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04.06.2020 | Research Article

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

verfasst von: J. Lars Kirkby, Duy Nguyen

Erschienen in: Annals of Finance | Ausgabe 3/2020

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Abstract

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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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.

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 \(a,b>0\).

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

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

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.

Accessed on September 20th 2017.

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.

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 \(\beta _2 = 4\).

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

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.

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)

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

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

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)

Boyarchenko, S., Levendorskii, S.: American options in Lévy models with stochastic volatility. Available at SSRN: http://ssrn.com/abstract=1031280 (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)

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)

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

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

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

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

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

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

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

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

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)

Corlay, S., Pagès, G., Printems, J.: The optimal quantization website. http://www.quantize.maths-fi.com (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)

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

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)

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)

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)

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)

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)

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

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

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)

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)

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

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

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)

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

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)

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)

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

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)

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

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)

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

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)

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

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)

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

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

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)

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)

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

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

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

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

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

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)

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

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

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. http://www.sciencedirect.com/science/article/pii/S187775031830228X (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)

Levendorskii, S., Xie, J.: Pricing discretely sampled Asian options under Levy processes. Available at SSRN: http://papers.ssrn.com/abstract=2088214 (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

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

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

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

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

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

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

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

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)

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

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

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

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

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

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

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)

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

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)

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)

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)

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

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

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

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)

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

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

Titel: Efficient Asian option pricing under regime switching jump diffusions and stochastic volatility models
verfasst von: J. Lars Kirkby
Duy Nguyen
Publikationsdatum: 04.06.2020
Verlag: Springer Berlin Heidelberg
Erschienen in: Annals of Finance / Ausgabe 3/2020
Print ISSN: 1614-2446
Elektronische ISSN: 1614-2454
DOI: https://doi.org/10.1007/s10436-020-00366-0

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