International Journal of Theoretical and Applied Finance

In this paper, we show how to construct dynamic forward rate models in terms of exogenously specified eigenfunctions (or factor loadings). We also show how to link forward rate models with different number of driving Brownian motions to each other in a way consistent with the implied eigenfunctions. Finally, we discuss how to best parameterize the models in the sense of maximizing the number of free parameters for a given set of eigenfunctions.

The goal of this work is to study binary market models with transaction costs, and to characterize their arbitrage opportunities. It has been already shown that the absence of arbitrage is related to the existence of λ-consistent price systems (λ-CPS), and, for this reason, we aim to provide conditions under which such systems exist. More precisely, we give a characterization for the smallest transaction cost λ_c (called "critical" λ) starting from which one can construct a λ-CPS. We also provide an expression for the set of all probability measures inducing λ-CPS. We show in particular that in the transition phase "λ = λ_c" these sets are empty if and only if the frictionless market admits arbitrage opportunities. As an application, we obtain an explicit formula for λ_c depending only on the parameters of the model for homogeneous and also for some semi-homogeneous binary markets.

In this paper, we develop the idea that firm sizes evolve as log Brownian motions dS_t = S_t(σdW_t + μdt) where the constants μ, σ are characteristics of the firm, chosen from some distribution, and that the firms are wound up at some random time. At any given time, we see a firm of a given size. What can we say about its characteristics given its size? How would we invest in such a market? What do these assumptions imply about the distribution of sizes? By making simple and well-chosen modeling assumptions, we are able to develop quite concrete forms of the dependence of firm characteristics on size, from which we are able to deduce optimal investment weights as a function of size alone. As in the approach of Fernholz [2002, Stochastic Portfolio Theory. Springer], this avoids the need to estimate growth rates of stocks in order to decide on investment strategy.

In this paper, we provide a new representation result for dynamic capital allocations and dynamic convex risk measures that are based on backward stochastic differential equations (BSDEs). We derive this representation from a classical differentiability result for BSDEs and the full allocation property of the Aumann–Shapley allocation. The representation covers BSDE-based dynamic convex and dynamic coherent risk measures. The results are applied to derive a representation for the dynamic entropic risk measure. Our results are also applicable in a specific way to the static case, where we are able to derive a new representation result for static convex risk measures that are Gâteaux-differentiable.

For prices of options with barrier and lookback features, defaultable bonds and credit default swaps (CDSs), and probability distribution functions in Lévy models, as well as for joint probability distributions of a Lévy process and its supremum or/and infimum, one can derive explicit analytical formulas in terms of inverse Laplace/Fourier transforms and the Wiener–Hopf factorization. Unless the characteristic exponent is rational, the main examples being Brownian motion, double exponential jump-diffusion and hyper-exponential jump-diffusion models, accurate numerical realization of these formulas is difficult or very time consuming, especially for options of very long and very short maturities. In this paper, a systematic approach to contour deformations in pricing formulas is developed, which greatly increases the accuracy and speed of calculations; the efficiency of the method is demonstrated with numerical examples. For options and CDSs of moderate and long maturities, much faster asymptotic formulas of comparable level of accuracy are developed.

Market making and optimal portfolio liquidation in the context of electronic limit order books are of considerably practical importance for high frequency (HF) market makers as well as more traditional brokerage firms supplying optimal execution services for clients. In general, the two problems are based on probabilistic models defined on certain reference probability spaces. However, due to uncertainty in model parameters or in periods of extreme market turmoil, ambiguity concerning the correct underlying probability measure may appear and an assessment of model risk, as well as the uncertainty on the choice of the model itself, becomes important, as for a market maker or a trader attempting to liquidate large positions, the uncertainty may result in unexpected consequences due to severe mispricing. This paper focuses on the market making and the optimal liquidation problems using limit orders, accounting for model risk or uncertainty. Both are formulated as stochastic optimal control problems, with the controls being the spreads, relative to a reference price, at which orders are placed. The models consider uncertainty in both the drift and volatility of the underlying reference price, for the study of the effect of the uncertainty on the behavior of the market maker, accounting also for inventory restriction, as well as on the optimal liquidation using limit orders.