Sample path generation of the stochastic volatility CGMY process and its application to path-dependent option pricing
aa r X i v : . [ q -f i n . C P ] J a n Sample path generation of the stochastic volatility CGMY processand its application to path-dependent option pricing
Young Shin Kim * January 28, 2021
Abstract
This paper proposes the sample path generation method for the stochastic volatility version of CGMYprocess. We present the Monte-Carlo method for European and American option pricing with the samplepath generation and calibrate model parameters to the American style S&P 100 index options market,using the least square regression method. Moreover, we discuss path-dependent options such as Asianand Barrier options.
Since Heston (1993) applied the CIR model by Cox et al. (1985) to option pricing, the model has beenthe standard framework for option pricing because it allows for stochastic volatility and volatility smileeffect that observed for the Black Scholes model (Black and Scholes (1973)). However, the L´evy processmodels with time-varying volatility have been used in option pricing in descrete time model, since empir-ical studies based on the stochastic volatility model show that the Brownian Motion is often rejected (SeeRachev and Mittnik (2000), Kim et al. (2010), Kim et al. (2011)). Carr et al. (2003) defined the class ofcontinuous time stochastic volatility model on L´evy processes (SVLP) using the time changed L´evy pro-cess. The SVLP has been successfully applied in European option pricing, but the absence of an efficientsample path generation method makes the SVLP model hard to be applied to path-dependent options suchas American, Barrier or Asian option. * College of Business, Stony Brook University, New York, USA ([email protected]) et al. (2010), and it is applied to Monte-Carlo simulation for CGMY market model witha GARCH volatility in Kim et al. (2010). Yet, the CGMYSV model is a continuous-time model differ-ent from the GARCH model with CGMY innovations. We develop an algorithm of the CGMYSV samplepath generation, and it will be applied Monte-Carlo simulation (MCS). The algorithm will be used to priceEuropean and American option and to calibrate risk-neutral parameters to the S&P 500 index option (Eu-ropean style) and S&P 100 option (American style) data. We will use the least square regression methodby Longstaff and Schwartz (2001) for American option pricing with MCS. We verify that the new samplepath generation method performs well in American option pricing by that empirical study. We also applythe algorithm to Asian and Barrier option pricing with MCS.The remainder of this paper is organized as follows. The CIR process and the CGMY process with theseries representation are presented in Section 2. The sample path generation method based on the seriesrepresentation is constructed in Section 3. In Section 4, we perform the CGMYSV model calibration forS&P 500 index option and S&P 100 index option. Also, the Asian and Barrier option prices are discussed.Finally, Section 6 concludes.
In this section, we briefly discuss CIR model and CGMY process.
The CIR model is given by dv t = κ ( η − v t ) dt + ζ √ v t dW t and v > , (1)2or κ, η, ζ > and Brownian motion { W t } t ≥ . Let F vt be a smallest σ -algebra generated by process { v s } ≤ s ≤ t then v t + ∆t | F vt d = ξ/ (2 c ) where c = κ (1 − e − κ∆t ) ζ and the random variable ξ is non-central χ -distributed with degrees of freedom κη/ζ and noncentrality parameter cv t e − κ∆t .Let V t = R t v s ds then the joint distribution of ( v t , , V t ) is characterized by the characteristic function Φ t ( a, b, x ) = E [exp( a V t + ibv t ) | v = x ] given as following (Proposition 6.2.5 in Lamberton and Lapeyre(1996)) Φ t ( a, b, x ) = A ( t, a, b ) exp ( B ( t, a, b ) x ) (2)with A ( t, a, b ) = exp (cid:16) κ ηtζ (cid:17)(cid:16) cosh (cid:0) γt (cid:1) + κ − ibζ γ sinh (cid:0) γt (cid:1)(cid:17) κη/ζ B ( t, a, b ) = ib (cid:0) γ cosh (cid:0) γt (cid:1) − κ sinh (cid:0) γt (cid:1)(cid:1) + 2 ia sinh (cid:0) γt (cid:1) γ cosh (cid:0) γt (cid:1) + ( κ − ibζ ) sinh (cid:0) γt (cid:1) γ = p κ − ζ ia. For α ∈ (0 , , C, λ + , λ − > , and µ ∈ R , an infinitely divisible random variable X with characteristicfunction (Ch.F) φ CGMY ( u ; α, C, λ + , λ − , µ ) = φ X ( u ) = E [ e iuX ]= exp (cid:0) ( µ − C Γ(1 − α )( λ α − − λ α − − )) iu − C Γ( − α ) (cid:0) ( λ + − iu ) α − λ α + + ( λ − + iu ) α − λ α − (cid:1)(cid:1) is referred to as the CGMY distributed random variable with parameters ( α , C , λ + , λ − , µ ) . In this case,we denote X ∼ CGMY ( α , C , λ + , λ − , µ ) .Let C = (Γ(2 − α )( λ α − + λ α − − )) − and µ = 0 . Then a CGMY random variable Z ∼ CGMY ( α , C , λ + , λ − , µ ) has zero mean ( E [ Z ] = 0 ) and unit variance (var ( Z ) = 1 ). In this case, we say that Z is The class of tempered stable processes has been introduced under different names including: “truncated L´evy flight” (Koponen(1995)), “KoBoL” process (Boyarchenko and Levendorski˘i (2000)), “CGMY” process (Carr et al. (2002)), and classical temperedstable process (Rachev et al. (2011)). Rosi´nski (2007) and Bianchi et al. (2010) generalized the notion of tempered stable processes. Z ∼ stdCGMY ( α , λ + , λ − ) . Moreover, the Ch.F of Z is given by φ stdCGMY ( u ; α, λ + , λ − ) = φ Z ( u ) = E [ e iuZ ]= exp λ α − − λ α − − ( α − λ α − + λ α − − ) iu + ( λ + − iu ) α − λ α + + ( λ − + iu ) α − λ α − α ( α − λ α − + λ α − − ) ! . (3)Since the CGMY distribution is purely non-Gaussian infinitely divisible, it generate a pure jump L´evy pro-cess { X t } t ≥ such that X ∼ CGMY ( α , C , λ + , λ − , µ ) . In this case, we say that ( X t ) t ≥ is CGMY processwith parameters ( α , C , λ + , λ − , µ ) . The characteristic function (Ch.F) of X t is φ X t ( u ) = exp( t log( φ CGMY ( u ; α, C, λ + , λ − , µ ))) . With the same argument, a pure jump L´evyprocess { Z t } t ≥ such that Z ∼ stdCGMY ( α , λ + , λ − ) is referredto as the standard CGMY process with parameters ( α , λ + , λ − ) . The CGMY and standard CGMY processesare characterized by their L´evy symbols ψ CGMY ( u ; α, C, λ + , λ − , µ ) = log φ CGMY ( u ; α, C, λ + , λ − , µ ) and ψ stdCGMY ( u ; α, λ + , λ − ) = log φ stdCGMY ( u ; α, λ + , λ − ) , respectively. Rosi´nski (2007) introduced the series representation form for the tempered stable random variable andprocess, and it can be used for the CGMY sample path generation. Assume that X ∼ CGMY ( α , C , λ + , λ − , . Let { U j } j =1 , , ··· be an independent and identically distributed (i.i.d.) sequence of uniform randomvariables on (0 , . Let { E j } j =1 , , ··· be i.i.d. sequences of exponential random variables with parameters1, and let { Γ j } j =1 , , ··· be a Poisson point process with parameter 1. Let { V j } j =1 , , ··· be an i.i.d. sequenceof random variables in { λ + , λ − } with P ( V j = λ + ) = P ( V j = λ − ) = 1 / . Suppose that { U j } j =1 , , ··· , { V j } j =1 , , ··· , { E j } j =1 , , ··· , and { Γ j } j =1 , , ··· are independent. Then X represented by the following series4orm: X = ∞ X j =1 "(cid:18) α Γ j C (cid:19) − /α ∧ E j U /αj | V j | − V j | V j | + b, where b = − C Γ(1 − α ) (cid:0) λ α − − λ α − − (cid:1) . Let { τ j } j =1 , , ··· be an i.i.d. sequence of uniform random variableson (0 , T ) independent of { U j } j =1 , , ··· , { V j } j =1 , , ··· , { E j } j =1 , , ··· , and { Γ j } j =1 , , ··· . Suppose X t = ∞ X j =1 "(cid:18) α Γ j CT (cid:19) − /α ∧ E j U /αj | V j | − V j | V j | τ j ≤ t + tb T , t ∈ [0 , T ] , where b T = − C Γ(1 − α ) (cid:0) λ α − − λ α − − (cid:1) . Then the the process { X t } t ∈ [0 ,T ] is the CGMY process withparameters ( α , C , λ + , λ − , for the time horizon T > . Suppose { Z t } t ≥ is the standard CGMY process with parameters ( α , λ + , λ − ) and { v t } is the stochasticvolatility process given by CIR model in(1). We define a process { L t } t ≥ by L t = Z V t + ρv t (4)where V t = R t v s ds , and { Z t } t ≥ is independent of the process { v t } t ≥ . The process { L t } t ≥ is referred toas the stochastic volatility version of the CGMY process or simply CGMYSV process with parameters ( α , λ + , λ − , κ , η , ζ , ρ , v ) . By (2), we obtain the characteristic function of L t as φ L t ( u ) = Φ t ( − iψ stdCGMY ( u ; α, λ + , λ − ) , ρu, v ) , (5)where φ stdCGMY ( u ; α, λ + , λ − ) is the characteristic function of Z defined in (3). Suppose that we have a CIR process { v t } t ≥ with parameters κ, η, and ζ as defined in (1), V t = R t v s ds for t > , and suppose that {F vt } t ≥ is natural filtration generated by { v t } t ≥ . Let P = { t < In Carr et al. (2003), { Z t } t ≥ is assumed to a CGMY process, but we assume a standard CGMY process in this paper tosimplify the model. The stochastic volatility L´evy process model is not a L´evy process in general. lgorithm 1: CGMYSV sample path generation
Result:
SVMYSV sample pathLet T be the time horizon ;Let M , J , and N be large positive integer ; ∆t = T /M , v n, = v , c = κ (1 − e − κ∆t ) ζ , C = (Γ(2 − α )( λ α − + λ α − − )) − ; n = 1 ; while n ≤ N do m = 1 ; while m ≤ M do ξ = non-central χ -distributed random variable with degrees of freedom κη/ζ andnoncentrality parameter cv n,m − e − κ∆t ; v n,m = ξ/ (2 c ) ; m = m + 1 end j = 1 , Γ = 0 ; while j ≤ N do U j = uniform random numbers on (0 , ; U ′ j = uniform random numbers on (0 , ; E j = exponential random numbers with parameter 1 ; E ′ j = exponential random numbers with parameter 1 ; Γ j = Γ j − + E ′ j ; if U ′ j ≤ . then V j = λ + else V j = λ − end τ j = uniform random numbers on (0 , T ) ; c ( τ j ) = C P Mk =1 v n,k − ( k − ∆t<τ j ≤ k∆t ; j = j + 1 ; end m = 1 , Y n, = 0 ; while m ≤ M do b m = − v n,m − ( λ α − − λ α − − ) (1 − α )( λ α − + λ α − − ) ; Y n,m = Y n,m − + P Jj =1 (cid:20)(cid:16) α Γ j c ( τ j ) T (cid:17) − /α ∧ E j U /αj | V j | − (cid:21) V j | V j | ( m − ∆t<τ j ≤ m∆t + b m ∆tL n,m = Y n,m + ρv n,m ; m = m + 1 ; end n = n + 1 ; end < · · · < t m < · · · < M = T } be the partition of time, ∆t m = t m − t m − for m ∈ { , , · · · , M } ,and let || P || = max { ∆t m | m = 1 , , · · · , M } . Suppose that { v t m } t m ∈ P and { L t m } t m ∈ P are discretesub-sequences of the CIR process and the CGMYSV process, respectively. Let ∆L t m = L t m − L t m − , ∆ V t m = V t m − V t m − , and ∆v t m = v t m − v t m − . Then we have ∆L t m | F vtm − d = ( Z V tm −V tm − ) | F vtm − + ρ ( v t m − v t m − ) | F vtm − = Z ∆ V tm | F vtm − + ρ ( ∆v t m | F vtm − ) , where Z ∆ V tm | F vtm − ∼ CGMY (cid:16) α, C ( ∆ V t m | F vtm − ) , λ + , λ − , (cid:17) , and C = (cid:0) Γ(2 − α )( λ α − + λ α − − ) (cid:1) − . Since we approximate ∆ V t m | F vtm − = Z t m t m − v t dt ≈ v t m − ∆t m , we have Z ∆ V tm | F vtm − ≈ Z v tm − ∆t m | F vtm − ∼ CGMY (cid:0) α, Cv t m − ∆t m , λ + , λ − , (cid:1) . Suppose { Y t m } t m ∈ P is a process defined by Y t m = Y t m − + Z v tm − ∆t m | F vtm − , m = 1 , , · · · , M with Y = 0 , then Y t m ≈ Z V tm | F vtm − and { Y t m } t m ∈ P is an approximation of the process { Z V tm | F vtm − } t m ∈ P .By the series representation, we have Z v tm − ∆t m | F vtm − d = ∞ X j =1 "(cid:18) α Γ j c m ∆t m (cid:19) − /α ∧ E j U /αj | V j | − V j | V j | + b m ∆t m where b m = − c m Γ(1 − α ) (cid:0) λ α − − λ α − − (cid:1) , c m = v t m − C , and { U j } j =1 , , ··· , { V j } j =1 , , ··· , { E j } j =1 , , ··· ,and { Γ j } j =1 , , ··· are given in Section 2.3. The same argument as the relation between series representationof the CGMY process presented in Section 2.3, we can define a series representation of Y t m as follows: Y t m = ∞ X j =1 "(cid:18) α Γ j c ( τ j ) T (cid:19) − /α ∧ E j U /αj | V j | − V j | V j | τ j ≤ t m + m X k =1 b k ∆t k (6)7here b k = − v t k − (cid:0) λ α − − λ α − − (cid:1) (1 − α )( λ α − + λ α − − ) ,c ( τ j ) = C M X m =1 v t m − t m − <τ j ≤ t m , and { τ j } j =1 , , ··· is an i.i.d. sequence of uniform random variables on (0 , T ) independent of { U j } j =1 , , ··· , { V j } j =1 , , ··· , { E j } j =1 , , ··· , and { Γ j } j =1 , , ··· . Therefore, we have L t m | F vtm − = Z V tm | F vtm − + ρv t m | F vtm − ≈ Y t m + ρv t m | F vtm − . (7)Combining equations (6) and (7), we can generate sample path of the CGMYSV process as Algorithm 1. In order to verify the performance of Algorithm 1, we generate a set of example sample paths of theCGMYSV process { L t } t ≥ with parameters α = 0 . , λ + = 25 . , λ − = 4 . , κ = 1 . , η = 0 . , ζ = 0 . , v = 0 . , and ρ = − . . We set M = 100 , J = 1024 , N = 10 , , and ∆t = 1 / which is the annual fraction of 1 day. Example 20 sample-paths are presented in the first plate of Figure 1.The second plate of the figure is for 20 sample path of CIR process. For goodness of fit test for the generatedpath, we perform Kolmogorov-Smirnov test. We compare the distribution of 10-days simulated randomnumbers { L n, | n = 1 , , · · · , N } with the distribution of L ∆t . The cumulative distribution functionof L ∆t can be obtained by the Ch.F of the CGMYSV using the inverse Fourier-Transform methos (SeeCarr et al. (2002) and Rachev et al. (2011) more details). Table 1 presents the result of the KS test, and ithas 70.29% p -value and it is not rejected at the 5% significant level. Using the same arguments, we performKS test for 25-days, 50-days, and 100-days simulated random numbers. They are not rejected at the 5%significant level, either. We graphically compare the empirical probability density function (pdf) of thesimulated sample path and the CGMYSV pdfs for those four cases. We draw empirical pdfs using gray barcharts and draw solid lines for CGMYSV pdfs in four plates in Figure 2.8 The CGMYSV Option Pricing Model
In this section we discuss the option pricing model on the CGMYSV model. We define the model andcalibrate parameters using European style S&P 500 index option (SPX option) and American style S&P 100index option (OEX option).Let r and q be the risk free rate of return and the continuous dividend rate of a given underlying asset,respectively. The risk-neutral price process { S t } t ≥ of a given underlying asset is assumed as S t = S exp(( r − q ) t + L t ) E [exp( L t )] (8)where { L t } t ≥ is the CGMYSV process with parameters ( α , λ + , λ − , κ , η , ζ , ρ , v ) . By (5), we also have S t = S exp(( r − q ) t + L t )Φ t ( − i log φ stdCT S ( − i ; α, λ + , λ − ) , − ρi, v ) . On the risk neutral price process { S t } t ≥ defined by (8), the European call and put prices with thestrike price K and the time to maturity T are equal to C ( K, T ) = e − rT E [( S t − K ) + ] and C ( K, T ) = e − rT E [( K − S t ) + ] , respectively. Moreover, the Fast-Fourier-Transform (FFT) method by Carr and Madan(1999) and Boyarchenko and Levendorski˘i (2000), we can calculate European call/put prices numerically.We calibrate the CGMYSV parameters ( α , λ + , λ − , κ , η , ζ , ρ , v ) using the SPX option prices on September11, 2017. We observed 247 call prices and 289 put prices on the day. The S&P 500 price S , risk-freerate of return, and continuous dividend rate at the day were S = 2488 . , r = 1 . , and q = 1 . respectively. The calibration results for SPX calls and puts are provided in Table 2. Figure 3 shows observedSPX call and put prices ( drawn by ‘ ◦ ’), and calibrated CGMYSV prices using FFT (drawn by ‘ + ’).We recalculate the European call and put prices using Monte-Carlo Simulation (MCS) method with thecalibrated parameters in Table 2. The sample paths of the MCS method are generated by Algorithm 1. Thenumber of sample paths is 10,000 in this investigation.To compare the MCS method with the FFT method, we use the four error estimators:the average abso-lute error (AAE), the average absolute error as a percentage of the mean price (APE), the average relative9ercentage error (ARPE), and the root mean square error (RMSE) (see Schoutens (2003)). Those fourerror estimators for the FFT method and the MCS method are in Table 3. Both call and put cases, the MCSmethod has larger error estimators than the FFT method. That is not surprising because we calibrate thoseparameters using the FFT method. The table says that the four error estimators of MCS method are similarto those of the FFT method. That means the sample path generation with Algorithm 1 performs well, andprices by the MCS are similar performance as FFT method.In this option pricing with MCS method, we also obtain standard error for each 247 call and 289 putoptions, but we do not provide them all because of the space limitation. Instead, we show MCS prices withthe 95% confidence interval in Figure 4 only for the case , < K < , and time to maturity 48 days.Finally, we perform the bootstrapping. We select an at-the-money call and an at-the-money put of K = 2 , and T = 28 days as an example, and calculate call and put prices with MCS parameters inTable 2, respectively. Table 4 shows that the MCS prices and their standard errors for 100, 1,000, 5,000,and 10,000 number of sample paths. We repeat this process 100 times and draw boxplots. Boxplots forcall and put for each number of sample paths are the up plate and the bottom plate of Figure 5. Stars inthose boxplots are the call/put prices using FFT method. We can observe that the number of sample pathsincreases, then the MCS prices close to the FFT price and dispersions are reduced. We see that the sample path generation method using the series representation works for MCS ofEuropean option pricing in the previous section. In this section, we discuss the American option pric-ing with the same sample path generation method. We use Least Square Regression Method (LSM) byLongstaff and Schwartz (2001) for American option pricing with MCS. When we do the regression for theexpected value of option, we use S t , S t , σ t , σ t and σ t S t as independent variables, following the idea inChapter 15 of Rachev et al. (2011).For empirical illustration, we use market prices of the OEX option, which is American style. We cali-brate parameters of the CGMYSV model with fixed seed numbers for each random number generation. That The measures are computes as follows: AAE = P Nj =1 | b P j − P j | N , APE = P Nj =1 | b P j − P j | /N P Nj =1 b P j /N , ARPE = N P Nj =1 | b P j − P j | b P j , andRMSE = q N P Nj =1 ( b P j − P j ) N , where N is the number of observations, and b P j and P j denote the model price and the observedmarket call/put prices, respectively. χ random number generator in the CIR process, and we generate uniform andexponential random numbers U j , U ′ j , E j , E ′ j and τ j with predefined seed numbers, and fix them. Then weset the model parameters, generate sample paths using Algorithm 1 with the fixed seed number and the fixedrandom number sets, and then calculate American option price using LSM. Repeat that process and find theoptimal parameters to minimize RMSE. As a benchmark, we calibrate the parameters of the CGMY optionpricing model (See Carr et al. (2002)) to the OEX option prices using LSM with sample path generated bythe series representation explained in Section 2.3.The calibration results are presented in Table 5. We calibrate the CGMY and the CGMYSV modelparameters for 12 Wednesdays in 2015 and 2016 exhibited in the table. The four error estimators, AAE,APE, ARPE, and RMSE, are also provided in Table 6. Since the smaller error estimator means the bettercalibration performance, smaller errors are written in bold letters for each day. This table shows that theCGMYSV calibration performs better than CGMY calibration except for the cases of February 10, 2016and June 10, 2015. On March 9, 2016, AAE and APE of the CGMY model are less than those of CGMYSV,but ARPE and RMSE of CGMY are larger than those of CGMYSV. ARPE of CGMY is smaller than thatof CGMYSV on November 10, 2015, but the other three error values of CGMY are larger than CGMYSV.Therefore, we can conclude that the CGMYSV option pricing model performs typically better than theCGMY option pricing model, except in a few cases in this investigation. Hence, LSM with Algorithm 1works well in the American option calibration.Finally, we perform the bootstrapping. We selected the at-the-money put for the strike price K = 910 and the days to maturity T = 31 days on April 6, 2016. Put prices are obtained by LSM using parameterscalibrated to the day provided in Table 5. On the day, the underlying S&P 100 index price was 918.21, andthe market put price was 13.95 for the strike price 910 and 31 days to maturity. Table 7 shows that the LSMprices and their standard errors for 100, 1,000, 5,000, and 10,000 number of sample paths. The LSM pricesapproach to the market price and the standard error decreases as the number of sample paths increases. Werepeat this process 100 times and present boxplots for those 100 prices, as Figure 6. Stars in those boxplotsare the market put prices. We can observe that the LSM prices close to the market price and dispersions arereduced as the number of sample paths increases. Additionally, Figure 7 provides a graphical illustration ofthe calibration for April 6, 2016. Calibrated CGMYSV prices are drawn by ‘ × ’, the market observed pricesare drawn by ‘ ◦ ’, and the 95% confidence intervals are marked by ’I’ shape. The day to maturities T are11ritten on the plate. The sample path generation method for the CGMYSV model can be used for Asian and Barrier optionpricing. In this section, we briefly show examples of Asian and Barrier option pricing using MCS with thesample path generated by Algorithm 1.We generate 10,000 sample path of the CGMYSV model with parameters α = 0 . , λ + = 25 . , λ − = 4 . , κ = 1 . , η = 0 . , ζ = 0 . , v = 0 . , and ρ = − . . Then we generate theunderlying price process { S t } t ≥ using (8) where S = 2 , , r = 0 . , d = 0 . .For the Asian option, we consider the arithmetic average call and put where the strike price K = 2 , and the time to maturity T = 25 days . Table 8 shows that the MCS prices for Asian call & put and theirstandard errors for 100, 1,000, 5,000, and 10,000 number of sample paths. The the standard error of theMCS prices decreases as the number of sample paths increases. We repeat this process 100 times andpresent boxplots for those 100 prices, as Figure 8. We can observe that the MCS prices converge, anddispersions are reduced as the sample paths increase.With the same argument, we find MCS price for the Barrier options. We consider the down-and-outcall and the up-and-out put Barrier options with the strike price K = 2 , and the time to maturity T = 25 days . Barrier of the down-and-out call and the up-and-out put are , and , , respectively.Table 9 shows that the MCS prices and their standard errors for 100, 1,000, 5,000, and 10,000 number ofsample paths. The standard error of the MCS prices decreases as the number of sample paths increases. Werepeat this process 100 times and present boxplots for those 100 prices, as Figure 9. We can also observethat the MCS prices converge, and dispersions are reduced as the sample paths increase. In this paper, we develop the CGMYSV sample path generation algorithm using the series representa-tion. The series representation method’s performance is tested by comparing the simulated distribution tothe pdf calculated by the inverse Fourier transform method. We apply the sample path generation methodto European and American option pricing with MCS and LSM. We compare the MCS method to the FFT12ethod in European option pricing with SPX option market data. Also, we calibrate the parameters ofthe CGMYSV model to the American style OEX option using LSM. We measure the performance of thecalibration using four error estimators and the boot-strapping method. We conclude that the sample pathgeneration method of CGMYSV model performs well, and it can be successfully applied to American op-tion pricing with LSM. Finally, we present Asian and Barrier option pricing examples with MCS methodusing the sample path generation algorithm.
Acknowledgments
The author is grateful to Professor Svetlozar T. Rachev for his valuable discussion onthis problem and encouragement. The author gratefully acknowledges the support of GlimmAnalytics LLCand Juro Instruments Co., Ltd.
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Stochastic Processes and Their Applications , (6),677–707.Schoutens, W. (2003). L´evy Processes in Finance: Pricing Financial Derivatives . John Wiley and Sons:Chichester. 14S statistic p -value10 days 0.0070 0.702925 days 0.0122 0.102650 days 0.0069 0.7296100 days 0.0110 0.1748Table 1: KS test for distributions of simulated sample pathsParameter Call Put α . . λ + . . λ − . . κ . . η . . ζ . . ρ − . − . v . . Table 2: Calibrated Parameters for the SPX option at September 11, 2017.Call PutError FFT MCS FFT MCSAAE . . . . APE . . . . ARPE . . . . RMSE . . . . Table 3: Error estimators for the parameter calibration to the call and put option market price at September11, 2017. Call PutNumber of Simulation Price Standard error Price Standard error100 15.9651 2.1001 34.3371 7.93561000 17.8790 0.7511 32.5601 2.23535000 19.6536 0.3634 32.7791 1.045810000 19.6840 0.2551 32.6914 0.7617FFT Price 19.6590 32.9541Market Price 19.05 31.50Table 4: MCS prices and standard errors for SPX call and put with the strike price K = 2500 and time tomaturity T = 28 days using calibrated parameters at September 11, 2017.15GMY CGMYSVDate α C λ + λ − α λ + λ − κ η ζ v ρ Apr. 6, 2016 . . . . . . . . . . . . Mar. 9, 2016 . . . . . . . . . . . . Feb. 10, 2016 . . . . . . . . . . . . Jan. 6, 20166 . . . . . . . . . . . . Dec. 9, 2015 . . . . . . . . . . . . Nov. 10, 2015 . . . . . . . . . . . . Oct. 7, 2015 . . . . . . . . . . . . Sep. 9, 2015 . . . . . . . . . . . . Aug. 12, 2015 . . . . . . . . . . . . Jul. 8, 2015 . . . . . . . . . . . − . Jun. 10, 2015 . . . . . . . . . . . . May. 6, 2015 . . . . . . . . . . . . Table 5: Parameter Calibration Results for the OEX Option market ate Model AAE APE ARPE RMSEApr. 6, 2016 CGMY 0.6623 0.1726 0.4587 0.8833CGMYSV Mar. 9, 2016 CGMY
Feb. 10, 2016 CGMY
CGMYSV 0.9267 0.0613 0.2075 1.2042Jan. 6, 2016 CGMY 0.6945 0.0644 0.1617 0.9379CGMYSV
Dec. 9, 2015 CGMY 0.7986 0.0931 0.1855 1.1230CGMYSV
Nov. 10, 2015 CGMY 0.3137 0.0576
Oct. 7, 2015 CGMY 0.5268 0.1018 0.3741 0.6891CGMYSV
Sep. 9, 2015 CGMY 0.8948 0.0880 0.1681 1.2499CGMYSV
Aug. 12, 2015 CGMY 0.6464 0.0941 0.1894 0.9097CGMYSV
Jul. 8, 2015 CGMY 0.7636 0.0800 0.1456 1.0079CGMYSV
Jun. 10, 2015 CGMY
CGMYSV 0.3180 0.0841 0.3676 0.4120May. 6, 2015 CGMY 0.7361 0.0765 0.1479 0.9776CGMYSV
Table 6: Error Estimates for the calibration of the OEX optionPutNumber of Simulation Price Standard error100 15.4565 2.72801000 15.0373 1.01735000 13.9616 0.399010000 13.8768 0.2856Market Price 13.950Table 7: MCS prices and standard errors for the OEX put with time to maturity of T = 31 days and strikeprice K = 910 using parameters calibrated at April 6, 2016.17all PutNumber of Simulation Price Standard error Price Standard error100 22.0834 1.7485 11.7400 3.44631000 21.0078 0.5866 9.7009 1.07595000 21.4664 0.2785 10.5634 0.544310000 21.6513 0.1937 9.9964 0.3679Table 8: MCS prices and standard errors for Asian call & put.Down & Out Call Up & Out PutNumber of Simulation Price Standard error Price Standard error100 12.9019 1.4535 36.2543 3.62541000 15.3738 0.5165 30.0687 0.95095000 16.6097 0.2460 31.8030 0.449810000 16.5518 0.1749 30.2590 0.3026Table 9: MCS prices and standard errors for the down-and-out call and the up-and-out put.18
10 20 30 40 50 60 70 80 90 100-0.25-0.2-0.15-0.1-0.0500.05
Figure 1: CGMYSV sample paths (left) and CIR sample paths (right).
10 days -0.5 -0.4 -0.3 -0.2 -0.1 0 0.10510152025303540
SimulationFFT
25 days -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2051015202530
SimulationFFT
50 days -1 -0.8 -0.6 -0.4 -0.2 002468101214161820
SimulationFFT
100 days -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2024681012
SimulationFFT
Figure 2: CGMYSV pdfs based on the simulated sample path (gray bar-plots) vs pdf using FFT method(solid curves). Distributions of X t are for t = 10 ∆t (top-left), t = 25 ∆t (top-right), t = 50 ∆t (bottom-left), and t = 10 ∆t (bottom-right), where ∆t = 1 / is one day of year fraction.19
000 2100 2200 2300 2400 2500 2600 2700 2800 2900
Strike Price C a ll P r i c e S&P 500 Call
T = 3 T = 28 T = 48 T = 68 T = 932000 2100 2200 2300 2400 2500 2600 2700
Strike Price P u t P r i c e S&P 500 Put
T = 3 T = 28 T = 48 T = 68 T = 93
Figure 3: Observed SPX option price and model prices calibrated to the market prices for Call (top) and put(bottom) on September 11, 2017. ‘ ◦ ’ stands for the market price and ‘+’ stands for the FFT price.20
420 2440 2460 2480 2500 2520 2540
Strike Prices C a ll P r i c e s Time to Maturity = 48 days
Strike Prices P u t P r i c e s Time to Maturity = 48 days
Figure 4: Confidence Intervals for the MCS option prices. Dots are observed market prices, dot-coves areMCS prices, and ‘I’ shape bars are 95% confidence intervals of MCS prices. The first (top) plate is for calloption pricing and the second (bottom) plate is for put option pricing.21
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Number of Simulation C a ll P r i c e s European Call
100 1000 5000 10000
Number of Simulation P u t P r i c e s European Put
Figure 5: Boot strapping for call (top) and put (bottom).22
00 1000 5000 100001015202530
American Put
Figure 6: Boot strapping for OEX put option with time to maturity of T = 31 days and strike price K = 910 .23
60 780 800 820 840 860 880 900 920 940
Strike Price P u t P r i c e OEX Put
T = 1 T = 6 T = 11 T = 16 T = 21 T = 31 T = 51 T = 116
Figure 7: OEX put prices on April 6, 2016 and LSM prices with their confidence intervals. Circles (‘ ◦ ’) are observed OEX put prices, ‘ × ’points are MCS prices, and ‘I’ shape bars are 95% confidence intervals of LMS prices.
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Asian Call
100 1000 5000 10000681012141618202224
Asian Put
Figure 8: Boot strapping for Asian call (top) & put (bottom).25
00 1000 5000 1000010121416182022
Barrier Option ( Down & Out Call)
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