Showing papers in "Mathematical Finance in 2003"
TL;DR: In this article, a mean-corrected exponential model is used to obtain a martingale in the filtration in which it was originally defined, and the important property of martingales in altered filtrations consistent with the one-dimensional marginal distributions of the level of the process at each future date.
Abstract: Three processes reflecting persistence of volatility are initially formulated by evaluating three Levy processes at a time change given by the integral of a mean-reverting square root process. The model for the mean-reverting time change is then generalized to include non-Gaussian models that are solutions to Ornstein-Uhlenbeck equations driven by one-sided discontinuous Levy processes permitting correlation with the stock. Positive stock price processes are obtained by exponentiating and mean correcting these processes, or alternatively by stochastically exponentiating these processes. The characteristic functions for the log price can be used to yield option prices via the fast Fourier transform. In general mean-corrected exponentiation performs better than employing the stochastic exponential. It is observed that the mean-corrected exponential model is not a martingale in the filtration in which it is originally defined. This leads us to formulate and investigate the important property of martingale marginals where we seek martingales in altered filtrations consistent with the one-dimensional marginal distributions of the level of the process at each future date.
749 citations
TL;DR: In this article, the authors investigated the Ornstein-Uhlenbeck type of stochastic volatility models from the viewpoint of derivative asset analysis and derived the price of simple European-style contracts in closed form.
Abstract: Stochastic volatility models of the Ornstein-Uhlenbeck type possess authentic capability of capturing some stylized features of financial time series. In this work we investigate this class of models from the viewpoint of derivative asset analysis. We discuss topics related to the incompleteness of this type of markets. In particular, for structure preserving martingale measures, we derive the price of simple European-style contracts in closed form. Furthermore, the range of viable prices is determined and an empirical application is presented.
296 citations
TL;DR: In this article, the authors present a new framework for fractional Brownian motion in which processes with all indices can be considered under the same probability measure, and develop option pricing in a fractional Black-Scholesmarket with a noise process driven by a sum of fractional brownian motions with various Hurst indices.
Abstract: We present a new framework for fractional Brownian motion in which processes with all indices can be considered under the same probability measure. Our results extend recent contributions by Hu, Oksendal, Duncan, Pasik-Duncan, and others. As an application we develop option pricing in a fractional Black-Scholesmarket with a noise process driven by a sum of fractional Brownian motions with various Hurst indices.
291 citations
TL;DR: In this article, the arbitrage-free dynamics of interest rates, in the presence of both jumps and diffusion, when the term structure is modeled through simple forward rates (i.e., through discretely compounded forward rates evolving continuously in time) or forward swap rates was analyzed.
Abstract: This paper characterizes the arbitrage-free dynamics of interest rates, in the presence of both jumps and diffusion, when the term structure is modeled through simple forward rates (i.e., through discretely compounded forward rates evolving continuously in time) or forward swap rates. Whereas instantaneous continuously compounded rates form the basis of most traditional interest rate models, simply compounded rates and their parameters are more directly observable in practice and are the basis of recent researchon “market models.” We consider very general types of jump processes, modeled through marked point processes, allowing randomness in jump sizes and dependence between jump sizes, jump times, and interest rates. We make explicit how jump and diffusion risk premia enter into the dynamics of simple forward rates. We also formulate reasonably tractable subclasses of models and provide pricing formulas for some derivative securities, including interest rate caps and options on swaps. Through these formulas, we illustrate the effect of jumps on implied volatilities in interest rate derivatives. This paper characterizes the arbitrage-free dynamics of interest rates, in the presence of both jumps and diffusion, when the term structure is modeled through simple forward rates—that is, through discretely compounded forward rates evolving continuously in time—or through forward swap rates. We consider very general types of jump processes (allowing randomness in jump sizes and dependence between jump sizes, jump times, and interest rates) and identify how jump and diffusion risk premia enter into the dynamics of simple forward rates. We also formulate a reasonably tractable subclass of models and provide pricing formulas for some term structure derivatives. Our investigation builds on several strands of research, in particular on the dynamics of instantaneous continuously compounded rates (as in Heath, Jarrow, and Morton 1992), option pricing withjumps (as in Merton 1976), LIBOR and swap rate market models (including Brace, Gatarek, and Musiela 1997; Jamshidian 1997; Miltersen,
125 citations
TL;DR: In this article, the authors study Merton's classical portfolio optimization problem for an investor who can trade in a risk-free bond and a stock, where the goal of the investor is to allocate money so that her expected utility from terminal wealth is maximized.
Abstract: We study Merton's classical portfolio optimization problem for an investor who can trade in a risk-free bond and a stock. The goal of the investor is to allocate money so that her expected utility from terminal wealth is maximized. The special feature of the problem studied in this paper is the inclusion of stochastic volatility in the dynamics of the risky asset. The model we use is driven by a superposition of non-Gaussian Ornstein-Uhlenbeck processes and it was recently proposed and intensively investigated for real market data by Barndorff-Nielsen and Shephard (2001). Using the dynamic programming method, explicit trading strategies and expressions for the value function via Feynman-Kac formulas are derived and verified for power utilities. Some numerical examples are also presented.
76 citations
TL;DR: In this paper, the authors consider simple models of financial markets with regular traders and insiders possessing some extra information hidden in a random variable that is accessible to the regular trader only at the end of the trading interval.
Abstract: We consider simple models of financial markets with regular traders and insiders possessing some extra information hidden in a random variable that is accessible to the regular trader only at the end of the trading interval. The problems we focus on are the calculation of the additional utility of the insider and a study of his free lunch possibilities. The information drift—that is, the drift to eliminate in order to preserve the martingale property in the insider's filtration—turns out to be the crucial quantity needed to answer these questions. It is most elegantly described by the logarithmic Malliavin trace of the conditional laws of the insider information with respect to the filtration of the regular trader. Several examples are given to illustrate additional utility and free lunch possibilities. In particular, if the insider has advance knowledge of the maximal stock price process, given by a regular diffusion, arbitrage opportunities exist.
76 citations
TL;DR: In this paper, the authors introduce the intensity-based defaultable Levy term structure model, based on the Heath-Jarrow-Morton approach of Bielecki and Rutkowski.
Abstract: We introduce the intensity-based defaultable Levy term structure model. It generalizes the default-free Levy term structure model by Eberlein and Raible, and the intensity-based defaultable Heath-Jarrow-Morton approach of Bielecki and Rutkowski. Furthermore, we include the concept of multiple defaults, based on Schonbucher, within this generalization.
72 citations
TL;DR: A general micromovement model that describes transactional price behavior and its recursive algorithm are presented in detail and a theorem on the convergence of conditional expectation of the model is proved.
Abstract: A general micromovement model that describes transactional price behavior is proposed. The model ties the sample characteristics of micromovement and macromovement in a consistent manner. An important feature of the model is that it can be transformed to a filtering problem with counting process observations. Consequently, the complete information of price and trading time is captured and then utilized in Bayes estimation via filtering for the parameters. The filtering equations are derived. A theorem on the convergence of conditional expectation of the model is proved. A consistent recursive algorithm is constructed via the Markov chain approximation method to compute the approximate posterior and then the Bayes estimates. A simplified model and its recursive algorithm are presented in detail. Simulations show that the computed Bayes estimates converge to their true values. The algorithm is applied to one month of intraday transaction prices for Microsoft and the Bayes estimates are obtained.
68 citations
TL;DR: In this article, the problem of minimal variance hedging in a market driven by a Levy martingale claim is studied, and an explicit formula for the minimal variance portfolio in terms of Malliavin derivatives is given.
Abstract: In a market driven by a Levy martingale, we consider a claim ξ. We study the problem of minimal variance hedging and we give an explicit formula for the minimal variance portfolio in terms of Malliavin derivatives. We discuss two types of stochastic (Malliavin) derivatives for ξ: one based on the chaos expansion in terms of iterated integrals with respect to the power jump processes and one based on the chaos expansion in terms of iterated integrals with respect to the Wiener process and the Poisson random measure components. We study the relation between these two expansions, the corresponding two derivatives, and the corresponding versions of the Clark-Haussmann-Ocone theorem.
64 citations
TL;DR: In this paper, the authors consider a financial market with an insider that has, at time t = 0, some additional information of the whole developing of the market, and use the forward integral, which is an anticipating integral, and the techniques of the Malliavin calculus so that they can take advantage of the privileged information to maximize the expected logarithmic utility from terminal wealth.
Abstract: In this paper we consider a financial market with an insider that has, at time t= 0, some additional information of the whole developing of the market. We use the forward integral, which is an anticipating integral, and the techniques of the Malliavin calculus so that we can take advantage of the privileged information to maximize the expected logarithmic utility from terminal wealth.
58 citations
TL;DR: Gobet and Kohatsu-Higa as mentioned in this paper derived additional weights that enable the numerical computation of the Greeks using Monte Carlo simulations, which is a multidimensional extension of previous results.
Abstract: In this paper, we consider the problem of the numerical computation of Greeks for a multidimensional barrier and lookback style options: the payoff function depends in a rather general way on the minima and maxima of the coordinates of the d-dimensional underlying asset process. Using Malliavin calculus techniques, we derive additional weights that enable computation of the Greeks using Monte Carlo simulations. Numerical experiments confirm the efficiency of the method. This work is a multidimensional extension of previous results (see Gobet and Kohatsu-Higa 2001).
TL;DR: In this paper, the authors reexamine the Malliavin weighting functions introduced by Fournie et al. as a new method for efficient and fast computations of the Greeks.
Abstract: This paper reexamines the Malliavin weighting functions introduced by Fournie et al. (1999) as a new method for efficient and fast computations of the Greeks. Reexpressing the weighting function generator in terms of its Skorohod integrand, we show that these weighting functions have to satisfy necessary and sufficient conditions expressed as conditional expectations. We then derive the weighting function with the smallest total variance. This is of particular interest as it bridges the method of Malliavin weights and the one of likelihood ratio, as introduced by Broadie and Glasserman (1996). The likelihood ratio is precisely the weighting function with the smallest total variance. We finally examine when to use the Malliavin method and when to prefer finite difference.
TL;DR: A new universal portfolio algorithm is presented that achieves almost the same level of wealth as could be achieved by knowing stock prices ahead of time by tracking the best in hindsight wealth achievable within target classes of linearly parameterized portfolio sequences.
Abstract: We present a new universal portfolio algorithm that achieves almost the same level of wealth as could be achieved by knowing stock prices ahead of time. Specifically the algorithm tracks the best in hindsight wealth achievable within target classes of linearly parameterized portfolio sequences. The target classes considered are more general than the standard constant rebalanced portfolio class and permit portfolio sequences to exhibit a continuous form of dependence on past prices or other side information. A primary advantage of the algorithm is that it is easily computable in a polynomial number of steps by way of simple closed-form expressions. This provides an edge over other universal algorithms that require both an exponential number of computations and numerical approximation.
TL;DR: In this paper, a first order scheme based on piecewise linear approximations is proposed to deal with the problem of pricing and hedging American option prices and a priori estimates of the errors are established.
Abstract: The numerical quantization method (see [B.P.1, B.P.2, B.P.P.1]) is a grid method which relies on the approximation of the solution of a nonlinear problem (e.g. backward Kolmogorov equation) by piecewise constant functions. Its purpose is to compute a large number of conditional expectations along the path of the associated diffusion process. We give here an improvement of this method by describing a first order scheme based on piecewise linear approximations. Main ingredients are correction terms in the transition probabilities weights. We emphasize the fact that in the case of optimal quantization, a non neglectable number of correction terms vanish. We think that this is a strong argument to use it. The problem of pricing and hedging American options is investigated and a priori estimates of the errors are established.
TL;DR: In this article, a geometric analysis of iterated crossvolatilities of asset prices is adopted to assess the stability of the (risk-free) measure under infinitesimal perturbations.
Abstract: Geometric analysis of iterated cross-volatilities of asset prices is adopted to assess the stability of the (risk-free) measure under infinitesimal perturbations. Perturbations of asset prices evolve through time according to an ordinary linear differential equation (hedged transfer). The decay (feedback) rate is explicitly computed through a Fourier series method implemented on high frequency time series.
TL;DR: In this paper, the quadratic error of a discrete time-hedging strategy in a complete multidimensional model was derived for any options with Lipschitz payoff and 1/n 1/4 for options with irregular payoff.
Abstract: The aim of this paper is to compute the quadratic error of a discrete time-hedging strategy in a complete multidimensional model. This result extends that of Gobet and Temam (2001) and Zhang (1999). More precisely, our basic assumption is that the asset prices satisfy the d-dimensional stochastic differential equation dXit=Xit(bi(Xt)dt+σi,j(Xt)dWjt). We precisely describe the risk of this strategy with respect to n, the number of rebalancing times. The rates of convergence obtained are for any options with Lipschitz payoff and 1/n1/4 for options with irregular payoff.
TL;DR: In this article, an error calculus based on Dirichlet forms is proposed for risk management in stochastic financial models, which is an extension of the Malliavin calculus.
Abstract: In the framework of risk management, for the study of the sensitivity of pricing and hedging in stochastic financial models to changes of parameters and to perturbations of the stock prices, we propose an error calculus which is an extension of the Malliavin calculus based on Dirichlet forms. Although useful also in physics, this error calculus is well adapted to stochastic analysis and seems to be the best practicable in finance. This tool is explained here intuitively and with some simple examples.
TL;DR: In this article, the convergence rate of the approximation of quantiles of the law of one component of (Xt) is investigated when one uses a Monte Carlo method combined with the Euler discretization scheme.
Abstract: In this paper we briefly present the results obtained in our paper (Talay and Zheng 2002a) on the convergence rate of the approximation of quantiles of the law of one component of (Xt), where (Xt) is a diffusion process, when one uses a Monte Carlo method combined with the Euler discretization scheme. We consider the case where (Xt) is uniformly hypoelliptic (in the sense of Condition (UH) below), or the inverse of the Malliavin covariance of the component under consideration satisfies the condition (M) below. We then show that Condition (M) seems widely satisfied in applied contexts. We particularly study financial applications: the computation of quantiles of models with stochastic volatility, the computation of the VaR of a portfolio, and the computation of a model risk measurement for the profit and loss of a misspecified hedging strategy.
TL;DR: In this article, the authors consider a Black and Scholes economy and investigate two approaches to hedge contingent claims: the general Malliavin calculus approach and the classical Delta-hedging formula under weaker conditions.
Abstract: In this paper we consider a Black and Scholes economy and investigate two approaches to hedging contingent claims. We show that the general Malliavin calculus approach can generate the classical Delta-hedging formula under weaker conditions.
TL;DR: In this article, the problem of computing hedging portfolios for options that may have discontinuous payoffs is considered in the framework of diffusion models in which the number of factors may be larger than the total number of Brownian motions driving the model.
Abstract: We consider the problem of computing hedging portfolios for options that may have discontinuous payoffs, in the framework of diffusion models in which the number of factors may be larger than the number of Brownian motions driving the model. Extending the work of Fournie et al. (1999), as well as Ma and Zhang (2000), using integration by parts of Malliavin calculus, we find two representations of the hedging portfolio in terms of expected values of random variables that do not involve differentiating the payoff function. Once this has been accomplished, the hedging portfolio can be computed by simple Monte Carlo. We find the theoretical bound for the error of the two methods. We also perform numerical experiments in order to compare these methods to two existing methods, and find that no method is clearly superior to others.
TL;DR: In this paper, a portfolio problem with control on downside losses is considered, and the optimal policy is equivalent to the hedging portfolio of a European option on a dynamic mutual fund that can be replicated by market primary assets.
Abstract: This paper considers a portfolio problem with control on downside losses. Incorporating the worst-case portfolio outcome in the objective function, the optimal policy is equivalent to the hedging portfolio of a European option on a dynamic mutual fund that can be replicated by market primary assets. Applying the Black-Scholes formula, a closed-form solution is obtained when the utility function is HARA and asset prices follow a multivariate geometric Brownian motion. The analysis provides a useful method of converting an investment problem to an option pricing model.
TL;DR: In this article, the authors give an explicit analytic solution for this optimization problem if the market price of the commodity is based on a simple Gaussian model, thereby replacing previously used incomplete approximations to the optimal strategy.
Abstract: The search for an optimal strategy to reduce the running risk in hedging a long-term supply commitment with short-dated futures contracts leads to a class of intrinsic optimization problems. We give an explicit analytic solution for this optimization problem if the market price of the commodity is based on a simple Gaussian model, thereby replacing previously used incomplete approximations to the optimal strategy.
TL;DR: In this paper, a generalization of the Greek called vega which is used to study the stability of option prices and hedging portfolios with respect to the volatility in various models is discussed.
Abstract: In this article we discuss a generalization of the Greek called vega which is used to study the stability of option prices and hedging portfolios with respect to the volatility in various models. We call this generalization the local vega index. We compute through Monte Carlo simulations this index in the cases of Asian options under the classical Black-Scholes setup. Simulation methods using Malliavin calculus and kernel density estimation are compared. Variance reduction methods are discussed.
TL;DR: In this article, a numerical procedure to price discrete European barrier options in the Black-Scholes model is proposed, where the pricing problem is divided into a series of initial value problems, one for each monitoring time.
Abstract: This paper designs a numerical procedure to price discrete European barrier options in Black-Scholes model. The pricing problem is divided into a series of initial value problems, one for each monitoring time. Each initial value problem is solved by replacing the driving Brownian motion by a lattice random walk. Some results from the theory of Besov spaces show that the convergence rate of lattice methods for initial value problems depends on two factors, namely the smoothness of the initial value (or the value function) and the moments for the increments of the lattice random walk. This fact is used to obtain an efficient method to price discrete European barrier options. Numerical examples and comparisons with other methods are carried out to show that the proposed method yields fast and accurate results.
TL;DR: In this article, the authors consider the variation of the hedging strategy of a European call option when the underlying asset follows a binomial tree and show that the asymptotic variation may be of any order.
Abstract: In this paper we consider the variation of the hedging strategy of a European call option when the underlying asset follows a binomial tree. In a binomial tree model the hedging strategy of a European call option converges to a continuous process when the number of time points increases so that the price process of the underlying asset converges to a Brownian motion, the Bachelier model. However, the variation of the hedging strategy need not converge to the variation of the limit process. In fact, it is shown that the asymptotic variation of the hedging strategy may be of any order.