- Timer Option: Implementation of time-discretization (Andersen's QE) scheme for Heston/SABR models [Report]
- Zhou Gongqi / Wu Zhuoqun
- Li M, Mercurio F (2015) Analytic Approximation of Finite‐Maturity Timer Option Prices. Journal of Futures Markets 35:245–273. https://doi.org/10.1002/fut.21659
- Li C (2016) Bessel Processes, Stochastic Volatility, and Timer Options. Mathematical Finance 26:122–148. https://doi.org/10.1111/mafi.12041
- SABR Model Low-bias Simulation Scheme [Report]
- Lu Tianzeng / Hu Zhijie
- Chen B, Oosterlee CW, Van Der Weide H (2012) A low-bias simulation scheme for the SABR stochastic volatility model. Int J Theor Appl Finan 15:1250016.
- Stochastic volatility inspired (SVI) model [Report]
- Liu Can
- Original Slides
- Gatheral J, Jacquier A (2013) Arbitrage-free SVI volatility surfaces. arXiv:12040646 [q-fin]
- Gatheral J, Jacquier A (2011) Convergence of Heston to SVI. Quantitative Finance 11:1129–1132. https://doi.org/10.1080/14697688.2010.550931
- Snowball Pricing with MC method
- Team 1: Wu Hao / Yang Shuming. [Report]
- Team 2: Chen Wanqing / Chen Ziying. [Report]
- Presentation: 10.27 (Thurs) 20 minutes by each group.
- Pull request deadline: 11.01 (Tues)
PyFENG
repository:- Core implementation (.py): python class and functions
- Make sure to add docstring in detail.
- Integrate into the
PyFENG
repository by making pull-requests (pr). - Do not place any
.ipynb
file.
- Your repository (
PHBS_ASP_2021
or any name is OK)- Test (.ipynb): one Jupyter notebook file briefly describing the method (base theory, equations, SDE, strength/weakness, etc).
- Include some test examples (e.g., the same parameter sets from the paper)
- See past years' proejct:
- Among the topics and HWs covered in the class, choose an in-depth research on one topic. You are also welcome to do the project on your own original idea. Otherwise, pick one from my suggestions which are basically understanding and implementing literatures. Topics includes
- Spread/Basket/Asian option pricing
- SV models (SABR, Heston, etc)
- Rough volatility
- Copula
- Ma J, Wu H (2021) A fast algorithm for simulation of rough volatility models. Quantitative Finance 0:1–16. https://doi.org/10.1080/14697688.2021.1970213
- Jaber EA (2019) Lifting the Heston model. Quantitative Finance 19:1995–2013. https://doi.org/10.1080/14697688.2019.1615113
- Horvath B, Jacquier A, Muguruza A (2019) Functional central limit theorems for rough volatility. arXiv:171103078 [math, q-fin] http://arxiv.org/abs/1711.03078
- McCrickerd R, Pakkanen MS (2018) Turbocharging Monte Carlo pricing for the rough Bergomi model. Quantitative Finance 18:1877–1886. https://doi.org/10.1080/14697688.2018.1459812
- Bennedsen, M., Lunde, A., Pakkanen, M.S., 2017. Hybrid scheme for Brownian semistationary processes. Finance Stoch 21, 931–965. https://doi.org/10.1007/s00780-017-0335-5
- J. Gatheral's python code
- [Partially implemented] Choi J (2018) Sum of all Black-Scholes-Merton models: An efficient pricing method for spread, basket, and Asian options. Journal of Futures Markets 38:627–644. https://doi.org/10.1002/fut.21909
- R implementation: https://github.com/PyFE/SumBSM-R
- [Most methods implemented] Krekel M, de Kock J, Korn R, Man T-K (2004) An analysis of pricing methods for basket options. Wilmott Magazine 2004:82–89
- Lord R, Kahl C (2010) Complex Logarithms in Heston-Like Models. Mathematical Finance 20:671–694. https://doi.org/10.1111/j.1467-9965.2010.00416.x
- Kahl C, Jäckel P (2005) Not-so-complex Logarithms in the Heston Model. Wilmott 19
- [Implemented but not perfect] OUSV: Schöbel R, Zhu J (1999) Stochastic Volatility With an Ornstein–Uhlenbeck Process: An Extension. Rev Financ 3:23–46. https://doi.org/10.1023/A:1009803506170
- [Implemented] Barone-Adesi G, Rasmussen H, Ravanelli C (2005) An option pricing formula for the GARCH diffusion model. Computational Statistics & Data Analysis 49:287–310. https://doi.org/10.1016/j.csda.2004.05.014
- Tubikanec I, Tamborrino M, Lansky P, Buckwar E (2021) Qualitative properties of numerical methods for the inhomogeneous geometric Brownian motion. arXiv:200310193 [cs, math] http://arxiv.org/abs/2003.10193
- [Implemented] Euler/Milstein/Log scheme (in class)
- [????] Capriotti L, Jiang Y, Shaimerdenova G (2018) Approximation methods for inhomogeneous geometric brownian motion. Int J Theor Appl Finan 22:1850055. https://doi.org/10.1142/S0219024918500553
- IGBM: Zhao B (2009) Inhomogeneous Geometric Brownian Motion. SSRN Electronic Journal. https://doi.org/10.2139/ssrn.1429449
- [Consider] PhD Thesis of Ravanelli, C., University of Lugano, Switzerland, https://doc.rero.ch/record/4229/files/1_2003ECO001.pdf
- [No method] Papadopoulos, Y.A., Lewis, A.L., 2018. A First Option Calibration of the GARCH Diffusion Model by a PDE Method. arXiv:1801.06141 [q-fin].
- [Implemented but not verified] Cai N, Song Y, Chen N (2017) Exact Simulation of the SABR Model. Oper Res 65:931–951. https://doi.org/10.1287/opre.2017.1617
- [Implemented but not verified] Leitao Á, Grzelak LA, Oosterlee CW (2017) On a one time-step Monte Carlo simulation approach of the SABR model: Application to European options. Applied Mathematics and Computation 293:461–479. https://doi.org/10.1016/j.amc.2016.08.030
* Heston Model (SV)
- [No need] Broadie M, Kaya Ö (2006) Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes. Operations Research 54:217–231. https://doi.org/10.1287/opre.1050.0247
- [Implemented] Glasserman P, Kim K-K (2011) Gamma expansion of the Heston stochastic volatility model. Finance Stoch 15:267–296. https://doi.org/10.1007/s00780-009-0115-y
- [Implemented] Andersen, L., 2008. Simple and efficient simulation of the Heston stochastic volatility model. The Journal of Computational Finance 11, 1–42. https://doi.org/10.21314/JCF.2008.189
- [Implemented / 2022 Thesis] Almost exact simulation
- [Implemented] Euler/Milstein
- [Implemented] Li C, Wu L (2019) Exact simulation of the Ornstein–Uhlenbeck driven stochastic volatility model. European Journal of Operational Research 275:768–779. https://doi.org/10.1016/j.ejor.2018.11.057
- [2021 Thesis] Almost exact simulation
- [Implemented but not verified] Baldeaux, J., 2012. Exact simulation of the 3/2 model. Int. J. Theor. Appl. Finan. 15, 1250032. https://doi.org/10.1142/S021902491250032X
- [Implemented but not verified] Almost exact simulation
- [Implemented but not verified] General SDE: Beskos, A., Roberts, G.O., 2005. Exact simulation of diffusions. Ann. Appl. Probab. 15, 2422–2444. https://doi.org/10.1214/105051605000000485
- [Implemented] Computing Moments from Laplace Transform: Choudhury, G.L., Lucantoni, D.M., 1996. Numerical Computation of the Moments of a Probability Distribution from its Transform. Operations Research 44, 368–381. https://doi.org/10.1287/opre.44.2.368
- Timer Option
- Implementation of time-discretization (Andersen's QE) scheme for Heston/SABR models
- Li M, Mercurio F (2015) Analytic Approximation of Finite‐Maturity Timer Option Prices. Journal of Futures Markets 35:245–273. https://doi.org/10.1002/fut.21659
- Li C (2016) Bessel Processes, Stochastic Volatility, and Timer Options. Mathematical Finance 26:122–148. https://doi.org/10.1111/mafi.12041
- (Other related papers)
- Bernard C, Cui Z (2011) Pricing timer options. JCF 15:69–104. https://doi.org/10.21314/JCF.2011.228
- Zeng P, Kwok YK, Zheng W (2015) Fast hilbert transform algorithms for pricing discrete timer options under stochastic volatility models. Int J Theor Appl Finan 18:1550046. https://doi.org/10.1142/S0219024915500466
- Zheng W, Zeng P (2016) Pricing timer options and variance derivatives with closed-form partial transform under the 3/2 model. Applied Mathematical Finance 23:344–373. https://doi.org/10.1080/1350486X.2017.1285242
- SABR Model Simulation
- Chen B, Oosterlee CW, Van Der Weide H (2012) A low-bias simulation scheme for the SABR stochastic volatility model. Int J Theor Appl Finan 15:1250016. https://doi.org/10.1142/S0219024912500161
- [Implemented but not verified] Leitao Á, Grzelak LA, Oosterlee CW (2017) On a one time-step Monte Carlo simulation approach of the SABR model: Application to European options. Applied Mathematics and Computation 293:461–479. https://doi.org/10.1016/j.amc.2016.08.030
- Stochastic volatility inspired (SVI) model (1-person project)
- Original Slides
- Gatheral J, Jacquier A (2013) Arbitrage-free SVI volatility surfaces. arXiv:12040646 [q-fin]
- Gatheral J, Jacquier A (2011) Convergence of Heston to SVI. Quantitative Finance 11:1129–1132. https://doi.org/10.1080/14697688.2010.550931
- Snowball
- Recently very popular in China. Link
- Almost Exact Simulation of the 3/2 Volatility Model [Report]
- @AHrmnd and @FLXE-Feidou
- Simulation of the GARCH Diffusion model [Report]
- Lifted Heston Model [Report]
- @jyyccc and @Cecilia525
- Jaber EA (2019) Lifting the Heston model. Quantitative Finance 19:1995–2013. https://doi.org/10.1080/14697688.2019.1615113
- Approximate Stochastic Volatility Model Option Pricing [Report]
- @zwc00098
- Ball CA, Roma A (1994) Stochastic Volatility Option Pricing. Journal of Financial and Quantitative Analysis 29:589–607. https://doi.org/10.2307/2331111
- Simulation of Rough Volatility Model [Report]
- @polarbluebear and @Baiwisher
- Ma J, Wu H (2021) A fast algorithm for simulation of rough volatility models. Quantitative Finance 0:1–16. https://doi.org/10.1080/14697688.2021.1970213