Homepage of Dr. Aleš Černý

 

Short intro

The theory of asset pricing and risk measurement in incomplete markets is concerned with the methodology and practical implementation of optimal hedging and pricing of derivative securities in the presence of hedging errors. The standard asset pricing theory assumes that all sources of risk are priced in the market; this assumption is most famously embedded in the Black-Scholes option pricing formula. In reality, even extremely frequent hedging leaves a significant amount of risk. In most models this risk is unaccounted for, as LTCM found to its own detriment. My work proposes standardized measurement of risk across different utility functions, allows for attribution of performance among different assets (for example stocks and options) in a dynamic framework, and provides extremely fast implementation of optimal dynamic hedge ratios and risk measurements using Fourier transform.

Simplified stochastic calculus

April 28th, 2023. Slides from the talk at LMU workshop .

April 12th, 2023. The final paper of the simplified calculus trilogy has just been accepted for publication in Stochastic Processes and Their Applications. This work illustrates how powerful and parsimonious the calculus can be, whether in integral transforms or in measure changes. The preprint is available here. More applications to come soon …

June 16th, 2022. A talk in memory of Peter Carr at the 11th Bachelier Congress. G-variations from Carr and Lee (2013, Finasto) are linked to the simplified stochastic calculus .

January 12th, 2022. The second paper of the series has appeared in the Electronic Journal of Probability. The paper provides a theoretical underpinning of the ‘calculus of predictable variations,’ which substantially simplifies and streamlines manipulation of general stochastic processes. Check out the very helpful tables in the introduction (thanks to the referees and AE) to get a feel for the new calculus. The preprint is available here.

April 7th, 2021. The first paper of the series has appeared in the European Journal of Operational Research . The referees made us think how to derive the Riccati equations in affine models à la Duffie, Pan, and Singleton (2000). The outcome of that exercise is available here.

January 12th, 2021. Pure-jump processes. A piece of research that originated in the development of the simplified stochastic calculus has just been accepted for publication in Bernoulli. The paper gives a better notion of a pure-jump process (i.e., sigma-locally finite variation pure-jump process), one that is closed under stochastic integration, composition, and smooth transformations.
[read more]

June 22nd, 2020. Just completed a suite of four papers dealing with the practical applications and the theoretical underpinning of simplified stochastic calculus. If you have never heard of this topic, you may wish to start here. There is more to come on the application front. Mathematicians will find something of interest here, here, and here. Joint work with Johannes Ruf (LSE Mathematics).

Mean–variance portfolio allocation without a risk-free asset

April 7th, 2023. This paper has now been accepted in Mathematics of Operations Research.

October 19th, 2021. A piece of research that has taken more than a decade to complete is now available on SSRN and arXiv. Have you ever wondered why the mean–variance theory looks more complicated when there is no risk-free asset? Somewhat surprisingly, it is about three times more demanding to compute the efficient frontier in the absence of a risk-free asset than in its presence.
[read more]

The law of one price in quadratic hedging and mean–variance portfolio selection

October 20th, 2024. The paper will appear in Finance & Stochastics in the second half of 2025.

November 2nd, 2022. The latest piece of research, available on SSRN and arXiv, disentangles the difference between no arbitrage and the lesser requirement of the law of one price (LOP). The latter broadly asserts that that identical financial flows should command the same price. We uncover a new mechanism through which LOP can fail in a continuous-time L2(P) setting without frictions, namely trading from just before a predictable stopping time,
[read more]

The Hansen ratio in mean–variance portfolio theory

August 17th, 2020. The Sharpe ratio is well known in the mean-variance portfolio theory. Most people do not know that the Hansen ratio is even better suited to the description of the efficient frontier. Simple concepts can be surprisingly difficult to spot; see here.

Latest research on the fundamentals of expected utility maximization

July 3rd, 2020. The January 2020 issue of Mathematical Finance features the paper Convex duality and Orlicz spaces in expected utility maximization co-authored with Sara Biagini (LUISS, Rome). One reviewer called it a quest. It certainly stands as the longest, most open-ended and nail- biting piece of research I have been involved in. I will try to give a flavour of the intricacies involved.
[read more]

Selected publications (full list here...)

[23]

with Christoph Czichowsky and Jan Kallsen, Numeraire-invariant quadratic hedging and mean–variance portfolio allocation, Mathematics of Operations Research, 49(2), 752–781, 2024

[22]

with Johannes Ruf, Simplified calculus for semimartingales: Multiplicative compensators and changes of measure, Stochastic Processes and Their Applications, 161, 572–602, 2023

[21]

with Johannes Ruf, Simplified stochastic calculus via semimartingale representations, Electronic Journal of Probability, 27, article no.3, pp. 1–32, 2022

[20]

with Johannes Ruf, Pure-jump semimartingales, Bernoulli, 27(4), pp. 2624–2648, 2021

[19]

with Johannes Ruf, Simplified stochastic calculus with applications in Economics and Finance, European Journal of Operational Research, 293(2), pp. 547–560, 2021

[18]

Semimartingale theory of monotone mean–variance portfolio allocation, Mathematical Finance, 30(3), pp. 1168–1178, 2020

[17]

with Igor Melicherčík, Simple explicit formula for near-optimal stochastic lifestyling, European Journal of Operational Research 284(2), 769–778, 2020

[16]

with Sara Biagini, Convex duality and Orlicz spaces in expected utility maximization, Mathematical Finance 30(1), 85–127, 2020

[15]

with Pavol Brunovský and Ján Komadel, Optimal trade execution under endogenous pressure to liquidate: Theory and numerical solutions, European Journal of Operational Research 264(3), 1159–1171, 2018

[14]

with Fabio Maccheroni, Massimo Marinacci and Aldo Rustichini, On the computation of optimal monotone mean–variance portfolios via truncated quadratic utility, Journal of Mathematical Economics 48(6), 386–395, 2012

[13]

with Chris Brooks and Joelle Miffre, Optimal hedging with higher moments, Journal of Futures Markets 32(10), 909–944, 2012

[12]

with Ioannis Kyriakou, An improved convolution algorithm for discretely sampled Asian options, Quantitative Finance 11(3), 381–389, 2011

[11]

with Sara Biagini, Admissible strategies in semimartingale portfolio optimization, SIAM Journal on Control and Optimization, 49(1), 42–72, 2011

[10] Mathematical Techniques in Finance: Tools for Incomplete Markets, Princeton University Press, 2nd edition, July 2009, pp. 416
  • hands-on introduction to asset pricing, optimal portfolio selection and evaluation of investment performance
  • simple EXCEL spreadsheets and MATLAB codes integrated in the text
  • large number of examples and solved exercises
  • more advanced topics include
    • fast Fourier transform
    • finite difference methods
    • multinomial lattices and Levy processes
[9]

with Jan Kallsen, Hedging by sequential regressions revisited, Mathematical Finance 19(4), 591–617, 2009

[8]

with Jan Kallsen, Mean–variance hedging and optimal investment in Heston's model with correlation, Mathematical Finance 18(3), 473–492, 2008

[7]

with Jan Kallsen, A counterexample concerning the variance-optimal martingale measure, Mathematical Finance 18(2), 305–316, 2008

[6]

with Jan Kallsen, On the structure of general mean–-variance hedging strategies, The Annals of Probability 35(4), 1479–1531, 2007

[5]

Optimal continuous-time hedging with leptokurtic returns, Mathematical Finance 17(2), 175–203, 2007.

[4]

with David K. Miles, Risk, return, and portfolio allocation under alternative pension systems with incomplete and imperfect financial markets, The Economic Journal 116(2), 529–557, 2006.

[3] Introduction to fast Fourier transform in finance, Journal of Derivatives 12(1), 73–88, 2004
[2] Generalized Sharpe ratios and asset pricing in incomplete markets, Review of Finance 7(2), 191–233, 2003. Presented at AFA Annual Meeting 2001, New Orleans.
[1]

with Stewart D. Hodges, The theory of good-deal pricing in financial markets, in Geman, Madan, Pliska, Vorst (eds.): Mathematical Finance — Bachelier Congress 2000, 175–202, Springer Verlag 2002.

Research Projects

  • with Prof. David Miles, 2000–2004, Economics of Social Security in Japan, £200,000+
  • with Prof. James Sefton, 2002–2004, Design of Behavioural Tax Model, £80,000

Selected refereed conferences and *invited talks

[24]
12/09
2024
*Developments in Pension Design and Investment Modelling,  Piraeus
Near–optimal stochastic lifestyling
[23]
11/09
2023
*Advances in Stochastic Analysis for Risk in Finance and Insurance, Luminy
Numeraire-invariant quadratic hedging and mean–variance portfolio allocation
[22]
16/06
2022
11th Bachelier Congress, Hong Kong
Simplified stochastic calculus: A talk in memory of Peter Carr
 
[21]
12/05
2022
*Talks in Financial and Insurance Mathematics, ETH Zurich
Simplified stochastic calculus via semimartingale representations
[20]
17/07
2018
10th Bachelier Congress, Dublin
Convex duality and Orlicz spaces in expected utility maximization
 
[19]
08/06
2017

*Convex Stochastic Optimization Workshop, Kings College London
Convex duality and Orlicz spaces in expected utility maximization

[18]
03/11
2016

*London Mathematical Finance Seminar, UCL
Optimal trade execution under endogenous pressure to liquidate

[17]
28/08
2015

*George Boole Mathematical Sciences Conference, Cork
Quadratic hedging with and without numeraire change

[16]
25/09
2014

*London-Paris Bachelier Workshop, Paris
Good-deal prices for log contract

[15]
04/06
2014

8th Bachelier Congress, Brussels
Asymptotics of quadratic hedging in Lévy models

[14]
26/08
2013
6th Summer School of Mathematical Finance, Vienna
Computation of optimal monotone mean–-variance portfolios
[13]
05/09
2012
*Finance and Actuarial Science Talks, ETH Zurich
Optimal hedging with higher moments
[12]
12/07
2010

AnStaP10, Conference in Honour of W. Schachermayer, Vienna
Admissible strategies for semimartingale portfolio optimization

[11]
24/06
2010

6th Bachelier Congress, Toronto
Admissible strategies for semimartingale portfolio optimization

[10]
18/07
2008

5th Bachelier Congress, London
Mean–variance hedging and optimal investment in Heston's model

[9]
24/08
2007

EFA 2007 Annual Meeting, Ljubljana
Optimal Hedging with Higher Moments

[8]
25/05
2007

*Stanford Unversity
Mean-Variance Hedging and Optimal Investment in Heston's Model

[7]
29/09
2005

*Courant Institute for Mathematical Sciences, NYU
On the structure of general mean–variance hedging strategies

[6]
28/09
2005

*Columbia University, New York
On the structure of general mean–variance hedging strategies

[5]
14/09
2005

*Summer School Bologna, Frontiers of Financial Mathematics, Bologna,
One-day workshop on the theory and applications of good-deal pricing

[4]
19/04
2005

*Developments in Quantitative Finance, Isaac Newton Institute, Cambridge
On the structure of general mean–variance hedging strategies

[3]
24/09
2004

ESF Exploratory Workshop, London Business School
The risk of optimal, continuously rebalanced hedging strategies

[2]
23/05
2002

*Workshop on Incomplete Markets, Carnegie Mellon University, Pittsburgh
Derivatives without differentiation

[1]
05/01
2001

AFA 2001 Annual Meeting, New Orleans
Generalized Sharpe ratios

Refereeing Activity

Applied Mathematical Finance, Annals of Operations Research, Automatica, Bernoulli, Economic Journal, European Financial Management, European Journal of Finance, European Journal of Operational Research, Finance and Stochastics, IEEE Transactions on Automatic Control, International Journal of Computer Mathematics, International Journal of Theoretical and Applied Finance, Journal of Computational and Applied Mathematics, Journal of Computational Finance, Journal of Finance, Journal of Financial Econometrics, Journal of Futures Markets, Mathematical Finance, Mathematics and Financial Economics,  Mathematical Reviews, Mathematics of Operations Research, Operations Research, Princeton University Press, Quantitative Finance, Review of Derivatives Research, Risk, SIAM Journal on Financial Mathematics, Statistics and Decisions

Editorial Appointments

06/2007- Review of Derivatives Research

PhD Supervision

[9]
09/2014
08/2018

Ján Komadel, Comenius University Bratislava
Optimization in financial mathematics

[8]
10/2011
10/2017

Juraj Špilda, Cass Business School
On Sources of Risk in Quadratic Hedging and Incomplete Markets

[7]
10/2013
09/2017

Xuecan Cui, Luxembourg School of Finance
Asset Pricing Models with underlying Lévy Processes

[6]
10/2008
10/2013

Nicolaos Karouzakis, Cass Business School
Three Essays on the Dynamic Evolution of Market Interest Rates…

[5]
10/2007
06/2012

Ka Kei Chan, Cass Business School
Theoretical essays on bank risk taking and financial stability

[4]
10/2006
11/2010

Ioannis Kyriakou, Cass Business School
Efficient valuation of exotic derivatives with path-dependence and early-exercise feature

[3]
10/2002
10/2006

Lubomír Schmidt, Imperial College Business School
Optimal life-cycle consumption and asset allocation with applications to pension finance and public economics

[2]
10/2001
09/2006

Mariam Harfush-Pardo, Imperial College Business School
An investigation on portfolio choice and wealth accumulation in fully funded pension systems with a guaranteed minimum benefit

[1]
10/2001
10/2004

Yung-Chih Wang, Imperial College Business School
Topics in investment appraisal and real options

Media Coverage

[1]
01/03
2011

Hospodárske Noviny
Slovák, ktorý vie vypočítať riziká pri obchodovaní na burze

Other

  • Erdős number: 4 (AČ → J. Ruf → V. Prokaj → L. Gerencsér → PE)
  • Kolmogorov number: 3 (AČ → J. Kallsen → A.N. Shiryaev → ANK)

Last revised June/23/2020