Mathematical Finance

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Return Dynamics when Persistence is Unobservable
Mathematical Finance - Tập 11 Số 4 - Trang 415-445 - 2001
Timothy C. Johnson

This paper proposes a new theory of the sources of time‐varying second (and higher) moments in financial time series. The key idea is that fully rational agents must infer the stochastic degree of persistence of fundamental shocks. Endogenous changes in their uncertainty determine the evolution of conditional moments of returns. The model accounts for the principal observed features of volatility dynamics and implies some new ones. Most strikingly, it implies a relationship between ex post trends, or momentum, and changes in volatility.

Static hedging and pricing of exotic options with payoff frames
Mathematical Finance - Tập 29 Số 2 - Trang 612-658 - 2019
Justin Kirkby, Shijie Deng
Abstract

We develop a general framework for statically hedging and pricing European‐style options with nonstandard terminal payoffs, which can be applied to mixed static–dynamic and semistatic hedges for many path‐dependent exotic options including variance swaps and barrier options. The goal is achieved by separating the hedging and pricing problems to obtain replicating strategies. Once prices have been obtained for a set of basis payoffs, the pricing and hedging of financial securities with arbitrary payoff functions is accomplished by computing a set of “hedge coefficients” for that security. This method is particularly well suited for pricing baskets of options simultaneously, and is robust to discontinuities of payoffs. In addition, the method enables a systematic comparison of the value of a payoff (or portfolio) across a set of competing model specifications with implications for security design.

BEHAVIORAL PORTFOLIO SELECTION IN CONTINUOUS TIME
Mathematical Finance - Tập 18 Số 3 - Trang 385-426 - 2008
Hanqing Jin, Xun Yu Zhou

This paper formulates and studies a general continuous‐time behavioral portfolio selection model under Kahneman and Tversky's (cumulative) prospect theory, featuring S‐shaped utility (value) functions and probability distortions. Unlike the conventional expected utility maximization model, such a behavioral model could be easily mis‐formulated (a.k.a. ill‐posed) if its different components do not coordinate well with each other. Certain classes of an ill‐posed model are identified. A systematic approach, which is fundamentally different from the ones employed for the utility model, is developed to solve a well‐posed model, assuming a complete market and general Itô processes for asset prices. The optimal terminal wealth positions, derived in fairly explicit forms, possess surprisingly simple structure reminiscent of a gambling policy betting on a good state of the world while accepting a fixed, known loss in case of a bad one. An example with a two‐piece CRRA utility is presented to illustrate the general results obtained, and is solved completely for all admissible parameters. The effect of the behavioral criterion on the risky allocations is finally discussed.

OPTIMAL INSURANCE DESIGN UNDER RANK‐DEPENDENT EXPECTED UTILITY
Mathematical Finance - Tập 25 Số 1 - Trang 154-186 - 2015
Carole Bernard, Xuedong He, Jia-an Yan, Xun Yu Zhou

We consider an optimal insurance design problem for an individual whose preferences are dictated by the rank‐dependent expected utility (RDEU) theory with a concave utility function and an inverse‐S shaped probability distortion function. This type of RDEU is known to describe human behavior better than the classical expected utility. By applying the technique of quantile formulation, we solve the problem explicitly. We show that the optimal contract not only insures large losses above a deductible but also insures small losses fully. This is consistent, for instance, with the demand for warranties. Finally, we compare our results, analytically and numerically, both to those in the expected utility framework and to cases in which the distortion function is convex or concave.

TRACTABLE ROBUST EXPECTED UTILITY AND RISK MODELS FOR PORTFOLIO OPTIMIZATION
Mathematical Finance - Tập 20 Số 4 - Trang 695-731
Karthik Natarajan, Melvyn Sim, Joline Uichanco
Explicit Representation of the Minimal Variance Portfolio in Markets Driven by Lévy Processes
Mathematical Finance - Tập 13 Số 1 - Trang 55-72 - 2003
Fred Espen Benth, Giulia Di Nunno, Arne Løkka, Frank Proske

In a market driven by a Lévy 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.

A DIFFUSION MODEL FOR ELECTRICITY PRICES
Mathematical Finance - Tập 12 Số 4 - Trang 287-298 - 2002
Takashi Kumagai

Starting from a simple supply/demand model for electricity, we obtain a diffusion (i.e., jumpless) model for spot prices which can exhibit price spikes. We estimate the parameters in the model using historical data from the Alberta and California markets. and compare this model with some others used for spot prices.

Coherent Measures of Risk
Mathematical Finance - Tập 9 Số 3 - Trang 203-228 - 1999
Philippe Artzner, Freddy Delbaen, Jean‐Marc Eber, David Heath

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.

Unspanned stochastic volatility in the multifactor CIR model
Mathematical Finance - Tập 29 Số 3 - Trang 827-836 - 2019
Damir Filipović, Martin Larsson, Francesco Statti
Abstract

Empirical evidence suggests that fixed‐income markets exhibit unspanned stochastic volatility (USV), that is, that one cannot fully hedge volatility risk solely using a portfolio of bonds. While Collin‐Dufresne and Goldstein (2002, Journal of Finance, 57, 1685–1730) showed that no two‐factor Cox–Ingersoll–Ross (CIR) model can exhibit USV, it has been unknown to date whether CIR models with more than two factors can exhibit USV or not. We formally review USV and relate it to bond market incompleteness. We provide necessary and sufficient conditions for a multifactor CIR model to exhibit USV. We then construct a class of three‐factor CIR models that exhibit USV. This answers in the affirmative the above previously open question. We also show that multifactor CIR models with diagonal drift matrix cannot exhibit USV.

Complete Models with Stochastic Volatility
Mathematical Finance - Tập 8 Số 1 - Trang 27-48 - 1998
David Hobson, L. C. G. Rogers

The paper proposes an original class of models for the continuous‐time price process of a financial security with nonconstant volatility. The idea is to define instantaneous volatility in terms of exponentially weighted moments of historic log‐price. The instantaneous volatility is therefore driven by the same stochastic factors as the price process, so that, unlike many other models of nonconstant volatility, it is not necessary to introduce additional sources of randomness. Thus the market is complete and there are unique, preference‐independent options prices.

We find a partial differential equation for the price of a European call option. Smiles and skews are found in the resulting plots of implied volatility.

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