Narrow Results

The problem of determining the European-style option price in incomplete markets is examined within the framework of stochastic optimization. An analytic method based on the stochastic optimization is developed that gives the general formalism for determining the option price and the optimal trading strategy (optimal feedback control) that reduces the total risk inherent in writing the option. The cases involving transaction costs, the stochastic volatility with uncertainty, stochastic adaptive process, and forecasting process are considered. A software package for the option pricing for incomplete markets is developed and the results of numerical simulations are presented.

This paper is devoted to the formulation of a model for the optimal asset-liability management for insurance companies. We focus on a typical guaranteed investment contract, by which the holder has the right to receive after T years a return that cannot be lower than a minimum predefined rate r_g. We take account of the rules that usually are imposed to insurance companies in the management of this funds as reserves and solvency margin. We formulate the problem as a stochastic optimization problem in a discrete time setting comparing this approach with the so-called hedging approach. The utility function to maximize depends on various parameters including specific goals of the company management.

Some preliminary numerical results are reported to ease the comparison between the two approaches.

A nonlinear Black-Scholes equation which models transaction costs arising in the hedging of portfolios is discretized semi-implicitly using high order compact finite difference schemes. A new compact scheme, generalizing the compact schemes of Rigal [29], is derived and proved to be unconditionally stable and non-oscillatory. The numerical results are compared to standard finite difference schemes. It turns out that the compact schemes have very satisfying stability and non-oscillatory properties and are generally more efficient than the considered classical schemes.

We investigate the growth optimal strategy over a finite time horizon for a stock and bond portfolio in an analytically solvable multiplicative Markovian market model. We show that the optimal strategy consists in holding the amount of capital invested in stocks within an interval around an ideal optimal investment. The size of the holding interval is determined by the intensity of the transaction costs and the time horizon.

In this paper, we analyze the exercise behavior of warrant holders and its impact on warrant values. For this purpose, we propose a parametric model to describing the exercise volume of warrants and calibrate it to exercise data of 40 warrants from the German market. We find that few too-early exercises but also a significant number of too-late exercises occur. This observed exercise behavior results in warrant values that are more than 3% below those under the optimal exercise strategy for at-the-money warrants and the differences are even much higher for in- and out-of-the-money warrants.

We consider an investor who has available a bank account (risk free asset) and a stock (risky asset). It is assumed that the interest rate for the risk free asset is zero and the stock price is modeled by a diffusion process. The wealth can be transferred between the two assets under a proportional transaction cost. Investor is allowed to obtain loans from the bank and also to short-sell the risky asset when necessary. The optimization problem addressed here is to maximize the probability of reaching a financial goal a before bankruptcy and to obtain an optimal portfolio selection policy. Our optimal policy is a combination of local-time processes and jumps. In the interesting case, it is determined by a non-linear switching curve on the state space. This work is a generalization of Weerasinghe [20], where this switching boundary is a vertical line segment.

We investigate the optimal strategy over a finite time horizon for a portfolio of stock and bond and a derivative in an multiplicative Markovian market model with transaction costs (friction). The optimization problem is solved by a Hamilton–Jacobi–Bellman equation, which by the verification theorem has well-behaved solutions if certain conditions on a potential are satisfied. In the case at hand, these conditions simply imply arbitrage-free ("Black–Scholes") pricing of the derivative. While pricing is hence not changed by friction allow a portfolio to fluctuate around a delta hedge. In the limit of weak friction, we determine the optimal control to essentially be of two parts: a strong control, which tries to bring the stock-and-derivative portfolio towards a Black–Scholes delta hedge; and a weak control, which moves the portfolio by adding or subtracting a Black–Scholes hedge. For simplicity we assume growth-optimal investment criteria and quadratic friction.

Considerable theoretical work has been devoted to the problem of option pricing and hedging with transaction costs. A variety of methods have been suggested and are currently being used for dynamic hedging of options in the presence of transaction costs. However, very little was done on the subject of an empirical comparison of different methods for option hedging with transaction costs. In a few existing studies the different methods are compared by studying their empirical performances in hedging only a plain-vanilla short call option. The reader is tempted to assume that the ranking of the different methods for hedging any kind of option remains the same as that for a vanilla call. The main goal of this paper is to show that the ranking of the alternative hedging strategies depends crucially on the type of the option position being hedged and the risk preferences of the hedger. In addition, we present and implement a simple optimization method that, in some cases, improves considerably the performance of some hedging strategies.

In this note, we study the infinite-dimensional conditional laws of Brownian semistationary processes. Motivated by the fact that these processes are typically not semimartingales, we present sufficient conditions ensuring that a Brownian semistationary process has conditional full support, a distributional property that has two important implications. It ensures, firstly, that the process admits no free lunches under proportional transaction costs, and secondly, that it can be approximated pathwise (in the sup norm) by semimartingales that admit equivalent martingale measures.

In this paper we present a theoretical framework for determining dynamic ask and bid prices of derivatives using the theory of dynamic coherent acceptability indices in discrete time. We prove a version of the First Fundamental Theorem of Asset Pricing using the dynamic coherent risk measures. We introduce the dynamic ask and bid prices of a derivative contract in markets with transaction costs. Based on these results, we derive a representation theorem for the dynamic bid and ask prices in terms of dynamically consistent sequence of sets of probability measures and risk-neutral measures. To illustrate our results, we compute the ask and bid prices of some path-dependent options using the dynamic Gain-Loss Ratio.

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.

We study the explicit calculation of the set of superhedging portfolios of contingent claims in a discrete-time market model for d assets with proportional transaction costs. The set of superhedging portfolios can be obtained by a recursive construction involving set operations, going backward in the event tree. We reformulate the problem as a sequence of linear vector optimization problems and solve it by adapting known algorithms. The corresponding superhedging strategy can be obtained going forward in the tree. Examples are given involving multiple correlated assets and basket options. Furthermore, we relate existing algorithms for the calculation of the scalar superhedging price to the set-valued algorithm by a recent duality theory for vector optimization problems. The main contribution of the paper is to establish the connection to linear vector optimization, which allows to solve numerically multi-asset superhedging problems under transaction costs.

We consider the problem of optimizing the expected logarithmic utility of the value of a portfolio in a binomial model with proportional transaction costs with a long time horizon. By duality methods, we can find expressions for the boundaries of the no-trade-region and the asymptotic optimal growth rate, which can be made explicit for small transaction costs (in the sense of an asymptotic expansion). Here we find that, contrary to the classical results in continuous time, see Janeček and Shreve (2004), Finance and Stochastics8, 181–206, the size of the no-trade-region as well as the asymptotic growth rate depend analytically on the level λ of transaction costs, implying a linear first-order effect of perturbations of (small) transaction costs, in contrast to effects of orders λ^1/3 and λ^2/3, respectively, as in continuous time models. Following the recent study by Gerhold et al. (2013), Finance and Stochastics17, 325–354, we obtain the asymptotic expansion by an almost explicit construction of the shadow price process.

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.

American options in a multi-asset market model with proportional transaction costs are studied in the case when the holder of an option is able to exercise it gradually at a so-called mixed (randomized) stopping time. The introduction of gradual exercise leads to tighter bounds on the option price when compared to the case studied in the existing literature, where the standard assumption is that the option can only be exercised instantly at an ordinary stopping time. Algorithmic constructions for the bid and ask prices and the associated superhedging strategies and optimal mixed stopping times for an American option with gradual exercise are developed and implemented, and dual representations are established.

We study the arbitrage opportunities in the presence of transaction costs in a sequence of binary markets approximating the fractional Black–Scholes model. This approximating sequence was constructed by Sottinen and named fractional binary markets. Since, in the frictionless case, these markets admit arbitrage, we aim to determine the size of the transaction costs needed to eliminate the arbitrage from these models. To gain more insight, we first consider only 1-step trading strategies and we prove that arbitrage opportunities appear when the transaction costs are of order . Next, we characterize the asymptotic behavior of the smallest transaction costs , called "critical" transaction costs, starting from which the arbitrage disappears. Since the fractional Black–Scholes model is arbitrage-free under arbitrarily small transaction costs, one could expect that converges to zero. However, the true behavior of is opposed to this intuition. More precisely, we show, with the help of a new family of trading strategies, that converges to one. We explain this apparent contradiction and conclude that it is appropriate to see the fractional binary markets as a large financial market and to study its asymptotic arbitrage opportunities. Finally, we construct a 1-step asymptotic arbitrage in this large market when the transaction costs are of order o(1/N^H), whereas for constant transaction costs, we prove that no such opportunity exists.

This paper studies Merton’s portfolio optimization problem with proportional transaction costs in a discrete-time finite horizon. Facing short-sale and borrowing constraints, investors have access to a risk-free asset and multiple risky assets whose returns follow a multivariate geometric Brownian motion. Lower and upper bounds for optimal solutions up to the problem with 20 risky assets and 40 investment periods are computed. Three lower bounds are proposed: the value function optimization (VF), the hyper-sphere and the hyper-cube policy parameterizations (HS and HC). VF attacks the conundrums in traditional value function iteration for high-dimensional dynamic programs with continuous decision and state spaces. HS and HC respectively approximate the geometry of the trading policy in the high-dimensional state space by two surfaces. To evaluate lower bounds, two new upper bounds are provided via a duality method based on a new auxiliary problem (OMG and OMG2). Compared with existing methods across various suites of parameters, new methods lucidly show superiority. The three lower bound methods always achieve higher utilities, HS and HC cut run times by a factor of 100, and OMG and OMG2 mostly provide tighter upper bounds. In addition, how the no-trading region characterizing the optimal policy deforms when short-sale and borrowing constraints bind is investigated.

This paper studies arbitrage pricing theory in financial markets with implicit transaction costs. We extend the existing theory to include the more realistic possibility that the price at which the investors trade is dependent on the traded volume. The investors in the market always buy at the ask and sell at the bid price. Implicit transaction costs are composed of two terms, one is able to capture the bid-ask spread, and the second the price impact. Moreover, a new definition of a self-financing portfolio is obtained. The self-financing condition suggests that continuous trading is possible, but is restricted to predictable trading strategies having cádlág (right-continuous with left limits) and cáglád (left-continuous with right limits) paths of bounded quadratic variation and of finitely many jumps. That is, cádlág and cáglád predictable trading strategies of infinite variation, with finitely many jumps and of finite quadratic variation are allowed in our setting. Restricting ourselves to cáglád predictable trading strategies, we show that the existence of an equivalent probability measure is equivalent to the absence of arbitrage opportunities, so that the first fundamental theorem of asset pricing (FFTAP) holds. It is also shown that the use of continuous and bounded variation trading strategies can improve the efficiency of hedging in a market with implicit transaction costs. To better understand how to apply the theory proposed we provide an example of an implicit transaction cost economy that is linear and nonlinear in the order size.

We consider so-called regular invertible Gaussian Volterra processes and derive a formula for their prediction laws. Examples of such processes include the fractional Brownian motions and the mixed fractional Brownian motions. As an application, we consider conditional-mean hedging under transaction costs in Black–Scholes type pricing models where the Brownian motion is replaced with a more general regular invertible Gaussian Volterra process.

We present an expansion for portfolio optimization in the presence of small, instantaneous, quadratic transaction costs. Specifically, the magnitude of transaction costs has a coefficient that is of the order $𝜖$ small, which leads to the optimization problem having an asymptotically-singular Hamilton–Jacobi–Bellman equation whose solution can be expanded in powers of $\sqrt{𝜖}$. In this paper, we derive explicit formulae for the first two terms of this expansion. Analysis and simulation are provided to show the behavior of this approximating solution.