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Efficient Asian option pricing under regime switching jump diffusions and stochastic volatility models

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Utilizing frame duality and a FFT-based implementation of density projection we develop a novel and efficient transform method to price Asian options for very general asset dynamics, including regime switching Lévy processes and other jump diffusions as well as stochastic volatility models with jumps. The method combines continuous-time Markov chain approximation, with Fourier pricing techniques. In particular, our method encompasses Heston, Hull-White, Stein-Stein, 3/2 model as well as recently proposed Jacobi, \(\alpha \)-Hypergeometric, and 4/2 models, for virtually any type of jump amplitude distribution in the return process. This framework thus provides a ‘unified’ approach to pricing Asian options in stochastic jump diffusion models and is readily extended to alternative exotic contracts. We also derive a characteristic function recursion by generalizing the Carverhill-Clewlow factorization which enables the application of transform methods in general. Numerical results are provided to illustrate the effectiveness of the method. Various extensions of this method have since been developed, including the pricing of barrier, American, and realized variance derivatives.

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Fig. 1
Fig. 2


  1. In between regime switching times, the asset price process follows a regular jump diffusion with constant drift rate and instantaneous volatility rate. Note that this formulation allows for a different jump distribution in each regime.

  2. Which can be seen by applying Cauchy’s inequality to \([a\sqrt{{v}_t}+\frac{b}{\sqrt{{v}_t}}]\ge 2\sqrt{a\sqrt{v_t} \frac{b}{\sqrt{v_t}}}=2\sqrt{ab}>0\) for \(a,b>0\).

  3. For more details on basis theory and its applications in finance, see Kirkby and Deng (2019).

  4. Details on the derivation of \( \widehat{\widetilde{\varphi }}(\xi )\) for the cubic basis can be found in Kirkby (2017b).

  5. For stochastic volatility models, the initial variance state \(v_0\) is not necessarily a member of the variance grid. One approach is to apply linear interpolation with equation (46) for \(j = j_0\) and \(j_0+1\), where \(v_{j_0}\le v_0 <v_{j_0+1}\). However, the grid can be easily adjusted so that \(v_0=v_{j_0}\) is a member.

  6. Accessed on September 20th 2017.

  7. We can increase \(\gamma \) to sufficiently cover the domain of \(v_t\). From numerical experimentation, we find that \(\gamma =4.5\) is sufficient for the models considered in this work.

  8. If moments of the variance process are unknown, the grid can be fixed using \(v_1 = \beta _1 v_0\) and \(v_{m_0} = \beta _2 v_0\). For example, \(\beta _1 = 10^{-3}\) and \(\beta _2 = 4\).

  9. This keeps an “anchor” at the boundary in the case where \(v_0 \approx 0\).

  10. To avoid excessive notation, we continue to suppress the superscript on \(\{\varphi _{a,k}^{m-1}\}=\{\varphi _{a,k}\} \), where the grid shift is understood.

  11. Note that for the purpose of this proof we have defined \(\beta ^j_{Y_{m-1},k}\) so that \( a^{1/2}\Upsilon _{a,N}\beta ^j_{Y_{m-1},k}=\langle f^j_{Y_{m-1}},\widetilde{\varphi }_{a,k} \rangle \).


  • Ackerer, D., Filipović, D., Pulido, S.: The Jacobi stochastic volatility model. Swiss Finance Institute Research Paper, pp. 16-35 (2016)

  • Andersen, L.: Simple and efficient simulation of the Heston stochastic volatility model. J Comput Finance 11, 1–42 (2008)

    Google Scholar 

  • Bates, D.: Jumps and stochastic volatility: exchange rate processes implicit in deutsche mark options. Rev Financ Stud 9, 69–107 (1996)

    Google Scholar 

  • Bayraktar, E., Xing, H.: Pricing Asian option for jump diffusion. Math Finance 21, 117–143 (2011)

    Google Scholar 

  • Boyarchenko, M., Levendorskii, S.: Ghost calibration and pricing double barrier options and credit default swaps in spectrally one-sided Lévy models: the parabolic Laplace inversion method. Quant Finance 15(3), 421–441 (2015)

    Google Scholar 

  • Boyarchenko, S., Levendorskii, S.: American options in Lévy models with stochastic volatility. Available at SSRN: (2007)

  • Boyarchenko, S., Levendorskii, S.: American options in regime-switching Lévy models with non-semibounded stochastic interest rates. In: American Control Conference, 2008, pp. 1023–1028. IEEE (2008)

  • Boyarchenko, S., Levendorskii, S.: American options in Lévy models with stochastic interest rates. J Comput Finance 12(4), 51–89 (2009)

    Google Scholar 

  • Boyarchenko, S., Levendorskii, S.: American options in the Heston model with stochastic interest rate and its generalizations. Appl Math Finance 20(1), 26–49 (2013)

    Google Scholar 

  • Boyle, P., Draviam, T.: Pricing exotic options under regime switching. Insur Math Econ 40, 267–282 (2007)

    Google Scholar 

  • Broadie, M., Glasserman, P.: Estimating security price derivatives using simulation. Manag Sci 42, 269–285 (1996)

    Google Scholar 

  • Broadie, M., Glasserman, P., Kou, P.: Connecting discrete and continuous path-dependent options. Finance Stoc 3, 55–82 (1999)

    Google Scholar 

  • Buffington, J., Elliot, R.: American options with regime switching models. Int J Theor Appl Finance 05, 1–26 (2002)

    Google Scholar 

  • Cai, N., Kou, S.G.: Pricing Asian options under a hyper-exponential jump diffusion model. Oper Res 60, 64–77 (2012)

    Google Scholar 

  • Cai, N., Song, Y., Kou, S.: A general framework for pricing Asian options under Markov processes. Oper Res 63, 540–554 (2015)

    Google Scholar 

  • Carverhill, A., Clewlow, L.: Flexible convolution: valuing average rate (Asian) options. Risk 3, 25–29 (1990)

    Google Scholar 

  • Cerny, A., Kyriakou, I.: An improved convolution algorithm for discretely sampled Asian options. Quant Finance 11, 381–389 (2011)

    Google Scholar 

  • Chourdakis, K.: Continuous Time Regime Switching models and applications in estimating processes with stochastic volatility and jumps. Working Paper 464, Queen Mary, University of London (2002)

  • Cont, R., Tankov, P.: Financial Modelling with Jump Processes: vol. 2. CRC Press: Boca Raton (2003)

    Google Scholar 

  • Corlay, S., Pagès, G., Printems, J.: The optimal quantization website. (2005)

  • Corsaro, S., Kyriakou, I., Marazzina, D., Marino, Z.: A general framework for pricing Asian options under stochastic volatility on parallel architectures. Eur J Oper Res 272(3), 1082–1095 (2019)

    Google Scholar 

  • Costabile, M., Leccadito, A., Massab, I., Russo, E.: Option pricing under regime switching jump diffusion models. J Comput Appl Math 256, 152–167 (2014)

    Google Scholar 

  • Cui, Z., Kirkby, J., Nguyen, D.: Equity-linked annuity pricing with cliquet-style guarantees in regime-switching and stochastic volatility models with jumps. Insur Math Econ 74, 46–62 (2017a)

    Google Scholar 

  • Cui, Z., Kirkby, J., Nguyen, D.: A general framework for discretely sampled realized variance derivatives in stochastic volatility models with jumps. Eur J Oper Res 262(1), 381–400 (2017b)

    Google Scholar 

  • Cui, Z., Kirkby, J., Nguyen, D.: A general valuation framework for SABR and stochastic local volatility models. SIAM J Financ Math 9(2), 520–563 (2018a)

    Google Scholar 

  • Cui, Z., Kirkby, J., Nguyen, D.: Continuous-time Markov Chain and regime switching approximations with applications to options pricing. In: Yin G., Zhang Q. (eds) Modeling, Stochastic Control, Optimization, and Applications. The IMA Volumes in Mathematics and its applications edition, vol. 164, Springer, Cham (2019a)

  • Cui, Z., Kirkby, J., Nguyen, D.: A general framework for time-changed Markov processes and applications. Eur J Oper Res 273(2), 785–800 (2019b)

    Google Scholar 

  • Cui, Z., Lee, C., Liu, Y.: Single-transform formulas for pricing asian options in a general approximation framework under markov processes. Eur J Oper Res 266(3), 1134–1139 (2018b)

    Google Scholar 

  • Cui, Z., Nguyen, D.: First hitting time of integral diffusions and applications. Stoch Models 33, 376–391 (2017)

    Google Scholar 

  • Da Fonseca, J., Martini, C.: The \(\alpha \)-Hypergeometric stochastic volatility model. Stoc Process Appl 126(5), 1472–1502 (2016)

    Google Scholar 

  • Dang, D., Nguyen, D., Sewell, G.: Numerical schemes for pricing Asian options under state-dependent regime switching jump diffusion models. Comput Math Appl 71, 443–458 (2016)

    Google Scholar 

  • d’Halluin, Y., Forsyth, P.A., Labahn, G.: A semi-Lagrangian approach for American Asian options under jump diffusion. SIAM J Sci Comput 27, 315–345 (2005)

    Google Scholar 

  • Dingeç, K.D., Sak, H., Hörmann, W.: Variance reduction for Asian options under a general model framework. Rev Finance 19(2), 907–949 (2015)

    Google Scholar 

  • Eberlein, E., Papapantoleon, A.: Equivalence of floating and fixed strike Asian and lookback options. Stoc Process Appl 115, 31–40 (2005)

    Google Scholar 

  • Elliott, R., Siu, T., Chan, L., Lau, J.: Pricing options under a generalized Markov modulated jump-diffusion model. Stoch Anal Appl 25, 821–843 (2007)

    Google Scholar 

  • Elliott, R.J., Chan, L., Siu, T.K.: Option pricing and esscher transform under regime switching. Ann Finance 1(4), 423–432 (2005)

    Google Scholar 

  • Fard, F.A., Siu, T.K.: Pricing and managing risks of european-style options in a markovian regime-switching binomial model. Ann Finance 9(3), 421–438 (2013)

    Google Scholar 

  • Florescu, I., Liu, R., Mariani, M., Sewell, G.: Numerical schemes for option pricing in regime-switching jump diffusion models. Int J Theor Appl Finance 16, 1–25 (2013)

    Google Scholar 

  • Funahashi, H., Kijima, M.: A unified approach for the pricing of options relating to averages. Rev Deriv Res 20, 203–229 (2017)

    Google Scholar 

  • Fusai, G., Kyriakou, I.: General optimized lower and upper bounds for discrete and continuous arithmetic Asian options. Math Oper Res 41(2), 377–744 (2016)

    Google Scholar 

  • German, H., Yor, M.: Bessel processes, Asian options, and perpetuities. Math Finance 3, 349–375 (1993)

    Google Scholar 

  • Grasselli, M.: The 4/2 stochastic volatility model: a unified approach for the Heston and the 3/2 model. Math Finance 27, 1013–1034 (2017)

    Google Scholar 

  • Hamilton, J.: Time Series Aalysis: Princeton University Press: New Jersey (1994)

    Google Scholar 

  • Henderson, V., Hobson, D., Shaw, W., Wojakowski, R.: Bounds for in-progress floating-strike asian options using symmetry. Ann Oper Res 151(1), 81–98 (2007)

    Google Scholar 

  • Henderson, V., Wojakowski, R.: On the equivalence of floating and fixed-strike Asian options. J Appl Probab 39, 391–394 (2002)

    Google Scholar 

  • Heston, S.: A closed-form solution for option pricing with stochastic volatility with application to bond and currency options. Rev Financ Stud 6, 327–343 (1993)

    Google Scholar 

  • Hull, J., White, A.: Pricing interest-rate derivative securities. Rev Financ Stud 3, 735–792 (1990)

    Google Scholar 

  • Ingersoll, J.E.: Theory of Financial Decision Making: Rowman & Littlefield: Lanham (1987)

    Google Scholar 

  • Jiang, J., Liu, R., Nguyen, D.: A recombining tree method for option pricing with state-dependent switching rates. Int J Theor Appl Finance 19, 1–26 (2016)

    Google Scholar 

  • Kemna, A.G.Z., Vorst, A.C.F.: A pricing method for options based on average asset values. J Bank Finance 14, 113–129 (1990)

    Google Scholar 

  • Kirkby, J.: Efficient option pricing by frame duality with the fast Fourier transform. SIAM J Financ Math 6(1), 713–747 (2015)

    Google Scholar 

  • Kirkby, J.: Robust barrier option pricing by frame projection under exponential Levy dynamics. Appl Math Finance 24(4), 337–386 (2017a)

    Google Scholar 

  • Kirkby, J.: Robust option pricing with characteristic functions and the B-spline order of density projection. J Comput Finance 21(2), 101–127 (2017b)

    Google Scholar 

  • Kirkby, J.: American and exotic option pricing with jump diffusions and other Levy processes. J Comput Finance 22(3), 89–148 (2018)

    Google Scholar 

  • Kirkby, J., Deng, S.: Static hedging and pricing of exotic options with payoff frames. Math Finance 29(2), 612–658 (2019)

    Google Scholar 

  • Kirkby, J., Nguyen, D., Cui, Z.: A unified approach to Bermudan and barrier options under stochastic volatility models with jumps. J Econ Dyn Control 80, 75–100 (2017)

    Google Scholar 

  • Kirkby, J.L.: An efficient transform method for Asian option pricing. SIAM J Financ Math 7(1), 845–892 (2016)

    Google Scholar 

  • Kou, S.: A jump-distribution model for option pricing. Manag Sci 48, 1086–1101 (2002)

    Google Scholar 

  • Leitao, A., Ortiz-Gracia, L., Wagner, E.I.: SWIFT valuation of discretely monitored arithmetic Asian options. Journal of Computational Science 28, 120–139. ISSN 1877-7503. (2018)

  • Leitao Rodriguez, A., Kirkby, J., Ortiz-Gracia, L.: The CTMC-Heston model: calibration and exotic option pricing with SWIFT. Working Paper (2019)

  • Levendorskii, S.: Pricing arithmetic asian options under lévy models by backward induction in the dual space. SIAM J Financ Math 9(1), 1–27 (2018)

    Google Scholar 

  • Levendorskii, S., Xie, J.: Pricing discretely sampled Asian options under Levy processes. Available at SSRN: (2012)

  • Li, L., Zhang, G.: Analysis of markov chain approximation for option pricing and hedging: Grid design and convergence behavior. Working Paper (2017a)

  • Li, L., Zhang, G.: Error analysis of finite difference and markov chain approximations for option pricing with non-smooth payoffs. Math Finance 28, 877–919 (2017b). forthcoming

    Google Scholar 

  • Linetsky, V.: Spectral expansions for Asian (average price) options. Oper Res 52, 856–867 (2004)

    Google Scholar 

  • Liu, R.: Regime-switching recombining tree for option pricing. Int J Theor Appl Finance 13(03), 479–499 (2010)

    Google Scholar 

  • Lo, C., Skindilias, K.: An improved Markov chain approximation methodology: derivatives pricing and model calibration. Int J Theor Appl Finance 17, 407–446 (2014)

    Google Scholar 

  • Merton, R.: Option pricing when underlying stock returns are discontinuous. J Financ Econ 3, 125–144 (1976)

    Google Scholar 

  • Mijatović, A., Pistorius, M.: Continuously monitored barrier options under Markov processes. Math Finance 23(1), 1–38 (2013)

    Google Scholar 

  • Pagès, G., Printems, J.: Functional quantization for numerics with an application to option pricing. Monte Carlo Methods Appl 11, 407–446 (2005)

    Google Scholar 

  • Pirjol, D., Lingjiong, Z.: Short maturity Asian options in local volatility models. SIAM J Financ Math 7(1), 947–992 (2016)

    Google Scholar 

  • Ramponi, A.: Fourier transform methods for regime-switching jump-diffusions and the pricing of forward starting options. Int J Theor Appl Finance 15, 1–26 (2012)

    Google Scholar 

  • Rogers, L., Shi, Z.: The value of Asian options. J Appl Probab 32, 1077–1088 (1995)

    Google Scholar 

  • Siu, T.: Bond pricing under a Markovian regime-switching jump-augmented Vasicek model via stochastic flows. Appl Math Comput 216, 3184–3190 (2010)

    Google Scholar 

  • Stein, E.M., Stein, J.: Stock price distributions with stochastic volatility: an analytic Approach. Rev Financ Stud 4, 272–752 (1991)

    Google Scholar 

  • Vecer, J.: A new pde approach for pricing arithmetic average Asian options. J Comput Finance 4, 105–113 (2001)

    Google Scholar 

  • Vecer, J., Xu, M.: Pricing Asian options in semi-martingale model. Quant Finance 4, 170–175 (2004)

    Google Scholar 

  • Weron, R., Bierbrauer, M., Trück, S.: Modeling electricity prices: jump diffusion and regime switching. Phys A Stat Mech Appl 336, 39–48 (2004)

    Google Scholar 

  • Yao, D., Zhang, Q., Zhou, Z.: A regime switching model for European options. In: Yan, H., Yin, G., Zhang, Q. (eds.) Stochastic Processes, Optimization, and Control Theory: Applications in Financial Engineering, Queueing Networks, and Manufacturing Systems. Springer, Berlin (2006)

    Google Scholar 

  • Yuen, F., Yang, H.: Option pricing in a jump-diffusion model with regime-switching. ASTIN Bull 39, 515–539 (2009)

    Google Scholar 

  • Yuen, F., Yang, H.: Pricing Asian option and equity-indexed annuities with regime switching by the trinomial tree method. North Am Actuary J 14, 256–272 (2012)

    Google Scholar 

  • Zeng, P., Kwok, Y.K.: Pricing bounds and approximations for discrete arithmetic Asian options under time-changed Lévy processes. Quant Finance 16(9), 1375–1391 (2016)

    Google Scholar 

  • Zhang, B., Oosterlee, C.: Efficient pricing of European-style Asian options under exponential Lévy processes based on Fourier cosine expansions. SIAM J Financ Math 4, 399–426 (2013)

    Google Scholar 

  • Zhang, J.: A semi-analytical method for pricing and hedging continuously sample arithmetic average rate options. J Comput Finance 5, 1–20 (2001)

    Google Scholar 

  • Zhang, J.: Pricing continuously sampled Asian option with perturbation method. J Futures Mark 23, 535–560 (2003)

    Google Scholar 

  • Zhang, M., Chan, L.: Saddlepoint approximations to option price in a regime-switching model. Ann Finance 12(1), 55–69 (2016)

    Google Scholar 

  • Zhang, X., Elliott, R.J., Siu, T.K.: Stochastic maximum principle for a Markov regime-switching jump-diffusion model and its application to finance. SIAM J Control Optim 2012, 964–990 (2012)

    Google Scholar 

  • Zhou, Z., Ma, J.: Second-order lattice Boltzmann methods for PDEs of Asian option pricing with regime switching. Comput Math Appl 71(7), 1448–1463 (2016). forthcoming

    Google Scholar 

  • Zvan, R., Forsyth, P., Vetzal, K.: Robust numerical methods for PDE models of Asian options. J Comput Finance 1, 39–78 (1998)

    Google Scholar 

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The usual disclaimer applies. The research of Duy Nguyen is partially supported by a Marist College summer research grant.

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Implementation details

1.1 Nonuniform variance grid

Table 15 Stable coefficient approximations derived for the cubic basis using a five point Newton-Cotes quadrature (Boole’s rule)

To apply the Markov chain approximation, we determine the variance grid \(\{v_j\}_{j=1}^{m_0}\) as follows. First, we fix \(t = T/2\) and center the grid about the mean of the variance process \(v_t\) by: \(v_1: = \max \{{{\bar{v}}}, {\bar{\mu }}(t) - \gamma {\bar{\sigma }}(t)\}\) if the domain of \(v_t\) is positive; otherwise \(v_1= {\bar{\mu }}(t) - \gamma {\bar{\sigma }}(t)\). We next choose \(v_{m_0}:= {\bar{\mu }}(t) + \gamma \bar{\sigma }(t)\). Here we have defined \({\bar{\mu }}(t) = {\mathbb {E}}[v_t |v_0]\) and \({\bar{\sigma }}(t)\) the standard deviation conditional on \(v_0\). Further, we define the constantFootnote 7\(\gamma = 4.5\) and \({{\bar{v}}} = 0.00001\).Footnote 8 Finally, we generate \(v_2,v_3,\ldots ,v_{m_0-1}\) using the procedure

$$\begin{aligned} v_i=v_0+{\bar{\alpha }}\sinh \left( c_2\frac{i}{m_0}+c_1\Big (1-\frac{i}{m_0}\Big ) \right) \end{aligned}$$


$$\begin{aligned} c_1={{\,\mathrm{arcsinh}\,}}\left( \frac{v_1-v_0}{{\bar{\alpha }}} \right) , \quad c_2= {{\,\mathrm{arcsinh}\,}}\left( \frac{v_{m_0}-v_0}{{\bar{\alpha }}} \right) \end{aligned}$$

for \({\bar{\alpha }}<(v_{m_0}-v_1)\). This transformation concentrates more grid points near the critical point \(v_0\), where the non-uniformity of the grid is determined by the parameter \({\bar{\alpha }}\): smaller \({\bar{\alpha }}\) results in a more nonuniform grid. For computations in this paper we choose \({\bar{\alpha }}= (v_{m_0} - v_1)/5\). Since \(v_0\) is not likely a member of the variance grid, find the bracketing index \(j_0\) such that \(v_{j_0}\le v_0 < v_{j_0+1}\). Holding the points \(v_1, v_2\) constant,Footnote 9 we shift the remaining points \(v_j, j\ge 2\) by \(v_0 -v_{j_0}\) so that \(v_{j_0}= v_0\) is a member of the adjusted grid.

Remark 3

We note that for Heston, 3/2 (with reformulated parameters), and 4/2 models

$$\begin{aligned} {\bar{\mu }}(t)= & {} e^{-\eta t}v_0 + \theta (1- e^{-\eta t}), \qquad {\bar{\sigma }}^2(t):=\frac{\sigma _v^2}{\eta }v_0(e^{-\eta t} - e^{-2\eta t}) \\&+ \frac{\theta \sigma ^2_v}{2\eta }(1- e^{-\eta t}+e^{-2\eta t}). \end{aligned}$$

In the Stein-Stein’s model, we have

$$\begin{aligned} {\bar{\mu }}(t) = e^{-\eta t}v_0 + \theta (1- e^{-\eta t}), \qquad {\bar{\sigma }}^2(t):=\frac{\sigma _v^2}{2\eta }(1- e^{-2\eta t}). \end{aligned}$$

For Hull-White’s model, the variance process is described by the moments

$$\begin{aligned} {\bar{\mu }}(t) = v_0 e^{a_vt}, \qquad {\bar{\sigma }}^2(t) = v_0^2 e^{2a_vt}(e^{\sigma _v^2t} -1). \end{aligned}$$

1.2 Approximation of \(\Psi \)

To approximate \({\bar{\Psi }} =\{{\bar{\Psi }}\}_{n,k}\) for \(n =1,\ldots , N\), \(k = 1,\ldots , N +N_{M-1}\), we apply Gaussian quadrature over each interval \(I_k^m =[x^*_{k}-2\Delta ,x^*_{k}+2\Delta ]\), where \(x_k^*\) is defined in Sect. 4.2. After a change of variables, this is accomplished by applying an \(N_q\)-point quadrature applied to each subinterval of \([-2,2]=[-2,-1]\cup [-1,0]\cup [0,1]\cup [1,2]\) as follows. Specifically,

$$\begin{aligned} \Psi (n,k)&: =a^{1/2} \int _{I^m_k}\left( e^y + 1\right) ^{i\xi _n}a^{1/2}\varphi (a(y-x_k^*))dy \\&= \int _{[-2,2]}\exp \left( \mathrm {i}\xi _n \ln \left( 1+\exp \left( x^*_k +\frac{y}{a}\right) \right) \right) \varphi (y)dy \\&\approx \sum _{l=1}^{4\cdot N_q} \omega _l\cdot \exp \left( \mathrm {i}\xi _n \ln \left( 1+\exp \left( x^*_k +\gamma _l\right) \right) \right) \varphi (\gamma _l), \end{aligned}$$

where \(\{(\gamma _l, \omega _l)\}_{l=1}^{4\cdot N_q}\) are the nodes and weights. If we define the sample grid

$$\begin{aligned} \eta _j, \quad j=1,...,N_\eta , \qquad N_\eta :=(N +N_{M-1} -1)\cdot N_q + 4 \cdot N_q, \end{aligned}$$

the Gaussian approximation of \(\Psi (n,k)\), for \(n=1,\ldots ,N\) and \(k=1,\ldots ,N+N_{M-1}\), is given by

$$\begin{aligned} {\bar{\Psi }} (n,k)&:= \sum _{l=1}^{4\cdot N_q} \theta ^n_{N_q(k-1)+l}\cdot \sigma _l = \sum _{l=1}^{2\cdot N_q} \left( \theta ^n_{N_q(k-1)+l} + \theta ^n_{N_q(k+3)+1-l}\right) \cdot \sigma _l , \end{aligned}$$

where \(\sigma _l := \varphi ^{[3]}(\gamma _l)\cdot \omega _l,\)\( l = 1,..., 4 \cdot N_q\), and

$$\begin{aligned} \theta _j^n := \exp \left( \mathrm {i}\xi _n \ln \left( 1+\exp \left( \eta _j\right) \right) \right) , \qquad n = 1,...,N, \quad j=1,...,N_\eta . \end{aligned}$$

To populate \({\bar{\Psi }} (n,k)\) efficiently, note that \(\theta _j^n = \theta _j^{n-1}\cdot \exp (\mathrm {i}\Delta _\xi \ln (1+\exp (\eta _j)))\). In this work, we utilize a five-point Gaussian quadrature, detailed in Appendix A.2.1.

1.2.1 Five-point Gaussian quadrature

A composite five point Gaussian quadrature is implemented by applying a five point rule over each interval \([r,r+1]\), for \(r = -2,-1,0,1\). By symmetry we only evaluate for \(r=-2,-1\). On \([-2,-1]\), we obtain the nodes and weights

$$\begin{aligned} \{\gamma _l\}_{l=1}^5= & {} \Big \{-\frac{3}{2}-g_3,-\frac{3}{2}- g_2, -\frac{3}{2},-\frac{3}{2}+g_2,-\frac{3}{2}+ g_3\Big \}, \\ \{\omega _l\}_{l=1}^5= & {} \frac{1}{2}\{ {{\hat{v}}}_3,{{\hat{v}}}_2,{{\hat{v}}}_1,{{\hat{v}}}_2,{{\hat{v}}}_3\}, \end{aligned}$$

while on \([-1,0]\) we have

$$\begin{aligned} \{\gamma _l\}_{l=6}^{10}= & {} \Big \{-\frac{1}{2}-g_3,-\frac{1}{2}- g_2, -\frac{1}{2},-\frac{1}{2}+g_2,-\frac{1}{2}+ g_3\Big \}, \\ \{\omega _l\}_{l=6}^{10}= & {} \frac{1}{2}\{{{\hat{v}}}_3,{{\hat{v}}}_2,{{\hat{v}}}_1, {{\hat{v}}}_2,{{\hat{v}}}_3\}, \end{aligned}$$

where we define the constants

$$\begin{aligned}&g_2: = \frac{1}{6}\sqrt{5 - 2 \sqrt{10/7}}, \quad g_3 : =\frac{1}{6}\sqrt{5 + 2 \sqrt{10/7}}\\&\quad {{\hat{v}}}_1: = 128/225, \quad {{\hat{v}}}_2: = (322 + 13\sqrt{70})/900, \quad {{\hat{v}}}_3:=(322-13\sqrt{70})/900. \end{aligned}$$

The final weights \(\sigma _l = \varphi ^{[3]}(\gamma _l)\cdot w_l\) are found by evaluating the cubic generator defined in equation (38) at each of \(\{\gamma _l\}_{l=1}^{10}\), and \(\sigma _l \) is stored for repeated use.

1.3 Continuous monitoring

Rather than set M extremely large to estimate the option value with continuous monitoring, we can utilize the now standard technique of Richardson extrapolation from values computed with modest levels of M. Let \({\mathcal {V}}_N(M)\) denote the discretely monitored value approximation with M monitoring dates, and with N fixed. By choosing \(d\in {\mathbb {N}}_+\), we can approximate the continuously monitored value with a four-point Richardson extrapolation:

$$\begin{aligned} {\mathcal {V}}_N^\infty (d) := \frac{1}{21}\left( 64{\mathcal {V}}_N(2^{d+3}) - 56 {\mathcal {V}}_N(2^{d+2})+14{\mathcal {V}}_N(2^{d+1})-{\mathcal {V}}_N(2^d) \right) . \end{aligned}$$

as applied in Zhang and Oosterlee (2013). An efficient APROJ extrapolation algorithm can be devised by reusing the matrix \({\bar{\Psi }}\) for each d. See Table 16 for an example of the continuous monitoring approximation

Table 16 Continuously monitored Asian option values by Richardson Extrapolation

Error analysis

This section demonstrates stability of the characteristic function recursion, and provides a bound on the rate of convergence of the APROJ value error.Footnote 10 Returning shortly to the case of \(m=2\), we have for \(m\ge 3\)

$$\begin{aligned} \epsilon ({\bar{\phi }}^j_{Z_{m-1}}(\xi ))&:= \phi ^j_{Z_{m-1}}(\xi )-\bar{\phi }^j_{Z_{m-1}}(\xi ) = \int _{\mathbb {R}}(e^y+1)^{\mathrm {i}\xi }(f^j_{Y_{m-1}}(y) -\bar{f}^j_{Y_{m-1}}(y))dy\\&= \int _{{\mathbb {R}}/G_m} (e^y+1)^{\mathrm {i}\xi }f^j_{Y_{m-1}}(y)dy \\&\quad + \left( \int _{G_m} (e^y+1)^{\mathrm {i}\xi }f^j_{Y_{m-1}}(y)dy - \Upsilon _{a,N}\sum _{k=1}^N\beta ^j_{Y_{m-1},k}\Psi _{m-1}(\xi ,k)\right) \\&\quad + \Upsilon _{a,N}\sum _{k=1}^N\beta ^j_{Y_{m-1},k}( \Psi _{m-1}(\xi ,k) - {\bar{\Psi }}_{m-1}(\xi ,k)) \\&\quad + \Upsilon _{a,N}\sum _{k=1}^N\bar{\Psi }_{m-1}(\xi ,k)(\beta ^j_{Y_{m-1},k}-{\bar{\beta }}^j_{Y_{m-1},k})\\&:=\left( \tau (G_{m-1}) + {\mathcal {E}}^j_{m-1,1}(\xi ) + {\mathcal {E}}^j_{m-1,2}(\xi )\right) + {\mathcal {E}}^j_{m-1}(\xi ), \end{aligned}$$

where \(G_m = [{\bar{\mu }}_m - {{\bar{a}}}/2, {\bar{\mu }}_m +{{\bar{a}}}/2]\), and the final term will be further split into two components.Footnote 11 The term \(\tau (G_{m-1})\) captures density truncation error and can be bounded by \(\tau (G)\) which converges exponentially in \({{\bar{a}}} = 2^{{{\bar{P}}}}\) for most processes of interest. If we define \(P_{{\mathcal {M}}_a} f^j_{Y_{m-1}}\) to be the true (untruncated) orthogonal projection of \(f^j_{Y_{m-1}}\) onto the space \({\mathcal {M}}_a\), the second term satisfies

$$\begin{aligned} {\mathcal {E}}^j_{m-1,1}(\xi )&:= \int _{G_m} (e^y+1)^{\mathrm {i}\xi }f^j_{Y_{m-1}}(y)dy - \Upsilon _{a,N}\sum _{k=1}^N\beta ^j_{Y_{m-1},k}\Psi _{m-1}(\xi ,k) \\&\le \left\Vert (e^y+1)^{\mathrm {i}\xi }\right\Vert ^{G_{m-1}}_2 \cdot \left\Vert f^j_{Y_{m-1}} - P_{{\mathcal {M}}_a} f^j_{Y_{m-1}}\right\Vert ^{{\mathbb {R}}}_2 \\&\le \sqrt{{{\bar{a}}}}\cdot C_1(f^j_{Y_{m-1}})\cdot \Delta ^4, \end{aligned}$$

where \(\Delta ^4\) is the theoretical convergence rate of cubic projection (in practice, the rate is much faster). The constant \(C_1(f^j_{Y_{m-1}})\) is bounded by a constant multiple of \(\left\Vert (-\mathrm {i}\xi )^4 \phi _{Y^j_{m-1}}(\xi )\right\Vert _2 \le \left\Vert \xi ^4\phi _{X_{\Delta _t}}^{{{\bar{k}}}}(\xi )\right\Vert _2\) for some \(1\le {{\bar{k}}} \le m_0\), where the inequality follows by Lemma 1. Hence, for a constant \(C_1\) we have the bound

$$\begin{aligned} {\mathcal {E}}^j_{m-1,1}(\xi ) \le C_1\cdot \sqrt{{{\bar{a}}}}\cdot \Delta ^4 \end{aligned}$$

which is independent of \(\xi , j, m\). This will govern the overall rate of convergence. Next define the numerical integration error

$$\begin{aligned} \epsilon ({\bar{\Psi }}):=\sup \{|{\bar{\Psi }}(\xi _n,k) - \Psi (\xi _n,k)|: 1\le n \le N, 1\le k\le N+N_{M-1}\}. \end{aligned}$$

For a constant \(C(\varphi )\), it follows from Lemma 5.2 of Kirkby (2016) that

$$\begin{aligned} {\mathcal {E}}^j_{m-1,2}(\xi )&= \Upsilon _{a,N}\sum _{k=1}^N\beta ^j_{Y_{m-1},k}(\Psi _{m-1}(\xi ,k) - {\bar{\Psi }}_{m-1}(\xi ,k)) \\&\le \sqrt{{{\bar{a}}}} \cdot \epsilon ({\bar{\Psi }})\cdot C(\varphi )\cdot \left\Vert f^j_{Y_{m-1}}\right\Vert _2 \le C_2 \cdot \sqrt{{{\bar{a}}}} \cdot \epsilon ({\bar{\Psi }}) \end{aligned}$$

for some \(C_2\), where the second inequality again follows from Lemma 1.

The remaining term, which provides the link between errors though time, is

$$\begin{aligned} {\mathcal {E}}^j_{m-1}(\xi )&:= \Upsilon _{a,N}\sum _{k=1}^N\bar{\Psi }_{m-1}(\xi ,k)(\beta ^j_{Y_{m-1},k}-{\bar{\beta }}^j_{Y_{m-1},k})\\&= a^{-1/2}\sum _{k=1}^N{\bar{\Psi }}_{m-1}(\xi ,k)\cdot \epsilon ({\bar{\beta }}^j_{Y_{m-1},k}), \end{aligned}$$

where \(\epsilon ({\bar{\beta }}^j_{Y_{m-1},k}):=a^{1/2} \Upsilon _{a,N}(\beta ^j_{Y_{m-1},k}-{\bar{\beta }}^j_{Y_{m-1},k})\). Since \({\bar{\beta }}^j_{Y_{m-1},k}\) contains several sources of error, we define \(\breve{\beta }^j_{Y_{m-1},k}\) to be the approximation using the true \(\phi ^j_{Y_{m-1}}\) rather than \({\bar{\phi }}^j_{Y_{m-1}}\). Hence

$$\begin{aligned} \epsilon ({\bar{\beta }}^j_{Y_{m-1},k})&=\left( \langle f^j_{Y_{m-1}},{\widetilde{\varphi }}_{a,k} \rangle - a^{1/2}\Upsilon _{a,N}\breve{\beta }^j_{Y_{m-1},k}\right) + a^{1/2}\Upsilon _{a,N}\left( \breve{\beta }^j_{Y_{m-1},k} - \bar{\beta }^j_{Y_{m-1},k}\right) \\&:=\epsilon _1({\bar{\beta }}^j_{Y_{m-1},k}) + \epsilon _2(\bar{\beta }^j_{Y_{m-1},k}), \end{aligned}$$

which yields \({\mathcal {E}}^j_{m-1}(\xi ) = {\mathcal {E}}^j_{m-1,3}(\xi ) + {\mathcal {E}}^j_{m-1,4}(\xi )\).

The first term can be captured by the Corollary 3.2 of Kirkby (2016), modified slightly to the present case. We assume that \(\widetilde{\phi }^j_{X_{\Delta _t}}\), \(j=1,\ldots ,m_0\) are analytic within a strip \({\mathcal {D}}_d:=\{z\in {\mathbb {C}}: \mathfrak {I}(z)\in (-d,d)\}\), for some \(d>0\), and satisfy equation (18). We further define a constant \(C_M\), which is bounded by a multiple of \(\max _{1\le m\le M, 1\le j\le m_0} \left\Vert \phi ^j_{Y_m}\right\Vert ^{\mathcal H_d}\), where \(\left\Vert f\right\Vert ^{{\mathcal {H}}_d}\) is the Hardy norm of f on \({\mathcal {D}}_d\).

Corollary 3

Fix \(a = 2^P\) and \(N = a \cdot {{\bar{a}}}\), where \({{\bar{a}}} = 2^{{{\bar{P}}}}\) for \({{\bar{P}}} >1 + \log _2|{\bar{\mu }}_M|\). Assume Lemma 1 holds for some \(c_{{{\bar{k}}}},\kappa _{{{\bar{k}}}} >0\) and \(\nu _{ {{\bar{k}}}} \in (0,2]\). Then for some \(0<\gamma \le d\)

$$\begin{aligned} \sup _{1\le k\le N}\left| a^{1/2}\Upsilon _{a,N}\cdot \breve{\beta }^j_{Y_{m},k} - \langle f^j_{Y_{m}},\widetilde{\varphi }_{a,k} \rangle \right|&\le \frac{a^{-1/2}}{\pi }\left( C_M\frac{e^{-({{\bar{a}}} - 2|\bar{\mu }_M|)\gamma /2}}{1-e^{-{{\bar{a}}} \gamma }} + \tau _{a}\left( \widetilde{\phi }_{X_{\Delta _t}}^{{{\bar{k}}}}\right) \right) \nonumber \\&:= \frac{a^{-1/2}}{\pi }\epsilon ^M(a,{{\bar{a}}}) \end{aligned}$$

independently of \(1\le m \le M\) and \(j=1,\ldots , m_0\) where \(\tau _{a}({\widetilde{\phi }}_{X_{\Delta _t}}^{{{\bar{k}}}}) =\mathcal O(a\exp (-\Delta t c_{{{\bar{k}}}} \cdot (2\pi a)^{\nu _{{{\bar{k}}}}}))\). For large enough \(a>0\), and \(d<\infty \), \(\gamma \) will approach d.

By Corollary 3 and \(|{\bar{\Psi }}_{m-1}(\xi ,k)|\le 1\) we have

$$\begin{aligned} |{\mathcal {E}}^j_{m-1,3}(\xi )| \le a^{-1/2}\sum _{k=1}^N|\bar{\Psi }_{m-1}(\xi ,k)| \cdot |\epsilon _1({\bar{\beta }}^j_{Y_{m-1},k})| \le \frac{{{\bar{a}}}}{\pi } \epsilon ^M(a, {{\bar{a}}}). \end{aligned}$$

The discretization error represented by the first term in \(\epsilon ^M(a,{{\bar{a}}})\) decays exponentially in \({{\bar{a}}}\), while the truncation error decays exponentially in a (for fixed \({{\bar{a}}}\)).

The final term to estimate is

$$\begin{aligned} {\mathcal {E}}^j_{m-1,4}(\xi ) :=a^{-1/2}\sum _{k=1}^N\bar{\Psi }_{m-1}(\xi ,k) \cdot \epsilon _2({\bar{\beta }}^j_{Y_{m-1},k}) \end{aligned}$$

where, with \(h^{m-1}_{a,k}(\xi ):=\widehat{\widetilde{\varphi }}(\xi /a)\exp (\mathrm {i}\xi x_k^{m-1})\),

$$\begin{aligned} \epsilon _2({\bar{\beta }}^j_{Y_{m-1},k})&= \frac{a^{-1/2}}{\pi } \mathfrak {R}\left( \Delta _\xi \sum _{n=1}^N{}^{'}\left( \phi ^j_{Y_{m-1}}(\xi _n) - {\bar{\phi }}^j_{Y_{m-1}}(\xi _n)\right) h^{m-1}_{a,k}(\xi _n)\right) \\&=\frac{a^{-1/2}}{\pi } \mathfrak {R}\left( \Delta _\xi \sum _{n=1}^N{}^{'}\left( \sum _{l=1}^{m_0}\left( \phi ^l_{Z_{m-2}}(\xi _n) - {\bar{\phi }}^l_{Z_{m-2}}(\xi _n)\right) {\mathcal {E}}_{l,j}(\xi _n)\right) h^{m-1}_{a,k}(\xi _n)\right) \\&=\frac{a^{-1/2}}{\pi } \mathfrak {R}\left( \Delta _\xi \sum _{n=1}^N{}^{'}\sum _{l=1}^{m_0}\epsilon ({\bar{\phi }}^l_{Z_{m-2}}(\xi _n))\mathcal E_{l,j}(\xi _n)\cdot h^{m-1}_{a,k}(\xi _n)\right) \end{aligned}$$

where \(\sum {}^{'}\) indicates that the first and last terms in the sum are halved.

Denoting \(\epsilon ({\bar{\phi }}_{Z_{m-2}}):= \max _{1\le n\le N, 1\le k\le m_0}| \epsilon ({\bar{\phi }}^{k}_{Z_{m-2}}(\xi _n))| \), and \(\tilde{\epsilon }({\bar{\phi }}^k_{Z_{m-2}}(\xi _n))\cdot \epsilon (\bar{\phi }_{Z_{m-2}})= \epsilon ({\bar{\phi }}^k_{Z_{m-2}}(\xi _n)), \)

$$\begin{aligned} \epsilon _2({\bar{\beta }}^j_{Y_{m-1},k})&=\epsilon ({\bar{\phi }}_{Z_{m-2}}) \frac{a^{-1/2}}{\pi } \mathfrak {R}\left( \Delta _\xi \sum _{n=1}^N{}^{'}h^{m-1}_{a,k}(\xi _n)\sum _{l=1}^{m_0}{{\tilde{\epsilon }}}({\bar{\phi }}^l_{Z_{m-2}}(\xi _n))\mathcal E_{l,j}(\xi _n) \right) \\&\le C_\epsilon \cdot \epsilon ({\bar{\phi }}_{Z_{m-2}}) \frac{a^{-1/2}}{\pi }, \end{aligned}$$

for N sufficiently large and some \(0<C_\epsilon <\infty \), since we can majorize the error term by an \(L^2\) function which admits an upper frame bound. Thus,

$$\begin{aligned} {\mathcal {E}}^j_{m-1,4}(\xi )&:=a^{-1/2}\sum _{k=1}^N\bar{\Psi }_{m-1}(\xi ,k) \cdot \epsilon _2({\bar{\beta }}^j_{Y_{m-1},k}) \\&=a^{-1/2}\epsilon ({\bar{\phi }}_{Z_{m-2}})\sum _{k=1}^N\bar{\Psi }_{m-1}(\xi ,k) \frac{a^{-1/2}}{\pi } \mathfrak {R}\left( \Delta _\xi \sum _{n=1}^N{}^{'}h^{m-1}_{a,k}(\xi _n)\sum _{l=1}^{m_0}{{\tilde{\epsilon }}}({\bar{\phi }}^l_{Z_{m-2}}(\xi _n))\mathcal E_{l,j}(\xi _n) \right) \\&= {\mathcal {O}}\left( \frac{\epsilon ({\bar{\phi }}_{Z_{m-2}})}{a^{1/2}} \left| \sum _{k=1}^N{\bar{\Psi }}_{m-1}(\xi ,k) \frac{a^{-1/2}}{\pi } \mathfrak {R}\left( \Delta _\xi \sum _{n=1}^N{}^{'}h^{m-1}_{a,k}(\xi _n)\sum _{l=1}^{m_0}\mathcal E_{l,j}(\xi _n) \right) \right| \right) \\&= {\mathcal {O}}\left( \frac{\epsilon ({\bar{\phi }}_{Z_{m-2}})}{a^{1/2}} \left| \sum _{k=1}^N{\bar{\Psi }}_{m-1}(\xi ,k) \frac{a^{-1/2}}{\pi } \mathfrak {R}\left( \Delta _\xi \sum _{n=1}^N{}^{'}h^{m-1}_{a,k}(\xi _n)\phi _{Y_1}^j(\xi _n) \right) \right| \right) \\&= {\mathcal {O}}\left( \frac{\epsilon ({\bar{\phi }}_{Z_{m-2}})}{a^{1/2}} \left| \sum _{k=1}^N{\bar{\Psi }}_{m-1}(\xi ,k) \cdot a^{1/2}\Upsilon _{a,N}{\bar{\beta }}_{Y_1,N_{m-1}+k}^j\right| \right) . \end{aligned}$$

Given the exponential decay of \({\bar{\beta }}_{Y_1,n}^j\) for processes satisfying the assumptions above, we have for some constants \(C^j\):

$$\begin{aligned} {\mathcal {E}}^j_{m-1,4}(\xi )&= {\mathcal {O}}\left( \frac{\epsilon (\bar{\phi }_{Z_{m-2}})}{a^{1/2}} \left| \sum _{k=1}^N{\bar{\Psi }}_{m-1}(\xi ,k) \cdot a^{1/2}\Upsilon _{a,N}{\bar{\beta }}_{Y_1,k}^j\right| \right) \\&\le C^j \epsilon ({\bar{\phi }}_{Z_{m-2}}) a^{-1/2}|{\bar{\phi }}_{Z_1}^j(\xi )|. \end{aligned}$$


$$\begin{aligned} \epsilon ({\bar{\phi }}^j_{Z_{m-1}}(\xi ))&\le \tau (G) + |\mathcal E^j_{m-1,1}(\xi )| + |{\mathcal {E}}^j_{m-1,2}(\xi )| + |\mathcal E^j_{m-1,3}(\xi )|+ |{\mathcal {E}}^j_{m-1,4}(\xi )|\\&\le \left( \tau (G) + C_1\cdot \sqrt{{{\bar{a}}}}\cdot \Delta ^4 + C_2 \cdot \sqrt{{{\bar{a}}}} \cdot \epsilon ({\bar{\Psi }}) + \frac{{{\bar{a}}}}{\pi } \epsilon ^M(a, {{\bar{a}}})\right) \\&\quad + a^{-1/2}C^j|{\bar{\phi }}_{Z_1}^j(\xi )| \epsilon ({\bar{\phi }}_{Z_{m-2}})\\&\le \gamma ^M(a,{{\bar{a}}}) + \Omega (a,\xi )\epsilon (\bar{\phi }_{Z_{m-2}}), \end{aligned}$$

where \( \gamma ^M(a,{{\bar{a}}})\) is the term in parentheses, and \( \Omega (a,\xi ):=a^{-1/2}\max _{1\le j\le m_0}C^j|{\bar{\phi }}_{Z_1}^j(\xi )|\). Using the same logic as above, we can show that \(\epsilon ({\bar{\phi }}_{Z_{1}})\le \gamma ^M(a,{{\bar{a}}}) \). Iterating from \(M-1\) we obtain

$$\begin{aligned} \epsilon ({\bar{\phi }}_{Z_{M-1}}(\xi ))&\le \gamma ^M(a,{{\bar{a}}}) \sum _{m=0}^{M-3} \Omega (a,\xi )^m + \Omega (a,\xi )^{M-2} \epsilon (\bar{\phi }_{Z_{1}})\\&= \gamma ^M(a,{{\bar{a}}}) \frac{1 - \Omega (a,\xi )^{M-2}}{1-\Omega (a,\xi )} + \Omega (a,\xi )^{M-2} \epsilon ({\bar{\phi }}_{Z_{1}})\\&\le \gamma ^M(a,{{\bar{a}}})\frac{1 - \Omega (a,\xi )^{M-1}}{1-\Omega (a,\xi )} \le 2\gamma ^M(a,{{\bar{a}}}), \end{aligned}$$

where the final inequality holds for a sufficiently large. By the definition of \(\phi _{Y_{M}}^j(\xi )\), and \(\epsilon (\bar{\phi }^j_{Y_{M}}(\xi )):=\phi ^j_{Y_{M}}(\xi ) - \bar{\phi }^j_{Y_{M}}(\xi )\) we have with \(\epsilon (\bar{\phi }^k_{Z_{M-1}}(\xi )) = \epsilon ({\bar{\phi }}_{Z_{M-1}}(\xi ))\cdot {{\tilde{\epsilon }}}({\bar{\phi }}^k_{Z_{M-1}}(\xi ))\)

$$\begin{aligned} |\epsilon ({\bar{\phi }}^j_{Y_{M}}(\xi ))|&= \left| \sum _{k =1,..., m_0}\epsilon ({\bar{\phi }}^k_{Z_{M-1}}(\xi )){\mathcal {E}}_{k,j}(\xi ) \right| \\&=|\epsilon ({\bar{\phi }}_{Z_{M-1}}(\xi ))| \left| \sum _{k =1,..., m_0}{{\tilde{\epsilon }}}({\bar{\phi }}^k_{Z_{M-1}}(\xi )){\mathcal {E}}_{k,j}(\xi ) \right| \\&\le {{\bar{C}}} \gamma ^M(a,{{\bar{a}}}), \end{aligned}$$

for some \({{\bar{C}}}\) independent of \(\xi \). Hence, the characteristic function recursion is stable, and the error in \({\bar{\phi }}_M^j\) can be made arbitrarily small. In particular, with the truncation error \(\tau (G)\) (which converges exponentially for processes of interest) is controlled by the choice of \({{\bar{a}}}\), \(\epsilon ({\bar{\Psi }})\) sufficiently small by Gaussian quadrature, and \(\epsilon ^M(a,\bar{a})\) exponentially convergent by Corollary 3, the error is dominated by that of cubic projection, which is \(\mathcal O(\Delta ^4)\). We now state the main result, whose proof is analogous to Section 5.2 in Kirkby (2016), which characterizes the final valuation error.

Proposition 4

Given a European-style payoff \(g(A_M)\) on the arithmetic average \(A_M\), suppose that the assumptions of Lemma 1 hold, and that \({{\bar{a}}}\) has been fixed sufficiently large to control truncation error. Define the value approximation \({\mathcal {V}}_N\circ g(S_0, \alpha _0)\) as in equation (46), where \(N:=a{{\bar{a}}}\). Then the value error, \({\mathcal {E}}({\mathcal {V}}_N) = |{\mathcal {V}}\circ g(S_0, \alpha _0) - {\mathcal {V}}_N\circ g(S_0, \alpha _0)|\) decays as \(\mathcal E({\mathcal {V}}_N) = {\mathcal {O}}(a^{-4})\) as the resolution parameter a is increased.

In practice, we typically observe a faster rate of error decay than this result would suggest, but it serves as a conservative estimate.

Additional stochastic volatility models

1.1 Heston and Bate’s models

Consider the Heston stochastic volatility model Heston (1993) with jumps (also known as the Bates model Bates (1996) in the literature):

$$\begin{aligned} \mathbf{Hes}: \left\{ \begin{array}{l} \frac{dS_t}{S_{t^-}}=(r- q-\lambda \kappa )dt+\sqrt{v_t}dW_t^1 + d\left( \sum _{i=1}^{N(t)} (e^{J_i}-1)\right) ,\\ dv_t=\eta (\theta -v_t)dt+\sigma _v\sqrt{v_t}dW^2_t, \end{array} \right. \end{aligned}$$

where \(\eta \) is the mean reversion rate, \(\theta \) is the equilibrium level, and \(\sigma _v>0\) is the volatility of volatility. Note that to ensure \(v_t>0\), the Feller’s condition \(2\eta \theta >\sigma _v^2\) is imposed (see Heston (1993)). Applying the transform in equation (21) yields

$$\begin{aligned} {\widetilde{X}}_t=\log \left( \frac{S_t}{S_0}\right) - \rho \int _{v_0}^{v_t}\frac{\varkappa (u)}{{{\hat{\sigma }}}(u)}du =\log \left( \frac{S_t}{S_0}\right) - \frac{\rho }{\sigma _v}(v_t-v_0), \end{aligned}$$

with \(f(v_t,v_0) = \frac{\rho }{\sigma _v}(v_t-v_0)\). It follows that

$$\begin{aligned} \left\{ \begin{array}{l} d{\widetilde{X}}_t=\left[ (\frac{\rho \eta }{\sigma _v}-\frac{1}{2})v_t + (r-q-\frac{\rho \eta \theta }{\sigma _v}-\lambda \kappa )\right] dt+\sqrt{(1-\rho ^2)v_t}dW^*_t +d \left( \sum _{i=1}^{N(t)}J_i\right) ,\\ dv_t=\eta (\theta -v_t)dt+\sigma _v\sqrt{v_t}dW^2_t. \end{array}\right. \end{aligned}$$

1.2 3/2 Model with jumps

Next consider the dynamics of the 3/2 stochastic volatility model with jumps

$$\begin{aligned} \mathbf 3/2: \left\{ \begin{array}{l} \frac{dS_t}{S_{t^-}}=(r-q-\lambda \kappa )dt+\sqrt{v_t}dW_t^1 +d\left( \sum _{i=1}^{N(t)} (e^{J_i}-1)\right) , \\ dv_t=v_t[\eta (\theta -v_t)dt+\sigma _v \sqrt{v_t}dW^2_t],\ v(0)=v_0, \end{array} \right. \end{aligned}$$

where in (56) \(v_t\) is the variance of the asset \(S_t\), r is the risk-free interest rate, \(\sigma _v>0\), \(\theta \in {\mathbb {R}}\) is the mean reversion level, \(\eta \) is given such that \(\eta v_t\)-a stochastic volatility quantity-is the speed of mean reversion.

While this form can be applied directly, we prefer an alternative formulation which nests the 3/2 model within the 4/2 model introduced in Sect. 3.1.1. Applying Ito’s formula we have

$$\begin{aligned}&d\left( \frac{1}{v_t}\right) =\eta \theta \left( \frac{\eta +\sigma _v^2}{\eta \theta }-\frac{1}{v_t} \right) dt-\displaystyle \frac{\sigma _v}{\sqrt{v_t}}dW_t^2 \end{aligned}$$
$$\begin{aligned}&\widehat{v_t}:=\frac{1}{v_t},\quad {\widehat{\eta }}:=\eta \theta ,\quad {\widehat{\theta }}:=\frac{\eta +\sigma _v^2}{\eta \theta },\quad {\widehat{\sigma }}_v:=-\sigma _v, \end{aligned}$$

from which (56) is reduced to

$$\begin{aligned} \left\{ \begin{array}{l} \frac{dS_t}{S_{t^-}}=(r- q -\lambda \kappa )dt+\displaystyle \frac{1}{\sqrt{{\widehat{v}}_t}}dW_t^1 +d\left( \sum _{i=1}^{N(t)} (e^{J_i}-1)\right) , \\ d{\widehat{v}}_t={\widehat{\eta }}[{\widehat{\theta }}-\widehat{v_t}]dt+{\widehat{\sigma }}_v \sqrt{{\widehat{v}}_t}dW_t^2,\quad {\widehat{v}}_0=1/v_0. \end{array} \right. \end{aligned}$$

Equation equation (21) prescribes the change of variables

$$\begin{aligned} \widetilde{X}_t=\log \Big (\frac{S_t}{S_0}\Big )-\frac{\rho }{{\widehat{\sigma }}_v}\log \Big (\frac{{\widehat{v}}_t}{{\widehat{v}}_0}\Big ), \end{aligned}$$

which results in the decorrelated dynamics

$$\begin{aligned} \left\{ \begin{array}{l} d{\widetilde{X}}_t=\displaystyle \Big [\frac{1}{2}\Big (\rho {\widehat{\sigma }}_v-2\frac{\rho {\widehat{\eta }}{\widehat{\theta }}}{{\widehat{\sigma }}_v}-1\Big )\frac{1}{{\widehat{v}}_t}+\frac{\rho {\widehat{\eta }}}{{\widehat{\sigma }}_v}+(r-q -\lambda \kappa )\Big ]dt+\sqrt{\frac{(1-\rho ^2)}{{\widehat{v}}_t}}dW^*_t +d \left( \sum _{i=1}^{N(t)}J_i\right) ,\\ d{\widehat{v}}_t={\widehat{\eta }}[{\widehat{\theta }}-\widehat{v_t}]dt+{\widehat{\sigma }}_v\sqrt{{\widehat{v}}_t} dW_t^2,\quad {\widehat{v}}_0=1/v_0. \end{array}\right. \end{aligned}$$

We see that the 4/2 model in (27) indeed contains the 3/2 model as a special case where \(a= 0\), \(b=1\), and the parameters for 3/2 are selected using the re-parameterization in equation (58).

1.3 Hull-White’s model with jumps

Augmenting the traditional Hull and White (1990) model with jumps produces

$$\begin{aligned} \mathbf{HW}: \left\{ \begin{array}{l} \frac{dS_t}{S_{t^-}}=(r-q -\lambda \kappa )dt+\sqrt{v_t}dW_t^1 +d\left( \sum _{i=1}^{N(t)} (e^{J_i}-1)\right) ,\\ dv_t=a_vv_tdt+\sigma _vv_tdW^2_t. \end{array} \right. \end{aligned}$$

The change of variable that will help us to remove the correlation, \(\rho \), between the two stochastic processes \(W_t^1\), \(W_t^2\) in (62) is given by

$$\begin{aligned} \widetilde{X}_t=\log \Big (\frac{S_t}{S_0}\Big )-\frac{2\rho }{\sigma _v}(\sqrt{v_t}-\sqrt{v_0}). \end{aligned}$$

The decorrelated dynamics satsify

$$\begin{aligned} \left\{ \begin{array}{l} d\widetilde{X}_t=\Big [\Big (\displaystyle \frac{\rho \sigma _v}{4}-\frac{a_v\rho }{\sigma _v}\Big )\sqrt{v_t}-\frac{1}{2}v_t +(r-q -\lambda \kappa )\Big ]dt+\sqrt{(1-\rho ^2)v_t}dW^*_t +d \left( \sum _{i=1}^{N(t)}J_i\right) ,\\ dv_t=a_v v_tdt+\sigma _v v_tdW^2_t. \end{array}\right. \end{aligned}$$

1.4 Stein-Stein’s model with jumps

Stein and Stein (1991) consider a stochastic volatility model where the two Brownian motions are independent. In this section, we extend their model by allowing for correlation and add a jump component. More specifically, the dynamics of the model is specified as follow:

$$\begin{aligned} \mathbf{SS}: \left\{ \begin{array}{l} \frac{dS_t}{S_{t^-}}=(r-q -\lambda \kappa )dt+v_tdW_t^1 +d\left( \sum _{i=1}^{N(t)} (e^{J_i}-1)\right) ,\\ dv_t=\eta (\theta -v_t)dt+\sigma _vdW^2_t, \end{array} \right. \end{aligned}$$

The change of variable that will help us to remove the correlation between the two stochastic processes \(W_t^1\), \(W_t^2\) in (65) is given by

$$\begin{aligned} \widetilde{X}_t=\log \Big (\frac{S_t}{S_0}\Big )-\frac{1}{2}\frac{\rho }{\sigma _v}(v_t^2-v_0^2). \end{aligned}$$

Then, we have

$$\begin{aligned} \left\{ \begin{array}{l} d{\widetilde{X}}_t=\Big [\Big (\displaystyle \frac{\rho \eta }{\sigma _v}-\frac{1}{2}\Big )v_t^2-\frac{\rho \eta \theta }{\sigma _v}v_t+r-q -\lambda \kappa -\frac{\rho \sigma _v}{2}\Big ]dt+v_t\sqrt{(1-\rho ^2)}dW^*_t +d \left( \sum _{i=1}^{N(t)}J_i\right) ,\\ dv_t=\eta (\theta -v_t)dt+\sigma _vdW^2_t. \end{array}\right. \end{aligned}$$

1.5 \(\alpha \)-Hypergeometric model

The \(\alpha \)-Hypergeometric model was recently proposed by Da Fonseca and Martini (2016). Unlike Heston model, for \(\alpha \)-Hypergeometric model the strict positivity of volatility is guaranteed. The dynamics of the stock price is given by

$$\begin{aligned} \left\{ \begin{array}{l} \frac{dS_t}{S_{t^-}}=(r-q -\lambda \kappa )dt+ e^{v_t}dW_t^1+d\left( \sum _{i=1}^{N(t)} (e^{J_i}-1)\right) , \\ dv_t=(\eta -\theta e^{a_v v_t})dt+\sigma _v dW^2_t,\ v(0)=v_0, \end{array} \right. \end{aligned}$$

where \(\eta ,v_0\in (-\infty ,+\infty ), \theta>0,\sigma _v>0,a_v>0\). Let

$$\begin{aligned} X_t=\log (\frac{S_t}{S_0})-\frac{\rho }{\sigma _v}(e^{v_t}-e^{v_0})-(r-q -\lambda \kappa )t, \end{aligned}$$

then we have

$$\begin{aligned} \left\{ \begin{array}{l} dX_t=\Big [\frac{\rho \theta }{\sigma _v}e^{(1+a_v)v_t}-\rho (\frac{\eta }{\sigma _v}+\frac{\sigma _v}{2})e^{v_t}-\frac{e^{2v_t}}{2}\Big ]dt +e^{v_t}\sqrt{1-\rho ^2}dW^*_t+d \left( \sum _{i=1}^{N(t)}J_i\right) , \\ dv_t=(\eta -\theta e^{a_v v_t})dt+\sigma _v dW^2_t. \end{array} \right. \end{aligned}$$

1.6 Jacobi model

An interesting recent SV model in the literature is the Jacobi model (without jump) of Ackerer et al. (2016), which specifies a bounded variance process Q(v), where \(v\in [v_{min},v_{max}]\) for \(0\le v_{min}<v_{max}\), defined by the quadratic function

$$\begin{aligned} Q(v)= \frac{(v-v_{min})(v_{max}-v)}{(\sqrt{v_{max}} - \sqrt{v_{min}})^2}, \quad 0\le Q(v) \le v, \quad v\in [v_{min},v_{max}]. \end{aligned}$$

With \(Z_t:=\log (S_t)\), the dynamics under the Jacobi model with jumps are

$$\begin{aligned} \left\{ \begin{array}{ll} dZ_t=(r-q -\lambda \kappa - v_t/2)dt + \sqrt{v_t -\rho ^2Q(v_t)}dW_t^{*} + \rho \sqrt{Q(v_t)}dW_t^{(2)} +d \left( \sum _{i=1}^{N(t)}J_i\right) ,\\ dv_t =\eta (\theta -v_t)dt+\alpha \sqrt{Q(v_t)} dW_t^{(2)}. \end{array} \right. \end{aligned}$$

where \(\eta \ge 0\), \(\theta \in [v_{min},v_{max}]\), and \(\alpha >0\). Note here that \({\mathbb {E}}[dW_t^{(2)}dW_t^{*}] = 0\), as the correlation structure is already incorporated with correlation parameter \( \rho \). From the equality

$$\begin{aligned} \rho \int _0^t \sqrt{Q(v_s)}dW_s^{(2)} = \frac{\rho }{\alpha }(v_t - v_0) -\frac{ \rho }{\alpha }\int _{0}^t\eta (\theta -v_s)ds, \end{aligned}$$

we can derive the auxiliary process \({\tilde{X}}_t := Z_t - \frac{ \rho }{\alpha }v_{t}\) with

$$\begin{aligned} d{\tilde{X}}_t&= \left( r-q -\lambda \kappa - \frac{v_{t}}{2} -\frac{\rho }{\alpha }\eta (\theta -v_{t})\right) dt + \sqrt{v_{t} -\rho ^2Q(v_{t})}dW_t^{*}+d \left( \sum _{i=1}^{N(t)}J_i\right) \\&= \left( \left( r-q -\lambda \kappa - \frac{ \rho }{\alpha }\eta \theta \right) + v_{t}\left( \frac{ \rho }{\alpha }\eta - \frac{1}{2} \right) \right) dt + \sqrt{v_{t} -\rho ^2Q(v_{t})}dW_t^{*}\\&\quad +d \left( \sum _{i=1}^{N(t)}J_i\right) . \end{aligned}$$

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Kirkby, J.L., Nguyen, D. Efficient Asian option pricing under regime switching jump diffusions and stochastic volatility models. Ann Finance 16, 307–351 (2020).

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