Showing papers in "Mathematical Finance in 1994"

Modeling stochastic volatility: a review and comparative study

Abstract: Diffusion models for volatility have been used to price options while ARCH models predominate in descriptive studies of asset volatility. This paper compares a discrete-time approximation of a popular diffusion model with ARCH models. These volatility models have many siimilarities but the models make different assumptions about how the magnitude of price responses to information alters volatility and the amount of subsequent information. Several volatility models are estimated for daily DM/ exchange rates from 1978 to 1990.

Maximum likelihood estimation using price data of the derivative contract

Abstract: This article develops a general methodology that uses the observed prices of a derivative contract to compute maximum likelihood parameter estimates for an unobserved asset value process. the use of this estimation methodology is demonstrated in two applications: Vasicek's term structure model and deposit insurance pricing. This methodology can also be useful in the empirical analysis of complex financial contracts involving embedded options.

Contingent claims valued and hedged by pricing and investing in a basis

Abstract: Contingent claims with payoffs depending on finitely many asset prices are modeled as elements of a separable Hilbert space. Under fairly general conditions, including market completeness, it is shown that one may change measure to a reference measure under which asset prices are Gaussian and for which the family of Hermite polynomials serves as an orthonormal basis. Basis pricing synthesizes claim valuation and basis investment provides static hedging opportunities. For claims written as functions of a single asset price we infer from observed option prices the implicit prices of basis elements and use these to construct the implied equivalent martingale measure density with respect to the reference measure, which in this case is the Black-Scholes geometric Brownian motion model. Data on S & P 500 options from the Wall Street Journal are used to illustrate the calculations involved. On this illustrative data set the equivalent martingale measure deviates from the Black-Scholes model by relatively discounting the larger price movements with a compensating premia placed on the smaller movements.

A multifactor gauss markov implementation of heath, jarrow, and morton

Abstract: Working within the Heath-Jarrow-Morton framework and using the theory of stochastic equations in infinite dimensions, a useful multifactor Gauss-Markov model for the movement of the whole of the yield curve is derived. Swaptions are priced. They are hedged by eliminating random terms between the semimartingale representations of the swaption and hedging instruments. Hedging efficiency is analyzed. the model is fitted to the swap/cap strips in Australia. Computation times on a 20-MHz laptop computer are acceptable.

The term structure of interest rates as a gaussian random field

Abstract: A simple model of the term structure of interest rates is introduced in which the family of instantaneous forward rates evolves as a continuous Gaussian random field. A necessary and sufficient condition for the associated family of discounted zero-coupon bond prices to be martingales is given, permitting the consistent pricing of interest rate contingent claims. Examples of the pricing of interest-rate caps and the situation when the Gaussian random field may be viewed as a deterministic time change of the standard Brownian sheet are discussed.

When is the short rate markovian

Abstract: We answer this question in the very general context of the n-factor Heath, Jarrow, and Morton model for the evolution of the term structure of interest rates, with nonrandom volatility. the answer is that a constraint is imposed on the behavior of the volatility structure. We explain the importance of this result for the design of efficient numerical algorithms for the valuation of options on the term structure.

Martingale measures for discrete‐time processes with infinite horizon

Abstract: Let (St)teI be an Rd-valued adapted stochastic process on (Ω, ?, (?t)teI, P). A basic problem occurring notably in the analysis of securities markets, is to decide whether there is a probability measure Q on ? equivalent to P such that (St)teI is a martingale with respect to Q. It is known (see the fundamental papers of Harrison and Kreps 1979; Harrison and Pliska 1981; and Kreps 1981) that there is an intimate relation of this problem with the notions of “no arbitrage” and “no free lunch” in financial economics. We introduce the intermediate concept of “no free lunch with bounded risk.” This is a somewhat more precise version of the notion of “no free lunch.” It requires an absolute bound of the maximal loss occurring in the trading strategies considered in the definition of “no free lunch.” We give an argument as to why the condition of “no free lunch with bounded risk” should be satisfied by a reasonable model of the price process (St)teI of a securities market. We can establish the equivalence of the condition of “no free lunch with bounded risk” with the existence of an equivalent martingale measure in the case when the index set I is discrete but (possibly) infinite. A similar theorem was recently obtained by Delbaen (1992) for continuous-time processes with continuous paths. We can combine these two theorems to get a similar result for the continuous-time case when the process (St)teR+ is bounded and, roughly speaking, the jumps occur at predictable times. In the infinite horizon setting, the price process has to be “almost a martingale” in order to allow an equivalent martingale measure.

Convergence of american option values from discrete‐ to continuous‐time financial models1

Abstract: Given a sequence of discrete-time option valuation models in which the sequence of processes defining the state variables converges weakly to a diffusion, we prove that the sequence of American option values obtained from these discrete-time models also converges to the corresponding value obtained from the continuous-time model for the standard models in the finance/economica literature. The convergence proof carries over to the case when the limiting risky asset price process follows a diffusion, except it pays discrete dividend\ on some fixed dates.

Threshold autoregressive modeling in finance: the price differences of equivalent assets

Abstract: Threshold autoregressive (TAR) models condition the first moment of a time series on lagged information using a step-function-type nonlinear structure. TAR techniques are expected to be relevant in financial time-series modeling in situations where deviations of prices from equilibrium values depend on discrete transaction costs and where market regulators follow intervention rules based on threshold values of control variables. an important finance application is in modeling the difference in prices of equivalent assets in the presence of transaction costs. the focus of this paper is on motivating the use of TAR models in this context and on the statistical estimation and testing procedures. the procedures are illustrated by modeling the difference between the prices of an index futures contract and the equivalent underlying cash index. It is found that the hypothesis of linearity is conclusively rejected in favor of threshold nonlinearity and that the estimated thresholds are largely consistent with arbitrage-related transaction costs.

Risk‐minimizing hedging strategies under restricted information

Abstract: We construct risk-minimizing hedging strategies in the case where there are restrictions on the available information. the underlying price process is a d-dimensional F-martingale, and strategies φ= (ϑ, η) are constrained to have η G-predictable and η G'-adapted for filtrations η G C G’C F. We show that there exists a unique (ηG, G')-risk-minimizing strategy for every contingent claim H e E 2 (T, P) and provide an explicit expression in terms of η G-predictable dual projections. Previous results of Follmer and Sondermann (1986) and Di Masi, Platen, and Runggaldier (1993) are recovered as special cases. Examples include a Black-Scholes model with delayed information and a jump process model with discrete observations.

Arbitrage and free lunch with bounded risk for unbounded continuous processes

Abstract: We give two examples showing that for unbounded continuous price processes, the no-free-lunch assumption and the existence of an equivalent martingale measure are not equivalent. In fact it turns out that the notion of an equivalent local martingale measure is natural in this context. Copyright 1994 Blackwell Publishers.

The statistics of long‐horizon regressions revisited1

Abstract: This paper compares commonly used approaches for estimating the relation between long-horizon returns and a predetermined variable X1, such as dividend yields. Specifically, we look at regression of (i) nonoverlapping multiperiod returns on Xt (ii) overlapping multiperiod returns on Xt, (iii) single-period returns on multiperiod Xt, and (iv) single-period returns on Xt and its implied long-horizon regression coefficient. We provide analytical formulae which quantify the efficiency of the estimators used in the various approaches. Using the formulae, as well as Monte Carlo simulations, we demonstrate that the relative efficiency of the estimators used in the various approaches differs remarkably, depending on the dynamic structure of the regressor. of special interest for financial economists, when the regressors are highly autocorrelated, we find that the regressions (ii) (iii), and (iv) provide only marginal efficiency gains above and beyond the nonoverlapping long-horizon regression.

ON COMPONENTWISE and VECTOR STOCHASTIC INTEGRATION

Abstract: We give a condition under which the componentwise stochastic integration with respect to a given Rd-valued continuous local martingale coincides with the more general vector stochastic integration defined by Jacod (1979). We then provide a result on the equivalence between the vector and the component completeness of a financial market in a special case.

The early exercise premium representation of foreign market American options

Abstract: The note deals with the pricing of American options related to foreign market equities. the form of the early exercise premium representation of the American option's price in a stochastic interest rate economy is established. Subsequently, the American fixed exchange rate foreign equity option and the American equity-linked foreign exchange option are studied in detail.

Consumption and portfolio policies with incomplete markets and short-sale constraints in the finite-dimensional case: some remarks

Abstract: This paper extends He and Pearson's (1991) martingale approach to the study of optimal intertemporal consumption and portfolio policies with incomplete markets and short-sale constraints to a framework in which no assumptions are made on the price process for the securities We show how both their characterization of the budget-feasible set and duality result can be extended to account for an unbounded set II of Arrow-Debreu state prices compatible with the arbitrage-free assumption We also supply a (fairly general) sufficient condition for II to be bounded, as required in their setting

Optimal consumption and portfolio selection with incomplete markets and upper and lower bound constraints

Abstract: We study an optimal consumption and portfolio selection problem for an investor by a martingale approach. We assume that time is a discrete and finite horizon, the sample space is finite and the number of securities is smaller than that of the possible securities price vector transitions. the investor is prohibited from investing stocks more (less, respectively) than given upper (lower) bounds at any time, and he maximizes an expected time additive utility function for the consumption process. First we give a set of budget feasibility conditions so that a consumption process is attainable by an admissible portfolio process. Also we state the existence of the unique primal optimal solutions. Next we formulate a dual control problem and establish the duality between primal and dual control problems. Also we show the existence of dual optimal solutions. Finally we consider the computational aspect of dual approach through a simple numerical example.

A pricing operator‐based testing foundation for a class of factor pricing models

Abstract: This paper develops a cross-market version of factor pricing models. It is shown that exact factor pricing holds across two submarkets with respect to their common factors if and only if the unique pricing operator for the first submarket is equal to that for the other submarket with probability 1. We define an APT measure as the squared distance between the two pricing operators. Then, testing whether this measure is zero is equivalent to testing exact factor pricing across the two submarkets. Since the estimation of this measure does not require parameterizing and extracting the underlying factors, one can test factor pricing models without knowing any factors. In addition, we present a randomization procedure so that one can use it to conduct a more comprehensive investigation on the empirical robustness of factor pricing models.

Asymptotic inference for the parameters of a discrete-time square-root process

Abstract: This paper is concerned with asymptotic properties of the maximum likelihood estimators for the discrete-time square-root process. This process and its generalizations are employed in financial literature as models for movements of asset prices. the considered process is nonergodic and therefore standard maximum likelihood theory does not apply. the nonstandard asymptotic theory is developed. Strong consistency of the estimators is established, joint asymptotic distribution of the properly normalized estimators is obtained and confidence intervals for the parameters are constructed. the results of the small simulation study are reported.

A note on the generalized multibeta capm

Abstract: The unified beta theory of Connor (1984) requires that the market portfolio be well diversified in a given factor structure. Wei (1988) extended Connor's results without relying on this assumption. This note provides an alternative to Wei's result by assuming that residuals from the projection of asset return on a set of k factors follow a joint elliptical distribution.

Efficiency gains in beta-pricing models1

Abstract: Recent literature shows that the risk premium is efficiently estimated in the usual two-pass procedure, estimating betas in the unrestricted model, and then regressing returns on estimated betas. This paper shows that this is not so when allowing for factor unobservability. Imposing the financial theory restriction from the outset leads to a strictly positive efficiency gain in the risk premium estimation. In addition, the role of an associated efficiency gain in the beta estimation is studied in the context of a zero-beta model.