Youri Kabanov - Academia.edu (original) (raw)

Papers by Youri Kabanov

Research paper thumbnail of Essential Supremum with Respect to a Random Partial Order

Social Science Research Network, 2013

Inspired by the theory of financial markets with transaction costs, we study a concept of essenti... more Inspired by the theory of financial markets with transaction costs, we study a concept of essential supremum in the framework where a random partial order in R d is lifted to the space L 0 (R d) of ddimensional random variables. In contrast to the classical definition, we define the essential supremum as a subset of random variables satisfying some natural properties. An application of the introduced notion to a hedging problem under transaction costs is given.

Research paper thumbnail of Optional Decomposition and Lagrange Multipliers

Social Science Research Network, Apr 21, 1998

Let Q be the set of equivalent martingale measures for a given process S, and let X be a process ... more Let Q be the set of equivalent martingale measures for a given process S, and let X be a process which is a local supermartingale with respect to any measure in Q. The optional decomposition theorem for X states that there exists a predictable integrand ϕ such that the difference X − ϕ • S is a decreasing process. In this paper we give a new proof which uses techniques from stochastic calculus rather than functional analysis, and which removes any boundedness assumption.

Research paper thumbnail of Bond Markets Where Prices are Driven by a General Marked Point Process

Social Science Research Network, Nov 1, 1995

We investigate the term structure of zero coupon bonds when interest rates are driven by a genera... more We investigate the term structure of zero coupon bonds when interest rates are driven by a general marked point process as well as by a Wiener process. Developing a theory which allows for measure-valued trading portfolios we study existence and uniqueness of a martingale measure. We also study completeness and its relation to the uniqueness of a martingale measure. For the case of a finite jump spectrum we give a fairly general completeness result and for a Wiener-Poisson model we prove the existence of a time-independent set of basic bonds. We also give sufficient conditions for the existence of an affine term structure.

Research paper thumbnail of On Leland's Strategy of Option Pricing with Transactions Costs

Social Science Research Network, Apr 24, 1997

We compute the limiting hedging error of the Leland strategy for the approximate pricing of the E... more We compute the limiting hedging error of the Leland strategy for the approximate pricing of the European call option in a market with transactions costs. It is not equal to zero in the case when the level of transactions costs is a constant, in contradiction with the claim in Leland (1985).

Research paper thumbnail of Hedging and Liquidation under Transaction Costs in Currency Markets

Social Science Research Network, Feb 25, 1999

ABSTRACT We consider a general semimartingale model of a currency market with transaction costs a... more ABSTRACT We consider a general semimartingale model of a currency market with transaction costs and give a description of the initial endowments which allow to hedge a contingent claim in various currencies by a self-financing portfolio. As an application we obtain a result on the structure of optimal strategies for the problem of maximizing expected utility from terminal wealth.

Research paper thumbnail of L'evy Processes in Credit Risk and Market Models

Preface During the last years banks have realized that default risk cannot be neglected. The subj... more Preface During the last years banks have realized that default risk cannot be neglected. The subjective assessment of credit worthiness has to be replaced with a more objective way of estimating default risk. Mathematical credit risk models in the literature are mainly models based on Brownian motion although it is known that real-life financial data provides a different statistical behavior than that implied by these models. Lévy processes are an appropriate tool to increase accuracy of models in finance. They have been used to model stock prices, and term structures of interest rates, thus allowing more accurate derivative pricing and risk management. In this study we discuss how general Lévy processes can be applied to credit risk and market models. There are also a few jump-diffusion models where the jump-components have sample paths of bounded variation, but some of the very important subclasses are special types of Lévy processes. The emphasis in this study lies on the application of Lévy processes in credit risk models. We will mainly deal with the question of pricing defaultable corporate bonds. However we will also consider a portfolio model where the main question is the modeling of the loss distribution. Furthermore, we present a Lévy Libor model which contains the classical Libor model as a special case. In Chapter 1 we briefly survey some aspects of credit risk, generalized hy-perbolic distributions and Lévy processes. In the overview of the structural approach, we show how Lévy processes can be used to generalize the classical structural approach due to Merton (1974). In Chapter 2 we present and generalize the approach of the commercial software package CreditRisk +TM by Credit Suisse Financial Products. This chapter might be of some interest to practitioners since we introduce an alternative distribution to CreditRisk +TM which allows for heavier tails and still guarantees analytical tractability. Then we introduce a credit risk Heath-Jarrow-Morton-framework based on Lévy processes in Chapter 3, where we generalize the Gaussian approach in Bi-elecki and Rutkowski (1999, 2000). With a view to more realistic modeling we also include the topic of reorganization within this framework, which is based on an idea of Schönbucher (1998, 2000a). In Chapter 4 we show how the market practice of pricing caplets with the Black formula can be pushed further to a model where the driving process is a I Lévy process. At this point I want to express my warmest thanks to …

Research paper thumbnail of Mean Square Error for the Leland-Lott Hedging Strategy: Convex Pay-Off

Cognition in Mathematics, 2012

Leland's approach to the hedging of derivatives under proportional transaction costs is based... more Leland's approach to the hedging of derivatives under proportional transaction costs is based on an approximate replication of the European-type contingent claim VT using the classical Black Scholes formulae with a suitably enlarged volatility. The formal mathematical framework is a scheme of series, i.e. a sequence of models with the transaction costs coefficients kn and n is the number of the portfolio revision dates. The enlarged volatility, in general, depends on n except the case which was investigated in details by Lott to whom belongs the first rigorous result on convergence of the approximating portfolio value to the pay-off. In this paper we consider only the Lott case alpha= 1/2. We prove first, for an arbitrary pay-off VT = G(ST ) where G is a convex piecewise smooth function, that the mean square approximation error converges to zero with rate n^1/2 in L2 and find the first order term of asymptotics. We are working in the setting with non-uniform revision intervals a...

Research paper thumbnail of Advances in Statistics of Random Processes Workshop in honor of Yury Kutoyants’ 70th birthday

Research paper thumbnail of Dynamic Models of Systemic Risk and Contagion

Informatics and Applications, 2017

Modern financial systems are complicated networks of interconnected financial institutions and de... more Modern financial systems are complicated networks of interconnected financial institutions and default of one of them may have serious consequences for others. The recent crises have shown that the complexity and interconnectedness are major factors of systemic risk which became a subject of intensive studies usually concentrated on static models. In this paper we develop a dynamic model based on the so-called structural approach where defaults are triggered by the exit of some stochastic process from a domain. In our case, this is a process defined by the evolution of bank's portfolios values. At the exit time a bank defaults and a cascade of defaults starts. We believe that the distribution of the exit time and the subsequent losses may serve as indicators allowing regulators to monitor the state of the system to take corrective actions to avoid the contagion in the financial system. We model the development of financial system as a random graph using the preferable attachment algorithm and provide results of numerical experiments on simulated data.

Research paper thumbnail of On Uniqueness of Clearing Vectors Reducing the Systemic Risk

Informatics and Applications, 2017

Clearing of financial system, i.e. of a network of interconnecting banks, is a procedure of simul... more Clearing of financial system, i.e. of a network of interconnecting banks, is a procedure of simultaneous repaying debts to reduce their total volume. The vector whose components are repayments of each bank is called clearing vector. In simple models considered by Eisenberg and Noe (2001) and, independently, by Suzuki (2002), it was shown that the clearing to the minimal value of debts accordingly to natural rules can be formulated as a fixpoint problems. The existence of their solutions, i.e. of clearing vectors, is rather straightforward and can be obtained by a direct reference to the Knaster-Tarski or Brouwer theorems. The uniqueness of clearing vectors is a more delicate problem which was solved by Eisenberg and Noe using a graph structure of the financial network. We prove uniqueness results in two generalizations of the Eisenberg-Noe model: in the Elsinger model with seniority of liabilities and in the Amini-Filipovic-Minca type model with several types of liquid assets whose firing sale has a market impact.

Research paper thumbnail of On Supremal and Maximal Sets with Respect to Random Partial Orders

SSRN Electronic Journal, 2013

The paper deals with definition of supremal sets in a rather general framework where deterministi... more The paper deals with definition of supremal sets in a rather general framework where deterministic and random preference relations (preorders) and partial orders are defined by continuous multi-utility representations. It gives a short survey of the approach developed in [5], [6] with some new results on maximal sets.

Research paper thumbnail of Consumption-Investment Optimization Problem in a LLvy Financial Model with Transaction Costs

SSRN Electronic Journal, 2013

We consider the consumption-investment optimization problem for the financial market model with c... more We consider the consumption-investment optimization problem for the financial market model with constant proportional transaction rates and Lévy price process dynamics. Contrarily to the recent work in [4], portfolio process trajectories are only left and right limited. This allows us to identify an optimal làdlàg strategy, e.g. in the two dimensional case, as it is possible to suitably rebalance the portfolio processes when they jump outside the no trade region of the solvency cone.

Research paper thumbnail of On the optimal portfolio for the exponential utility maximization: remarks to the six-author paper

Mathematical Finance, 2002

This note contains ramifications of results of Delbaen et al. (2002). Assuming that the price pro... more This note contains ramifications of results of Delbaen et al. (2002). Assuming that the price process is locally bounded and admits an equivalent local martingale measure with finite entropy, we show, without further assumption, that in the case of exponential utility the optimal portfolio process is a martingale with respect to each local martingale measure with finite entropy. Moreover, the optimal value always can be attained on a sequence of uniformly bounded portfolios.

Research paper thumbnail of Bond Market Structure in the Presence of Marked Point Processes

Mathematical Finance, 1997

We investigate the term structure of zero coupon bonds when interest rates are driven by a genera... more We investigate the term structure of zero coupon bonds when interest rates are driven by a general marked point process as well as by a Wiener process. Developing a theory which allows for measure-valued trading portfolios we study existence and uniqueness of a martingale measure. We also study completeness and its relation to the uniqueness of a martingale measure. For the case of a finite jump spectrum we give a fairly general completeness result and for a Wiener-Poisson model we prove the existence of a time-independent set of basic bonds. We also give sufficient conditions for the existence of an affine term structure.

Research paper thumbnail of Hedging under Transaction Costs in Currency Markets: a Discrete-Time Model

Mathematical Finance, 2002

We consider a discrete-time model of a currency market with transaction costs and give a descript... more We consider a discrete-time model of a currency market with transaction costs and give a description of initial endowments that allow the investor to hedge a contingent claim in various currencies by a self-®nancing portfolio.

Research paper thumbnail of Essential supremum and essential maximum with respect to random preference relations

Journal of Mathematical Economics, 2013

Research paper thumbnail of Essential supremum with respect to a random partial order

Journal of Mathematical Economics, 2013

Inspired by the theory of financial markets with transaction costs, we study a concept of essenti... more Inspired by the theory of financial markets with transaction costs, we study a concept of essential supremum in the framework where a random partial order in R d is lifted to the space L 0 (R d) of ddimensional random variables. In contrast to the classical definition, we define the essential supremum as a subset of random variables satisfying some natural properties. An application of the introduced notion to a hedging problem under transaction costs is given.

Research paper thumbnail of Consistent price systems and arbitrage opportunities of the second kind in models with transaction costs

Finance and Stochastics, 2011

In contrast with the classical models of frictionless financial markets, market models with propo... more In contrast with the classical models of frictionless financial markets, market models with proportional transaction costs, even satisfying usual no-arbitrage properties, may admit arbitrage opportunities of the second kind. This means that there are self-financing portfolios with initial endowments laying outside the solvency region but ending inside. Such a phenomenon was discovered by M. Rásonyi in the discrete-time framework. In this note we consider a rather abstract continuous-time setting and prove necessary and sufficient conditions for the property which we call No Free Lunch of the 2nd Kind, NFL2. We provide a number of equivalent conditions elucidating, in particular, the financial meaning of the property B which appeared as an indispensable "technical" hypothesis in previous papers on hedging (super-replication) of contingent claims under transaction costs. We show that it is equivalent to another condition on the "richness" of the set of consistent price systems, close to the condition PCE introduced by Rásonyi. In the last section we deduce the Rásonyi theorem from our general result using specific features of discrete-time models.

Research paper thumbnail of Hedging of American options under transaction costs

Finance and Stochastics, 2008

We consider a continuous-time model of financial market with proportional transaction costs. Our ... more We consider a continuous-time model of financial market with proportional transaction costs. Our result is a dual description of the set of initial endowments of self-financing portfolios super-replicating Americantype contingent claim. The latter is a right-continuous adapted vector process describing the number of assets to be delivered at the exercise date. We introduce a specific class of price systems, called coherent, and show that the hedging endowments are those whose "values" are larger than the expected weighted "values" of the pay-off process for every coherent price system used for the "evaluation" of the assets.

Research paper thumbnail of Mean square error for the Leland–Lott hedging strategy: convex pay-offs

Finance and Stochastics, 2010

The Leland strategy of approximate hedging of the call-option under proportional transaction cost... more The Leland strategy of approximate hedging of the call-option under proportional transaction costs prescribes to use, at equidistant instants of portfolio revisions, the classical Black-Scholes formula but with a suitably enlarged volatility. An appropriate mathematical framework is a scheme of series, i.e. a sequence of models M n with the transaction costs coefficients k n depending on n, the number of the revision intervals. The enlarged volatility σ n , in general, also depends on n. Lott investigated in detail the particular case where the transaction costs coefficients decrease as n −1/2 and where the Leland formula yields σ n not depending on n. He proved that the terminal value of the portfolio converges in probability to the pay-off g(S T) where G(x) = (x − K) +. In the present note we consider the case of much more general convex piecewise smooth pay-off functions G. We show that the convergence holds also in L 2 and find the first order term of asymptotics for the mean square error. We are working in the setting with non-uniform revision intervals and establish the asymptotic expansion when the revision dates are t n i = g(i/n) where the strictly increasing scale function g : [0, 1] → [0, 1] and its inverse f are continuous with their first and second derivatives on the whole interval or g(t) = 1 − (1 − t) β , β ≥ 1.

Research paper thumbnail of Essential Supremum with Respect to a Random Partial Order

Social Science Research Network, 2013

Inspired by the theory of financial markets with transaction costs, we study a concept of essenti... more Inspired by the theory of financial markets with transaction costs, we study a concept of essential supremum in the framework where a random partial order in R d is lifted to the space L 0 (R d) of ddimensional random variables. In contrast to the classical definition, we define the essential supremum as a subset of random variables satisfying some natural properties. An application of the introduced notion to a hedging problem under transaction costs is given.

Research paper thumbnail of Optional Decomposition and Lagrange Multipliers

Social Science Research Network, Apr 21, 1998

Let Q be the set of equivalent martingale measures for a given process S, and let X be a process ... more Let Q be the set of equivalent martingale measures for a given process S, and let X be a process which is a local supermartingale with respect to any measure in Q. The optional decomposition theorem for X states that there exists a predictable integrand ϕ such that the difference X − ϕ • S is a decreasing process. In this paper we give a new proof which uses techniques from stochastic calculus rather than functional analysis, and which removes any boundedness assumption.

Research paper thumbnail of Bond Markets Where Prices are Driven by a General Marked Point Process

Social Science Research Network, Nov 1, 1995

We investigate the term structure of zero coupon bonds when interest rates are driven by a genera... more We investigate the term structure of zero coupon bonds when interest rates are driven by a general marked point process as well as by a Wiener process. Developing a theory which allows for measure-valued trading portfolios we study existence and uniqueness of a martingale measure. We also study completeness and its relation to the uniqueness of a martingale measure. For the case of a finite jump spectrum we give a fairly general completeness result and for a Wiener-Poisson model we prove the existence of a time-independent set of basic bonds. We also give sufficient conditions for the existence of an affine term structure.

Research paper thumbnail of On Leland's Strategy of Option Pricing with Transactions Costs

Social Science Research Network, Apr 24, 1997

We compute the limiting hedging error of the Leland strategy for the approximate pricing of the E... more We compute the limiting hedging error of the Leland strategy for the approximate pricing of the European call option in a market with transactions costs. It is not equal to zero in the case when the level of transactions costs is a constant, in contradiction with the claim in Leland (1985).

Research paper thumbnail of Hedging and Liquidation under Transaction Costs in Currency Markets

Social Science Research Network, Feb 25, 1999

ABSTRACT We consider a general semimartingale model of a currency market with transaction costs a... more ABSTRACT We consider a general semimartingale model of a currency market with transaction costs and give a description of the initial endowments which allow to hedge a contingent claim in various currencies by a self-financing portfolio. As an application we obtain a result on the structure of optimal strategies for the problem of maximizing expected utility from terminal wealth.

Research paper thumbnail of L'evy Processes in Credit Risk and Market Models

Preface During the last years banks have realized that default risk cannot be neglected. The subj... more Preface During the last years banks have realized that default risk cannot be neglected. The subjective assessment of credit worthiness has to be replaced with a more objective way of estimating default risk. Mathematical credit risk models in the literature are mainly models based on Brownian motion although it is known that real-life financial data provides a different statistical behavior than that implied by these models. Lévy processes are an appropriate tool to increase accuracy of models in finance. They have been used to model stock prices, and term structures of interest rates, thus allowing more accurate derivative pricing and risk management. In this study we discuss how general Lévy processes can be applied to credit risk and market models. There are also a few jump-diffusion models where the jump-components have sample paths of bounded variation, but some of the very important subclasses are special types of Lévy processes. The emphasis in this study lies on the application of Lévy processes in credit risk models. We will mainly deal with the question of pricing defaultable corporate bonds. However we will also consider a portfolio model where the main question is the modeling of the loss distribution. Furthermore, we present a Lévy Libor model which contains the classical Libor model as a special case. In Chapter 1 we briefly survey some aspects of credit risk, generalized hy-perbolic distributions and Lévy processes. In the overview of the structural approach, we show how Lévy processes can be used to generalize the classical structural approach due to Merton (1974). In Chapter 2 we present and generalize the approach of the commercial software package CreditRisk +TM by Credit Suisse Financial Products. This chapter might be of some interest to practitioners since we introduce an alternative distribution to CreditRisk +TM which allows for heavier tails and still guarantees analytical tractability. Then we introduce a credit risk Heath-Jarrow-Morton-framework based on Lévy processes in Chapter 3, where we generalize the Gaussian approach in Bi-elecki and Rutkowski (1999, 2000). With a view to more realistic modeling we also include the topic of reorganization within this framework, which is based on an idea of Schönbucher (1998, 2000a). In Chapter 4 we show how the market practice of pricing caplets with the Black formula can be pushed further to a model where the driving process is a I Lévy process. At this point I want to express my warmest thanks to …

Research paper thumbnail of Mean Square Error for the Leland-Lott Hedging Strategy: Convex Pay-Off

Cognition in Mathematics, 2012

Leland's approach to the hedging of derivatives under proportional transaction costs is based... more Leland's approach to the hedging of derivatives under proportional transaction costs is based on an approximate replication of the European-type contingent claim VT using the classical Black Scholes formulae with a suitably enlarged volatility. The formal mathematical framework is a scheme of series, i.e. a sequence of models with the transaction costs coefficients kn and n is the number of the portfolio revision dates. The enlarged volatility, in general, depends on n except the case which was investigated in details by Lott to whom belongs the first rigorous result on convergence of the approximating portfolio value to the pay-off. In this paper we consider only the Lott case alpha= 1/2. We prove first, for an arbitrary pay-off VT = G(ST ) where G is a convex piecewise smooth function, that the mean square approximation error converges to zero with rate n^1/2 in L2 and find the first order term of asymptotics. We are working in the setting with non-uniform revision intervals a...

Research paper thumbnail of Advances in Statistics of Random Processes Workshop in honor of Yury Kutoyants’ 70th birthday

Research paper thumbnail of Dynamic Models of Systemic Risk and Contagion

Informatics and Applications, 2017

Modern financial systems are complicated networks of interconnected financial institutions and de... more Modern financial systems are complicated networks of interconnected financial institutions and default of one of them may have serious consequences for others. The recent crises have shown that the complexity and interconnectedness are major factors of systemic risk which became a subject of intensive studies usually concentrated on static models. In this paper we develop a dynamic model based on the so-called structural approach where defaults are triggered by the exit of some stochastic process from a domain. In our case, this is a process defined by the evolution of bank's portfolios values. At the exit time a bank defaults and a cascade of defaults starts. We believe that the distribution of the exit time and the subsequent losses may serve as indicators allowing regulators to monitor the state of the system to take corrective actions to avoid the contagion in the financial system. We model the development of financial system as a random graph using the preferable attachment algorithm and provide results of numerical experiments on simulated data.

Research paper thumbnail of On Uniqueness of Clearing Vectors Reducing the Systemic Risk

Informatics and Applications, 2017

Clearing of financial system, i.e. of a network of interconnecting banks, is a procedure of simul... more Clearing of financial system, i.e. of a network of interconnecting banks, is a procedure of simultaneous repaying debts to reduce their total volume. The vector whose components are repayments of each bank is called clearing vector. In simple models considered by Eisenberg and Noe (2001) and, independently, by Suzuki (2002), it was shown that the clearing to the minimal value of debts accordingly to natural rules can be formulated as a fixpoint problems. The existence of their solutions, i.e. of clearing vectors, is rather straightforward and can be obtained by a direct reference to the Knaster-Tarski or Brouwer theorems. The uniqueness of clearing vectors is a more delicate problem which was solved by Eisenberg and Noe using a graph structure of the financial network. We prove uniqueness results in two generalizations of the Eisenberg-Noe model: in the Elsinger model with seniority of liabilities and in the Amini-Filipovic-Minca type model with several types of liquid assets whose firing sale has a market impact.

Research paper thumbnail of On Supremal and Maximal Sets with Respect to Random Partial Orders

SSRN Electronic Journal, 2013

The paper deals with definition of supremal sets in a rather general framework where deterministi... more The paper deals with definition of supremal sets in a rather general framework where deterministic and random preference relations (preorders) and partial orders are defined by continuous multi-utility representations. It gives a short survey of the approach developed in [5], [6] with some new results on maximal sets.

Research paper thumbnail of Consumption-Investment Optimization Problem in a LLvy Financial Model with Transaction Costs

SSRN Electronic Journal, 2013

We consider the consumption-investment optimization problem for the financial market model with c... more We consider the consumption-investment optimization problem for the financial market model with constant proportional transaction rates and Lévy price process dynamics. Contrarily to the recent work in [4], portfolio process trajectories are only left and right limited. This allows us to identify an optimal làdlàg strategy, e.g. in the two dimensional case, as it is possible to suitably rebalance the portfolio processes when they jump outside the no trade region of the solvency cone.

Research paper thumbnail of On the optimal portfolio for the exponential utility maximization: remarks to the six-author paper

Mathematical Finance, 2002

This note contains ramifications of results of Delbaen et al. (2002). Assuming that the price pro... more This note contains ramifications of results of Delbaen et al. (2002). Assuming that the price process is locally bounded and admits an equivalent local martingale measure with finite entropy, we show, without further assumption, that in the case of exponential utility the optimal portfolio process is a martingale with respect to each local martingale measure with finite entropy. Moreover, the optimal value always can be attained on a sequence of uniformly bounded portfolios.

Research paper thumbnail of Bond Market Structure in the Presence of Marked Point Processes

Mathematical Finance, 1997

We investigate the term structure of zero coupon bonds when interest rates are driven by a genera... more We investigate the term structure of zero coupon bonds when interest rates are driven by a general marked point process as well as by a Wiener process. Developing a theory which allows for measure-valued trading portfolios we study existence and uniqueness of a martingale measure. We also study completeness and its relation to the uniqueness of a martingale measure. For the case of a finite jump spectrum we give a fairly general completeness result and for a Wiener-Poisson model we prove the existence of a time-independent set of basic bonds. We also give sufficient conditions for the existence of an affine term structure.

Research paper thumbnail of Hedging under Transaction Costs in Currency Markets: a Discrete-Time Model

Mathematical Finance, 2002

We consider a discrete-time model of a currency market with transaction costs and give a descript... more We consider a discrete-time model of a currency market with transaction costs and give a description of initial endowments that allow the investor to hedge a contingent claim in various currencies by a self-®nancing portfolio.

Research paper thumbnail of Essential supremum and essential maximum with respect to random preference relations

Journal of Mathematical Economics, 2013

Research paper thumbnail of Essential supremum with respect to a random partial order

Journal of Mathematical Economics, 2013

Inspired by the theory of financial markets with transaction costs, we study a concept of essenti... more Inspired by the theory of financial markets with transaction costs, we study a concept of essential supremum in the framework where a random partial order in R d is lifted to the space L 0 (R d) of ddimensional random variables. In contrast to the classical definition, we define the essential supremum as a subset of random variables satisfying some natural properties. An application of the introduced notion to a hedging problem under transaction costs is given.

Research paper thumbnail of Consistent price systems and arbitrage opportunities of the second kind in models with transaction costs

Finance and Stochastics, 2011

In contrast with the classical models of frictionless financial markets, market models with propo... more In contrast with the classical models of frictionless financial markets, market models with proportional transaction costs, even satisfying usual no-arbitrage properties, may admit arbitrage opportunities of the second kind. This means that there are self-financing portfolios with initial endowments laying outside the solvency region but ending inside. Such a phenomenon was discovered by M. Rásonyi in the discrete-time framework. In this note we consider a rather abstract continuous-time setting and prove necessary and sufficient conditions for the property which we call No Free Lunch of the 2nd Kind, NFL2. We provide a number of equivalent conditions elucidating, in particular, the financial meaning of the property B which appeared as an indispensable "technical" hypothesis in previous papers on hedging (super-replication) of contingent claims under transaction costs. We show that it is equivalent to another condition on the "richness" of the set of consistent price systems, close to the condition PCE introduced by Rásonyi. In the last section we deduce the Rásonyi theorem from our general result using specific features of discrete-time models.

Research paper thumbnail of Hedging of American options under transaction costs

Finance and Stochastics, 2008

We consider a continuous-time model of financial market with proportional transaction costs. Our ... more We consider a continuous-time model of financial market with proportional transaction costs. Our result is a dual description of the set of initial endowments of self-financing portfolios super-replicating Americantype contingent claim. The latter is a right-continuous adapted vector process describing the number of assets to be delivered at the exercise date. We introduce a specific class of price systems, called coherent, and show that the hedging endowments are those whose "values" are larger than the expected weighted "values" of the pay-off process for every coherent price system used for the "evaluation" of the assets.

Research paper thumbnail of Mean square error for the Leland–Lott hedging strategy: convex pay-offs

Finance and Stochastics, 2010

The Leland strategy of approximate hedging of the call-option under proportional transaction cost... more The Leland strategy of approximate hedging of the call-option under proportional transaction costs prescribes to use, at equidistant instants of portfolio revisions, the classical Black-Scholes formula but with a suitably enlarged volatility. An appropriate mathematical framework is a scheme of series, i.e. a sequence of models M n with the transaction costs coefficients k n depending on n, the number of the revision intervals. The enlarged volatility σ n , in general, also depends on n. Lott investigated in detail the particular case where the transaction costs coefficients decrease as n −1/2 and where the Leland formula yields σ n not depending on n. He proved that the terminal value of the portfolio converges in probability to the pay-off g(S T) where G(x) = (x − K) +. In the present note we consider the case of much more general convex piecewise smooth pay-off functions G. We show that the convergence holds also in L 2 and find the first order term of asymptotics for the mean square error. We are working in the setting with non-uniform revision intervals and establish the asymptotic expansion when the revision dates are t n i = g(i/n) where the strictly increasing scale function g : [0, 1] → [0, 1] and its inverse f are continuous with their first and second derivatives on the whole interval or g(t) = 1 − (1 − t) β , β ≥ 1.