Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets

Abstract. In this paper we consider the valuation of an option with time to expiration T and pay-off function g which is a convex function (as is a European call option), and constant interest rate r = 0, for a variety of underlying price process models constructed from two independent Poisson processes, and an independent Brownian motion. This gives rise to incomplete market models with an infinite number of risk neutral measures. The collection of risk neutral measures gives rise to different prices, which comprise intervals that we calculate. The intervals can vary dramatically depending on the model parameters.

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“…We also have the following result of Kramkov [16,Theorem 3.2], unfortunately under an additional assumption on the price process S:…”

Section: The Incomplete Casementioning

confidence: 93%

Options Prices in Incomplete Markets

Jacod

Protter

Crépey³

et al. 2017

ESAIM: ProcS

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“…Application in the pricing of contingent claims in incomplete markets It is well known that, in incomplete security markets, for a given contingent claim £, the maximum price of the contingent claim at time t is given by El Karoui and Quenez [9] and Kramkov [17] showed that under the assumption £ > 0 and s u p^ £ e »£ < oo for each v € D {V,} is a (2"-supermartingale and has the following optional decomposition theorem:…”

Section: Some Applications In Security Markets and Economic Theorymentioning

confidence: 99%

Infinite time interval BSDEs and the convergence of g-martingales

Chen¹,

Wang

2000

J Aust Math Soc A

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In this paper, we first give a sufficient condition on the coefficients of a class of infinite time interval backward stochastic differential equations (BSDEs) under which the infinite time interval BSDEs have a unique solution for any given square integrable terminal value, and then, using the infinite time interval BSDEs, we study the convergence of g-martingales introduced by Peng via a kind of BSDEs. Finally, we study the applications of g-expectations and g-martingales in both finance and economics.2000 Mathematics subject classification: primary 60H10, 60G48. Keywords and phrases: infinite time interval BSDEs, g-expectation, g-martingale, upcrossing inequality of g-martingale, convergence of g-martingale.The adapted solution for a linear BSDE which appears as the adjoint process for a stochastic control problem was first introduced by Bismut in 1973, then by Bensoussan and others, while the first result for the existence and uniqueness of an adapted solution to a nonlinear BSDE with finite time interval and Lipschitzian coefficient was obtained by Pardoux and Peng [20]. Later many researchers developed the theory and its applications in a series of papers (see for example Darling [5] Karoui, Peng and Quenez [8] and the references therein) under some other assumptions on coefficients but for fixed terminal time. From these papers, the basic theorem is that, for a fixed terminal time T > 0, under the suitable assumptions on terminal value £, coefficient g and driving process M, the following BSDE has a solution pair (y,, z,)

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“…In his paper [5] Kraxnkov considers two notions of superhedging (in fact he uses the word «hedging», not «superhedging»), superhedging and minimal superhedging. The set ting of the paper is a very general, continuous time model.…”

Section: Thenmentioning

confidence: 99%

Risk averse asymptotics and the optional decomposition

Grandits¹,

Summer²

2006

Teor. Veroyatnost. i Primenen.

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Ключевые слова и фразы:хеджирование, экспоненциальная функ ция полезности, неприятие риска, опциональное разложение.1. Introduction. In this paper we consider a financial market with an agent who has to satisfy a contingent claim, e.g., an option, at final time T. The agent, who has the possibility of investing in an asset of the underlying financial market, has to decide on a trading strategy and there are, at least, two natural, but different possibilities: utility maximization and superhedging. We will investigate the relationship between these two concepts and show how superhedging can be interpreted as the limit of utility maximiza tion. This interpretation will give some new insight into the notion of superhedging.For example the notion of superhedging in general does not yield a unique outcome, i.e., there are different superhedging strategies that lead to different terminal wealth at time T. But by looking at superhedging as a limit of utility maximization one is naturally led to a special, unique outcome.Let us assume we are given a finite probability space Q = {tt>i,... ,u>jv}, a finite time horizon T G N, a filtration (.^t)t=o,i,...,r, and a probability measure P. For finite CI we always assume that P[CJ»] > 0 for all i = 1,..., N. Furthermore we assume, as is typically done in arbitrage pricing theory, that &o is trivial. The market consists of a stock S = (St)t=o,i,...,T, i-e., an R-valued adapted process, and a riskless bond B. We assume without loss of generality that the stock S represents the discounted price process, i.e., that it is denoted in terms of units of the riskless bond, and that the bond is constant equal to 1.The stock process S can be represented by an event tree. Each node in the tree corresponds to an atom'F €«ft,t = 0,l,...,T. By an atom F G &t we mean an element F G & u

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Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets

Cited by 276 publications

References 13 publications

Options Prices in Incomplete Markets

Options Prices in Incomplete Markets

Infinite time interval BSDEs and the convergence of g-martingales

Risk averse asymptotics and the optional decomposition

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