M6—On Minimal Market Models and Minimal Martingale Measures

Hulley, Hardy; Schweizer, Martin

doi:10.1007/978-3-642-03479-4_3

Cited by 41 publications

(48 citation statements)

References 22 publications

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“…Note that

{false(μ_{t} false)}_{t \geq 0}

given by satisfies the so‐called structure condition (see Schweizer, ) (because of and the fact that the drift part is of form

\int_{0}^{t} c false(μ_{s} false) λ false(μ_{s} false) d s

). This structural condition characterizes the condition of “no unbounded profit with bounded risk” (NUPBR) in the case of continuous semimartingales (see, e.g., Hulley & Schweizer, ).…”

Section: The Continuous Time Case With Functionally Generated Portfoliosmentioning

confidence: 99%

“…Note that ( ) ≥0 given by (47) satisfies the so-called structure condition (see Schweizer, 1995) (because of (48) and the fact that the drift part is of form ∫ 0 ( ) ( ) ). This structural condition characterizes the condition of "no unbounded profit with bounded risk" (NUPBR) in the case of continuous semimartingales (see, e.g., Hulley & Schweizer, 2010). In this setting, the proportions of current (relative) wealth invested in each of the assets are described by processes in the following set:…”

Section: Functionally Generated Log-optimal Portfoliosmentioning

confidence: 99%

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Cover's universal portfolio, stochastic portfolio theory, and the numéraire portfolio

Cuchiero

Schachermayer

Wong

2018

Mathematical Finance

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Cover's celebrated theorem states that the long‐run yield of a properly chosen “universal” portfolio is almost as good as that of the best retrospectively chosen constant rebalanced portfolio. The “universality” refers to the fact that this result is model‐free , that is, not dependent on an underlying stochastic process. We extend Cover's theorem to the setting of stochastic portfolio theory: the market portfolio is taken as the numéraire, and the rebalancing rule need not be constant anymore but may depend on the current state of the stock market. By fixing a stochastic model of the stock market this model‐free result is complemented by a comparison with the numéraire portfolio. Roughly speaking, under appropriate assumptions the asymptotic growth rate coincides for the three approaches mentioned in the title of this paper. We present results in both discrete and continuous time.

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“…Note that

{false(μ_{t} false)}_{t \geq 0}

given by satisfies the so‐called structure condition (see Schweizer, ) (because of and the fact that the drift part is of form

\int_{0}^{t} c false(μ_{s} false) λ false(μ_{s} false) d s

). This structural condition characterizes the condition of “no unbounded profit with bounded risk” (NUPBR) in the case of continuous semimartingales (see, e.g., Hulley & Schweizer, ).…”

Section: The Continuous Time Case With Functionally Generated Portfoliosmentioning

confidence: 99%

Section: Functionally Generated Log-optimal Portfoliosmentioning

confidence: 99%

Cover's universal portfolio, stochastic portfolio theory, and the numéraire portfolio

Cuchiero

Schachermayer

Wong

2018

Mathematical Finance

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“…Hence, the so-called structure condition (see e.g. [30,Chapter 3]) holds true and [30, Theorem 3.4] thus implies that (NUPBR) holds for the process µ and thus in turn for µ as well. As (NFLVR) ⇔ (NUPBR) + (NA) (see [10]) and since (NFLVR) does not hold, this thus means that relative arbitrages necessarily exist.…”

mentioning

confidence: 99%

Polynomial processes in stochastic portfolio theory

Cuchiero

2019

Stochastic Processes and their Applications

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We introduce polynomial processes in the sense of [8] in the context of stochastic portfolio theory to model simultaneously companies' market capitalizations and the corresponding market weights. These models substantially extend volatility stabilized market models considered by Robert Fernholz and Ioannis Karatzas in [18], in particular they allow for correlation between the individual stocks. At the same time they remain remarkably tractable which makes them applicable in practice, especially for estimation and calibration to high dimensional equity index data. In the diffusion case we characterize the joint polynomial property of the market capitalizations and the corresponding weights, exploiting the fact that the transformation between absolute and relative quantities perfectly fits the structural properties of polynomial processes. Explicit parameter conditions assuring the existence of a local martingale deflator and relative arbitrages with respect to the market portfolio are given and the connection to non-attainment of the boundary of the unit simplex is discussed. We also consider extensions to models with jumps and the computation of optimal relative arbitrage strategies.2000 Mathematics Subject Classification. 60J25,91G10, 60H30.

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“…[39], [41], [42]). As pointed out by [28], this assumption equals the no unbounded profit with bounded risk condition, which is weaker than the classic risk-neutral condition of no free-lunch with vanishing risk (see e.g. [16]) equivalent to the existence of a martingale measure.…”

Section: Introductionmentioning

confidence: 99%

Polynomial diffusion models for life insurance liabilities

Biagini

Zhang²

2016

Insurance: Mathematics and Economics

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In this paper we study the pricing and hedging problem of a portfolio of life insurance products under the benchmark approach, where the reference market is modelled as driven by a state variable following a polynomial diffusion on a compact state space. Such a model can be used to guarantee not only the positivity of the OIS short rate and the mortality intensity, but also the possibility of approximating both pricing formula and hedging strategy of a large class of life insurance products by explicit formulas.JEL Classification: C02, G10, G19

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M6—On Minimal Market Models and Minimal Martingale Measures

Cited by 41 publications

References 22 publications

Cover's universal portfolio, stochastic portfolio theory, and the numéraire portfolio

Cover's universal portfolio, stochastic portfolio theory, and the numéraire portfolio

Polynomial processes in stochastic portfolio theory

Polynomial diffusion models for life insurance liabilities

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