Showing papers in "Finance and Stochastics in 1997"
TL;DR: A number of stochastic processes with normal inverse Gaussian marginals and various types of dependence structures are discussed, including Ornstein-Uhlenbeck type processes, superpositions of such processes and Stochastic volatility models in one and more dimensions.
Abstract: With the aim of modelling key stylized features of observational series from finance and turbulence a number of stochastic processes with normal inverse Gaussian marginals and various types of dependence structures are discussed. Ornstein-Uhlenbeck type processes, superpositions of such processes and stochastic volatility models in one and more dimensions are considered in particular, and some discussion is given of the feasibility of making likelihood inference for these models.
1,323 citations
TL;DR: Stochastic differential equations are derived for term structures of forward libor and swap rates, and shown to have a unique positive solution when the percentage volatility function is bounded, implying existence of an arbitrage-free model with such volatility specification.
Abstract: A self-contained theory is presented for pricing and hedging LIBOR and swap derivatives by arbitrage. Appropriate payoff homogeneity and mea- surability conditions are identified which guarantee that a given payoff can be attained by a self-financing trading strategy. LIBOR and swap derivatives satisfy this condition, implying they can be priced and hedged with a finite number of zero-coupon bonds, even when there is no instantaneous saving bond. No- tion of locally arbitrage-free price system is introduced and equivalent criteria established. Stochastic differential equations are derived for term structures of forward libor and swap rates, and shown to have a unique positive solution when the percentage volatility function is bounded, implying existence of an arbitrage- free model with such volatility specification. The construction is explicit for the lognormal LIBOR and swap "market models", the former following Musiela and Rutkowski (1995). Primary examples of LIBOR and swap derivatives are discussed and appropriate practical models suggested for each.
715 citations
TL;DR: The stylized facts surveyed in this paper shed new light on the market structure that appears composed of heterogeneous agents and poses several challenges such as the definition of price and of the time-scale, the concepts of risk and efficiency, the modeling of the markets and the learning process.
Abstract: This paper presents stylized facts concerning the spot intra-daily for- eign exchange markets. It first describes intra-daily data and proposes a set of definitions for the variables of interest. Empirical regularities of the foreign ex- change intra-daily data are then grouped under three major topics: the distribution of price changes, the process of price formation and the heterogeneous structure of the market. The stylized facts surveyed in this paper shed new light on the market structure that appears composed of heterogeneous agents. It also poses several challenges such as the definition of price and of the time-scale, the con- cepts of risk and efficiency, the modeling of the markets and the learning process.
566 citations
TL;DR: It is shown that a market is approximately complete iff an equivalent martingale measure is unique and two constructions of stochastic integrals with respect to processes taking values in a space of continuous functions are suggested.
Abstract: To the memory of our friend and colleague Oliviero Lessi. Abstract. The main purpose of the paper is to provide a mathematical back- ground for the theory of bond markets similar to that available for stock markets. We suggest two constructions of stochastic integrals with respect to processes taking values in a space of continuous functions. Such integrals are used to define the evolution of the value of a portfolio of bonds corresponding to a trad- ing strategy which is a measure-valued predictable process. The existence of an equivalent martingale measure is discussed and HJM-type conditions are derived for a jump-diffusion model. The question of market completeness is considered as a problem of the range of a certain integral operator. We introduce a concept of approximate market completeness and show that a market is approximately complete iff an equivalent martingale measure is unique.
253 citations
TL;DR: This paper considers the valuation of an option with time to expiration and pay-off function which is a convex function, and constant interest rate, and finds that, for “most” such models, the range of the values of the option is the interval, this interval being the biggest interval in which the values must lie, whatever model is used.
Abstract: In this paper we consider the valuation of an option with time to expiration \(T\) and pay-off function \(g\) which is a convex function (as is a European call option), and constant interest rate \(r\), in the case where the underlying model for stock prices \((S_t)\) is a purely discontinuous process (hence typically the model is incomplete). The main result is that, for “most” such models, the range of the values of the option, using all possible equivalent martingale measures for the valuation, is the interval \((e^{-rT}g(e^{rT}S_0),S_0)\), this interval being the biggest interval in which the values must lie, whatever model is used.
182 citations
TL;DR: This paper gives a new proof which uses techniques from stochastic calculus rather than functional analysis, and which removes any boundedness assumption.
Abstract: Let \({\cal Q}\) 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 \({\cal Q}\). The optional decomposition theorem for \(X\) states that there exists a predictable integrand \(\varphi\) such that the difference \(X-\varphi\cdot S\) 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.
177 citations
TL;DR: The problem of term structure of interest rates modelling is considered in a continuous-time framework, with the emphasis on the bond prices, forward bond prices and so-called LIBOR rates, rather than on the instantaneous continuously compounded rates.
Abstract: The problem of term structure of interest rates modelling is considered in a continuous-time framework. The emphasis is on the bond prices, forward bond prices and so-called LIBOR rates, rather than on the instantaneous continuously compounded rates as in most traditional models. Forward and spot probability measures are introduced in this general set-up. Two conditions of no-arbitrage between bonds and cash are examined. A process of savings account implied by an arbitrage-free family of bond prices is identified by means of a multiplicative decomposition of semimartingales. The uniqueness of an implied savings account is established under fairly general conditions. The notion of a family of forward processes is introduced, and the existence of an associated arbitrage-free family of bond prices is examined. A straightforward construction of a lognormal model of forward LIBOR rates, based on the backward induction, is presented.
169 citations
TL;DR: The main results give necessary and/or sufficient conditions for the closedness of G_T(\Theta ) in ${\cal L} ^2(P)$.
Abstract: Let \(X\) be an \({\Bbb R}^d\)-valued special semimartingale on a probability space \((\Omega , {\cal F} , ({\cal F} _t)_{0 \leq t \leq T} ,P)\) with canonical decomposition \(X=X_0+M+A\). Denote by \(G_T(\Theta )\) the space of all random variables \((\theta \cdot X)_T\), where \(\theta \) is a predictable \(X\)-integrable process such that the stochastic integral \(\theta \cdot X\) is in the space \({\cal S} ^2\) of semimartingales. We investigate under which conditions on the semimartingale \(X\) the space \(G_T(\Theta )\) is closed in \({\cal L} ^2(\Omega , {\cal F} ,P)\), a question which arises naturally in the applications to financial mathematics. Our main results give necessary and/or sufficient conditions for the closedness of \(G_T(\Theta )\) in \({\cal L} ^2(P)\). Most of these conditions deal with BMO-martingales and reverse Holder inequalities which are equivalent to weighted norm inequalities. By means of these last inequalities, we also extend previous results on the Follmer-Schweizer decomposition.
131 citations
TL;DR: Filtering and parameter estimation techniques from hidden Markov Models are applied to a discrete time asset allocation problem and for the commonly used mean-variance utility explicit optimal strategies are obtained.
Abstract: Filtering and parameter estimation techniques from hidden Markov Models are applied to a discrete time asset allocation problem. For the commonly used mean-variance utility explicit optimal strategies are obtained.
97 citations
TL;DR: In this paper, the limiting hedging error of the Leland strategy for the approximate pricing of the European call option in a market with transactions costs was shown to be not equal to zero in the case when the level of transactions costs is a constant, in contradiction with the claim in Leland.
Abstract: 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).
95 citations
TL;DR: This article shows that it is not necessary to make clever choices of the grid positioning, and by interpreting the nature of the binomial approximation appropriately, it is able to derive very quick and accurate pricings of barrier options.
Abstract: We discuss here an alternative interpretation of the familiar binomial lattice approach to option pricing, illustrating it with reference to pricing of barrier options, one- and two-sided, with fixed, moving or partial barriers, and also the pricing of American put options. It has often been observed that if one tries to price a barrier option using a binomial lattice, then one can find slow convergence to the true price unless care is taken over the placing of the grid points in the lattice; see, for example, the work of Boyle & Lau [2]. The placing of grid points is critical whether one uses a dynamic programming approach, or a Monte Carlo approach, and this can make it difficult to compute hedge ratios, for example. The problems arise from translating a crossing of the barrier for the continuous diffusion process into an event for the binomial approximation. In this article, we show that it is not necessary to make clever choices of the grid positioning, and by interpreting the nature of the binomial approximation appropriately, we are able to derive very quick and accurate pricings of barrier options. The interpretation we give here is applicable much more widely, and helps to smooth out the ‘odd-even’ ripples in the option price as a function of time-to-go which are a common feature of binomial lattice pricing.
TL;DR: A probabilistic change-of-numeraire technique to compute closed-form prices of European options to exchange one asset against another when the relative price of the underlying assets follows a diffusion process with natural boundaries and a quadratic diffusion coefficient is used.
Abstract: This paper uses a probabilistic change-of-numeraire technique to compute closed-form prices of European options to exchange one asset against another when the relative price of the underlying assets follows a diffusion process with natural boundaries and a quadratic diffusion coefficient. The paper shows in particular how to interpret the option price formula in terms of exercise probabilities which are calculated under the martingale measures associated with two specific numeraire portfolios. An application to the pricing of bond options and certain interest rate derivatives illustrates the main results.
TL;DR: The existence of a unique equivalent measure up to an explosion time is proved by means of results that give a handle on situations where an equivalent martingale measure cannot exist.
Abstract: Simple sufficient conditions for the existence of a unique equivalent martingale measure are provided. Furthermore, these conditions give us a handle on situations where an equivalent martingale measure cannot exist. The existence of a unique equivalent martingale measure is of relevance to problems in mathematical finance. Two examples of models for which the question of existence was unresolved are studied. By means of our results existence of a unique equivalent measure up to an explosion time is proved.
TL;DR: A general result is proved relating the minimal pricing error one incurs in pricing all bonds with one term structure to the maximal arbitrage profit one can achieve with restricted portfolios.
Abstract: This paper proposes a methodology for simultaneously computing a smooth estimator of the term structure of interest rates and economically justified bounds for it. It unifies existing estimation procedures that apply regression, smoothing and linear programming methods. Our methodology adjusts for possibly asymmetric transaction costs. Various regression and smoothing techniques have been suggested for estimating the term structure. They focus on what functional form to choose or which measure of smoothness to maximize, mostly neglecting the discussion of the appropriate measure of fit. Arbitrage theory provides insight into how small the pricing error will be and in which sense, depending on the structure of transaction costs. We prove a general result relating the minimal pricing error one incurs in pricing all bonds with one term structure to the maximal arbitrage profit one can achieve with restricted portfolios.
TL;DR: The law of $x_t$ under the forward measure is identified when x_t is a diffusion process or, more generally, a continuous-path Markov process.
Abstract: For a Markov process \(x_t\), the forward measure \(P^T\) over the time interval \([0,T]\) is defined by the Radon-Nikodym derivative \(dP^T/dP = M\exp(-\int_0^Tc(x_s)ds)\), where \(c\) is a given non-negative function and \(M\) is the normalizing constant. In this paper, the law of \(x_t\) under the forward measure is identified when \(x_t\) is a diffusion process or, more generally, a continuous-path Markov process.
TL;DR: The closed form pricing formulae for contracts written on zero coupon bonds for the lognormal forward LIBOR rates are derived and are purely probabilistic in contrast with the earlier results obtained.
Abstract: We derive the closed form pricing formulae for contracts written on zero coupon bonds for the lognormal forward LIBOR rates. The method is purely probabilistic in contrast with the earlier results obtained by Miltersen et al. (1997).