| Černý, A. (2009), Mathematical Techniques in Finance: Tools for Incomplete Markets, 2nd ed., Princeton University Press | |||
| textbook website | preface | table of contents | |
| Černý, A. (2004), Mathematical Techniques in Finance: Tools for Incomplete Markets, Princeton University Press | |||
| textbook website | preface | table of contents | |
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.
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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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.
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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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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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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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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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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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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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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.
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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 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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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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| 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 pricing
model. 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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binomial.gss
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dft.gss
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fft.gss
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czt.gss
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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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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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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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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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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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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 Saa-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), Cerny (1999) and Hodges (1998). The theory presented here shows that any of these techniques can be seen as a generalization of no-arbitrage pricing.
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 mean-variance portfolios (cf. Maccheroni et al.2007), thus simplifying the computation of the latter.
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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 the hedging decision. The approach is applied to a set of 20 commodities that are hedged with futures contracts. 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 small to moderate commodity exposure, the performance of hedges constructed allowing for non-zero higher moments is only very slightly better than the performance of the much simpler OLS hedge ratio. For high commodity exposures, higher moments do matter and their relative weights in the utility function affect the optimal decision. We support our empirical findings by 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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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.
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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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