doi: 10.3934/dcdss.2022026
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Stochastic local volatility models and the Wei-Norman factorization method

1. 

University of Jaen - Department of Applied Mathematics, Campus Las Lagunillas, Jaen, 23071, Spain

2. 

University of Bari - Department of Economics and Finance, Via C. Rosalba 53, Bari, 70124, Italy

* Corresponding author: Giuseppe Orlando

Received  September 2021 Revised  December 2021 Early access February 2022

Fund Project: The first author is supported by the Spanish MICINN grant PGC2018-097831-B-I00 and Junta de Andalucía grant FEDER/UJA-1381026

In this paper, we show that a time-dependent local stochastic volatility (SLV) model can be reduced to a system of autonomous PDEs that can be solved using the heat kernel, by means of the Wei-Norman factorization method and Lie algebraic techniques. Then, we compare the results of traditional Monte Carlo simulations with the explicit solutions obtained by said techniques. This approach is new in the literature and, in addition to reducing a non-autonomous problem into an autonomous one, allows for shorter time in numerical computations.

Citation: Julio Guerrero, Giuseppe Orlando. Stochastic local volatility models and the Wei-Norman factorization method. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022026
References:
[1]

C. Alexander and L. Nogueira, Stochastic Local Volatility, Technical report, Henley Business School, Reading University, 2008.

[2]

J. ArmstrongM. FordeM. Lorig and H. Zhang, Small-time asymptotics under local-stochastic volatility with a jump-to-default: Curvature and the heat kernel expansion, SIAM J. Financial Math., 8 (2017), 82-113.  doi: 10.1137/140971397.

[3]

P. Balland and Q. Tran, SABR goes normal, Risk, 26 (2013), 72. 

[4]

W. Barger and M. Lorig, Approximate pricing of European and barrier claims in a local-stochastic volatility setting, Int. J. Financ. Eng., 4 (2017), 1750018, 31 pp. doi: 10.1142/S2424786317500189.

[5]

M. Bufalo and G. Orlando, An improved Barone-Adesi Whaley formula for turbulent markets, J. Comput. Appl. Math., 406 (2022), 113993, 16 pp. doi: 10.1016/j.cam.2021.113993.

[6]

P. Carr and D. B. Madan, A note on sufficient conditions for no arbitrage, Finance Research Letters, 2 (2005), 125-130.  doi: 10.1016/j.frl.2005.04.005.

[7]

B. Chen, C. W. Oosterlee and H. van der Weide, A low-bias simulation scheme for the SABR stochastic volatility model, Int. J. Theor. Appl. Finance, 15 (2012), 1250016, 37 pp. doi: 10.1142/S0219024912500161.

[8]

Z. CuiJ. L. Kirkby and D. Nguyen, A general valuation framework for SABR and stochastic local volatility models, SIAM J. Financial Math., 9 (2018), 520-563.  doi: 10.1137/16M1106572.

[9]

M. DaiL. Tang and X. Yue, Calibration of stochastic volatility models: A Tikhonov regularization approach, J. Econom. Dynam. Control, 64 (2016), 66-81.  doi: 10.1016/j.jedc.2016.01.002.

[10]

M. H. A. Davis and D. G. Hobson, The range of traded option prices, Math. Finance, 17 (2007), 1-14.  doi: 10.1111/j.1467-9965.2007.00291.x.

[11]

E. Derman and I. Kani, Riding on a smile, Risk, 7 (1994), 32-39. 

[12]

E. DermanI. Kani and N. Chriss, Implied trinomial tress of the volatility smile, The Journal of Derivatives, 3 (1996), 7-22.  doi: 10.3905/jod.1996.407952.

[13]

D. Diavatopoulos and O. Sokolinskiy, Stochastic volatility models: Faking a smile, in Handbook of Financial Econometrics, Mathematics, Statistics, and Machine Learning, World Scientific, 2020, 1271–1293. doi: 10.1142/9789811202391_0033.

[14]

P. Doust, No-arbitrage SABR, The Journal of Computational Finance, 15 (2012), 3. 

[15]

B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20. 

[16]

A. Friedmann, Partial Differential Equations of Parabolic type, R. C. Krieger PC, Florida, 1983.

[17]

J. Guerrero and M. Berrondo, Semiclassical interpretation of Wei-Norman factorization for $SU(1, 1)$ and its related integral transforms, J. Math. Phys., 61 (2020), 082107, 20 pp. doi: 10.1063/1.5143586.

[18]

P. S. HaganD. KumarA. S. Lesniewski and D. E. Woodward, Managing smile risk, The Best of Wilmott, 1 (2002), 249-296. 

[19]

S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud., 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327.

[20]

J. Hull and A. White, The pricing of options on assets with stochastic volatilities, The Journal of Finance, 42 (1987), 281-300.  doi: 10.1111/j.1540-6261.1987.tb02568.x.

[21]

B. Izgi and A. Bakkaloglu, Fundamental solution of bond pricing in the ho-lee stochastic interest rate model under the invariant criteria, New Trends in Mathematical Sciences, 5 (2017), 196-203.  doi: 10.20852/ntmsci.2017.138.

[22]

P. Jäckel, Stochastic volatility models: Past, present and future, The Best of Wilmott, 1 (2004), 355-377. 

[23]

M. Kamal and J. Gatheral, Implied volatility surface, Encyclopedia of Quantitative Finance. doi: 10.1002/9780470061602.eqf08004.

[24]

Á. LeitaoL. A. Grzelak and C. W. Oosterlee, On a one time-step Monte Carlo simulation approach of the SABR model: Application to European options, Appl. Math. Comput., 293 (2017), 461-479.  doi: 10.1016/j.amc.2016.08.030.

[25]

M. MininniG. Orlando and G. Taglialatela, Challenges in approximating the Black and Scholes call formula with hyperbolic tangents, Decis. Econ. Finance, 44 (2021), 73-100.  doi: 10.1007/s10203-020-00305-8.

[26]

G. Orlando and M. Bufalo, Empirical evidences on the interconnectedness between sampling and asset returns' distributions, Risks, 9 (2021), 88. 

[27]

G. Orlando and G. Taglialatela, A review on implied volatility calculation, J. Comput. Appl. Math., 320 (2017), 202-220.  doi: 10.1016/j.cam.2017.02.002.

[28]

G. Orlando and G. Taglialatela, On the approximation of the Black and Scholes call function, J. Comput. Appl. Math., 384 (2021), Paper No. 113154, 14 pp. doi: 10.1016/j.cam.2020.113154.

[29]

M. Rubinstein, Implied binomial trees, The Journal of Finance, 49 (1994), 771-818.  doi: 10.1111/j.1540-6261.1994.tb00079.x.

[30]

E. M. Stein and J. C. Stein, Stock price distributions with stochastic volatility: An analytic approach, The Review of Financial Studies, 4 (1991), 727-752.  doi: 10.1093/rfs/4.4.727.

[31]

M. C. Thomsett, The Complete Options Trader: A Strategic Reference for Derivatives Profits, Springer, 2018.

[32]

A. W. van der Stoep, L. A. Grzelak and C. W. Oosterlee, The Heston stochastic-local volatility model: Efficient Monte Carlo simulation, Int. J. Theor. Appl. Finance, 17 (2014), 1450045, 30 pp. doi: 10.1142/S0219024914500459.

[33]

J. Wei and E. Norman, Lie algebraic solution of linear differential equations, J. Mathematical Phys., 4 (1963), 575-581.  doi: 10.1063/1.1703993.

[34]

J. Wei and E. Norman, On global representations of the solutions of linear differential equations as a product of exponentials, Proc. Amer. Math. Soc., 15 (1964), 327-334.  doi: 10.1090/S0002-9939-1964-0160009-0.

show all references

References:
[1]

C. Alexander and L. Nogueira, Stochastic Local Volatility, Technical report, Henley Business School, Reading University, 2008.

[2]

J. ArmstrongM. FordeM. Lorig and H. Zhang, Small-time asymptotics under local-stochastic volatility with a jump-to-default: Curvature and the heat kernel expansion, SIAM J. Financial Math., 8 (2017), 82-113.  doi: 10.1137/140971397.

[3]

P. Balland and Q. Tran, SABR goes normal, Risk, 26 (2013), 72. 

[4]

W. Barger and M. Lorig, Approximate pricing of European and barrier claims in a local-stochastic volatility setting, Int. J. Financ. Eng., 4 (2017), 1750018, 31 pp. doi: 10.1142/S2424786317500189.

[5]

M. Bufalo and G. Orlando, An improved Barone-Adesi Whaley formula for turbulent markets, J. Comput. Appl. Math., 406 (2022), 113993, 16 pp. doi: 10.1016/j.cam.2021.113993.

[6]

P. Carr and D. B. Madan, A note on sufficient conditions for no arbitrage, Finance Research Letters, 2 (2005), 125-130.  doi: 10.1016/j.frl.2005.04.005.

[7]

B. Chen, C. W. Oosterlee and H. van der Weide, A low-bias simulation scheme for the SABR stochastic volatility model, Int. J. Theor. Appl. Finance, 15 (2012), 1250016, 37 pp. doi: 10.1142/S0219024912500161.

[8]

Z. CuiJ. L. Kirkby and D. Nguyen, A general valuation framework for SABR and stochastic local volatility models, SIAM J. Financial Math., 9 (2018), 520-563.  doi: 10.1137/16M1106572.

[9]

M. DaiL. Tang and X. Yue, Calibration of stochastic volatility models: A Tikhonov regularization approach, J. Econom. Dynam. Control, 64 (2016), 66-81.  doi: 10.1016/j.jedc.2016.01.002.

[10]

M. H. A. Davis and D. G. Hobson, The range of traded option prices, Math. Finance, 17 (2007), 1-14.  doi: 10.1111/j.1467-9965.2007.00291.x.

[11]

E. Derman and I. Kani, Riding on a smile, Risk, 7 (1994), 32-39. 

[12]

E. DermanI. Kani and N. Chriss, Implied trinomial tress of the volatility smile, The Journal of Derivatives, 3 (1996), 7-22.  doi: 10.3905/jod.1996.407952.

[13]

D. Diavatopoulos and O. Sokolinskiy, Stochastic volatility models: Faking a smile, in Handbook of Financial Econometrics, Mathematics, Statistics, and Machine Learning, World Scientific, 2020, 1271–1293. doi: 10.1142/9789811202391_0033.

[14]

P. Doust, No-arbitrage SABR, The Journal of Computational Finance, 15 (2012), 3. 

[15]

B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20. 

[16]

A. Friedmann, Partial Differential Equations of Parabolic type, R. C. Krieger PC, Florida, 1983.

[17]

J. Guerrero and M. Berrondo, Semiclassical interpretation of Wei-Norman factorization for $SU(1, 1)$ and its related integral transforms, J. Math. Phys., 61 (2020), 082107, 20 pp. doi: 10.1063/1.5143586.

[18]

P. S. HaganD. KumarA. S. Lesniewski and D. E. Woodward, Managing smile risk, The Best of Wilmott, 1 (2002), 249-296. 

[19]

S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud., 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327.

[20]

J. Hull and A. White, The pricing of options on assets with stochastic volatilities, The Journal of Finance, 42 (1987), 281-300.  doi: 10.1111/j.1540-6261.1987.tb02568.x.

[21]

B. Izgi and A. Bakkaloglu, Fundamental solution of bond pricing in the ho-lee stochastic interest rate model under the invariant criteria, New Trends in Mathematical Sciences, 5 (2017), 196-203.  doi: 10.20852/ntmsci.2017.138.

[22]

P. Jäckel, Stochastic volatility models: Past, present and future, The Best of Wilmott, 1 (2004), 355-377. 

[23]

M. Kamal and J. Gatheral, Implied volatility surface, Encyclopedia of Quantitative Finance. doi: 10.1002/9780470061602.eqf08004.

[24]

Á. LeitaoL. A. Grzelak and C. W. Oosterlee, On a one time-step Monte Carlo simulation approach of the SABR model: Application to European options, Appl. Math. Comput., 293 (2017), 461-479.  doi: 10.1016/j.amc.2016.08.030.

[25]

M. MininniG. Orlando and G. Taglialatela, Challenges in approximating the Black and Scholes call formula with hyperbolic tangents, Decis. Econ. Finance, 44 (2021), 73-100.  doi: 10.1007/s10203-020-00305-8.

[26]

G. Orlando and M. Bufalo, Empirical evidences on the interconnectedness between sampling and asset returns' distributions, Risks, 9 (2021), 88. 

[27]

G. Orlando and G. Taglialatela, A review on implied volatility calculation, J. Comput. Appl. Math., 320 (2017), 202-220.  doi: 10.1016/j.cam.2017.02.002.

[28]

G. Orlando and G. Taglialatela, On the approximation of the Black and Scholes call function, J. Comput. Appl. Math., 384 (2021), Paper No. 113154, 14 pp. doi: 10.1016/j.cam.2020.113154.

[29]

M. Rubinstein, Implied binomial trees, The Journal of Finance, 49 (1994), 771-818.  doi: 10.1111/j.1540-6261.1994.tb00079.x.

[30]

E. M. Stein and J. C. Stein, Stock price distributions with stochastic volatility: An analytic approach, The Review of Financial Studies, 4 (1991), 727-752.  doi: 10.1093/rfs/4.4.727.

[31]

M. C. Thomsett, The Complete Options Trader: A Strategic Reference for Derivatives Profits, Springer, 2018.

[32]

A. W. van der Stoep, L. A. Grzelak and C. W. Oosterlee, The Heston stochastic-local volatility model: Efficient Monte Carlo simulation, Int. J. Theor. Appl. Finance, 17 (2014), 1450045, 30 pp. doi: 10.1142/S0219024914500459.

[33]

J. Wei and E. Norman, Lie algebraic solution of linear differential equations, J. Mathematical Phys., 4 (1963), 575-581.  doi: 10.1063/1.1703993.

[34]

J. Wei and E. Norman, On global representations of the solutions of linear differential equations as a product of exponentials, Proc. Amer. Math. Soc., 15 (1964), 327-334.  doi: 10.1090/S0002-9939-1964-0160009-0.

Figure 1.  Setting: $ S_0 = 100 $, $ v_0 = 0.2 $, $ \beta = 1 $, $ \rho = 0.5 $, $ \alpha = 0.2 $. Notice that in absence of dividends and discount factors there is no drift (if we consider the drift coefficients $ k_\omega $ and $ k_\mu $ zero), therefore, the expected value should be equal to the initial value
Figure 2.  (a) Wei-Norman SLV prices. (b) Wei-Norman SLV/SABR implied volatility
Figure 5.  Scenarios: $ \alpha $ low = 0.1, $ \alpha $ high = 0.5, $ \alpha $ base = 0.2
Figure 6.  Scenarios: $ \alpha $ low = 0.1, $ \alpha $ high = 0.5, $ \alpha $ base = 0.2
Figure 7.  Scenarios: $ \rho $ low = -0.8, $ \rho $ high = 0.8, $ \rho $ base = 0.5
Figure 8.  Scenarios: $ \rho $ low = -0.8, $ \rho $ high = 0.8, $ \rho $ base = 0.5
Table 1.  Monte Carlo simulations: SABR vs Wei-Norman, case $ c = 0 $
Monte Carlo CPU time (sec.) $ \rho $
SABR 0.19040 1
Wei-Norman 0.10700
SABR 0.31250 0.5
Wei-Norman 0.18840
SABR 0.15460 0
Wei-Norman 0.10170
SABR 0.14600 -0.5
Wei-Norman 0.08960
SABR 0.16660 -1
Wei-Norman 0.12870
Simulations=10,000 for each model; Setting: T = 1, S0 = 100, v0 = 0.2, β = 1, α = 0.2
Monte Carlo CPU time (sec.) $ \rho $
SABR 0.19040 1
Wei-Norman 0.10700
SABR 0.31250 0.5
Wei-Norman 0.18840
SABR 0.15460 0
Wei-Norman 0.10170
SABR 0.14600 -0.5
Wei-Norman 0.08960
SABR 0.16660 -1
Wei-Norman 0.12870
Simulations=10,000 for each model; Setting: T = 1, S0 = 100, v0 = 0.2, β = 1, α = 0.2
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