Books |
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| [2]
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| Černý, A. (2009), Mathematical
Techniques in Finance: Tools for Incomplete Markets, 2nd ed., Princeton University
Press
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| [1] |
| Černý, A. (2004), Mathematical
Techniques in Finance: Tools for Incomplete Markets, Princeton University
Press
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Peer reviewed publications |
| [31] |
A. Černý and C. Czichowsky (2025),
The law of one price in quadratic hedging and mean–variance portfolio selection. To appear in Finance & StochasticsAbstract: The law of one price (LOP) broadly asserts that identical financial flows should command the same price. We show that, when properly formulated, LOP is the minimal condition for a well-defined mean–variance portfolio selection framework without degeneracy. Crucially, the paper identifies a new mechanism through which LOP can fail in a continuous-time L2 setting without frictions, namely trading from just before a predictable stopping time, which surprisingly identifies LOP violations even for continuous price processes.
Closing this loophole allows to give a version of the Fundamental Theorem of Asset Pricing appropriate in the quadratic context, establishing the equivalence of the economic concept of LOP with the probabilistic property of the existence of a local E-martingale state price density. The latter provides unique prices for all square-integrable claims in an extended market and subsequently plays an important role in quadratic hedging and mean-variance portfolio selection.
Mathematically, we formulate a novel variant of the uniform boundedness principle for conditionally linear functionals on the L0 module of conditionally square-integrable random variables. We then study the representation of time-consistent families of such functionals in terms of stochastic exponentials of a fixed local martingale.
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| [30] |
M. Halická, M. Trnovská, and A. Černý (2025), On
indication, strict monotonicity, and efficiency of projections in a
general class of path-based data envelopment models. European Journal of Operational Research, 320(1),
175–187
Abstract:
Data envelopment analysis (DEA) theory formulates a number of desirable properties that DEA models should satisfy. Among these,
indication, strict monotonicity, and strong efficiency of projections tend to be grouped together in the sense that, in individual
models, typically, either all three are satisfied or all three fail at the same time. Specifically, in slacks-based graph models, the
three properties are always met; in path-based models, such as radial models, directional distance function models, and the
hyperbolic function model, the three properties, with some minor exceptions, typically all fail.
Motivated by this observation, the article examines relationships among indication, strict monotonicity, and strong efficiency
of projections in the class of path-based models over variable returns-to-scale technology sets. Under mild assumptions, it is shown
that the property of strict monotonicity and strong efficiency of projections are equivalent, and that both properties imply
indication. This paper also characterises a narrow class of technology sets and path directions for which the three properties hold
in path-based models.
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| [29] |
A. Černý, C. Czichowsky, and J.Kallsen (2024),
Numeraire-invariant quadratic hedging and mean–variance portfolio
allocation. Mathematics of Operations Research,
49(2), 752–781
Abstract: The paper investigates quadratic hedging in a general semimartingale market
that does not necessarily contain a risk-free asset. An equivalence result for hedging with and without numeraire change is established. This
permits direct computation of the optimal strategy without choosing a reference asset and/or performing a numeraire change. New explicit
expressions for optimal strategies are obtained, featuring the use of oblique projections that provide unified treatment of the case with and
without a risk-free asset. The main result advances our understanding of the efficient frontier formation in the most general case where a
risk-free asset may not be present. Several illustrations of the numeraire-invariant approach are given.
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| [28] |
M. Halická, M. Trnovská, and A. Černý (2024),
A unified approach to radial, hyperbolic, and directional efficiency
measurement in Data Envelopment Analysis. European Journal of Operational Research, 312(1), 298–314.
Abstract:
The paper analyzes properties of a large class of "path-based" Data Envelopment Analysis models through a unifying general
scheme. The scheme includes the well-known oriented radial models, the hyperbolic distance function model, the directional distance
function models, and even permits their generalisations. The modelling is not constrained to non-negative data and is flexible enough
to accommodate variants of standard models over arbitrary data.
Mathematical tools developed in the paper allow systematic analysis of the models from the point of view of ten desirable
properties. It is shown that some of the properties are satisfied (resp., fail) for all models in the general scheme, while others
have a more nuanced behaviour and must be assessed individually in each model. Our results can help researchers and practitioners
navigate among the different models and apply the models to mixed data.
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| [27] |
A. Černý and J. Ruf (2023),
Simplified calculus for semimartingales: Multiplicative compensators and
changes of measure. Stochastic Processes and Their Applications,
161, 572–602
Abstract: The paper develops multiplicative compensation for complex-valued
semimartingales and studies some of its consequences. It is shown that the stochastic exponential of any complex-valued semimartingale with
independent increments becomes a true martingale after multiplicative compensation when such compensation is meaningful. This generalization of
the Lévy–Khintchin formula fills an existing gap in the literature. It allows, for example, the computation of the Mellin transform of a
signed stochastic exponential, which in turn has practical applications in mean--variance portfolio theory. Girsanov-type results based on
multiplicatively compensated semimartingales simplify treatment of absolutely continuous measure changes. As an example, we obtain the
characteristic function of log returns for a popular class of minimax measures in a Lévy setting.
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| [26] |
A. Černý and J. Ruf (2022), Simplified
stochastic calculus via semimartingale representations,
Electronic Journal of Probability,
27, article no. 3, 1–33
Abstract: We develop a stochastic calculus that makes it easy to capture a variety of predictable transformations of semimartingales such as changes of variables, stochastic integrals, and their compositions. The framework offers a unified treatment of real-valued and complex-valued semimartingales. The proposed calculus is a blueprint for the derivation of new relationships among stochastic processes with specific examples provided in the paper.
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| [25] |
A. Černý and J. Ruf (2021), Pure-jump semimartingales, Bernoulli, 27(4), 2624–2648
Abstract: A new integral with respect to an integer-valued random measure is introduced. In contrast to the finite variation integral ubiquitous in semimartingale theory (Jacod and Shiryaev 2003, II.1.5), the new integral is closed under stochastic integration, composition, and smooth transformations. The new integral gives rise to a previously unstudied class of pure-jump processes — the sigma-locally finite variation pure-jump processes. As an application, it is shown that every semimartingale X has a unique decomposition
X = X0 + Xqc + Xdp,
where Xqc is quasi-left-continuous and Xdp is a sigma-locally finite variation pure-jump process that jumps only at predictable times, both starting at zero. The decomposition mirrors the classical result for local martingales (Yoeurp, 1976, Theoreme 1.4) and gives a rigorous meaning to the notions of continuous-time and discrete-time components of a semimartingale. Against this backdrop, the paper investigates a wider class of processes that are equal to the sum of their jumps in the semimartingale topology and constructs a taxonomic hierarchy of pure-jump semimartingales.
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| [24] |
A. Černý and J. Ruf
(2021), Simplified stochastic calculus with applications in Economics
and Finance, European Journal of Operational Research, 293(2), 547–560
[Supplementary material: streamlined calculation of Riccati equations in affine models]
Abstract: The paper introduces a simple way of recording and manipulating general
stochastic processes without explicit reference to a probability measure. In the new calculus, operations traditionally presented in a
measure-specific way are instead captured by tracing the behaviour of jumps (also when no jumps are physically present). The calculus is fail-
safe in that, under minimal assumptions, all informal calculations yield mathematically well-defined stochastic processes. The calculus is also
intuitive as it allows the user to pretend all jumps are of compound Poisson type. The new calculus is very effective when it comes to
computing drifts and expected values that possibly involve a change of measure. Such drift calculations yield, for example, partial
integro-differential equations, Hamilton–Jacobi–Bellman equations, Feynman–Kac formulae, or exponential moments needed in
numerous applications. We provide several illustrations of the new technique, among them a novel result on the Margrabe option to exchange one
defaultable asset for another.
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| [23] |
A. Černý (2020), Semimartingale theory of monotone mean–variance portfolio allocation, Mathematical Finance, 30(3), 1168–1178
Abstract: We study dynamic optimal portfolio allocation for monotone mean–variance preferences in a general semimartingale model. Armed with new results in this area we revisit the work of Cui, Li, Wang, and Zhu (2012, MAFI) and fully characterize the circumstances under which one can set aside a non-negative cash flow while simultaneously improving the mean–variance efficiency of the left-over wealth. The paper analyzes, for the first time, the monotone hull of the Sharpe ratio and highlights its relevance to the problem at hand.
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| [22] |
A. Černý and I. Melicherčík (2020), Simple explicit formula for near-optimal stochastic lifestyling. Supplementary table for Section 3.4 [html]. European Journal of Operational Research 284(2), 769–778
Abstract: In lifecycle economics the Samuelson paradigm (Samuelson, 1969) states that optimal investment is in constant proportions out of lifetime wealth (including current savings and present value of future income known as human capital). It is well known that in the presence of credit constraints this paradigm no longer applies. Instead, the optimal investment gives rise to so-called stochastic lifestyling (Cairns, Blake, and Dowd, 2006), whereby for low levels of accumulated capital it is optimal to invest fully in stocks and then gradually switch to safer assets as the level of savings increases. In stochastic lifestyling not only does the ratio between risky and safe assets change but also the mix of risky assets varies over time. While the existing literature relies on complex numerical algorithms to quantify optimal lifestyling the present paper provides a simple formula that captures the main essence of the lifestyling effect with remarkable accuracy.
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| [21] |
S. Biagini and A. Černý (2020), Convex duality and Orlicz spaces in expected utility maximization, Mathematical Finance 30(1), 85–127
Abstract: In this paper, we report further progress toward a complete theory of
state-independent expected utility maximization with semimartingale price processes for arbitrary utility function. Without any technical
assumptions, we establish a surprising Fenchel duality result on conjugate Orlicz spaces, offering a new economic insight into the nature of
primal optima and providing a fresh perspective on the classical papers of Kramkov and Schachermayer. The analysis points to an intriguing
interplay between no-arbitrage conditions and standard convex optimization and motivates the study of the fundamental theorem of asset pricing
for Orlicz tame strategies.
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| [20] |
P. Brunovský, A. Černý and J. Komadel
(2018), Optimal trade execution under endogenous pressure to liquidate: Theory and numerical solutions, European Journal of Operational Research 264(3), 1159–1171
Abstract: We study optimal liquidation of a
trading position (so-called block order or meta-order) in a market with a linear temporary price
impact (Kyle, 1985). We endogenize the pressure to liquidate by introducing a downward drift in the
unaffected asset price while simultaneously ruling out short sales. In this setting the liquidation
time horizon becomes a stopping time determined endogenously, as part of the optimal strategy. We find
that the optimal liquidation strategy is consistent with the square-root law which states that the
average price impact per share is proportional to the square root of the size of the meta-order
(Bershova and Rakhlin, 2013; Farmer et al., 2013; Donier et al., 2015; Tóth et al., 2016).
Mathematically, the Hamilton-Jacobi-Bellman equation of our
optimization leads to a severely singular and numerically unstable ordinary differential equation
initial value problem. We provide careful analysis of related singular mixed boundary value problems
and devise a numerically stable computation strategy by re-introducing time dimension into an
otherwise time-homogeneous task.
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| [19] |
A. Tsanakas, M.V. Wüthrich and
A. Černý (2013) Market value margin via mean–variance hedging, ASTIN Bulletin 43(3), 301–322
Abstract:
We use mean-variance hedging in discrete time, in order to value a terminal insurance liability.
The prediction of the liability is decomposed into claims development results, that is,
yearly deteriorations in its conditional expected value. We assume the existence of a tradeable
derivative with binary pay-off, written on the claims development result and available in each
period. In simple scenarios, the resulting valuation formulas become very similar to regulatory
cost-of-capital-based formulas. However, adoption of the mean-variance framework improves upon the
regulatory approach, by allowing for potential calibration to observed market prices, inclusion of other
tradeable assets, and consistent extension to multiple periods. Furthermore, it is shown that the
hedging strategy can also lead to increased capital efficiency and consistency of market valuation with
Euler-type capital allocations.
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| [18] |
P. Brunovský, A. Černý and M. Winkler
(2013), A singular differential equation stemming from an optimization problem in financial economics, Applied Mathematics and Optimization
68(2), 255–274
Abstract: We consider the ordinary differential equation
x2u'' = axu' + bu - c(u' - 1)2, 0 < x < x0,
with c > 0 and the singular initial condition u(0) = 0, which in financial economics describes optimal disposal of an asset in a market with
liquidity effects. It is shown in the paper that if a + b < 0 then no solutions exist, whereas if a + b > 0 then there are infinitely many
solutions with indistinguishable asymptotics near 0. Moreover, it is proved that in the latter case there is
precisely one solution u corresponding to the choice x0 = 8, which is such that 0 <
u(x) < x for all x > 0, and that this solution is strictly
increasing and concave.
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| [17] |
A. Černý, F. Maccheroni, M. Marinacci and A.
Rustichini (2012), On the computation of optimal monotone mean–variance portfolios via truncated quadratic utility, Journal of Mathematical Economics 48(6), 386–395
Abstract: We report a surprising link between optimal portfolios generated by
a special type of variational preferences called divergence preferences (cf. Maccheroni et al. 2006) and optimal portfolios generated
by classical expected utility. As a special case we connect optimization of truncated quadratic utility
(cf. Černý, 2003) to the optimal monotone mean-variance portfolios (cf. Maccheroni et al., 2007), thus
simplifying the computation of the latter.
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| [16] |
Brooks, C., A. Černý and J. Miffre (2012),
Optimal hedging with higher moments, Journal of Futures Markets 32(10), 909–944
Abstract:
This study proposes a utility-based framework for the determination of optimal hedge ratios
that can allow for the impact of higher moments on hedging decisions. We examine the entire
hyperbolic absolute risk aversion
(HARA) family of utilities which include quadratic, logarithmic, power and exponential utility
functions. We find that for both moderate and large spot (commodity) exposures, the performance of
out-of-sample hedges constructed allowing for non-zero higher moments is better than the performance
of the simpler OLS hedge ratio. The picture is, however, not uniform throughout our seven spot
commodities as there is one instance (cotton) for which the modeling of higher moments decreases
welfare out-of-sample relative to the simpler OLS. We support our empirical findings by a theoretical
analysis of optimal hedging decisions and we uncover a novel link between optimal hedge ratios and the
minimax hedge ratio, that is the ratio which minimizes the largest loss of the hedged position.
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| [15] |
Černý, A. and I. Kyriakou (2011), An improved convolution algorithm for discretely sampled Asian options, Quantitative Finance 11(3), 381–389
Abstract:
We suggest an improved FFT pricing algorithm for discretely sampled Asian options with
general independently distributed returns in the underlying. Our work complements the studies
of Carverhill and Clewlow (1992), Benhamou (2000), and Fusai and Meucci (2008), and, if we restrict
our attention only to lognormally distributed returns, also Večeř (2002). While the existing
convolution algorithms compute the density of the underlying state variable by moving forward
on a suitably defined state space grid our new algorithm uses backward price convolution, which
resembles classical lattice pricing algorithms. For the first time in the literature we provide an
analytical upper bound for the pricing error caused by the truncation of the state space grid
and by the curtailment of the integration range. We highlight the benefits of the new scheme and
benchmark its performance against existing finite difference, Monte Carlo, and forward density
convolution algorithms.
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| [14] |
Biagini, S. and A. Černý (2011),
Admissible strategies in semimartingale portfolio selection, SIAM Journal on Control and Optimization 49(1), 42–72
Abstract:
The choice of admissible trading strategies in mathematical modelling of financial markets is a
delicate issue, going back to Harrison and Kreps (1979). In the context of optimal portfolio selection
with expected utility preferences this question has been the focus of considerable attention over
the last twenty years.
We propose a novel notion of admissibility that has many pleasant features - admissibility is
characterized purely under the objective measure P; each admissible strategy can be approximated by
simple strategies using finite number of trading dates; the wealth of any admissible strategy is a
supermartingale under all pricing measures; local boundedness of the price process is not required;
neither strict monotonicity, strict concavity nor differentiability of the utility function
are necessary; the definition encompasses both the classical mean-variance preferences and the monotone
expected utility.
For utility functions finite on the whole real line, our class represents a minimal set containing
simple strategies which also contains the optimizer, under conditions that are milder than the
celebrated reasonable asymptotic elasticity condition on the utility function.
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| [13] |
Černý, A., D. K. Miles and Ľ. Schmidt (2010), The impact of changing demographics and pensions on the demand for housing and financial assets, Journal of Pension Economics and Finance 9(3), 393–420
Abstract:
The main aim of this paper is to to analyse the impact of shifting demographics and changes in pension arrangements in a model which includes housing both as an investment asset and a consumption good. We consider the impact on welfare, and on macroeconomic aggregates, of some specific pension reforms. Using a calibrated OLG model with several sources of uncertainty we find that the impact of ageing and of reform of social security upon the demand for housing and the level of owner occupation is substantial. We find that pension reform has a very significant impact on the demand for, and price of, housing. The interaction between pension reform and housing is a neglected subject and one which the results we present suggest is important.
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| [12] |
A. Černý (2009), Characterization of the oblique projector U(VU)+V with application to constrained least squares, Linear Algebra and Its Applications, 431(9), 1564–1570
Abstract:
We provide a full characterization of the oblique projector U(VU)+V in the general case
where the range of U and the null space of V are not complementary subspaces. We discuss the new result
in the context of constrained least squares minimization.
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| [11] |
Černý, A. and J. Kallsen (2009), Hedging by sequential regressions revisited, Mathematical Finance 19(4), 591–617
Abstract:
Almost 20 years ago Föllmer and Schweizer (1989) suggested a simple and influential scheme for
the computation of hedging strategies in an incomplete market. Their approach of local risk minimization
results in a sequence of one-period least squares regressions running recursively backwards in time. In
the meantime there have been significant developments in the global risk minimization theory for
semimartingale price processes. In this paper we revisit hedging by sequential regression in the context
of global risk minimization, in the light of recent results obtained by Černý and Kallsen
(2007). A number of illustrative numerical examples is given.
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[doi]
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| [10] |
Černý, A. and J. Kallsen (2008),
Mean–variance hedging and optimal investment in Heston's model with correlation, Mathematical Finance 18(3), 473–492
Abstract:This paper solves the mean–variance hedging problem in Heston’s model with a
stochastic opportunity set moving systematically with the volatility of stock returns. We allow for
correlation between stock returns and their volatility (so-called leverage effect). Our contribution is
threefold: using a new concept of opportunity-neutral measure we present a simplified strategy for
computing a candidate solution in the correlated case. We then go on to show that this candidate
generates the true variance-optimal martingale measure; this step seems to be partially missing
in the literature. Finally, we derive formulas for the hedging strategy and the hedging
error.
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| [9] |
Černý, A. and J. Kallsen (2008), A counterexample concerning the variance-optimal martingale measure, Mathematical Finance 18(2),
305–316
Abstract:
The present note addresses an open question concerning a sufficient characterization of the
variance-optimal martingale measure. Denote by S the discounted price process of an asset and suppose
that Q* is an equivalent martingale measure whose density is a multiple of 1
−φ•ST for some S-integrable process φ. We show that Q*
does not necessarily coincide with the variance-optimal martingale measure, not even if φ•S
is a uniformly integrable Q*-martingale.
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| [8] |
Černý, A. and J. Kallsen (2007), On the structure of general mean–variance hedging strategies, The Annals of Probability 35(4), 1479–1531
Abstract: We provide a new characterization of mean-variance hedging strategies
in a general semimartingale market. The key point is the introduction of a
new probability measure P* which turns the dynamic asset allocation problem
into a myopic one. The minimal martingale measure relative to P* coincides
with the variance-optimal martingale measure relative to the original
probability measure P. |
[doi]
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| [7] |
Černý, A. (2007) Optimal continuous-time hedging with leptokurtic returns, Mathematical Finance, 17(2),
175–203
Abstract: We examine the behavior of optimal mean–variance hedging strategies at high rebalancing
frequencies in a model where stock prices follow a discretely sampled exponential
Lévy process and one hedges a European call option to maturity. Using elementary
methods we show that all the attributes of a discretely rebalanced optimal hedge, i.e.,
the mean value, the hedge ratio, and the expected squared hedging error, converge
pointwise in the state space as the rebalancing interval goes to zero. The limiting formulae
represent 1-D and 2-D generalized Fourier transforms, which can be evaluated
much faster than backward recursion schemes, with the same degree of accuracy. In the
special case of a compound Poisson process we demonstrate that the convergence results
hold true if instead of using an infinitely divisible distribution from the outset one
models log returns by multinomial approximations thereof. This result represents an important extension of Cox, Ross, and Rubinstein to markets with leptokurtic returns.
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| [6] |
Miles, D. K. and A. Černý (2006),
Risk, return and portfolio allocation under alternative pension systems
with incomplete and imperfect financial markets, Economic Journal, 116(2),
529–557
Abstract: This article uses stochastic simulations on a calibrated model to assess the impact of different
pension reform strategies where financial markets are less than perfect. We investigate the optimal
split between funded and unfunded systems when there are sources of uninsurable risk that are
allocated in different ways by different types of pension system when there are imperfections in
financial markets. This article calculates the expected welfare of agents of different cohorts under
various policy scenarios. We estimate how the optimal level of unfunded, state pensions depends on
rate of return and income risks and also upon preferences.
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| [5] |
Černý, A. (2004), Introduction to fast Fourier transform in finance, Journal of Derivatives, 12(1), 73–88
Abstract: The Fourier transform is an important tool in financial
economics. It delivers real-time pricing while allowing for a realistic structure of asset returns, taking into account excess
kurtosis and stochastic volatility. Fourier transform is also rather abstract and thus intimidating to many practitioners. This
article explains the working of the fast Fourier transform in the familiar binomial option pricingmodel. In fact, a good
understanding of FFT requires no more than some high school mathematics and familiarity with roulette, or a bicycle wheel, or a
similar circular object divided into equally sized segments. The returns to such a small intellectual investment are overwhelming.
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| [4] |
Černý, A. (2004), Dynamic programming
and mean–variance hedging in discrete time, Applied Mathematical Finance 11(1), 1–25
Abstract: In this paper the general discrete time mean-variance hedging problem is solved by dynamic programming. Thanks
to its simple recursive structure the solution is well suited to computer implementation. On the theoretical side, it is shown how the variance-optimal measure arises in the dynamic programming solution and how one can define conditional expectations under this (generally non-equivalent) measure. The result is then related to the results of previous studies in continuous time.
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| [3] |
Černý, A. (2003), Generalized
Sharpe ratios and asset pricing in incomplete markets, Review of Finance, 7(2), 191–233
Abstract: The paper presents an incomplete market pricing methodology generating asset
price bounds conditional on the absence of attractive investment opportunities in
equilibrium. The paper extends and generalises the seminal article of Cochrane and
Saá-Requejo who pioneered option pricing based on the absence of arbitrage and
high Sharpe Ratios. Our contribution is threefold:
We base the equilibrium restrictions on an arbitrary utility function, obtaining
the Cochrane and Saá-Requejo analysis as a special case with truncated quadratic
utility. We extend the definition of Sharpe Ratio from quadratic utility to the entire
family of CRRA utility functions and restate the equilibrium restrictions in terms
of Generalised Sharpe Ratios which, unlike the standard Sharpe Ratio, provide a
consistent ranking of investment opportunities even when asset returns are highly
non-normal. Last but not least, we demonstrate that for Itô processes the Cochrane
and Saá-Requejo price bounds are invariant to the choice of the utility function, and
that in the limit they tend to a unique price determined by the minimal martingale
measure.
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| [2] |
Černý, A. (1999), Currency crises:
Introduction of spot speculators, International Journal of Finance and
Economics, 4(1), 1999, 75–89
Abstract: The present paper studies a fixed exchange rate regime subjected to a speculative attack by spot speculators. In light of recent developments in the ERM it has become apparent that the original concept of speculative attack by Krugman (1979) does not suffice because it only allows for one time shift in portfolio and therefore excludes spot speculators who wish to sell back their holdings of foreign currency on a later date, thus restoring their original position in domestic currency. Unlike previous literature, my model indicates that the collapse of a fixed exchange rate can be accompanied with a discrete depreciation of the domestic currency, a phenomenon commonly observed in real currency crises, but absent from the previous models.
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| [1] |
Černý, A. and N. Schmitt (1995)
Antidumping constraints and trade, Swiss Journal of Economics and Statistics,
131 (3), 441–452 http://www.sgvs.ch/papers/1995-III-10.pdf
Abstract: We analyze the Bertrand–Nash equilibrium in a two-firm-two-country model of product differentiation. We show that, when both countries impose antidumping constraints, Nash equilibria exist where both firms continue to trade, none of them trades, or only one firm trades. In each case, we identify the ranges of parameters for which each of these equilibria holds. We show that these equilibria critically depend on the initial tariff rate (or transport cost) and the degree of substitution between products.
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Book chapters, conference
proceedings
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| [4] |
Černý, A. (2016), Discrete-time quadratic hedging of barrier options in exponential Lévy model, in J. Kallsen and A. Papapantoleon (eds.), Advanced Modeling in Mathematical Finance, 257-275, Springer, ISBN 978-3-319-45873-1.
Abstract: We examine optimal quadratic hedging of barrier options in a discretely
sampled exponential Lévy model that has been realistically calibrated to reflect the leptokurtic nature
of equity returns. Our main finding is that the impact of hedging errors on prices is several times
higher than the impact of other pricing biases studied in the literature.
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| [3] |
Černý, A. (2010), Fourier transform, in Cont
R. (ed.), Encyclopedia of Quantitative Finance, 782-786, Wiley: Chichester, ISBN
978-0-470-05756-8.
Abstract: This
article is a concise introduction to applications of Fourier transform and fast Fourier transform (FFT)
in option pricing. The first section defines the discrete Fourier transform (DFT) and states its most
important properties. The second section explains how the binomial asset pricing model can be
implemented by means of a circular convolution and how circular convolution can in turn be computed
using three DFTs. This algorithm is generalized to the multinomial model. The third section discusses
fast implementation of DFT and in particular it analyzes how the length of the input vector can
(adversely) affect the speed of computation. We also discuss continuous Fourier transform and option
pricing in continuous-time affine models. We show how continuous-time pricing formulae can be
efficiently implemented for multiple strike values using FFT. Additional references and suggestions for
further reading are provided.
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| [2] |
Miles, D. K. and A. Černý (2004), Alternative pension reform strategies for Japan, Toshiaki Tachibanaki (ed.), The Economics of Social Security in
Japan, ESRI Studies on Ageing, 75-135, Edward Elgar, ISBN 978-1-843-76682-7.
Abstract: This
report summarises the research we have undertaken into the
implications of various pension reform strategies in Japan. Reform is essential because ageing will
generate extreme pressures on the public, unfunded pension system. We consider the macroeconomic, or
aggregate, and the distributional implications of reforms that, to varying degrees, would increase
reliance upon funded pensions. We also estimate the welfare implications of reforms by calculating the
expected gains and losses to households of various generations. We take as a point of reference a
scenario where unfunded pensions provide an income to the retired worth a high proportion of salaries
at the end of their working life; we take that proportion to be 50% of gross (or around 70% of net)
salaries.
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[pdf]
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| [1] |
Černý, A. and S. D. Hodges (2002), The theory
of good-deal pricing in financial markets, in Geman H., Madan D., Pliska S., Vorst T.(eds.):
Mathematical Finance – Bachelier Congress 2000, 175-202,Springer, ISBN: 978-3-540-67781-9.
Abstract: The term "no-good-deal pricing" in this paper encompasses
pricing techniques based on the absence of attractive investment opportunities — good deals — in equilibrium. We borrowed the term from Cochrane and Saá-Requejo (2000) who pioneered the calculation of price bounds conditional on the absence of high Sharpe ratios. Alternative methodologies for calculating tighter-than-no-arbitrage price bounds have been suggested by Bernardo and Ledoit (2000), Černý (1999), and Hodges (1998). The theory presented here shows that any of these techniques can be seen as a generalization of no-arbitrage pricing.
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[doi]
[pdf]
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Working papers |
| [40] |
M. Halická, M. Trnovská,
and A. Černý (2023, November),
On indication, strict monotonicity, and efficiency of projections in a general class of path-based data envelopment models,
https://arxiv.org/abs/2311.16382.
Appeared in European Journal of Operational Research.
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[pdf]
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| [39] |
A. Černý and C. Czichowsky (2022, October), The law of one price
in quadratic hedging and mean–variance portfolio selection, https://arxiv.org/abs/2210.15613. To appear in
Finance & Stochastics.
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[pdf]
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| [38] |
M. Halická, M. Trnovská,
and A. Černý (2022, October),
A unified approach to radial, hyperbolic, and directional efficiency measurement in Data Envelopment Analysis,
https://arxiv.org/abs/2210.03687.
Appeared in European Journal of Operational Research.
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[pdf]
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| [37] |
A. Černý, C. Czichowsky, and J.Kallsen (2021, October), Numeraire-invariant quadratic hedging and
mean–variance portfolio allocation,
https://arxiv.org/abs/2110.09416.
Appeared in Mathematics of Operations Research.
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[pdf]
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| [36] |
A. Černý (2020, July), The Hansen ratio in mean–variance portfolio theory, https://arxiv.org/abs/2007.15980.
Abstract: It is shown that the ratio between the mean and the L2–norm leads to a particularly parsimonious description of the mean–variance efficient frontier and the dual pricing kernel restrictions known as the Hansen–Jagannathan bounds. Because this ratio has not appeared in economic theory previously, it seems appropriate to name it the Hansen ratio. The initial treatment of the mean–variance theory via the Hansen ratio is extended in two directions, to monotone mean–variance preferences and to arbitrary Hilbert space setting. A multiperiod example with IID returns is also discussed. |
[pdf]
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| [35] |
A. Černý and J. Ruf (2020, June), Simplified calculus for semimartingales: Multiplicative compensators and changes of measure,
https://arxiv.org/abs/2006.12765.
Appeared in Stochastic Processes and Their Applications.
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[pdf]
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| [34] |
A. Černý and J. Ruf (2020, June), Simplified stochastic calculus via semimartingale representations,
https://arxiv.org/abs/2006.11914. Appeared in Electronic Journal of Probability.
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[pdf] |
| [33] |
A. Černý and J. Ruf (2019, December), Simplified stochastic calculus with applications in Economics and Finance, https://arxiv.org/abs/1912.03651. An earlier version was circulated under the title "Finance without Brownian motions: An introduction to simplified stochastic calculus."
Appeared in European Journal of Operational Research.
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[pdf]
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| [32] |
A. Černý and J. Ruf (2019, September), Pure-jump semimartingales, https://arxiv.org/abs/1909.03020.
Appeared in Bernoulli.
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[pdf]
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| [31] |
A. Černý (2019, January), Semimartingale theory of monotone mean–variance portfolio allocation, https://arxiv.org/abs/1903.06912. Appeared in Mathematical Finance.
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[pdf]
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| [30] |
A. Černý and I. Melicherčík (2018, January), Simple explicit formula for near-optimal stochastic lifestyling, https://arxiv.org/abs/1801.00980. Supplementary table for Section 3.4 [html].
Appeared in European Journal of Operational Research.
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[pdf]
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| [29] |
S. Biagini and A. Černý (2017, November), Convex duality and Orlicz spaces in expected utility maximization, https://arxiv.org/abs/1711.09121. Appeared in Mathematical Finance. |
[pdf]
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| [28] |
P. Brunovský, A. Černý and J. Komadel
(2017, April), Optimal trade execution under endogenous pressure to liquidate: Theory and numerical solutions,
http://ssrn.com/abstract=2946755. Appeared in European Journal of Operational Research. |
[pdf]
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| [27] |
Černý, A. (2016), Discrete-time quadratic hedging of barrier options in exponential Lévy model, https://ssrn.com/abstract=2746572.
Appeared in J. Kallsen and A. Papapantoleon (eds.), Advanced Modeling in Mathematical Finance, 257-275,
Springer, ISBN 978-3-319-45873-1.
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| [26] |
A. Černý, S. Denkl, and J. Kallsen (2013,
September), Hedging in Lévy models and time step equivalent of jumps,
http://arxiv.org/abs/1309.7833.
Abstract:We
consider option hedging in a model where the underlying follows an exponential Lévy process. We
derive approximations to the variance-optimal and to some suboptimal strategies as well as to their mean
squared hedging errors. The results are obtained by considering the Lévy model as a perturbation
of the Black-Scholes model. The approximations depend on the first four moments of logarithmic stock
returns in the Lévy model and option price sensitivities (greeks) in the limiting Black-Scholes
model. We illustrate numerically that our formulas work well for a variety of Lévy models
suggested in the literature. From a theoretical point of view, it turns out that jumps have a similar
effect on hedging errors as discrete-time hedging in the Black-Scholes model.
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[pdf]
[+]
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| [25] |
A. Černý and I. Melicherčík (2013,
September), A simple formula for optimal management of individual pension accounts. A substantially revised version appeared in European Journal of Operational Research under the title "Simple explicit formula for near-optimal stochastic lifestyling". 10.1016/j.ejor.2019.12.032
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[pdf]
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| [24] |
P. Brunovský, A. Černý, and M. Winkler (2012, September), A singular differential equation stemming from an optimal control problem in financial economics, http://arxiv.org/abs/1209.5027. Appeared in Applied Mathematics and Optimization.
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| [23] |
A. Tsanakas, M.V. Wuethrich, and A. Černý (2012, September), Market value margin via mean–variance hedging, http://ssrn.com/abstract=2148911. Appeared in ASTIN Bulletin.
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| [22] |
A. Černý and J. Špilda (2012, April), A note on 'Discrete time hedging errors for options with irregular
payoffs', SSRN Working Paper, http://ssrn.com/abstract=2042519.
Abstract: This note provides correction to the main
results in the article Discrete time hedging errors for options with irregular payoffs (Finance and Stochastics, 5, 357-367, 2001).
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[pdf]
[+]
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| [21] |
S. Biagini and A. Černý (2009, October), Admissible strategies in semimartingale portfolio selection,
http://ssrn.com/abstract=1491707. Appeared in SIAM Journal on Control and Optimization.
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[pdf]
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| [20] |
A. Černý and I. Kyriakou (2009, January), An improved convolution algorithm for discretely sampled Asian options,
http://ssrn.com/abstract=1098367. Appeared in Quantitative Finance. |
[pdf]
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| [19] |
A. Černý, F. Maccheroni, M. Marinacci and A. Rustichini (2008, October), On the computation of optimal monotone mean–variance portfolios via truncated quadratic utility, http://ssrn.com/abstract=1278623. Appeared in Journal of Mathematical Economics.
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[pdf]
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| [18] |
A. Černý (2008, September), Characterization of the oblique projector U(VU)+V with application to constrained least squares, http://arXiv.org/abs/0809.4500. Appeared in Linear Algebra and Its Applications. |
[pdf] |
| [17] |
A. Černý (2008, February), Fast Fourier
transform and option pricing, http://ssrn.com/abstract=1098367. Appeared as Fourier Transform, in Cont
R. (ed.), Encyclopedia of Quantitative Finance. |
[pdf] |
| [16] |
A. Černý and J. Kallsen (2007, August),
Hedging by sequential regressions revisited, http://ssrn.com/abstract=1004706. Appeared in Mathematical Finance. |
[pdf] |
| [15] |
Brooks, C., A. Černý, and J. Miffre (2007, February), Optimal hedging with higher moments, http://ssrn.com/abstract=945807. Appeared in Journal of Futures Markets.
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[pdf]
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| [14] |
Černý, A. and J. Kallsen (2006, July), A counterexample concerning the variance-optimal martingale measure,
http://ssrn.com/abstract=912952. Appeared in Mathematical Finance. |
[pdf] |
| [13] |
Černý, A. and J. Kallsen (2006, June), Mean–variance edging and optimal investment in Heston's model with
correlation, http://ssrn.com/abstract=909305. Appeared in Mathematical
Finance. |
[pdf]
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| [12] |
Černý, A. (2006, January), Performance
of option hedging strategies: The tale of two trading desks, http://ssrn.com/abstract=877912. |
[pdf] |
| [11] |
Černý, A., Miles, D., and Ľ. Schmidt (2005,
June), The impact of changing demographics and pensions on the demand for housing and financial assets,
CEPR Discussion Paper 5143. Appeared in The
Journal of Pension Economics and Finance. |
[pdf]
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| [10] |
Černý, A. (2004, May), Optimal continuous-time hedging with leptokurtic returns, http://ssrn.com/abstract=713361. Appeared in Mathematical Finance. |
[pdf]
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| [9] |
Černý, A and J. Kallsen (2005,
May), On the structure of general mean–variance hedging strategies, http://ssrn.com/abstract=712743. Appeared in Annals
of Probability. |
[pdf]
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| [8] |
Černý, A. (2004, June), Introduction
to fast Fourier transform in finance, http://ssrn.com/abstract=559416. Appeared in Journal of Derivatives. |
[pdf] |
| [7] |
Černý, A. (2003, October), The risk of optimal, continuously rebalanced hedging strategies and its efficient evaluation via Fourier transform, http://ssrn.com/abstract=559417. |
[pdf]
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| [6] |
Miles, D. K. and A. Černý (2001, April), Risk, return, and portfolio allocation under alternative pension systems
with imperfect financial markets, http://ssrn.com/abstract=268968. Appeared in Economic Journal. |
[pdf] |
| [5] |
Černý, A. (2000, February),
Generalized Sharpe ratios and asset pricing in incomplete markets, http://ssrn.com/abstract=244731. Appeared in Review of Finance.< |
[pdf] |
| [4] |
Černý, A. (1999, June), Dynamic programming and mean–variance hedging in discrete time, http://ssrn.com/abstract=561223. Appeared in Applied Mathematical Finance.
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[pdf] |
| [3] |
Černý, A. (1999, April), Minimal martingale measure, CAPM and representative agent pricing in incomplete
markets, http://ssrn.com/abstract=851188.
Abstract: The minimal martingale measure (MMM) was introduced and studied by Föllmer and Schweizer (1990) in the context of mean square hedging in incomplete markets. Recently, the theory of no-good-deal pricing gave further evidence that the MMM plays a prominent role in security valuation in an incomplete market when security prices follow a diffusion process. Namely, it was shown that the price defined by the MMM lies in the centre of no-good-deal price bounds. In the first part of the paper we examine the relationship between the MMM and the optimal portfolio problem in diffusion environment and show that the MMM arises in equilibrium with log-utility maximizing representative agent. A puzzling property of the MMM is that outside the diffusion environment it easily becomes negative. As we show in the second part of the paper this fact can be explained from the link between the MMM and the CAPM risk-neutral measure. |
[pdf]
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| [2] |
Černý, A. (1998, September) Currency crises: Strategic game between central bank and speculators. http://ssrn.com/abstract=1428928.
Abstract: The paper studies an optimal switching policy between fixed and floating exchange rate regimes when the central bank dislikes losing reserves. We show that the optimal central bank intervention rule is not fully transparent in that the central bank will choose to randomize the devaluation over a range of the shadow exchange rate values to prevent a massive loss of reserves at one point in time. As a result, the collapse of the exchange rate becomes unpredictable even under perfect information and common knowledge. However, unlike in models with multiple equilibria we can determine the probability of the collapse within our model. The collapse probability is endogenously determined from the interaction between the central bank and the speculators as a unique function of the shadow exchange rate. The model is therefore able to predict how unpredictable the currency devaluation is.
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[pdf]
[+]
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| [1] |
Černý, A. and S. D. Hodges (1998, June), The theory of good-deal pricing in financial markets, http://ssrn.com/abstract=560682. Appeared in Mathematical Finance - Bachelier Congress 2000, Springer Verlag. |
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