Mathematical Finance

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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
AbstractEmpirical 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.
A YIELD‐FACTOR MODEL OF INTEREST RATES
Mathematical Finance - Tập 6 Số 4 - Trang 379-406 - 1996
Darrell Duffie, Rui Kan
This paper presents a consistent and arbitrage‐free multifactor model of the term structure of interest rates in which yields at selected fixed maturities follow a parametric muitivariate Markov diffusion process with “stochastic volatility.” the yield of any zero‐coupon bond is taken to be a maturity‐dependent affine combination of the selected “basis” set of yields. We provide necessary and sufficient conditions on the stochastic model for this affine representation. We include numerical techniques for solving the model, as well as numerical techniques for calculating the prices of term‐structure derivative prices. the case of jump diffusions is also considered.
MAXIMUM LIKELIHOOD ESTIMATION USING PRICE DATA OF THE DERIVATIVE CONTRACT
Mathematical Finance - Tập 4 Số 2 - Trang 155-167 - 1994
Jin‐Chuan Duan
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.
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