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Constantinos Kardaras - One of the best experts on this subject based on the ideXlab platform.
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the Numeraire property and long term growth optimality for drawdown constrained investments
LSE Research Online Documents on Economics, 2017Co-Authors: Constantinos Kardaras, Jan Obloj, Eckhard PlatenAbstract:We consider the portfolio choice problem for a long-run investor in a general continuous semimartingale model. We combine the decision criterion of pathwise growth optimality with a flexible specification of attitude towards risk, encoded by a linear drawdown constraint imposed on admissible wealth processes. We define the constrained numraire property through the notion of expected relative return and prove that drawdown-constrained Numeraire portfolio exists and is unique, but may depend on the investment horizon. However, when sampled at the times of its maximum and asymptotically as the time-horizon becomes distant, the drawdown-constrained Numeraire portfolio is given explicitly through a model-independent transformation of the unconstrained Numeraire portfolio. The asymptotically growth-optimal strategy is obtained as limit of Numeraire strategies on finite horizons.
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No arbitrage of the first kind and local martingale numéraires
Finance and Stochastics, 2016Co-Authors: Yuri Kabanov, Constantinos Kardaras, Shiqi SongAbstract:A supermartingale deflator (resp. local martingale deflator) multiplicatively transforms nonnegative wealth processes into supermartingales (resp. local martingales). A supermartingale Numeraire (resp. local martingale Numeraire) is a wealth process whose reciprocal is a supermartingale deflator (resp. local martingale deflator). It has been established in previous works that absence of arbitrage of the first kind (\(\mbox{NA}_{1}\)) is equivalent to the existence of the (unique) supermartingale Numeraire, and further equivalent to the existence of a strictly positive local martingale deflator; however, under \(\mbox{NA}_{1}\), a local martingale Numeraire may fail to exist. In this work, we establish that under \(\mbox{NA}_{1}\), a supermartingale Numeraire under the original probability \(P\) becomes a local martingale Numeraire for equivalent probabilities arbitrarily close to \(P\) in the total variation distance.
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maximality and Numeraires in convex sets of nonnegative random variables
Journal of Functional Analysis, 2015Co-Authors: Constantinos KardarasAbstract:We introduce the concepts of max-closedness and Numeraires of convex subsets of L+0, the nonnegative orthant of the topological vector space L0 of all random variables built over a probability space, equipped with a topology consistent with convergence in probability. Max-closedness asks that maximal elements of the closure of a set already lie on the set. We discuss how Numeraires arise naturally as strictly positive optimisers of certain concave monotone maximisation problems. It is further shown that the set of Numeraires of a convex, max-closed and bounded set of L+0 that contains at least one strictly positive element is dense in the set of its maximal elements.
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No arbitrage and local martingale deflators
arXiv: Probability, 2015Co-Authors: Yuri Kabanov, Constantinos Kardaras, Shiqi SongAbstract:A supermartingale deflator (resp., local martingale deflator) multiplicatively transforms nonnegative wealth processes into supermartingales (resp., local martingales). The supermartingale Numeraire (resp., local martingale Numeraire) is the wealth processes whose reciprocal is a supermartingale deflator (resp., local martingale deflator). It has been established in previous literature that absence of arbitrage of the first kind (NA1) is equivalent to existence of the supermartingale Numeraire, and further equivalent to existence of a strictly positive local martingale deflator; however, under NA1, the local martingale Numeraire may fail to exist. In this work, we establish that, under NA1, any total-variation neighbourhood of the original probability has an equivalent probability under which the local martingale Numeraire exists. This result, available previously only for single risky-asset models, is in striking resemblance with the fact that any total-variation neighbourhood of a separating measure contains an equivalent $\sigma$-martingale measure. The presentation of our main result is relatively self-contained, including a proof of existence of the supermartingale Numeraire under NA1. We further show that, if the Levy measures of the asset-price process have finite support, NA1 is equivalent to existence of the local martingale Numeraire with respect to the original probability.
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the Numeraire property and long term growth optimality for drawdown constrained investments
2012Co-Authors: Constantinos Kardaras, Jan Obloj, Eckhard PlatenAbstract:We consider the portfolio choice problem for a long-run investor in a general continuous semimartingale model. We suggest to use path-wise growth optimality as the decision criterion and encode preferences through restrictions on the class of admissible wealth processes. Specifically, the investor is only interested in strategies which satisfy a given linear drawdown constraint. The paper introduces the Numeraire property through the notion of expected relative return and shows that drawdown-constrained strategies with the Numeraire property exist and are unique, but may depend on the financial planning horizon. However, when sampled at the times of its maximum and asymptotically as the time-horizon becomes distant, the drawdown-constrained \num\ portfolio is given explicitly through a model-independent transformation of the unconstrained \num\ portfolio. Further, it is established that the asymptotically growth-optimal strategy is obtained as limit of Numeraire strategies on finite horizons.
Eckhard Platen - One of the best experts on this subject based on the ideXlab platform.
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the fundamental theorem of asset pricing for self financing portfolios
Research Paper Series, 2020Co-Authors: Eckhard Platen, Stefan TappeAbstract:Consider a financial market with nonnegative semimartingales which does not need to have a Numeraire. We are interested in the absence of arbitrage in the sense that no self-financing portfolio gives rise to arbitrage opportunities, where we are allowed to add a savings account to the market. We will prove that in this sense the market is free of arbitrage if and only if there exists an equivalent local martingale deator which is a multiplicative special semimartingale. In this case, the additional savings account relates to the finite variation part of the multiplicative decomposition of the deflator. By focusing on self-financing portfolios, this result clarifies links between previous results in the literature and makes the respective concepts more realistic.
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the Numeraire property and long term growth optimality for drawdown constrained investments
LSE Research Online Documents on Economics, 2017Co-Authors: Constantinos Kardaras, Jan Obloj, Eckhard PlatenAbstract:We consider the portfolio choice problem for a long-run investor in a general continuous semimartingale model. We combine the decision criterion of pathwise growth optimality with a flexible specification of attitude towards risk, encoded by a linear drawdown constraint imposed on admissible wealth processes. We define the constrained numraire property through the notion of expected relative return and prove that drawdown-constrained Numeraire portfolio exists and is unique, but may depend on the investment horizon. However, when sampled at the times of its maximum and asymptotically as the time-horizon becomes distant, the drawdown-constrained Numeraire portfolio is given explicitly through a model-independent transformation of the unconstrained Numeraire portfolio. The asymptotically growth-optimal strategy is obtained as limit of Numeraire strategies on finite horizons.
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pricing of long dated equity linked life insurance contracts
Published Paper Series, 2016Co-Authors: Leunglung Chan, Eckhard PlatenAbstract:This article adopts an approach to pricing of equity-linked life insurance contracts, which only requires the existence of the Numeraire portfolio. An equity-linked life insurance contract is equivalent to a sum of the guaranteed amount and the value of an option on the equity index with some mortality risk attached. The Numeraire portfolio equals the growth optimal portfolio and is used as Numeraire or benchmark, where the real-world probability measure is taken as pricing measure. To obtain tractable solutions the short rate is modelled as a quadratic form of some Gaussian factor processes. Furthermore, the dynamics of the mortality rate is modelled as a threshold life table. The dynamics of the discounted equity market index or benchmark is modelled by a time transformed squared Bessel process. The equity-linked life insurance contracts are evaluated analytically.
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the Numeraire property and long term growth optimality for drawdown constrained investments
2012Co-Authors: Constantinos Kardaras, Jan Obloj, Eckhard PlatenAbstract:We consider the portfolio choice problem for a long-run investor in a general continuous semimartingale model. We suggest to use path-wise growth optimality as the decision criterion and encode preferences through restrictions on the class of admissible wealth processes. Specifically, the investor is only interested in strategies which satisfy a given linear drawdown constraint. The paper introduces the Numeraire property through the notion of expected relative return and shows that drawdown-constrained strategies with the Numeraire property exist and are unique, but may depend on the financial planning horizon. However, when sampled at the times of its maximum and asymptotically as the time-horizon becomes distant, the drawdown-constrained \num\ portfolio is given explicitly through a model-independent transformation of the unconstrained \num\ portfolio. Further, it is established that the asymptotically growth-optimal strategy is obtained as limit of Numeraire strategies on finite horizons.
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the Numeraire property and long term growth optimality for drawdown constrained investments
arXiv: Portfolio Management, 2012Co-Authors: Constantinos Kardaras, Jan Obloj, Eckhard PlatenAbstract:We consider the portfolio choice problem for a long-run investor in a general continuous semimartingale model. We suggest to use path-wise growth optimality as the decision criterion and encode preferences through restrictions on the class of admissible wealth processes. Specifically, the investor is only interested in strategies which satisfy a given linear drawdown constraint. The paper introduces the Numeraire property through the notion of expected relative return and shows that drawdown-constrained strategies with the Numeraire property exist and are unique, but may depend on the financial planning horizon. However, when sampled at the times of its maximum and asymptotically as the time-horizon becomes distant, the drawdown-constrained Numeraire portfolio is given explicitly through a model-independent transformation of the unconstrained Numeraire portfolio. Further, it is established that the asymptotically growth-optimal strategy is obtained as limit of Numeraire strategies on finite horizons.
Tetsuya Saito - One of the best experts on this subject based on the ideXlab platform.
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formation of symmetric free trade blocs optimal tariff structure and world welfare
Journal of Asian Economics, 2021Co-Authors: Winston W Chang, Tailiang Chen, Tetsuya SaitoAbstract:Abstract We present a new tariff-game rule and a new Numeraire rule in Krugman's celebrated model to form symmetric trading blocs. We hold that to maintain logical consistency in a world of symmetric trading blocs, an individual bloc should act on the actions of other individual external blocs in a one-to-one fashion, rather than to the actions of the rest of the world as a whole as assumed by Krugman, and show that Krugman's seemingly innocuous choice of the world price of a given good as the Numeraire will produce asymmetry in the optimum Nash equilibrium tariff. We prove that the optimal tariff schedule is monotonically decreasing in our relative bloc size, and that the world welfare increases with our new relative bloc size as the latter grows beyond the lowest-welfare pessimal number, which is rather small by our simulations. Though confined to symmetric trading blocs, this paper fortifies the analytical foundation of Krugman's model. In some sense, it reinforces Kemp-Wan-Shimomura's and Ohyama-Panagariya-Krishna's results with the provisos that countries are symmetric and interact mutually in a symmetric fashion without a compensation scheme. It strengthens the case of regionalism as a stepping stone (building bloc) toward a complete world economic integration.
Jean-charles Rochet - One of the best experts on this subject based on the ideXlab platform.
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changes of Numeraire changes of probability measure and option pricing
Journal of Applied Probability, 1995Co-Authors: Helyette Geman, Nicole El Karoui, Jean-charles RochetAbstract:The use of the risk-neutral probability measure has proved to be very powerful for computing the prices of contingent claims in the context of complete markets, or the prices of redundant securities when the assumption of complete markets is relaxed. We show here that many other probability measures can be defined in the same way to solve different asset-pricing problems, in particular option pricing. Moreover, these probability measure changes are in fact associated with Numeraire changes, this feature, besides providing a financial interpretation, permits efficient selection of the Numeraire appropriate for the pricing of a given contingent claim and also permits exhibition of the hedging portfolio, which is in many respects more important than the valuation itself. The key theorem of general Numeraire change is illustrated by many examples, among which the extension to a stochastic interest rates framework of the Margrabe formula, Geske formula, etc.
Winston W Chang - One of the best experts on this subject based on the ideXlab platform.
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formation of symmetric free trade blocs optimal tariff structure and world welfare
Journal of Asian Economics, 2021Co-Authors: Winston W Chang, Tailiang Chen, Tetsuya SaitoAbstract:Abstract We present a new tariff-game rule and a new Numeraire rule in Krugman's celebrated model to form symmetric trading blocs. We hold that to maintain logical consistency in a world of symmetric trading blocs, an individual bloc should act on the actions of other individual external blocs in a one-to-one fashion, rather than to the actions of the rest of the world as a whole as assumed by Krugman, and show that Krugman's seemingly innocuous choice of the world price of a given good as the Numeraire will produce asymmetry in the optimum Nash equilibrium tariff. We prove that the optimal tariff schedule is monotonically decreasing in our relative bloc size, and that the world welfare increases with our new relative bloc size as the latter grows beyond the lowest-welfare pessimal number, which is rather small by our simulations. Though confined to symmetric trading blocs, this paper fortifies the analytical foundation of Krugman's model. In some sense, it reinforces Kemp-Wan-Shimomura's and Ohyama-Panagariya-Krishna's results with the provisos that countries are symmetric and interact mutually in a symmetric fashion without a compensation scheme. It strengthens the case of regionalism as a stepping stone (building bloc) toward a complete world economic integration.
