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ASP Course Project

Repository

List of Projects

References

  • Ballotta L, Kyriakou I (2014) Monte Carlo Simulation of the CGMY Process and Option Pricing. Journal of Futures Markets 34:1095–1121. https://doi.org/10.1002/fut.21647
  • El Euch O, Rosenbaum M (2019) The characteristic function of rough Heston models. Mathematical Finance 29:3–38. https://doi.org/10.1111/mafi.12173
  • Henry-Labordère P (2005) A general asymptotic implied volatility for stochastic volatility models. arXiv:cond-mat/0504317
  • 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
  • Medvedev A, Scaillet O (2007) Approximation and Calibration of Short-Term Implied Volatilities Under Jump-Diffusion Stochastic Volatility. The Review of Financial Studies 20:427–459. https://doi.org/10.1093/rfs/hhl013
  • Tubikanec I, Tamborrino M, Lansky P, Buckwar E (2021) Qualitative properties of numerical methods for the inhomogeneous geometric Brownian motion. arXiv:200310193 [cs, math]
  • Zhao B (2009) Inhomogeneous Geometric Brownian Motion. SSRN Electronic Journal. https://doi.org/10.2139/ssrn.1429449

Presentation and Deadline

  • Presentation: 4.19 (Friday) 10 minutes by each group.
  • Pull request deadline: 4.27 (Sat)

File requirements

  • PyFENG repository:
    • Core implementation (.py): python class and functions
    • Make sure to add docstring in detail.
      • Example: bsm.py
      • Specify equation/formula number or page in the original paper
      • The best examples of docstring are from numpy documentation: example
    • 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' projects:

Project Topics

  • 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
    • Simulation methods for the SV models (SABR, Heston, Rough volatility, etc)
    • Pricing of derivatives (Timer option, Variance Swap, Spread/Basket/Asian options)

* Rough Volatility

* Equivalent BS volatility of SV models [New]

  • Several studies on the short-time approximation of IV under general SV model.
    • Medvedev A, Scaillet O (2007) Approximation and Calibration of Short-Term Implied Volatilities Under Jump-Diffusion Stochastic Volatility. The Review of Financial Studies 20:427–459. https://doi.org/10.1093/rfs/hhl013
    • Lorig M, Pagliarani S, Pascucci A (2017) Explicit implied volatilities for multifactor local-stochastic volatility models. Mathematical Finance 27:926–960. https://doi.org/10.1111/mafi.12105
    • [2023 Theiss] Implemented for GARCH model. Medvedev A, Scaillet O (2007) and Lorig M, Pagliarani S, Pascucci A (2017)
    • Armstrong John, Forde Martin, Lorig Matthew, Zhang Hongzhong (2017) Small-Time Asymptotics under Local-Stochastic Volatility with a Jump-to-Default: Curvature and the Heat Kernel Expansion. SIAM J Finan Math 8:82–113. https://doi.org/10.1137/140971397
    • [Implemented in PyfengForPapers. Uncorrelated case only.] Ball CA, Roma A (1994) Stochastic Volatility Option Pricing. Journal of Financial and Quantitative Analysis 29:589–607.
  • Can we apply them to other SV models? 3/2 SV model?
  • Can they be used for the simulation of asset price?

* European Option pricing with Fourier Transform

* GARCH-diffusion model

  • [Implemented in PyFENG] Euler/Milstein/Log scheme (in class)
  • [Implemented in PyfengForPapers. Uncorrelated case only] 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
  • [????] 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].

* SABR model

  • [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)

* OUSV Model

  • [Implemented in PyFENG] Euler/Milstein, Prof Choi's exact simulation scheme based on KL expansion.
  • [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 by Chang Xinlei] Almost exact simulation based on the two moments obtained from the numerical derivatives

* 3/2 SV model

  • [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

* Other Simulation-related Papers

  • [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

Exotic Derivatives

  • Spread/Basket/Asian Option
  • 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
    • [Heston] 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)
    • [MC, Perpetual] 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)
  • Snowball
    • Recently very popular in China. Link

Old Topics