European Contingent Claims Research Articles

Utility based option evaluation with proportional transaction costs

We examine the problem of finding investors reservation prices of European contingent claims in a continuous time finite horizon economy with proportional transaction costs. We derive analytically numerous properties of reservation prices reflecting their dependency on the size of the option position, the risk aversion, and the initial wealth. For the special case of HARA utility functions, we consider the so-called marginal reservation prices, i.e. unit reservation prices of infinitesimal positions of the contingent claim, and prove that these are independent of the initial wealth and the parameters of the utility function, except for the exponent. Furthermore, we present numerical examples suggesting that the marginal reservation prices are, for reasonable parameter values, insensitive to the drift of the underlying security and the exponent of the HARA utility function.

A TAYLOR FORMULA TO PRICE AND HEDGE EUROPEAN CONTINGENT CLAIMS

We present the no-arbitrage price and the hedging strategy of an European contingent claim through a representation formula which is an extension of the Clark-Ocone formula. Our formula can be interpreted as a second order Taylor formula of the no arbitrage price of a contingent claim. The zero order term is given by the mean of the contingent claim payoff, the first order term by the stochastic integral of the mean of its Malliavin derivative and the second order term by the stochastic integral of the conditional expectation of the second Malliavin derivative. A Taylor series expansion is also provided together with a bound to the approximation error obtained by neglecting the second order term in the Taylor formula.

Local Expected Shortfall-Hedging

This paper proposes a self-financing trading strategy that minimizes the expected shortfall locally when hedging a European contingent claim. A positive shortfall occurs if the hedger is not willing to follow a perfect hedging or a superhedging strategy. In contrast to the classical variance criterion, the expected shortfall criterion depends only on undesirable outcomes where the terminal value of the written option exceeds the terminal value of the hedge portfolio. Searching a strategy which minimizes the expected shortfall is equivalent to the iterative solution of linear programs whose number increases exponentially with respect to the number of trading dates. Therefore, we partition this complex overall problem into several one-period problems and minimize the expected shortfall only locally, i.e. only over the next trading period. This approximation is quite accurate and the number of linear programs to be solved increases only linearly with respect to the number of trading dates.

A Martingale Characterization of Consumption Choices and Hedging Costs with Margin Requirements

This paper examines optimal consumption and investment choices and the cost of hedging contingent claims in the presence of margin requirements or, more generally, of nonlinear wealth dynamics and constraints on the portfolio policies. Existence of optimal policies is established using martingale and duality techniques under general assumptions on the securities? price process and the investor?s preferences. As an illustration, explicit solutions are provided for an agent with logarithmic utility. A PDE characterization of the cost of hedging a nonnegative path-independent European contingent claim is also provided.

Direct characterization of the value of super-replication under stochastic volatility and portfolio constraints

We study the problem of minimal initial capital needed in order to hedge a European contingent claim without risk. The financial market presents incompleteness arising from two sources: stochastic volatility and portfolio constraints described by a closed convex set. In contrast with previous literature which uses the dual formulation of the problem, we use an original dynamic programming principle stated directly on the initial problem, as in Soner and Touzi (1998. SIAM J. Control Optim.; 1999. Preprint). We then recover all previous known results under weaker assumptions and without appealing to the dual formulation. We also prove a new characterization result of the value of super-replication as the unique continuous viscosity solution of the associated Hamilton–Jacobi–Bellman equation with a suitable terminal condition.

A Martingale Characterization of Consumption Choices and Hedging Costs with Margin Requirements

This paper examines optimal consumption and investment choices and the cost of hedging contingent claims in the presence of margin requirements or, more generally, of nonlinear wealth dynamics and constraints on the portfolio policies. Existence of optimal policies is established using martingale and duality techniques under general assumptions on the securities' price process and the investor's preferences. As an illustration, explicit solutions are provided for an agent with ‘logarithmic’ utility. A PDE characterization of the cost of hedging a nonnegative path‐independent European contingent claim is also provided.

Utility Based Option Evaluation with Proportional Transaction Costs

We examine the problem of finding investors' reservation prices of European contingent claims in a continuous time finite horizon economy with proportional transaction costs. We derive analytically numerous properties of reservation prices reflecting their dependency on the size of the option position, the risk aversion, and the initial wealth. For the special case of HARA utility functions, we consider the so-called marginal reservation prices, i.e. unit reservation prices of infinitesimal positions of the contingent claim, and prove that these are independent of the initial wealth and the parameters of the utility function, except for the exponent. Furthermore, we present numerical examples suggesting that the marginal reservation prices are, for reasonable parameter values, insensitive to the drift of the underlying security and the exponent of the HARA utility function.

Valuation of European options in the market with daily price limit

A valuation problem of the European style contingent claim in the market with daily price movement limit is studied. Unlike the one leading to the well known Black-Scholes formula, this problem depicts considerable conceptual difficulty and anomaly created by the presence of various arbitrage opportunities inherently built in the model due to the daily price movement limit. The presence of arbitrage makes it go against the grain of the well established arbitrage pricing theory. In this paper, how these complications arise are discussed and then a valuation approach devised, which is called the ‘vanishing transaction cost technique,’ of getting around the difficulty.

Super-replication under proportional transaction costs: From discrete to continuous-time models

In this paper, we study the problem of finding the minimal initial capital (i.e. super-replication value) needed in order to hedge (without risk) European contingent claims in a Markov setting under proportional transaction costs. The main result is that the cheapest (trivial) buy-and-hold strategy is optimal. Such a negative result has been derived previously in different contexts. First, we focus on discrete-time binomial models. We prove that the continuous-time limit of the super-replication value is the cost of the cheapest buy-and-hold strategy. Then, the result is proved in a multivariate continuous-time model with Brownian filtration. As a direct consequence, we obtain an explicit characterization of the hedging set, i.e. the set of all initial positions in the market assets from which the contingent claim can be hedged through some admissible portfolio strategy.

The European option with hereditary price structures: Basic theory

In this paper, we propose a new model for the ( B, S)-market in which the stock price and the asset in the riskless bank account have hereditary price structures. Specifically, the dynamics of the stock price and the bank account are described by linear stochastic functional differential equations. The pricing of the European contingent claims is studied and the corresponding trading strategy is derived.

Fast valuation of financial derivatives

A method for pricing European contingent claims (options) based on Monte Carlo simulation with variance reduction is presented. The evolution of the option price can be formulated as a Kolmogorov final value problem and thus be calculated numerically either by solving the deterministic partial differential equation or by simulating a large number of trajectories of the corresponding stochastic differential equation. The authors discuss a Monte Carlo simulation method combined with variance reduction obtained from a Girsanov transformation of the stochastic differential equation by a correction term that is obtained as a rough solution of the partial differential equation computed by a classical numerical method. The trade-off between these methods is investigated and it is shown that the composite method is more efficient than either the standard Monte Carlo or the classical numerical method.

Valuation and hedging of contingent claims in the HJM model with deterministic volatilities

The aim of the present paper is mostly expository, namely, we intend to provide a concise presentation of arbitrage pricing and hedging of European contingent claims within the Heath, Jarrow and Morton frame-work introduced in Heath et al. (1992) under deterministic volatilities. Such a special case of the HJM model, frequently referred to as the Gaussian HJM model, was studied among others in Amin and Jarrow (1992), Jamshidian (1993), Brace and Musiela (1994a, 1994b). Here, we focus mainly on the partial differential equations approach to the valuation and hedging of derivative securities in the HJM framework. For the sake of completeness, the risk neutral methodology (more specifically, the forward measure technique) is also exposed.

Numerical Valuation of High Dimensional Multivariate European Securities

We consider the problem of pricing a contingent claim whose payoff depends on several sources of uncertainty. Using classical assumptions from the Arbitrage Pricing Theory, the theoretical price can be computed as the discounted expected value of future cash flows under the modified risk-neutral information process. Although analytical solutions have been developed elsewhere for a few particular option pricing problems, computing the arbitrage prices of securities under several sources of uncertainty is still an open problem in many instances. In this paper, we present efficient numerical techniques based upon Monte Carlo simulation for pricing European contingent claims depending on an arbitrary number of risk sources. We introduce in particular the method of quadratic resampling (QR), a new powerful error reduction technique for Monte Carlo simulation. Quadratic resampling can be efficiently combined with classical variance reduction methods such as importance sampling. Our numerical experiments show that the method is practical for pricing claims depending on up to one hundred underlying assets.

Residual risks and hedging strategies in Markovian markets

We prove two explicit formulae for the quadratic residual risk and for the optimal hedging portfolio of a European contingent claim when the underlying stock prices are functions of a Markov process. These expressions allow the practical handling of a great deal of non classical models which are less optimistic than Black and Scholes's one