Financial mathematics and quantitative finance

Main groups of results and related publications

1. Exended Koponen family of Lévy processes (a.k.a. CGMY and KoBoL models)

In 1995, Koponen constructed a family of Lévy processes with exponentially decaying symmetric jump densities. In

S.Z. Levendorskiĭ, S.I. Boyarchenko, Option Pricing for Truncated Lévy processes, International Journal of Theoretical and Applied Finance, 3:3 (2000), 549-552 pdf file
S.Z. Levendorskiĭ, S.I. Boyarchenko, Non-Gaussian Merton-Black-Scholes theory, World Scientific, Singapore, 2002, xxi+398 pp.

this family was generalized, which was necessary for applications in Finance.

2. Fourier transform technique with applications to pricing European contingent claims was introduced in

S.Z. Levendorskiĭ, S.I. Boyarchenko, On Rational Pricing of Derivative Securities for a Family of Non-Gaussian Processes, Preprint 98/7, Institut für Mathematik, Uni Potsdam, 1998. PostScript File (zipped)
S.I. Boyarchenko, S.Z. Levendorskiĭ, Option Pricing for Truncated Lévy processes, International Journal of Theoretical and Applied Finance, 3:3 (2000), 549-552 pdf file

3. Difficulties of the straightforaward application of Fast Fourier Transform (FFT) and inverse Fast Fourier Transform (iFFT) was explained, and certain remedies were suggested in

Chapter 12 in S.I. Boyarchenko, S.Z. Levendorskiĭ, Non-Gaussian Merton-Black-Scholes Theory, World Scientific, Singapore 2002
N.Boyarchenko, S. Levendorskiĭ, On errors and bias of Fourier transform methods in quadratic term structure models, International Journal of Theoretical and Applied Finance 10, No. 2 (2007) 273-306 http://ssrn.com/abstract=874835

4. Generalized Black-Scholes equation for prices of contingent claims in Lévy models was derived in

S.I. Boyarchenko, S.Z. Levendorskiĭ, Option Pricing for Truncated Lévy processes, International Journal of Theoretical and Applied Finance, 3:3 (2000), 549-552 pdf file
Barrier Options and Touch-and-out Options under regular Lévy processes of Exponential Type, Annals of Applied Probability, 12:4 (2002), 1261--1298. pdf file
S.I. Boyarchenko, S.Z. Levendorskiĭ, Non-Gaussian Merton-Black-Scholes Theory, World Scientific, Singapore 2002

The equation was derived in the sense of generalized functions, which was sufficient for derivation of

5. Pricing formulas for barrier options in wide classes Lévy models and study of the asymptotics of the price near the boundary. It was shown that in many cases of interest, the derivative of the price of a barrier option w.r.t. underlying is not bounded in a neighborhood of the barrier, and asymptotic pricing formulas were derived in op. cit. and in

O.E.Kudryavzev, S.Z. Levendorskiĭ, Pricing of first touch digitals under Normal Inverse Gaussian processes, International Journal of Theoretical and Applied Finance , Vol. 9, No. 6 (2006) 915-949 http://ssrn.com/abstract=520045

6. Efficient pricing of barrier options and Refined and Enhanced FFT-iFFT techniques . The methods listed above are not computationally efficient. Efficient methods are developed in

O.E.Kudryavzev, S.Z. Levendorskiĭ, Fast and accurate pricing of barrier options under Lévy processes, to appear in Special Issue of Finance and Stochastics Dec 2007 http://ssrn.com/abstract=1040061
Mitya Boyarchenko and Sergei Levendorskiĭ, Refined and Enhanced Fast Fourier Transform Techniques, with an Application to the Pricing of Barrier Options, May 10, 2008, http://ssrn.com/abstract=1142833
Mitya Boyarchenko and Sergei Levendorskiĭ, Prices and Sensitivities of Barrier and First-Touch Digital Options in Levy-Driven Models, July 3, 2008, http://ssrn.com/abstract=1155149
Mitya Boyarchenko and Sergei Levendorskiĭ, Valuation of Continuously Monitored Double Barrier Options and Related Securities, August 14, http://ssrn.com/abstract=1227065

In particular, Ref 4 contains an efficient procedure for double barrier options, and Ref 2-4 are based on new efficient variations of FFT-iFFT techniques

7. Efficient pricing procedures for American options in Lévy models with finite time horizon were constructed in

Sergei Levendorskiĭ, Pricing of the American put under Lévy processes, International Journal of Theoretical and Applied Finance 7:3 (2004), 303-336 pdf file
Sergei Levendorskiĭ,O.Kudryavtsev and V.Zherder, The relative efficiency of numerical methods for pricing American options under Lévy processes, Journal of Computational Finance, 9, Winter 2005/06, 69-98 http://ssrn.com/abstract=610542

Ref 1 uses an extremely efficient reduction to solution to ODEs of first order with exponential free term, which allows one to reduce the pricing problem to a sequence of algebraic problems.

8. Generalization of Kou's model, later rediscovered under the name Hyper Exponential Jump Diffusion model (HEJD), was constructed in

Sergei Levendorskiĭ, Pricing of the American put under Lévy processes, International Journal of Theoretical and Applied Finance 7:3 (2004), 303-336 pdf file

9. Irregularities of prices and early exercise boundaries in models with jumps

The irregularity of prices of barrier options was proved and demonstrated in

S.I. Boyarchenko, S.Z. Levendorskiĭ, Non-Gaussian Merton-Black-Scholes Theory, World Scientific, Singapore 2002
S.Z. Levendorskiĭ,Early exercise boundary and option pricing in Lévy driven models, Quantitative Finance, 4 (2004), 525-547 http://www.eco.utexas.edu/~leven/qfinrev2a.pdf
O.E.Kudryavzev, S.Z. Levendorskiĭ, Pricing of first touch digitals under Normal Inverse Gaussian processes, International Journal of Theoretical and Applied Finance , Vol. 9, No. 6 (2006) 915-949 http://ssrn.com/abstract=520045

The failure of the smooth pasting principle was proved in

S.I. Boyarchenko, S.Z. Levendorskiĭ,Perpetual American options under Lévy processes, SIAM Journal on Control and Optimization, 40:6 (2002), 1663--1696 pdf file
S.I. Boyarchenko, S.Z. Levendorskiĭ, Non-Gaussian Merton-Black-Scholes Theory, World Scientific, Singapore 2002

The gap between the early exercise boundary and strike, at expiry, for American options on non-dividend stock in Lévy models and more general Markov processes with jumps

S.I. Boyarchenko, S.Z. Levendorskiĭ, Non-Gaussian Merton-Black-Scholes Theory, World Scientific, Singapore 2002
Sergei Levendorskiĭ, Pricing of the American put under Lévy processes, International Journal of Theoretical and Applied Finance 7:3 (2004), 303-336 pdf file
S.Z. Levendorskiĭ, The American Put and European Options Near Expiry, Under Lévy Processes (February 26, 2004). http://ssrn.com/abstract=520062
S.Z. Levendorskiĭ, Early exercise boundary and option pricing in Lévy driven models, Quantitative Finance, 4 (2004), 525-547 http://www.eco.utexas.edu/~leven/qfinrev2a.pdf
S.Levendorskiĭ, American and European Options in Multi-Factor Jump-Diffusion Models, Near Expiry, Finance and Stochastics, 12:4 (2008), pp.~541-560

10. Pricing of American options in regime-switching models. A general efficient iteration method for pricing American options in regime-switching Lévy models based on Carr's randomization and the Wiener-Hopf method (modifications for barrier options and discrete time models are straightforward) is developed in

S.I. Boyarchenko, S.Z. Levendorskiĭ,American Options in Regime-Switching Models (September 6, 2006) http://ssrn.com/abstract=929215
S.I. Boyarchenko, S.Z. Levendorskiĭ,Carr's randomization for American options in regime-switching models, ICIAM07-Proceedings, Volume 7, Issue 1, Date: December 2007, Pages: 1081303-1081304
S.I. Boyarchenko, S.Z. Levendorskiĭ,Pricing American Options in Regime-Switching Models: FFT Realization, to appear in SIAM J Control and Optimization, April, 2008, http://ssrn.com/abstract=1127562

This method is used to approximate

11. American options in models with stochastic volatility and/or stochastic interest rates. Several models have been studied; extensions in several directions, in particular, to barrier options and options in discrete time models are straightforward

S.I. Boyarchenko, S.Z. Levendorskiĭ, American options in Lévy models with stochastic interest rates, to appear in Computational Finance Available at SSRN: http://ssrn.com/abstract=1015409
S.I. Boyarchenko, S.Z. Levendorskiĭ, American options in regime-switching models with non-semibounded stochastic interest rates, Proceedings of 2008 American Control Conference (2008), pp.~1023-1028 http://ssrn.com/abstract=1015410
S.I. Boyarchenko, S.Z. Levendorskiĭ, American options in the Heston model with stochastic interest rates, November 2007, http://ssrn.com/abstract=1031282
S.I. Boyarchenko, S.Z. Levendorskiĭ, American options in Lévy models with stochastic volatility, November 2007, http://ssrn.com/abstract=1031280
S.I. Boyarchenko, S.Z. Levendorskiĭ, American options in Lévy models with stochastic interest rates of CIR type, November 2007, http://ssrn.com/abstract=1032716

12. Asymptotic pricing

Ole E. Barndorff-Nielsen and Sergei Levendorskiĭ, Feller Processes of Normal Inverse Gaussian Type, Quantitative Finance, 1 (2001), 318-331
Sergei Levendorskiĭ, Pseudo-diffusions and Quadratic Term Structure Models; Mathematical Finance (2005), 15, 393-424. http://ssrn.com/abstract=520044
Nina Boyarchenko and Sergei Levendorskiĭ, Asymptotic Pricing in Term Structure Models Driven by Jump-Diffusions of Ornstein-Uhlenbeck Type (March 14, 2006) http://ssrn.com/abstract=890725