Analysis of Error with Malliavin Calculus: Application to Hedging

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   dXⁱ_t = Xⁱ_t ( bⁱ ( X_t ) dt +σ _{i , j} ( X_t ) dW^j_t )  . 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/ n ^1/4  for options with irregular payoff.     

Quantiles of the Euler Scheme for Diffusion Processes and Financial Applications

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  ( X_t )  , where  ( X_t )  is a diffusion process, when one uses a Monte Carlo method combined with the Euler discretization scheme. We consider the case where  ( X_t )  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.     

Monte Carlo Evaluation of Greeks for Multidimensional Barrier and Lookback Options

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 ).     

Hedging Options: The Malliavin Calculus Approach versus the Δ-Hedging Approach

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 Δ-hedging formula under weaker conditions.     

First-Order Schemes in the Numerical Quantization Method

The numerical quantization method is a grid method that relies on the approximation of the solution to a nonlinear problem 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 probability weights. We emphasize the fact that in the case of optimal quantization, many of these correcting 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 proposed.     

Efficient Computation of Hedging Portfolios for Options with Discontinuous Payoffs

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 Fournié 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.     

A General Approach to Hedging Options: Applications to Barrier and Partial Barrier Options

In this paper we consider a Black and Scholes economy and show how the Malliavin calculus approach can be extended to cover hedging of any square integrable contingent claim. As an application we derive the replicating portfolios of some barrier and partial barrier options.     

Optimal Malliavin Weighting Function for the Computation of the Greeks

This paper reexamines the Malliavin weighting functions introduced by Fournié 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.     

An Anticipating Calculus Approach to the Utility Maximization of an Insider

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.     

UTILITY MAXIMIZATION IN AN INSIDER INFLUENCED MARKET

We study a controlled stochastic system whose state is described by a stochastic differential equation with anticipating coefficients. This setting is used to model markets where insiders have some influence on the dynamics of prices. We give a characterization theorem for the optimal logarithmic portfolio of an investor with a different information flow from that of the insider. We provide explicit results in the partial information case that we extend in order to incorporate the enlargement of filtration techniques for markets with insiders. Finally, we consider a market with an insider who influences the drift of the underlying price asset process. This example gives a situation where it makes a difference for a small agent to acknowledge the existence of an insider in the market.     

DIFFUSION COEFFICIENT ESTIMATION AND ASSET PRICING WHEN RISK PREMIA AND SENSITIVITIES ARE TIME VARYING: A COMMENT

The purpose of this note is to analyze the diffusion coefficient estimator suggested by Chesney, Elliott, Madan, and Yang (1993). I start by correcting their formula (4.1), and by showing that their procedure is a member of a class of estimators sharing the same Milstein approximation. I then show how to select the minimum variance estimator (for constant μσ) within a two-parameter subclass of procedures which do not depend on the current realization of the process. I also show that if μ is small the best procedure only allows moderate reduction in variance with respect to the classical quadratic variation estimator (which is a member of the same class). the note concludes by highlighting the fact that the empirical use of the filtered volatilities poses an error in variables problem which can be addressed using instrumental variables methods.     

Equilibrium Models With Singular Asset Prices

General equilibrium models in which economic agents have finite marginal utility from consumption at the origin lead to financial assets having continuous prices with singular components. In particular, there is no bona fide "interest rate" in such models, although asset prices can be determined by equilibrium considerations (and uniquely, up to the formation of mutual funds). the singularly continuous processes in question charge precisely the set of time points at which some agent "drops out" of the economy, or "comes back" into it, between intervals of zero consumption. Not surprisingly, these processes are governed by local time.     

PORTFOLIO OPTIMIZATION WITH DOWNSIDE CONSTRAINTS

We consider the portfolio optimization problem for an investor whose consumption rate process and terminal wealth are subject to downside constraints. In the standard financial market model that consists of d risky assets and one riskless asset, we assume that the riskless asset earns a constant instantaneous rate of interest,   r > 0  , and that the risky assets are geometric Brownian motions. The optimal portfolio policy for a wide scale of utility functions is derived explicitly. The gradient operator and the Clark–Ocone formula in Malliavin calculus are used in the derivation of this policy. We show how Malliavin calculus approach can help us get around certain difficulties that arise in using the classical "delta hedging" approach.     

Error Calculus and Path Sensitivity in Financial Models

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 that 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.     

Backward Stochastic Differential Equations in Finance

We are concerned with different properties of backward stochastic differential equations and their applications to finance. These equations, first introduced by Pardoux and Peng (1990), are useful for the theory of contingent claim valuation, especially cases with constraints and for the theory of recursive utilities, introduced by Duffie and Epstein (1992a, 1992b).