Books 

[2] 


[1] 


Peer reviewed publications 
[20] 
A. Tsanakas, M.V. Wüthrich and A. Černý (2013) Market Value Margin via MeanVariance Hedging, ASTIN Bulletin, 43(3), 301322
Abstract: We use meanvariance 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 payoff, written on the claims development result and available in each period. In simple scenarios, the resulting valuation formulas become very similar to regulatory costofcapitalbased formulas. However, adoption of the meanvariance 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 Eulertype capital allocations. 
[doi][pdf]
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[19] 
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), 255274
Abstract:
We consider the ordinary differential equation
x^{2}u'' = axu' + bu − c(u' − 1)^{2} for 0 < x < x_{0}
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 x_{0} = ∞, which is such that 0 < u(x) < x for all x > 0, and that this
solution is strictly increasing and concave. 
[doi][pdf]
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[18] 
A. Černý, F. Maccheroni, M. Marinacci and A. Rustichini (2012), On the Computation of Optimal Monotone MeanVariance Portfolios Via Truncated Quadratic Utility, Journal of Mathematical Economics 48(6), 386395
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. Cerny 2003) to the optimal monotone meanvariance portfolios (cf. Maccheroni et al.2007), thus simplifying the computation of the latter. 
[doi][pdf]
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[17] 
Brooks, C., A. Černý and J. Miffre (2012), Optimal Hedging With Higher Moments, Journal of Futures Markets 32 (10), 909944
Abstract: This study proposes a utilitybased 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 outofsample hedges constructed allowing for nonzero 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 outofsample 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. 
[doi][pdf]
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[16] 
Černý, A. and I. Kyriakou (2011), An Improved Convolution Algorithm for Discretely Sampled Asian Options, Quantitative Finance 11(3), 381389
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 Vecer (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. 
[doi][pdf]
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[15] 
Biagini, S. and A. Černý (2011), Admissible Strategies in Semimartingale Portfolio Selection, SIAM Journal on Control and Optimization 49(1), 4272
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
meanvariance 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. 
[doi][pdf]
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[14] 
Černý, A., D. K. Miles and L. 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), 393420
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. 
[doi][pdf]
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[13] 
A. Černý (2009), Characterization of the oblique projector U(VU)^{+}V with application to constrained least squares, Linear Algebra and Its Applications, 431(9), 15641570
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. 
[doi][pdf]
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[12] 
Bank, P. and A. Černý (2009), Preface to a special issue on meanvariance hedging, Review of Derivatives Research, 12(1), 12, 
[doi][pdf] 
[11] 
Černý, A. and J. Kallsen (2009), Hedging by Sequential Regressions Revisited, Mathematical Finance 19(4), 591617
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 oneperiod
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. 
[doi][pdf]
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[10] 
Černý, A. and J. Kallsen (2008), MeanVariance
Hedging and Optimal Investment in Heston's Model With Correlation,
Mathematical Finance 18(3), 473492
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 (socalled leverage effect).
Our contribution is threefold: using a new concept of opportunityneutral
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 varianceoptimal martingale measure; this
step seems to be partially missing in the literature. Finally, we derive
formulas for the hedging strategy and the hedging error. 
[doi] [pdf]
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[9] 
Černý, A. and J. Kallsen (2008), A
Counterexample Concerning The VarianceOptimal Martingale Measure,
Mathematical Finance 18(2), 305316
Abstract: The present note addresses an open question concerning a sufficient characterization
of the varianceoptimal 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•S_{T} for some Sintegrable process φ. We show that Q^{*} does not necessarily coincide with the varianceoptimal martingale measure, not even
if φ•S is a uniformly integrable Q^{*}martingale. 
[doi] [pdf]
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[8] 
Černý, A. and J. Kallsen (2007), On
The Structure of General MeanVariance Hedging Strategies, The Annals of Probability 35(4), 14791531
Abstract: We provide a new characterization of meanvariance 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 varianceoptimal martingale measure relative to the original
probability measure P. 
[doi] [pdf]
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[7] 
Černý, A. (2007) Optimal ContinuousTime
Hedging with Leptokurtic Returns, Mathematical Finance, 17(2),
175203
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
Levy 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 1D and 2D 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. 
[doi] [pdf]
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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),
529557
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. 
[doi] [pdf]
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[5] 

[doi] [pdf]
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[4] 
Černý, A. (2004), Dynamic Programming
and MeanVariance Hedging in Discrete Time, Applied Mathematical Finance
11(1), 125
Abstract: In this paper the general discrete time meanvariance 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 varianceoptimal measure arises in the dynamic programming solution and how one can define conditional expectations under this (generally nonequivalent) measure. The result is then related to the results of previous studies in continuous time. 
[doi] [pdf]
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[3] 
Černý, A. (2003), Generalized
Sharpe Ratios and Asset Pricing in Incomplete Markets, European Finance
Review, 7(2), 191233
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
SaáRequejo 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 SaáRequejo 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
nonnormal. Last but not least, we demonstrate that for Itô processes the Cochrane
and SaáRequejo 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. 
[doi] [pdf]
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[2] 
Černý, A. (1999), Currency crises:
Introduction of spot speculators, International Journal of Finance and
Economics, 4(1), 1999, 7589
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. 
[doi] [pdf]
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[1] 
Černý, A. and N. Schmitt (1995)
Antidumping Constraints and Trade, Swiss Journal of Economics and Statistics,
131 (3), 441452 http://www.sjes.ch/papers/1995III10.pdf
Abstract: We analyze the BertrandNash equilibrium in a twofirmtwocountry 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 
[3] 
Černý, A. (2010),
Fourier Transform, in Cont R. (ed.),
Encyclopedia of Quantitative Finance,
782786, Wiley: Chichester, ISBN 9780470057568 

[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, 75135,
Edward Elgar, ISBN 9781843766827
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. 
[pdf]
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[1] 
Černý, A. and S. D. Hodges (2002),
The Theory of GoodDeal Pricing in Financial Markets, in Geman H., Madan D., Pliska S.,
Vorst T.(eds.): Mathematical Finance  Bachelier Congress 2000, 175202,
Springer, ISBN: 9783540677819
Abstract: The term "nogooddeal 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 SaaRequejo (2000) who pioneered the calculation of price bounds conditional on the absence of high Sharpe ratios. Alternative methodologies for calculating tighterthannoarbitrage price bounds have been suggested by Bernardo and Ledoit (2000), Cerny (1999) and Hodges (1998). The theory presented here shows that any of these techniques can be seen as a generalization of noarbitrage pricing. 
[pdf]
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Working papers 
[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 varianceoptimal 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 BlackScholes 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 BlackScholes 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 discretetime hedging in the BlackScholes model.

[pdf]
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[25] 
A. Černý and I. Melicherčík (2013, September), A Simple Formula for Optimal Management of Individual Pension Accounts, Cass Business School Working Paper
Abstract: We consider optimal investment for an individual pension savings plan in receipt of
gradual contributions against which one cannot borrow, using expected power utility as
the optimality criterion. It is well known that in the presence of credit constraints the
Samuelson paradigm of investment in constant proportions out of total wealth (including
current savings and future contributions) no longer applies. Instead, the optimal
investment gives rise to socalled stochastic lifestyling, 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
leverage between risky and safe assets change but also the actual mix of the risky assets
varies over time. While the existing literature relies on complex numerical computations
to quantify optimal lifestyling the present paper provides a simple formula that
captures the main essence of the lifestyling effect.

[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


[23] 
A. Tsanakas, M.V. Wuethrich and A. Černý (2012, September), Market Value Margin via MeanVariance Hedging, http://ssrn.com/abstract=2148911. To appear in the ASTIN Bulletin


[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:
We consider the ordinary differential equation
x^{2}u'' = axu' + bu − c(u' − 1)^{2} for 0 < x < x_{0}
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 x_{0} = ∞, which is such that 0 < u(x) < x for all x > 0, and that this
solution is strictly increasing and concave. 
[pdf]
[+]

[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

[pdf]

[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]

[19] 
A. Černý, F. Maccheroni, M. Marinacci and A. Rustichini (2008, October), On the Computation of Optimal Monotone MeanVariance Portfolios Via Truncated Quadratic Utility, Appeared in the Journal of Mathematical Economics

[pdf]

[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

[pdf]

[14] 
Černý, A. and J. Kallsen (2006,
July), A Counterexample Concerning The VarianceOptimal Martingale Measure,
http://ssrn.com/abstract=912952, appeared in Mathematical Finance 
[pdf] 
[13] 
Černý, A. and J. Kallsen (2006,
June), MeanVariance Hedging and Optimal Investment in Heston's Model With
Correlation, http://ssrn.com/abstract=909305, appeared in Mathematical
Finance 
[pdf] 
[12] 
Černý, A. (2006, January), Performance
of Option Hedging Strategies: The Tale of Two Trading Desks, SSRN working
paper, http://ssrn.com/abstract=877912 
[pdf] 
[11] 
Černý, A., Miles, D. and L.
Schmidt (2005, June), The Impact of Changing Demographics and Pensions on
The Demand for Housing and Financial Assets, CEPR Discussion Paper 5143, to appear in The Journal of Pension Economics and Finance 
[pdf] 
[10] 
Černý, A. (2004, May), Optimal
ContinuousTime Hedging with Leptokurtic Returns, SSRN working paper, http://ssrn.com/abstract=713361,
appeared in Mathematical Finance 
[pdf] 
[9] 
Černý, A and J. Kallsen (2005,
May), On The Structure of General MeanVariance Hedging Strategies, SSRN
working paper, http://ssrn.com/abstract=712743, appeared in The Annals
of Probability 
[pdf] 
[8] 
Černý, A. (2004, June), Introduction
to Fast Fourier Transform in Finance, SSRN working paper, 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, SSRN working paper, http://ssrn.com/abstract=559417 
[pdf] 
[6] 
Miles, D. K. and A. Černý (2001,
April), Risk Return and Portfolio Allocation under Alternative Pension Systems
with Imperfect Financial Markets, CEPR Discussion Paper 2779, CESifo Working
Paper Series No. 441, SSRN working paper http://ssrn.com/abstract=268968,
appeared in The Economic Journal 
[pdf] 
[5] 
Černý, A. (2000, February),
Generalized Sharpe Ratios and Asset Pricing in Incomplete Markets, SSRN
working paper, http://ssrn.com/abstract=244731, appeared in European
Finance Review 
[pdf] 
[4] 
Černý, A. (1999, June), Dynamic
Programming and MeanVariance Hedging in Discrete Time, SSRN working paper,
http://ssrn.com/abstract=561223, appeared in Applied Mathematical Finance

[pdf] 
[3] 
Černý, A. (1999, April), Minimal
martingale measure, CAPM and representative agent pricing in incomplete
markets, Imperial College Management School Discussion Paper, SWP9901/F,
SSRN working paper, 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 nogooddeal 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 nogooddeal 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 logutility 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 riskneutral measure. 
[pdf]
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[2] 
Černý, A. (1998, September)
Currency crises: Strategic game between central bank and speculators.
Imperial College Management School Discussion Paper SWP9814/F, 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.

[pdf]
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[1] 
Černý, A. and S. D. Hodges (1998,
June), The Theory of GoodDeal Pricing in Financial Markets, FORC Preprint
98/90, SSRN working paper http://ssrn.com/abstract=560682, appeared in Mathematical Finance  Bachelier Congress 2000, Springer Verlag 
[pdf] 