusions
In the paper, we propose the MS-SVCJ model to better capture volatility clustering. Under theproposed model, we derive an analytical solution for the price of European options. Due to thegeneral nature of the model, we can apply the analytical solution to some special cases, such as theMS-SVJ model with general jump size distribution or the MS-SV model. An analytical solutionfor the MS-SV model avoids solving ordinary differential equations using numerical inverse Fourier ption Pricing under MS-SVCJ Model Table 8
Call Option Prices
Market ModelStrike Bid Ask Mid-Price Price Bias125 15.05 16.85 15.95 15.83 − . − . − . − . − . . . .
125 130 135 140 145 150 155 160
Strike P r i c e Market PriceModel Price
Figure 6
Comparison for Option Price under Market and Model transform methods. We also consider an approximation approach to price American-style optionsby leveraging our analytical solution for European options. To efficiently compute option prices, wepropose the RR algorithm to derive the probability distribution of AIV, analyze its computationalcomplexity, and verify its effectiveness numerically. ption Pricing under MS-SVCJ Model The empirical case study discussed in Section 5.4 uses asset prices to estimate the MS andPEA model parameters, while calibrating the jump process parameters by market option prices. Inpractice, market data on actual option prices can be used to calibrate all of the model parametersof any option pricing model. Although not the focus of this work, a complete calibration procedurebased on only market option prices would make our algorithm more relevant to practitioners.We briefly suggest one possible approach, which adopts a two-stage calibration procedure, cf.Galluccio and Lecam (2008), Clark (2011), Tan (2012), Homescu (2014); however, determining agood procedure is definitely a critical need for further research.The two-stage calibration procedure calibrates the MS process and jump process (the compoundPoisson process and the PEA process) separately. Specifically, at the first stage, we calibratethe MS-SV model to market option prices. The approach in Britten-Jones and Neuberger (2000)appears to be well suited to our MS-SV model, since it models volatility following a MS process.In this approach, they first determine a base model reflecting one’s prior information on market,then adjust the base model to fit option prices.At the second stage, we can calibrate CJ component of the MS-SVCJ model with the calibratedMS-SV process at the first stage. The calibration is to determine the compound Poisson processand the PEA process by minimizing the sum of an in-sample quadratic pricing error and a con-vex penalization term. In practice, the key is to select the convex penalization, which consistsof two terms, one due to the compound Poisson process and the other due to the PEA process.Cont and Tankov (2004) used relative entropy (or Kullback-Leibler divergence) from a prior distri-bution as a convex penalization term. For the PEA process, since parameters for the PEA processform a Hilbert space, a quadratic function (or Tikhonov regularization) is appropriate and applied.The methodology developed here should also be applicable in other contexts beyond optionpricing, e.g., variance swap pricing, which depends highly on the AIV. Other open problems forfuture research include hedging under the proposed model, as well as extensions to more generalmodels, e.g., models with general jump size distribution. ption Pricing under MS-SVCJ Model Appendix A: Proof of Proposition 1
We denote l k as the number of states { u k } of sample paths of the MS process from step 1 tostep L − k ∈ { , , . . . , m } . Thus, for each sample path of the MS process, we can get a tuple( l , l , . . . , l m ). According to the definition of the weight | ω | of a sample path of the MS process,since σ i , i ∈ { , , · · · , L − } , take value in state set { u , · · · , u m } , after merging the same statesin sample path, the weight can be rewritten as | ω | = σ +( l u + ... + l m u m ) L where σ is the initialstate of the MS process and is pre-determined. This leads to a surjection from tuple ( l , l , . . . , l m )to weight | ω | rather than an injection due to the possibility of the different tuples generatingthe same weights. Thus, the number of distinct values for v is less than the number of distincttuples, which satisfies the equation L − l + . . . + l m , for which the solution is a combinatorialproblem given by (cid:0) L − m − m − (cid:1) , which is the number of ways to select m − { , , , . . . , L − m − } . Hence, we have | Ψ | ≤ (cid:0) L + m − m − (cid:1) . (cid:3) Appendix B: Proof of Lemma 1
According to the arbitrage-free pricing theory, the European option price is the expectation ofthe terminal payoff under the risk-neutral probability measure. At maturity T , the payoff of aEuropean call option is ( S T − K ) + , so the European call option price is given by C = e − rT E ( S T − K ) + = + ∞ X n =0 p ( N T = n ) e − rT E ( S T ( n ) − K ) + = + ∞ X n =0 p ( N T = n ) X ω ∈ Ω p ( ω ) e − rT E ( S T ( n, ω ) − K ) + = + ∞ X n =0 p ( N T = n ) X ω ∈ Ω p ( ω ) E J n ( e − rT E ( S T ( J n , ω ) − K ) + )where N T is the number of jumps in the asset price up to T ; J n := ( J , · · · , J n ) is an n -dimensionalrandom vector of n jump sizes; Ω is the sample path space of { σ k } ; S T ( J n , ω ) is the asset price atthe maturity T given J n and ω . (cid:3) Appendix C: Details of Derivation for W ( J n , ω )Here, we consider the conditional probability distribution of the asset price at T , given n jumps J n := ( J , J , ..., J n ) and the sample path ω of { σ k } . Correspondingly, the asset price S T ( J n , ω )is the solution to the following stochastic differential equation: dS t S t − = ( r − λζ ) dt + ˆ σ t dB t + ( J t − dN t where the volatility process { ˆ σ t } is a deterministic function of time t . ption Pricing under MS-SVCJ Model In terms of the analysis in Hull and White (1987), the solution to above equation is given by S T ( J n , ω ) = S n Y i =1 J i e ( r − λζ − V ( J n,ω )2 ) T + √ V ( J n ,ω ) B T where V ( J n , ω ) = T R T ˆ σ t ( J n , ω ) dt . Hence, we have W ( J n , ω ) = ln S T ( J n , ω ) S n Q i =1 J i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( J n , ω ) ∼ N (( r − λζ − V ( J n , ω )2 ) T, V ( J n , ω ) T ) . (cid:3) Appendix D: Proof of Theorem C = + ∞ X n =0 p ( N T = n ) X ω ∈ Ω p ( ω ) E J n ( e − rT E ( S T ( J n , ω ) − K ) + )= + ∞ X n =0 p ( N T = n ) X ω ∈ Ω p ( ω ) E J n ( BS ( S e − λζT + n P i =1 ln( J i ) , | ω | + ˆ b n X i =1 ln ( J i ) , r, T, K ))= + ∞ X n =0 p ( N T = n ) X ω ∈ Ω p ( ω ) E Ξ n ( BS ( S e − λζT + X n , | ω | + ˆ bY n , r, T, K ))= + ∞ X n =0 p ( N T = n ) X ω ∈ Ω p ( ω ) C n ( | ω | ) = + ∞ X n =0 p ( N T = n ) X v ∈ Ψ X ω ∈ Ω: | ω | = v p ( ω ) C n ( | ω | )= + ∞ X n =0 p ( N T = n ) X v ∈ Ψ p V ( v ) C n ( v ) , where C n ( Z ) = E Ξ n ( BS ( S e − λζT + X n , Z +ˆ bY n , r, T, K )), Ξ n := ( X n , Y n ) := ( n P i =1 ln( J i ) , n P i =1 ln ( J i )). (cid:3) Appendix E: Proof of Proposition 2
For notational convenience, we denote: x i = ln( J i )( x, y ) = ( n X i =1 x i , n X i =1 x i ) (11)where n is the number of jumps during [0 , T ] and the jump size distribution ln( J i ) i.i.d ∼ N ( µ, ε ).When n = 1, the proof is obvious and omitted.Next, we suppose n ≥
2. Before determining the joint probability density g ( x, y ), we determinethe support set of the bivariate random variable ( x, y ), D = { ( x, y ) : g ( x, y ) > } . By the Cauchy-Schwarz inequality, we have: (cid:0) x + x + . . . + x n (cid:1) (cid:0) + 1 + . . . + 1 (cid:1) ≥ ( x + x + . . . + x n ) , (12) ption Pricing under MS-SVCJ Model so substituting the definition of ( x, y ) in Equation (11) into Equation (12), we have y ≥ x n . Thus,the support set D = { ( x, y ) : −∞ < x < + ∞ , x n ≤ y < + ∞} , on which we derive the joint probabilitydensity g ( x, y ). We have: y = n X i =1 x i = n X i =1 (cid:16) ( x i − x ) + 2 x i x − x (cid:17) = n X i =1 ( x i − x ) + 2 x n X i =1 x i − nx = ( n − s + 2 xn x − n (cid:16) xn (cid:17) = ( n − s + x n ⇒ yε = ( n − s ε + x nε where x = 1 n n X i =1 x i , s = 1 n − n X i =1 ( x i − x ) . According to Theorem 5.3.1 in Casella and Berger (2002), ( n − s ε and x are mutually indepen-dent with probability distributions χ ( n −
1) and N ( nµ, nε ), respectively. By conditional proba-bility, we decompose g ( x, y ) = g ( x ) g ( y | x ), where g ( x ) = 1 √ π √ nε e − ( x − nµ )22 nε g ( y | x ) = 1Γ (cid:0) n − (cid:1) n − y − x n ε ! n − − e − y − x n ε ε , which leads to the desired result. (cid:3) Appendix F: Proof of Corollary f ( J i , t, t i ) = 0, we have ˆ b = 0 and C n ( Z ) = E X n ( BS ( S e − λζT + X n , Z, r, T, K )).Since C n ( Z ) does not require the probability distribution of jump size J t , the expression holdsfor the jump size J t following a general distribution, so the European call option price is C = + ∞ X n =0 p ( N T = n ) X v ∈ Ψ p V ( v ) C n ( v )= X v ∈ Ψ p V ( v ) + ∞ X n =0 p ( N T = n ) E X n ( BS ( S e − λζT + X n , v, r, T, K )) = X v ∈ Ψ p V ( v ) C jd ( v )where C jd ( v ) = + ∞ P n =0 p ( N T = n ) E X n ( BS ( S e − λζT + X n , v, r, T, K )) is the European call option priceunder the jump-diffusion model. (cid:3) Appendix G: Complexity of RR Algorithm
To prove the complexity of RR algorithm, we first provide the following lemma.
Lemma 2.
The number of distinct triples [ x, l, σ l ] at step l is less than m (cid:0) l + m − m − (cid:1) .Proof. At step l , the number of distinct triples [ x, l, σ l ] can be decomposed into the product ofthe number of distinct values of x and the number of distinct values of σ l , which is a combinatorial ption Pricing under MS-SVCJ Model problem. Obviously, the number of distinct values of σ l is m . To determine the number of distinctvalues of x , we denote l k as the number of states { u k } of subsample paths of the MS process fromstep 1 to step l − k ∈ { , , . . . , m } . Thus, for each subsample path of the MS process, we can geta tuple ( l , l , . . . , l m ). Similar to Proposition 1, we have: | ω l | = σ + ( l u + . . . + l m u m ) ll − l + . . . + l m where σ is the initial state of the MS process and is pre-determined.According to Proposition 1, the number of distinct values of x is less than (cid:0) l − m − m − (cid:1) . Hence, thenumber of distinct triples [ x, l, σ l ] at step l is less than m (cid:0) l + m − m − (cid:1) . (cid:3) Proposition 5.
The total number of distinct triples [ x, l, σ l ] from step to step L is less than m (cid:0) L + m − m (cid:1) , hence the complexity of the RR algorithm is O ( L m ) .Proof. The total number of distinct triples [ x, l, σ l ] from step 1 to step L is a summation ofthe number of distinct triples [ x, l, σ l ] at every step 1 ≤ l ≤ L . According to Lemma 2 providing anupper bound for the number of distinct triples [ x, l, σ l ] at step l , carrying out a summation for L steps, we will provide an upper bound for the total number of distinct triples [ x, l, σ l ] from step 1to step L , L X i =1 m (cid:18) i − m − m − (cid:19) = m L − X i =0 (cid:18) i + m − m − (cid:19) = m "(cid:18) m − m − (cid:19) + L − X i =1 (cid:18) i + m − m − (cid:19) = m "(cid:18) mm (cid:19) + L − X i =1 (cid:18) i + m − m − (cid:19) = m "(cid:18) m + 1 m (cid:19) + L − X i =2 (cid:18) i + m − m − (cid:19) . . . = m (cid:18) L − mm (cid:19) . Hence, the total number of distinct triples [ x, l, σ l ] from step 1 to step L is less than m (cid:0) L + m − m (cid:1) .In addition, since L ≫ m in settings of practical interest, we have: m (cid:18) L − mm (cid:19) = m ( L − m )! m !( L − L + m − m − · L + m − m − · · · L + 11 · L = O ( L m ) . (cid:3) ption Pricing under MS-SVCJ Model Appendix H: MATLAB Code for RR Algorithm function [ l e f t v a r i a n c e , l e f t p r o b ]= AveStdTest5 ( i n i p r o b , v a r i a n c e , mat rix , n ) % i n i p r o b : t h e i n i t i a l s t a t e o f t h e MS p r o c e s s , e . g . , [ 0 1 0 0 ] ;% v a r i a n c e : t h e s t a t e s p a c e o f v a r i a n c e i n a s c e n d i n g o r d e r , e . g . , [ 0 . 0 20 . 0 4 0 . 0 6 0 . 0 8 ]% m a t r i x : t h e t r a n s i t i o n m a t r i x P ;% n : t h e t o t a l number o f t i m e s t e p s L ; t i c ;l e f t v a r i a n c e=t r a n s p o s e ( v a r i a n c e )+ dot ( i n i p r o b , v a r i a n c e ) ;l e f t p r o b=t r a n s p o s e ( i n i p r o b ∗ m a t r i x ) ;n s t e p=n − s i z e ( ma t rix , 1 ) ; f o r i =2: n s t e pl e f t v a r i a n c e=l e f t v a r i a n c e+v a r i a n c e ;t r a n s m a t=repmat ( ma t rix , s i z e ( l e f t v a r i a n c e , 1 ) / n s t a t e , 1 ) ;l e f t p r o b=l e f t p r o b . ∗ t r a n s m a t ;gro up=f i n d g r o u p s ( round ( l e f t v a r i a n c e ( : , 1 ) , 1 0 ) ) ;l e f t p r o b m a k e= zeros ( max ( g ro up ) , n s t a t e ) ; l e f t v a r i a n c e m a k e= zeros ( max (group ) , n s t a t e ) ; f o r j =1: n s t a t el e f t p r o b m a k e ( : , j )=ac c uma rra y ( group , l e f t p r o b ( : , j ) , [ ] , @sum) ;l e f t v a r i a n c e m a k e ( : , j )=a c c umarray ( group , l e f t v a r i a n c e ( : , j ) , [ ] ,@min ) ; end l e f t p r o b=t r a n s p o s e ( l e f t p r o b m a k e ) ;l e f t v a r i a n c e=t r a n s p o s e ( l e f t v a r i a n c e m a k e ) ;l e f t p r o b=l e f t p r o b ( : ) ;l e f t v a r i a n c e=l e f t v a r i a n c e ( : ) ; end group=f i n d g r o u p s ( round ( l e f t v a r i a n c e ( : , 1 ) , 1 0 ) ) ;l e f t p r o b m a k e=a c c umarra y ( group , l e f t p r o b , [ ] , @sum) ;l e f t v a r i a n c e m a k e=a c c uma rray ( group , l e f t v a r i a n c e , [ ] , @min ) ;l e f t v a r i a n c e=l e f t v a r i a n c e m a k e /n ;l e f t p r o b=l e f t p r o b m a k e ; toc ; end ption Pricing under MS-SVCJ Model Appendix I: Impact of Assumption on Jump Time
We discuss the impact of the assumption that all jumps in the small interval occur at the beginningof the interval on the asset price and AIV. Since the cumulative impact on asset price does notrelate to the actual times of the jumps, the asset price at T does not change under the assumption.In what follows, considering the probability of a jump, we analyze the expectation bias ( EB ) causedby the assumption on AIV.First, we denote the jump probability and the expectation bias as P l and EB l , respectively, whenthere are l jumps up to maturity T , given by EB = + ∞ X l =1 P l ∗ EB l ,P l = ( λT ) l l ! e − λT . Second, since the sample path of the MS process and the jump during the interval [0 , T − ∆] donot cause the bias, we only investigate the bias caused by a jump during the interval [ T − ∆ , T ].Given l jumps up to maturity T , we denote the conditional jump probability and the expectationbias as P lj and EB lj , respectively, for 1 ≤ j ≤ l jumps during the interval [ T − ∆ , T ], given by EB l = l X j =1 P lj ∗ EB lj P lj = ( λ ∆) j j ! e − λ ∆ ∗ ( λ ( T − ∆)) l − j ( l − j )! e − λ ( T − ∆) P l where P lj = p ( N ( T − ∆ ,T ) = j | N (0 ,T ) = l ) = p ( N ( T − ∆ ,T ) = j, N (0 ,T ) = l ) p ( N (0 ,T ) = l )= p ( N ( T − ∆ ,T ) = j, N (0 ,T − ∆) = l − j ) p ( N (0 ,T ) = l ) = p ( N ( T − ∆ ,T ) = j ) p ( N (0 ,T − ∆) = l − j ) p ( N (0 ,T ) = l ) . Third, we derive the detailed expression for EB lj . For the i th(1 ≤ i ≤ j ) jump J i at time t i ∈ [ T − ∆ , T ], without or with the assumption, the cumulative effects until expiration date T are,respectively: Z Tt i b ln ( J i ) e − β ( s − t i ) ds = b ln ( J i ) β (1 − e − β ( T − t i ) ) , Without the assumption , Z TT − ∆ b ln ( J i ) e − β ( s − ( T − ∆)) ds = b ln ( J i ) β (1 − e − β ∆ ) , With the assumption . ption Pricing under MS-SVCJ Model Hence, EB lj = 1 T E ( j X i =1 ( b ln ( J i ) β (1 − e − λ ∆ ) − b ln ( J i ) β (1 − e − λ ( T − t i ) )))= bηβT E ( j X i =1 (1 − e − β ∆ ) − (1 − e − β ( T − t i ) ))= bηβT E ( j X i =1 ( e − βY i − e − β ∆ ))= jbηβT ( 1 − e − β ∆ β ∆ − e − β ∆ )where η = E (ln ( J i )) = µ + ε and T − t i = Y i ∼ U [0 , ∆], where U [0 , ∆] is a uniform distribution,since for the Poisson process with intensity λ , conditioned on N t = n , the joint probability distri-bution of the ordered arrival times of jumps t < t < · · · < t n is the same as the joint probabilitydistribution of the order statistics U (1) < U (2) < · · · < U ( n ) with U i i.i.d. ∼ U [0 , t ], i = 1 , , · · · , n .Since EB lj → EB l → EB , we have EB = + ∞ X l =1 l X j =1 ( λ ∆) j j ! e − λ ∆ ∗ ( λ ( T − ∆)) l − j ( l − j )! e − λ ( T − ∆) jbηβT − (1 + β ∆) e − β ∆ β ∆ . Taking N max = 10, T = 0 . λ = 3, β = 250, ∆ = 0 . µ = − . ε = 0 .
005 in Table 2 as anexample, EB = 2 . × − . The option price C = 0 . σ imp = 0 . q σ imp − q σ imp − EB = 4 . × − , which is less than 0.002%. Appendix J: Estimating Parameters in the PEA Process
We describe estimation of the parameters of the PEA process: proportional coefficient b , attenuatingfactor β and duration ∆. For this purpose, we adopt the approach of Todorov (2011), in whichthe modeling of co-jumps is similar to ours, viz., the jump in variance is also proportional to thesquared jump in return and exponentially decays over time.Once a jump in return occurs, the proportional coefficient b determines the corresponding incre-ment of variance. In terms of the expressions of m c and m d in Todorov (2011), we derive theproportional coefficient b = 2.The function f ( u ) in Todorov (2011) describes the evolving pattern of jump in variance, whichcorresponds to our function f ( · ). For β , through sampling points { ( u i , f ( u i )) } ni =1 from f ( u ) inTodorov (2011) and implementing the least squares method, we estimated the attenuating factor β = 250 with the goodness of fit, R = 0 . .
02, which means once there is jump in variance, our model cancover 99 . ≈ − e − ∗ . e − ∗ of this increment over the next 5 days. We assume a year includes 252trading days, hence 252 ∗ . ≈ ption Pricing under MS-SVCJ Model Appendix K: Additional Empirical Results on Computational Complexity for CE and RR
We confirm the theoretical computational complexity of the CE and RR algorithms with a largernumber of states m ∈ { , , , , } . Specifically, the computation time as a function of the numberof time steps L is shown in Figure 7, in line with the theoretical results and numerical experimentsin the main body of the main manuscript.
15 20 25 30-6-4-20246 (a) CE with m = 2 , (b) RR Algorithm with m = 2 , , , , Figure 7
Computation Time as a Function of the Number of Time Steps L (Log Scales) Appendix L: Estimation of the MS Process for the Application Example in Section 5.4
Following the standard risk premia assumptions in the literature, the asset price without jumpsunder the objective probability measure follows geometric Brownian motion with drift ϑ and MSstochastic volatility σ t . To estimate the MS process, we consider the discrete version of the assetprice described by S t + a − S t S t = ϑa + σ t N √ a = ⇒ ˜ r at = S t + a − S t √ aS t = ϑ √ a + σ t N , where N follows a standard normal distribution. Using this result with existing MATLAB codesprovided in Perlin (2015) to the diffusion subsample generating ˜ r at , the estimated parameters forMS process are easily obtained. (cid:3) Appendix M: Estimation of the PEA Process for the Application Example in Section 5.4
Given the path of the MS process { σ t } , we derive closed-form expressions for variance, skewness,kurtosis of asset log-return: E ( r at − E ( r at )) = aσ t + a (1 + bδ ) M E ( r at − E ( r at )) = ( a + 3 bδ ( δa − e − δa )) M E ( r at − E ( r at )) = 3( aσ t ) + 6 a σ t (1 + bδ ) M +( a + 6 bδ ( δa − e − δa )) M + ( 3 a b ( b + 2) δ + 3 a ) M (13) ption Pricing under MS-SVCJ Model where M i = λm i , and m i , i = 2 , , i th moments of the log-jump distribution.Specifically, given the path of the MS process { σ t } , our original model in Equation (1) has similarprobability characteristics as the model in Todorov (2011), and Equation (13) can be justified byTheorem 1 of Todorov (2011). Applying the generalized method of moments (GMM) estimationto the jump subsample, the estimated parameters for the PEA process are easily obtained. Acknowledgments
Fu gratefully acknowledges financial support from the U.S. National Science Foundation [Grant CMMI-1434419]. Li gratefully acknowledges financial support from the Natural Science Foundation of China[Grant 71671094], and the Fundamental Research Funds for the Central Universities [Grants 63185019 and63172308]. The views and opinions expressed in this article are solely the authors own and do not reflect thebusiness and positions of R. Wu’s affiliation.
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