Limit distributions for the error in approximations of stochastic integrals by Riemann sums with stochastic partitions are studied. The integrands and integrators are supposed to be one-dimensional continuous semimartingales. Lower bounds for asymptotic conditional variance of the error are given and effective discretization schemes which attain the bounds are explicitly constructed. Two examples of their applications are given; efficient delta hedging strategies under fixed or linear transaction costs and effective discretization schemes for the Euler–Maruyama approximation are constructed.

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Discretization error of stochastic integrals

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 lying outside the solvency region but ending inside Such a phenomenon was discovered by M Rasonyi in the discrete-time framework In this note, we consider a rather abstract continuous-time setting and prove necessary and sufficient conditions for a property which we call no free lunch of the second 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 (superreplication) 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 Rasonyi In the last section, we deduce the Rasonyi theorem from our general result by using specific features of discrete-time models

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

We consider weak convergence of the rescaled error processes arising from Riemann discretizations of certain stochastic integrals and relate the $L_p$-integrability of the weak limit to the fractional smoothness in the Malliavin sense of the stochastic integral.

Weak convergence of error processes in discretizations of stochastic integrals and Besov spaces

Introduced by Artzner et al. (1998) the axiomatic characterization of a static coherent risk measure was extended by Jouini et al. (2004) in a multi-dimensional setting to the concept of vector-valued risk measures. In this paper, we propose a dynamic version of the vector-valued risk measures in a continuous-time framework. Particular attention is devoted to the choice of a convenient risk space. We provide dual characterization results, we study different notions of time consistency and we give examples of vector-valued risk measure processes.

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Vector-valued coherent risk measure processes

We present a novel arbitrage-related notion for markets with transaction costs in discrete time and characterize it in terms of price systems. Pertinence of this concept is demonstrated. A discussion of the case with one risky asset and an outlook on continuous-time models complement the main result.

Arbitrage Under Transaction Costs Revisited

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 formula with a suitably enlarged volatility. The formal mathematical framework is a scheme of series, i.e., a sequence of models with transaction cost coefficients kn=k0n−α, where α∈[0,1/2] and n is the number of portfolio revision dates. The enlarged volatility $\widehat{\sigma}_{n}$ in general depends on n except for the case which was investigated in detail by Lott, to whom belongs the first rigorous result on convergence of the approximating portfolio value $V^{n}_{T}$ to the pay-off VT. In this paper, we consider only the Lott case α=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 the asymptotics. We are working in a setting with non-uniform revision intervals and establish the asymptotic expansion when the revision dates are $t_{i}^{n}=g(i/n)$, 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. We show that the sequence $n^{1/2}(V_{T}^{n}-V_{T})$ converges in law to a random variable which is the terminal value of a component of a two-dimensional Markov diffusion process and calculate the limit. Our central result is a functional limit theorem for the discrepancy process.

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Mean square error for the Leland–Lott hedging strategy: convex pay-offs

We consider a continuous-time model of a financial market with proportional transaction costs. Our result is a dual description of the set of initial endowments of self-financing portfolios super-replicating an American-type 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 payoff process for every coherent price system used for the “evaluation” of the assets.

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Hedging of American options under transaction costs

Leland’s approach to the hedging of derivatives under proportional transaction costs is based on an approximate replication of the European-type contingent claim V T using the classical Black–Scholes formula with a suitably enlarged volatility. The formal mathematical framework is a scheme of series, i.e., a sequence of models with transaction cost coefficients k n =k 0 n −α , where α∈[0,1/2] and n is the number of portfolio revision dates. The enlarged volatility $\widehat{\sigma}_{n}$ in general depends on n except for the case which was investigated in detail by Lott, to whom belongs the first rigorous result on convergence of the approximating portfolio value $V^{n}_{T}$ to the pay-off V T . In this paper, we consider only the Lott case α=1/2. We prove first, for an arbitrary pay-off V T =G(S T ) where G is a convex piecewise smooth function, that the mean square approximation error converges to zero with rate n −1/2 in L 2 and find the first order term of the asymptotics. We are working in a setting with non-uniform revision intervals and establish the asymptotic expansion when the revision dates are $t_{i}^{n}=g(i/n)$, 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. We show that the sequence $n^{1/2}(V_{T}^{n}-V_{T})$ converges in law to a random variable which is the terminal value of a component of a two-dimensional Markov diffusion process and calculate the limit. Our central result is a functional limit theorem for the discrepancy process.

Mean square error for the Leland-Lott hedging strategy: convex pay-offs

We study the Leland model for hedging portfolios in the presence of a constant proportional transaction costs coecient. The modified Leland's strategy defined in (2), contrarily to the classical one, ensures the asymptotic replication of a large class of payo. In this setting, we prove a limit theorem for the deviation between the real portfolio and the payo. As Pergamenshchikov did in the framework of the usual Leland's strategy (11), we identify the rate of convergence and the associated limit distribution. This rate turns out to be improved using the modified strategy and non periodic revision dates.

https://hal.archives-ouvertes.fr/hal-00467704/document

Emmanuel Denis

Papers

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

Mean square error for the Leland–Lott hedging strategy: convex pay-offs

Hedging of American options under transaction costs

Mean square error for the Leland-Lott hedging strategy: convex pay-offs

Limit Theorem for a Modified Leland Hedging Strategy under Constant Transaction Costs rate