Showing papers in "Mathematical Finance in 1999"

Coherent Measures of Risk

Abstract: In this paper we study both market risks and nonmarket risks, without complete markets assumption, and discuss methods of measurement of these risks. We present and justify a set of four desirable properties for measures of risk, and call the measures satisfying these properties “coherent.” We examine the measures of risk provided and the related actions required by SPAN, by the SEC=NASD rules, and by quantile-based methods. We demonstrate the universality of scenario-based methods for providing coherent measures. We offer suggestions concerning the SEC method. We also suggest a method to repair the failure of subadditivity of quantile-based methods.

Term Structure Models Driven by General Lévy Processes

Abstract: As a generalization of the Gaussian Heath–Jarrow–Morton term structure model, we present a new class of bond price models that can be driven by a wide range of Levy processes. We deduce the forward and short rate processes implied by this model and prove that, under certain assumptions, the short rate is Markovian if and only if the volatility structure has either the Vasicek or the Ho–Lee form. Finally, we compare numerically forward rates and European call option prices in a model driven by a hyperbolic Levy motion with those in the Gaussian model.

Interest Rate Dynamics and Consistent Forward Rate Curves

Abstract: We consider as given an arbitrage-free interest rate model M, and a parametrized family of forward rate curves G. We study the question as to when the given family G is consistent with the dynamics of the interest rate model M, in the sense that M actually will produce forward rate curves belonging to G. We allow the interest rate model to be driven by a multidimensional Wiener process, as well as by a marked point process, and we give necessary and sufficient conditions for consistency. As test cases, we study some popular models, obtaining both positive and negative results about consistency. We also introduce a natural exponential-polynomial family of forward rate curves, and for this family we give necessary and sufficient conditions for the existence of consistent interest rate models with deterministic volatility functions.

Asymptotically Optimal Importance Sampling and Stratification for Pricing Path‐Dependent Options

Abstract: This paper develops a variance reduction technique for Monte Carlo simulations of path-dependent options driven by high-dimensional Gaussian vectors. The method combines importance sampling based on a change of drift with stratified sampling along a small number of key dimensions. The change of drift is selected through a large deviations analysis and is shown to be optimal in an asymptotic sense. The drift selected has an interpretation as the path of the underlying state variables which maximizes the product of probability and payoff—the most important path. The directions used for stratified sampling are optimal for a quadratic approximation to the integrand or payoff function. Indeed, under differentiability assumptions our importance sampling method eliminates variability due to the linear part of the payoff function, and stratification eliminates much of the variability due to the quadratic part of the payoff. The two parts of the method are linked because the asymptotically optimal drift vector frequently provides a particularly effective direction for stratification. We illustrate the use of the method with path-dependent options, a stochastic volatility model, and interest rate derivatives. The method reveals novel features of the structure of their payoffs.

Controlling Risk Exposure and Dividends Payout Schemes:Insurance Company Example

Abstract: The paper represents a model for financial valuation of a firm which has control of the dividend payment stream and its risk as well as potential profit by choosing different business activities among those available to it. This model extends the classical Miller–Modigliani theory of firm valuation to the situation of controllable business activities in a stochastic environment. We associate the value of the company with the expected present value of the net dividend distributions (under the optimal policy). The example we consider is a large corporation, such as an insurance company, whose liquid assets in the absence of control fluctuate as a Brownian motion with a constant positive drift and a constant diffusion coefficient. We interpret the diffusion coefficient as risk exposure, and drift is understood as potential profit. At each moment of time there is an option to reduce risk exposure while simultaneously reducing the potential profit—for example, by using proportional reinsurance with another carrier for an insurance company. Management of a company controls the dividends paid out to the shareholders, and the objective is to find a policy that maximizes the expected total discounted dividends paid out until the time of bankruptcy. Two cases are considered: one in which the rate of dividend payout is bounded by some positive constant M, and one in which there is no restriction on the rate of dividend payout. We use recently developed techniques of mathematical finance to obtain an easy understandable closed form solution. We show that there are two levels u0 and u1 with u0≤u1. As a function of currently available reserve, the risk exposure monotonically increases on (0,u0) from 0 to the maximum possible. When the reserve exceeds u1 the dividends are paid at the maximal rate in the first case and in the second case every excess above u1 is distributed as dividend. We also show that for M small enough u0=u1 and the optimal risk exposure is always less than the maximal.

A Note on the Nelson–Siegel Family

Abstract: We study a problem posed in Bjork and Christensen (1999): does there exist any nontrivial interest rate model which is consistent with the Nelson-Siegel family? They show that within the HJM framework with deterministic volatility structure the answer is no.In this paper we give a generalized version of this result including stochastic volatility structure. For that purpose we introduce the class of consistent state space processes, which have the property to provide an arbitrage-free interest rate model when representing the parameters of the Nelson-Siegel family. We characterize the consistent state space Ito processes in terms of their drift and diffusion coefficients. By solving an inverse problem we find their explicit form. It turns out that there exists no nontrivial interest rate model driven by a consistent state space Ito process.

Bounds on European Option Prices under Stochastic Volatility

Abstract: In this paper we consider the range of prices consistent with no arbitrage for European options in a general stochastic volatility model. We give conditions under which the infimum and the supremum of the possible option prices are equal to the intrinsic value of the option and to the current price of the stock, respectively, and show that these conditions are satisfied in most of the stochastic volatility models from the financial literature. We also discuss properties of Black–Scholes hedging strategies in stochastic volatility models where the volatility is bounded.

Generalized Hyperbolic Diffusion Processes with Applications in Finance

Abstract: A special class of diffusion processes, the generalized hyperbolic diffusion processes, is introduced. As a byproduct we present a technique for the construction of one-dimensional ergodic diffusion processes with a predetermined stationary density. We specifically study the application of this new type of diffusion process to financial data, especially U.S. stock prices. It is seen that in addition to confirming stylized features of the financial market, a key explanation concerning “thick-”tailed log returns is provided.

Pricing General Barrier Options: A Numerical Approach Using Sharp Large Deviations

Abstract: In this paper we develop simulation techniques in order to evaluate single and double barrier options with general features. Our method is based on Sharp Large Deviation estimates, which allow one to improve the usual Monte Carlo procedure. Numerical results are provided and show the validity of the proposed simulation algorithm.

Pricing American Stock Options by Linear Programming

Abstract: We investigate numerical solution of finite difference approximations to American option pricing problems, using a new direct numerical method: simplex solution of a linear programming formulation. This approach is based on an extension to the parabolic case of the equivalence between linear order complementarity problems and abstract linear programs known for certain elliptic operators. We test this method empirically, comparing simplex and interior point algorithms with the projected successive overrelaxation (PSOR) algorithm applied to the American vanilla and lookback puts. We conclude that simplex is roughly comparable with projected SOR on average (faster for fine discretizations, slower for coarse), but is more desirable for robustness of solution time under changes in parameters. Furthermore, significant speedups over the results given here have been achieved and will be published elsewhere.

Currency Prices, the Nominal Exchange Rate, and Security Prices in a Two-Country Dynamic Monetary Equilibrium

Abstract: This paper examines a continuous-time two-country dynamic monetary equilibrium in which countries with possibly heterogeneous tastes and endowments hold their own money for the purpose of transaction services formulated via money in the utility function. Given a price system, no-arbitrage pricing results are provided for the price of each money and the nominal exchange rate. Characterizations are provided for equilibrium prices for general time-additive preferences and non-Markovian exogenous processes. Under a Markovian structure of model primitives, the currency prices are shown to solve a bivariate system of partial differential equations. Assuming that each country is endowed with heterogeneous separable power utility and the exogenous quantities all follow geometric Brownian motions, an equilibrium is shown to exist and additional characterization is provided. A further example of nonseparable Cobb–Douglas preferences is investigated. The additional features over the customary environment of homogeneous logarithmic preferences are emphasized.

The Second Fundamental Theorem of Asset Pricing

Abstract: This paper presents a resolution of the paradox proposed by the example of an economy with complette markets and a multiplicityof martingale measures constructed by Artzner and Heath (1995). The resolution lies in noting that completeness is with respect to a topology on the space of cash flows and is connected with uniqueness of the price functional in the topological dual space. Uniqueness may be lost outside the dual and this is what occurs in the counterexample of Artzner and Heath.

Viability and Equilibrium in Securities Markets with Frictions

Abstract: In this paper we study some foundational issues in the theory of asset pricing with market frictions. We model market frictions by letting the set of marketed contingent claims (the opportunity set) be a convex set, and the pricing rule at which these claims are available be convex. This is the reduced form of multiperiod securities price models incorporating a large class of market frictions. It is said to be viable as a model of economic equilibrium if there exist price-taking maximizing agents who are happy with their initial endowment, given the opportunity set, and hence for whom supply equals demand. This is equivalent to the existence of a positive lineaar pricing rule on the entirespace of contingent claims—an underlying frictionless linear pricing rule—that lies below the convex pricing rule on the set of marketed claims. This is also equivalent to the absence of asymptotic free lunches—a generalization of opportunities of arbitrage. When a market for a nonmarketed contingent claim opens, a bid-ask price pair for this claim is said to be consistent if it is a bid-ask price pair in at least a viable economy with this extended opportunity set. If the set of marketed contingent claims is a convex cone and the pricing rule is convex and sublinear, we show that the set of consistent prices of a claim is a closed interval and is equal (up to its boundary) to the set of its prices for all the underlying frictionless pricing rules. We also show that there exists a unique extended consistent sublinear pricing rule—the supremum of the underlying frictionless linear pricing rules—for which the original equilibrium does not collapse when a new market opens, regardless of preferences and endowments. If the opportunity set is the reduced form of a multiperiod securities market model, we study the closedness of the interval of prices of a contingent claim for the underlying frictionless pricing rules.

Self‐Financing Trading Strategies for Sliding, Rolling‐Horizon, and Consol Bonds

Abstract: The time evolution of a sliding bond is studied in discrete- and continuous-time setups. By definition, a sliding bond represents the price process of a discount bond with a fixed time to maturity. Examples of measure-valued trading strategies (introduced by Bj"ork et al. 1997a, 1997b) which are based on the price process of a sliding bond are discussed. In particular, a self-financing strategy that involves holding at any time one unit of a sliding bond is examined (the wealth process of this strategy is referred to as the rolling-horizon bond). In contrast to the sliding bond, which does not represent a tradable security, the rolling-horizon bond (or the rolling-consol bond) may play the role of a fixed-income security with infinite lifespan in portfolio management problems.

European‐Type Contingent Claims in an Incomplete Market with Constrained Wealth and Portfolio

Abstract: This paper considers the problem of hedgeability and replicability of European-type contingent claims in an incomplete market with the wealth and the portfolio possibly being constrained. For the case of no constraint, using the idea of a Four Step Scheme (Ma, Protter, and Yong 1994), we prove the replicability of a class of contingent claims (including European call and put options) without assuming ad hoc technical conditions. For the case with the wealth and portfolio being constrained, several positive and negative results concerning hedgeability and replicability are presented.