Archil Gulisashvili

Archil Gulisashvili, portrait
Professor
Morton 555
Mathematics
gulisash@ohio.edu
740-593-1281

Personal Website

Education

Dr.Sc., Tbilisi State University

Research Interests

Financial Mathematics

General stochastic asset price models; classical stochastic volatility models (Hull-White, Stein-Stein, Heston); Gaussian models and models with reflection; fractional, rough, and super rough models; scaling properties of stochastic volatility models; sample path and small-noise large deviation principles for stochastic volatility models; models with jumps; option pricing theory; asymptotic behavior of stock price distribution densities, option pricing functions, and the implied volatility; moment explosions; geometrical methods in financial mathematics; Heston geometry; large and moderate deviation principles; the Gärtner-Ellis theorem. 

Stochastic Processes 

Volterra Gaussian processes; non-homogeneous Markov processes; time-reversal and duality theory for Markov processes; applications of Markov processes to parabolic initial and final value problems; reflecting diffusions.

Semigroup Theory and Propagator Theory

Schrödinger semigroups and Feynman-Kac propagators; non-autonomous Kato classes of functions and measures; smoothing properties of Schrödinger semigroups.

Selected Publications

Books

Large Deviations and Asymptotic Methods in Finance, Springer Proceedings in Mathematics and Statistics, Vol 110, P. K. Friz, J. Gatheral, A. Gulisashvili, A. Jacquier, J. Teichmann (Eds.), Springer International Publishing Switzerland, 2015.

A Gulisashvili, Analytically Tractable Stochastic Stock Price Models, Springer Finance, 2012.

A. Gulisashvili and J. A. van Casteren, Non-Autonomous Kato Classes and Feynman-Kac Propagators, World Scientific, Singapore, 2006.

Papers

  • Multivariate Stochastic Volatility Models and Large Deviation Principles, submitted for publication, available at arXiv:2203.09015, 2022.
  • Time-inhomogeneous Gaussian stochastic volatility models: Large deviations and super-roughness, Stochastic Processes and their Applications, 139, September 2021, 37-79; available at arXiv:2002.05143, 2020.
  • Large deviation principles for stochastic volatility models with reflection and three faces of the Stein and Stein model, submitted for publication, available at arXiv:2006.15431, 2020.
  • (With S. Gerhold and C. Gerstenecker), Large deviations for fractional volatility models with non-Gaussian volatility driver, Stochastic Processes and their Applications, 142 (2021), 580-600; available at arXiv:2003.12825.
  • Gaussian stochastic volatility models: Scaling regimes, large deviations, and moment explosions, Stochastic Processes and their Applications, 130 (2020), 3648-3686; available at arXiv:1808.00421 and SSRN:https://ssrn.com/ abstract=3367829
  • (With M. Lagunas, R. Merino, and J. Vives), High order approximations to option prices in the Heston model, Journal of Computational Finance, 24 (2020), 1-20; available at arXiv:1905.06315.
  • (With Ch. Bayer, P. K. Friz, B. Horvath, and B. Stemper) Short-time near-the-money skew in rough fractional volatility models, Quantitative Finance, 19 (2019), 779-798; available at arXiv:1703.05132
  • Large deviation principle for Volterra type fractional stochastic volatility models, SIAM Journal on Financial Mathematics, 9 (2018), 1102-1136; available at arXiv:1710.10711
  • (With F. Viens and X. Zhang) Small-time asymptotics for Gaussian self-similar stochastic volatility models, Applied Mathematics and Optimization, 82 (2018), 183-223, available at arXiv:1505.05256
  • (With F. Viens and X. Zhang) Extreme-strike asymptotics for general Gaussian stochastic volatility models, Annals of Finance 15 (2018), 59-101; available at arXiv:1502. 05442v3
  • Distance to the line in the Heston model, Journal of Mathematical Analysis and Applications, 450 (2017), 197-228; available at arXiv:1409.6027.
  • (With P. K. Friz, B. Gess, and S. Riedel), The Jain-Monrad criterion for rough paths and applications to Random Fourier Series and Non-Markovian Hörmander Theory, Annals of Probability, 44 (2016), 684-738.
  • (With P. Tankov) Tail behavior of sums and differences of log-normal random variables, Bernoulli, Volume 22, Number 1 (February 2016), 444-493.
  • (With B. Horvath and A. Jacquier), On the probability of hitting the boundary for Brownian motions on the SABR plane, Electronic Communications in Probability, 21, No 75 (2016), 1-13.
  • (With P. Tankov), Implied volatility of basket options at extreme strikes, in: Large Deviations and Asymptotic Methods in Finance, Springer Proceedings in Mathematics
    and Statistics, Vol 110, P. K. Friz, J. Gatheral, A. Gulisashvili, A. Jacquier, J. Teichmann (Eds.), Springer International Publishing Switzerland, 2015, pp. 175-
    212.
  • (With J. Teichmann), The Gärtner-Ellis theorem, homogenization, and affine processes, in: Large Deviations and Asymptotic Methods in Finance, Springer Proceedings in Mathematics and Statistics, Vol 110, P. K. Friz, J. Gatheral, A. Gulisashvili, A. Jacquier, J. Teichmann (Eds.), Springer International Publishing Switzerland, 2015, pp. 287-320.
  • (With P. Laurence), The Heston Riemannian distance function, Journal de mathématiques pures et appliquées, 101, (2014) 303-329.
  • Asymptotic equivalence in Lee’s moment formulas for the implied volatility, asset price models with moment explosions, and Piterbarg’s conjecture, International Journal of Theoretical and Applied Finance, 15 (2012), 1250020 (34 pages).
  • (With P. K. Friz, S. Gerhold, and S. Sturm), On refined volatility smile expansion in the Heston model, Quantitative Finance, 11 (2011), 1151-1164.
  • (With E. M. Stein) Asymptotic behavior of the distribution densities in models with stochastic volatility, I , Mathematical Finance, 20 (2010), 447-477.
  • (With E. M. Stein) Asymptotic behavior of the stock price distribution density and implied volatility in stochastic volatility models, Applied Mathematics and Optimization, 61 (2010), 287-315.
  • Asymptotic formulas with error estimates for call pricing functions and the implied volatility at extreme strikes, SIAM Journal on Financial Mathematics, 1 (2010), 609-641.
  • (With E. M. Stein) Implied volatility in the Hull-White model, Mathematical Finance, 19 (2009), 303-327.
  • Markov processes, classes of time-dependent measures, and Feynman-Kac propagators, Transactions of the American Mathematical Society, 360 (2008), 4063-4098.
  • Nonautonomous Kato classes of measures and Feynman-Kac propagators, Transactions of the American Mathematical Society, 357 (2005), 4607-4632.
  • On the heat equation with a time-dependent singular potential, Journal of Functional Analysis, 194, No 1 (2002), 17-52.
  • Sharp estimates in smoothing theorems for Schrödinger semigroups, Journal of Functional Analysis, 170 (2000), 161-187.
  • (With M. Kon) Exact smoothing properties of Schrödinger semigroups, American Journal of Mathematics, 118, No 6 (1996), 1215-1248.