More Robust Pricing of European Options Based on Fourier Cosine Series Expansions
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More Robust Pricing of European Options Based onFourier Cosine Series Expansions
Fabien Le Floc’h (v1.1 released June 2020)
Abstract
We present an alternative formula to price European options through cosine series expansions, under modelswith a known characteristic function such as the Heston stochastic volatility model. It is more robust across strikes andas fast as the original COS method.
Key Words:
COS method, Heston, stochastic volatility, characteristic function, quantitative finance
1. Introduction
Fang and Oosterlee (2008) describe a novel approach to the pricing of European options un-der models with a known characteristic function, based on Fourier cosine series expansions,referred to as the COS method hereafter. The method is very fast but its accuracy is notalways reliable for far out-of-the-money options with the proposed truncation range.We illustrate this issue, explain the root of the error, and derive an alternative pricingformula that stays accurate for out-of-the-money options while staying as fast, even in thecase of pricing multiple options of same maturity and different strike prices.Confusion around the truncation range is not new, a Wilmott forum post from 2015(Zukimaten, 2015) suggests the use of an alternative range, and VBA code from 2010(Schmellow, 2010) contains the same alternative truncation range setup, although it is notenabled.While we investigate only the pricing of European options, which is particularly usefulfor the calibration of stochastic volatility models, the COS method has also been applied tothe pricing of Bermudan and Barrier options (Fang and Oosterlee, 2009), of Asian options(Zhang and Oosterlee, 2013) and of two-assets options (Ruijter and Oosterlee, 2012).The improvement proposed here is also applicable to the Shannon wavelet SWIFT methodfrom Ortiz-Gracia and Oosterlee (2016).
2. Quick overview of the COS method
We consider an asset F with a known (normalized) characteristic function φ ( x ) = E h e ix ln F ( T,T ) F (0 ,T ) i . (1) Feedback from Matthias Thul, who also naturally implemented the alternative truncation range independently, is grate-fully acknowledged.
ISSN: Print/ISSN Online/YY/issnum1stp–ppcnt c (cid:13) une 4, 2020 1:4 cos˙method˙improved Fabien Le Floc’h F (0 , T ) is typically the forward price to maturity T of an underlying asset S . For example, foran equity with spot price S , dividend rate q and interest rate r , we have F (0 , T ) = S (0) e ( r − q ) T .The price of a Put option with the COS method is P ( K, T ) = B ( T ) " ℜ ( φ (0)) V Put + N − X k =1 ℜ (cid:18) φ (cid:18) kπb − a (cid:19) V Put k e ikπ − x − ab − a (cid:19) (2)with V Put = Kb − a ( e a − − a ) and for k ≥ V Put k = 2 Kb − a (cid:20)
11 + η k ( e a + η k sin ( η k a ) − cos ( η k a )) − η k sin ( η k a ) (cid:21) (3)where B ( T ) is the discount factor to maturity, x = ln KF (0 ,T ) and η k = kπb − a .The truncation range [ a, b ] is chosen according to the first two cumulants c and c of themodel considered using the rule a = c − L p | c | , b = c + L p | c | and L is a truncation level.The Call option price is obtained through the Put-Call parity relationship.
3. The problem
In the COS method, it is particularly important to always compute the Put option price and torely on the Put Call price parity to keep a high accuracy in general because the absolute valueof the cosine coefficients of the Call option increase exponentially with the time to maturitywhile those of the Put option are constant. We however found out that very in-the-money putoption could still be severely mis-priced, depending on the truncation parameter L . This isespecially visible for very short maturities, even with the recommended value L = 12. Figure1 shows that the absolute error in the Call option price increases significantly with its strikeunder the Heston stochastic volatility model. The reference price is obtained by the optimal α method of Lord and Kahl (2007). Note that when the strike K is beyond the truncation, thatis when ln KF ≥ b , the pricing formula is not really applicable anymore and the discountedintrinsic value B ( T ) | K − F | + should be used instead. Table 1 gives the limit strike for differenttruncation levels L .Figure 1.: Error of in-the-money put option prices of maturity 2 days with Hestonparameters κ = 1 . , θ = 0 . , σ = 1 . , ρ = − . , v = 0 . , F = 1 . L . Strike ab s ( E rr o r) L une 4, 2020 1:4 cos˙method˙improved Draft One way to mitigate the error is to use the discounted intrinsic value before the truncationends, for example when x ≥ b or x ≤ a . But this means it will not be possible to solve theimplied volatility is x is too large or too small. In practice, this happens for short maturitiesand some sets of Heston parameters, even if L is relatively large. We found that including thefourth cumulant numerically did not improve the results. Ideally one would like a larger L forvery short maturities and a smaller L for long maturities.Table 1.: Strike limits for the parameters of Figure 1 L b
Strike at b Strike at b/
212 0.2810 1.32 1.1516 0.3747 1.45 1.2124 0.5622 1.75 1.32Why does this happen? The root of this inaccuracy can be found in how the cosine coef-ficients V Put k are computed. The coefficients correspond to the cosine transform of the Putpayoff: V Put k = 2 b − a Z ba K | − e y | + cos (cid:18) kπ y − ab − a (cid:19) dy = 2 b − a Z a K (1 − e y ) cos (cid:18) kπ y − ab − a (cid:19) dy = 2 b − a K ( − χ k ( a,
0) + ψ k ( a, y = ln S T K and the functions χ and ψ defined in (Fang and Oosterlee, 2008, p. 6) equations(22) and (23).In particular, the cosine coefficients for the Put option are computed relatively to the strikeprice K but the truncation range is relative to the spot price.
4. An alternative pricing formula
An alternative is to compute the cosine coefficients of the Put option relative to the forwardprice F : V Put k = 2 b − a Z ba F (cid:12)(cid:12)(cid:12)(cid:12) KF − e y (cid:12)(cid:12)(cid:12)(cid:12) + cos (cid:18) kπ y − ab − a (cid:19) dy (4)= 2 Fb − a Z za ( e z − e y ) cos (cid:18) kπ y − ab − a (cid:19) dy (5)= 2 Fb − a ( − χ k ( a, z ) + e z ψ k ( a, z )) (6)= 2 b − a ( − F χ k ( a, z ) + Kψ k ( a, z )) (7)where z = ln KF and y = ln S T F . une 4, 2020 1:4 cos˙method˙improved Fabien Le Floc’h
We have V Put ( z ) = 2 F e a − e z + e z ( z − a ) b − a , (8) V Put k ( z ) = 2 F ( b − a ) (cid:0) η k (cid:1) [ e a − cos ( η k ( z − a )) e z − η k sin ( η k ( z − a )) e z ]+ 2 F ( b − a ) η k sin ( η k ( z − a )) e z for k = 1 , ..., N − η k = kπb − a .The Put option price is then obtained by the usual formula, but with x = 0. P ( F, K, T ) = B ( T ) " ℜ ( φ (0)) V Put ( z ) + N − X k =1 ℜ (cid:18) φ (cid:18) kπb − a (cid:19) e − ikπ ab − a (cid:19) V Put k ( z ) . (10)Contrary to the original COS method, the V Put k coefficients now depend on the strike and thusneed to be recomputed for each strike. In the evaluation of ψ and χ , the costliest operationis to compute the cos and sin functions. This needs to be done for each k . But now the term ℜ (cid:16) φ (cid:16) kπb − a (cid:17) e − ikπ ab − a (cid:17) is fully independent of the strike and can be pre-computed, for eachmaturity. This saves one cos and one sin function evaluation per k . The total cost is thus thesame as the original COS method.It can be verified that this alternative formula is equivalent to shifting the truncation rangefrom [ a, b ] to [ a − ln KF , b − ln KF ] in the original COS method, but using equations (9) and (10)is much more efficient to compute option prices for a range of strikes.When the strike K is such that z < a , the Put option value should be set directly to 0.Similarly When z > b , the Put option value should be set to its intrinsic value.
5. Numerical Example
We consider the same short maturity options as in Figure 1, and we compute the absoluteerror in the price of a Call option with a truncation level L = 12 of the classic and theimproved method for N = 256, varying the strike. We stop at strike K = 1 .
32 since thenFigure 2.: Absolute error of in-the-money put option prices of maturity 2 days using κ = 1 . , θ = 0 . , σ = 1 . , ρ = − . , v = 0 . , F = 1 . L = 12. Strike ab s ( E rr o r) Method
ClassicImproved une 4, 2020 1:4 cos˙method˙improved
Draft ln KF > b . Figure 2 shows that the error of the improved method stays below 10 − , close tomachine epsilon while the error of the classic method can be as high as 1 . · − .For longer maturities, and well behaved Heston parameters, it turns out that the newformula has a constant error over the range of strikes, while the classic formula has a lowererror for K < F (Puts) and much larger for
K > F (Calls). In appendix A, we show that theerror of the COS method e ( z ) is composed of two terms: e ( z ) = Z R \ [ a,b ] ( v ( y, T ) − ˆ v ( y, T )) f ( y | x ) dy + ∞ X k = N ℜ (cid:20) φ (cid:18) kπb − a (cid:19) e ikπ − ab − a (cid:21) V k (11)where ˆ v is the cosine expansion of the payoff v and f is the probability density. The first termis the payoff approximation error beyond the boundaries and the second term is the seriestruncation error. When N is sufficiently large, the first error will dominate.In Figure 4(a), we plot the payoff error v − ˆ v by strike both the classic and the new methodfor K < F e a . We plot the payoff error separately for large strikes K > F b in Figures 4(b) andFigure 3.: Error in the Put payoff approximation v ( z, T ) − ˆ v ( z, T ) for a = − . , b = 3 .
025 and F = 2016, which corresponds to the interval for the Blackmodel with volatility σ = 55%, maturity T = 1 . L = 6. -5e-070e+005e-071e-06 20 40 60 Strike E rr o r Method
ClassicNew (a) Low strikes z < a . Strike E rr o r (b) Classic, high strikes z > b . -0.0040.0000.0040.008 1e+05 2e+05 3e+05 4e+05 Strike E rr o r (c) New, high strikes z > b . une 4, 2020 1:4 cos˙method˙improved Fabien Le Floc’h since it oscillates around zero, which will results some cancellation. But its amplitude forlarge strikes is alarming and explains why the overall error of the classic method degrades for
K > F . In contrast, the payoff error of the new method oscillates around zero for high strikes.
We consider an option of maturity T = 1 and strike K = 0 .
25 on an asset following theHeston stochastic volatility model with parameters v = 0 . , κ = 0 . , θ = 0 . , σ = 2 . , ρ =0 . , F = 1. The option is therefore very out of the money. The reference price is given by theLord and Kahl optimal α method (Lord and Kahl, 2007) and we compute prices with theclassic and the improved Cos method with a truncation level L = 12 and a large N = 16384so that series truncation is not an issue. On this example, we include the fourth cumulant( c = 0 . c = − . c = 0 . L = 8.The results presented in Table 2 show that the Cos method with the new coefficients pre-sented in this paper significantly reduces the maximum error over a range of strikes. Indeed,the absolute error stays relatively constant then. In comparison, the original formula leads tohigher accuracy for low strikes and much worse error for high strikes, as the truncation rangeis effectively shifted.Table 2.: out-of-the-money 1Y option of notional 1,000,000 with Heston parameters v = 0 . , κ = 0 . , θ = 0 . , σ = 2 . , ρ = 0 . modlob Lord-Kahl 1e-8 7907 119.38532 0.00000Cos (Classic) L=12 16384 119.38531 0.00002Cos (Improved) L=12 16384 119.38418 0.0011550% modlob
Lord-Kahl 1e-8 4497 834.40773 0.00000Cos (Classic) L=12 16384 834.40756 0.00017Cos (Improved) L=12 16384 834.40657 0.00116100% modlob
Lord-Kahl 1e-8 467 20511.93508 0.00000Cos (Classic) L=12 16384 20511.93388 0.00120Cos (Improved) L=12 16384 20511.93388 0.00120200% modlob
Lord-Kahl 1e-8 3787 6563.82888 0.00000Cos (Classic) L=12 16384 6563.82102 0.00786Cos (Improved) L=12 16384 6563.82773 0.00115400% modlob
Lord-Kahl 1e-8 6537 3951.92085 0.00000Cos (Classic) L=12 16384 3951.86703 0.05382Cos (Improved) L=12 16384 3951.91908 0.00177
6. Conclusion
For the same computational cost as the original COS method, a more uniform accuracy acrossstrikes is obtained with the alternate COS formula presented in this paper. This is particularly In a previous version, we made an error in the calculation of the truncation range, which led to a much more accurateprice for lower strikes on this example. This is corrected here.une 4, 2020 1:4 cos˙method˙improved
REFERENCES visible for deep out-of-the-money options. References
Fang, F. (2010) The COS Method: An Efficient Fourier Method for Pricing Financial Derivatives. PhD thesis. TU Delft,Delft University of Technology. (accessed ????).Fang, F. and Oosterlee, C. W. (2008) A novel pricing method for European options based on Fourier-cosine seriesexpansions,
SIAM Journal on Scientific Computing , 31(2), pp. 826–848.Fang, F. and Oosterlee, C. W. (2009) Pricing early-exercise and discrete barrier options by Fourier-cosine series expan-sions,
Numerische Mathematik , 114(1), p. 27.Lord, R. and Kahl, C. (2007) Optimal Fourier inversion in semi-analytical option pricing,
SSRN pa-pers.ssrn.com/abstract=921336 .Neidinger, R. D. (2013) Efficient recurrence relations for univariate and multivariate Taylor series coefficients. In:
Con-ference Publications , Vol. 2013, p. 587.Ortiz-Gracia, L. and Oosterlee, C. W. (2016) A highly efficient Shannon wavelet inverse Fourier technique for pricingEuropean options,
SIAM Journal on Scientific Computing , 38(1), pp. B118–B143.Ruijter, M. J. and Oosterlee, C. W. (2012) Two-dimensional Fourier cosine series expansion method for pricing financialoptions,
SIAM Journal on Scientific Computing , 34(5), pp. B642–B671.Schmellow (2010) Heston model - put option price calculated by FFT,
Wilmott forum https://forum.wilmott.com/viewtopic.php?f=34&t=78554 .Thul, M. (2016) Automatic (Differentiation) for the COS Method,
Blog .Zhang, B. and Oosterlee, C. W. (2013) Efficient pricing of European-style Asian options under exponential L´evy processesbased on Fourier cosine expansions,
SIAM Journal on Financial Mathematics , 4(1), pp. 399–426.Zukimaten (2015) Entering the COS method differently,
Wilmott forum https://forum.wilmott.com/viewtopic.php?f=8&t=98655&p=752930&hilit=cos+method . Appendix A. Error estimate
Let v ( x, t ) be the undiscounted option price at time t and f ( y | x ) the probability density ofbeing at y starting from x . At maturity T the payoff is v ( x, T ). For a European (non-pathdependent) option we can price the option by integrating over the density: v ( x, t ) = Z + ∞−∞ v ( y, T ) f ( y | x ) dy (A1)In order to evaluate the integral, we will truncate it to an interval [ a, b ]: v ( x, t ) = Z R \ [ a,b ] v ( y, T ) f ( y | x ) dy + Z ba v ( y, T ) f ( y | x ) dy (A2)We now use the cosine expansion on the interval [ a, b ]: f ( y | x ) = P ∞ k =0 ′ A k ( x ) cos (cid:16) kπ y − ab − a (cid:17) with A k ( x ) = 2 b − a Z ba f ( y | x ) cos (cid:18) kπ y − ab − a (cid:19) dy (A3)This leads to v ( x, t ) = Z R \ [ a,b ] v ( y, T ) f ( y | x ) dy + Z ba v ( y, T ) ∞ X k =0 ′ A k cos (cid:18) kπ y − ab − a (cid:19) dy (A4)Numerically, the sum stops at a finite N ∈ N . We thus split the sum in two parts: v ( x, t ) = Z R \ [ a,b ] v ( y, T ) f ( y | x ) dy + Z ba v ( y, T ) ∞ X k = N A k cos (cid:18) kπ y − ab − a (cid:19) dy + Z ba v ( y, T ) N − X k =0 ′ A k cos (cid:18) kπ y − ab − a (cid:19) dy (A5) une 4, 2020 1:4 cos˙method˙improved REFERENCES
The characteristic function φ corresponding to the density f is φ ( x ) = R + ∞−∞ e iux f ( u ) du . Wethus have the identity ℜ (cid:20) φ (cid:18) kπb − a (cid:19) e ikπ − ab − a (cid:21) = Z + ∞−∞ cos (cid:18) kπ u − ab − a (cid:19) f ( u ) du (A6)We will use it in the definition of A k to obtain A k = 2 b − a Z R f ( y | x ) cos (cid:18) kπ y − ab − a (cid:19) dy − b − a Z R \ [ a,b ] f ( y | x ) cos (cid:18) kπ y − ab − a (cid:19) dy (A7)= 2 b − a ℜ (cid:20) φ (cid:18) kπb − a (cid:19) e ikπ − ab − a (cid:21) − b − a Z R \ [ a,b ] f ( y | x ) cos (cid:18) kπ y − ab − a (cid:19) dy (A8)We replace A k in the last integral of v ( x, t ) to obtain v ( x, t ) = Z R \ [ a,b ] v ( y, T ) f ( y | x ) dy + Z ba v ( y, T ) ∞ X k = N A k cos (cid:18) kπ y − ab − a (cid:19) dy − N − X k =0 ′ b − a "Z R \ [ a,b ] f ( y | x ) cos (cid:18) kπ y − ab − a (cid:19) dy ba v ( y, T ) cos (cid:18) kπ y − ab − a (cid:19) dy + N − X k =0 ′ ℜ (cid:20) φ (cid:18) kπb − a (cid:19) e ikπ − ab − a (cid:21) b − a Z ba v ( y, T ) cos (cid:18) kπ y − ab − a (cid:19) dy (A9)The cosine expansion of the payoff is ˆ v ( y, T ) = P ∞ k =0 V k cos (cid:16) kπ y − ab − a (cid:17) with V k = 2 b − a Z ba v ( y, T ) cos (cid:18) kπ y − ab − a (cid:19) dy (A10)For y ∈ [ a, b ] we have v ( y ) = ˆ v ( y ). This leads to v ( x, t ) = Z R \ [ a,b ] v ( y, T ) f ( y | x ) dy + Z ba v ( y, T ) ∞ X k = N A k cos (cid:18) kπ y − ab − a (cid:19) dy − N − X k =0 ′ "Z R \ [ a,b ] f ( y | x ) cos (cid:18) kπ y − ab − a (cid:19) dy V k + N − X k =0 ′ ℜ (cid:20) φ (cid:18) kπb − a (cid:19) e ikπ − ab − a (cid:21) V k (A11)Contrary to what is done in (Fang and Oosterlee, 2008, equation (43)) (corrected in (Fang,2010)), we can not expand v ( y, T ) as a cosine serie in the first term since the expansionmatches the original payoff v ( y, T ) only inside the interval [ a, b ]. Outside, it is periodic, whilethe original payoff is not. Instead,we will decompose v as v − ˆ v + ˆ v . une 4, 2020 1:4 cos˙method˙improved REFERENCES v ( x, t ) = Z R \ [ a,b ] ( v ( y, T ) − ˆ v ( y, T )) f ( y | x ) dy + Z R \ [ a,b ] ˆ v ( y, T ) f ( y | x ) dy + Z ba v ( y, T ) ∞ X k = N A k cos (cid:18) kπ y − ab − a (cid:19) dy − N − X k =0 ′ "Z R \ [ a,b ] f ( y | x ) cos (cid:18) kπ y − ab − a (cid:19) dy V k + N − X k =0 ′ ℜ (cid:20) φ (cid:18) kπb − a (cid:19) e ikπ − ab − a (cid:21) V k (A12)Now we expand ˆ v in the second integral, this simplifies with the fourth integral to obtain v ( x, t ) = Z R \ [ a,b ] ( v ( y, T ) − ˆ v ( y, T )) f ( y | x ) dy + Z R \ [ a,b ] f ( y | x ) ∞ X k = N V k cos (cid:18) kπ y − ab − a (cid:19) dy + Z ba v ( y, T ) ∞ X k = N A k cos (cid:18) kπ y − ab − a (cid:19) dy + N − X k =0 ′ ℜ (cid:20) φ (cid:18) kπb − a (cid:19) e ikπ − ab − a (cid:21) V k (A13)We recognize the definition of V k in the third integral, this means: v ( x, t ) = Z R \ [ a,b ] ( v ( y, T ) − ˆ v ( y, T )) f ( y | x ) dy + Z R \ [ a,b ] f ( y | x ) ∞ X k = N V k cos (cid:18) kπ y − ab − a (cid:19) dy + ∞ X k = N A k b − a V k + N − X k =0 ′ ℜ (cid:20) φ (cid:18) kπb − a (cid:19) e ikπ − ab − a (cid:21) V k (A14)We now use the definition of A k in the third integral to obtain v ( x, t ) = Z R \ [ a,b ] ( v ( y, T ) − ˆ v ( y, T )) f ( y | x ) dy + Z R \ [ a,b ] f ( y | x ) ∞ X k = N V k cos (cid:18) kπ y − ab − a (cid:19) dy + ∞ X k = N V k Z ba f ( y | x ) cos (cid:18) kπ y − ab − a (cid:19) dy + N − X k =0 ′ ℜ (cid:20) φ (cid:18) kπb − a (cid:19) e ikπ − ab − a (cid:21) V k (A15)We can now combine the second and third integrals together: une 4, 2020 1:4 cos˙method˙improved REFERENCES v ( x, t ) = Z R \ [ a,b ] ( v ( y, T ) − ˆ v ( y, T )) f ( y | x ) dy + ∞ X k = N V k Z R f ( y | x ) cos (cid:18) kπ y − ab − a (cid:19) dy + N − X k =0 ′ ℜ (cid:20) φ (cid:18) kπb − a (cid:19) e ikπ − ab − a (cid:21) V k (A16)We recognize the characteristic function identity in the second integral and we obtain v ( x, t ) = Z R \ [ a,b ] ( v ( y, T ) − ˆ v ( y, T )) f ( y | x ) dy + ∞ X k = N ℜ (cid:20) φ (cid:18) kπb − a (cid:19) e ikπ − ab − a (cid:21) V k + N − X k =0 ′ ℜ (cid:20) φ (cid:18) kπb − a (cid:19) e ikπ − ab − a (cid:21) V k (A17) Appendix B. First Cumulants
B.1 First cumulants for Heston
The two first cumulants c and c of log (cid:0) FK (cid:1) are used to define the integration boundaries a and b of the COS method. The cumulant generating function is: g ( u ) = log( φ ( − iu )) (B1)We have c = g ′ (0) (B2) c = g ′′ (0) (B3)Those can be computed numerically. Analytic formulas are given in (Fang and Oosterlee,2008), unfortunately their formula for c is wrong. Here are our own derived formulas from aTaylor expansion: c = (1 − e − κt ) θ − v κ − θt (B4) c = v κ { κ (cid:0) ρσt − e − κt (cid:1) + κ (cid:0) ρσ ( e − κt − − σ te − κt (cid:1) + σ (1 − e − κt ) } + θ κ { κ t − κ (cid:0) ρσt + ( ρσt − e − κt (cid:1) + 2 κ (cid:0) (1 + 2 e − κt ) σ t + 8(1 − e − κt ) ρσ (cid:1) + σ ( e − κt + 4 e − κt − } (B5) une 4, 2020 1:4 cos˙method˙improved REFERENCES The fourth cumulant, derived by a computer algebra system, reads c = c A + c B with c B = 2 σ v κ (cid:0)(cid:0)(cid:0) T ρ σ − T ρ (cid:1) κ − / T (cid:0) T ρ σ − T ρ (cid:0) ρ + 2 (cid:1) σ + 4 ρ + 2 (cid:1) κ + (cid:0) / T ρ σ − T (cid:0) ρ + 3 / (cid:1) σ + 6 T ρ (cid:0) ρ + 2 (cid:1) σ − ρ (cid:1) κ − / σ (cid:0) T σ − T ρ σ + (cid:0) T ρ + 18 T (cid:1) σ − ρ (cid:1) κ − / σ (cid:0) T σ − T σ ρ − (cid:1) κ − / σ ( T σ + 10 ρ ) κ + 3 / σ (cid:1) e − κ T + (cid:0)(cid:0) − / − T ρ σ + 6 T σ ρ (cid:1) κ + 3 σ (cid:0) T ρ σ + (cid:0) − T ρ − / T (cid:1) σ + 3 ρ (cid:1) κ − / σ (cid:0) T σ − T σ ρ + 12 ρ + 3 (cid:1) κ − (9 T σ − ρ ) σ κ − / σ (cid:19) e − κ T + 9 σ (cid:0) ( T σ ρ − κ + (cid:0) − / σ T + 5 / ρ σ (cid:1) κ − / σ (cid:1) e − κ T − σ e − κ T − (cid:0) κ ρ σ − / σ − κ (cid:1) (cid:18)(cid:0) ρ + 1 / (cid:1) κ − / κ ρ σ + 5 σ (cid:19)(cid:19) and c A = − σ θκ (cid:0)(cid:0)(cid:0) T ρ σ − T ρ (cid:1) κ − / T (cid:0) T ρ σ − T ρ (cid:0) ρ + 1 (cid:1) σ + 8 ρ + 2 (cid:1) κ + (cid:0) / T ρ σ − / T (cid:0) ρ + 3 / (cid:1) σ + (cid:0) T ρ + 24 T ρ (cid:1) σ − ρ (cid:1) κ − / (cid:0) T σ − T ρ σ + (cid:0) T ρ + 54 T (cid:1) σ − ρ − ρ (cid:1) σ κ − / σ (cid:18) T σ − T σ ρ ρ + 15 / (cid:19) κ − σ κ (cid:18) T σ − ρ (cid:19) − σ (cid:19) e − κ T + (cid:0)(cid:0) − / − / T ρ σ + 3 T σ ρ (cid:1) κ + 3 / σ (cid:0) T ρ σ + (cid:0) − T ρ − / T (cid:1) σ + 4 ρ (cid:1) κ − / σ (cid:0) T σ − T σ ρ + 20 ρ + 6 (cid:1) κ + (cid:18) − T σ
16 + 9 / ρ σ (cid:19) κ − σ (cid:19) e − κ T +3 / σ (cid:0) ( T σ ρ − κ + (cid:0) − / σ T + 2 ρ σ (cid:1) κ − / σ (cid:1) e − κ T − σ e − κ T (cid:0) − / T − T ρ (cid:1) κ + (cid:18)(cid:0) T ρ + 9 T ρ (cid:1) σ + 18 ρ + 154 (cid:19) κ − σ (cid:0) T (cid:0) ρ + 1 / (cid:1) σ + 8 / ρ + 10 / ρ (cid:1) κ + 15 σ κ (cid:18) T σ ρ + 10 ρ + 115 (cid:19) + (cid:18) − ρ σ − T σ (cid:19) κ + 279 σ (cid:19) Alternatively, Taylor series algorithmic differentiation (Neidinger, 2013) may be used to com-pute the fourth cumulant as suggested in (Thul, 2016). une 4, 2020 1:4 cos˙method˙improved REFERENCES
B.2 First cumulants for stochastic volatility with jump (SVJ)