Equivalent martingale measures and no-arbitrage in stochastic securities market models

Dalang, Robert C.; Morton, Anthony; Willinger, Walter

doi:10.1080/17442509008833613

Cited by 350 publications

(276 citation statements)

References 8 publications

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“…Then it follows from the Dalang-Morton-Willinger theorem (Dalang et al 1990) that there exists a t ≤ T − 1 and a one-step trading strategy ϑ t+1 ∈ L 0 (F t ) J+K such that ϑ R t+1 · ∆R t+1 + ϑ S t+1 · ∆S t+1 is nonnegative and strictly positive with positive probability. The same is true for ε t (ϑ R t+1 · ∆R t+1 + ϑ S t+1 · ∆S t+1 ) for arbitrary F t -measurable ε t > 0.…”

Section: B Proofs Of Sectionmentioning

confidence: 99%

“…Since the price process (R t ) T t=0 satisfies (NA), one obtains from the Dalang-Morton-Willinger theorem (Dalang et al 1990) that there exists an equivalent martingale measure Q ∼ P such that…”

Section: B Proofs Of Sectionmentioning

confidence: 99%

“…By the Dalang-Morton-Willinger theorem (see Dalang et al, 1990) this is equivalent to the existence of a probability measure Q equivalent to P such that R j t = E Q R j T | F t for all j and t.…”

Section: The Financial Marketmentioning

confidence: 99%

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Equilibrium Pricing in Incomplete Markets Under Translation Invariant Preferences

et al. 2016

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We provide results on the existence and uniqueness of equilibrium in dynamically incomplete financial markets in discrete time. Our framework allows for heterogeneous agents, unspanned random endowments and convex trading constraints. In the special case where all agents have preferences of the same type and all random endowments are replicable by trading in the financial market we show that a one-fund theorem holds and give an explicit expression for the equilibrium pricing kernel. If the underlying noise is generated by finitely many Bernoulli random walks, the equilibrium dynamics can be described by a system of coupled backward stochastic difference equations, which in the continuous-time limit becomes a multi-dimensional backward stochastic differential equation. If the market is complete in equilibrium, the system of equations decouples, but if not, one needs to keep track of the prices and continuation values of all agents to solve it. As an example we simulate option prices in the presence of stochastic volatility, demand pressure and short-selling constraints.JEL classification: C62, D52, D53

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Section: B Proofs Of Sectionmentioning

confidence: 99%

Section: B Proofs Of Sectionmentioning

confidence: 99%

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Equilibrium Pricing in Incomplete Markets Under Translation Invariant Preferences

et al. 2016

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“…The following theorem (see [10] or [20,22] for more recent approaches) is fundamental in discrete time mathematics of finance. In what follows we shall denote by Q the family of all martingale measures.…”

Section: Portfolio and Absence Of Arbitragementioning

confidence: 99%

Problems of Mathematical Finance by Stochastic Control Methods

Stettner

2009

IFIP Advances in Information and Communication Technology

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Abstract. The purpose of this paper is to present main ideas of mathematics of finance using the stochastic control methods. There is an interplay between stochastic control and mathematics of finance. On the one hand stochastic control is a powerful tool to study financial problems. On the other hand financial applications have stimulated development in several research subareas of stochastic control in the last two decades. We start with pricing of financial derivatives and modeling of asset prices, studying the conditions for the absence of arbitrage. Then we consider pricing of defaultable contingent claims. Investments in bonds lead us to the term structure modeling problems. Special attention is devoted to historical static portfolio analysis called Markowitz theory. We also briefly sketch dynamic portfolio problems using viscosity solutions to HamiltonJacobi-Bellman equation, martingale-convex analysis method or stochastic maximum principle together with backward stochastic differential equation. Finally, long time portfolio analysis for both risk neutral and risk sensitive functionals is introduced. Pricing of Financial DerivativesOne of the fundamental problems of mathematics of finance is pricing of the derivative securities (shortly derivatives) i.e. securities the value of which depends on the basic securities such as stocks or bonds. In this section we restrict ourselves to stocks, although similar problems (unfortunately much harder) concern also derivatives of bonds. Modeling of Asset PricesWe start with modeling of asset prices (stocks). We assume that we are given d assets on the market and denote the price of the i-th asset at time t by S i (t). We shall consider in parallel way two approaches: in discrete and in continuous time. We assume a given probability space (Ω , F , (F t ), P). In the case of discrete time the asset prices satisfy the relation S i (t + 1) S i (t) = ζ i (t, z(t + 1), ξ (t + 1)),Research supported by MNiSzW grant no. 1 P03A 013 28.

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“…In the celebrated papers [9,10,18] the financial market is modelled through a probability measure P that describes the future movements of the stock prices in the time interval [0, T ]. The stock price process S and the measure P are defined on a probability space Ω and a filtration F = {F t } {t∈[0,T ]} .…”

Section: Introductionmentioning

confidence: 99%

Convex Duality with Transaction Costs

Dolinsky

Soner

2017

Mathematics of OR

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Abstract. Convex duality for two two different super-replication problems in a continuous time financial market with proportional transaction cost is proved. In this market, static hedging in a finite number of options, in addition to usual dynamic hedging with the underlying stock, are allowed. The first one of the problems considered is the model-independent hedging that requires the super-replication to hold for every continuous path. In the second one the market model is given through a probability measure P and the inequalities are understood P almost surely. The main result, using the convex duality, proves that the two super-replication problems have the same value provided that P satisfies the conditional full support property. Hence, the transaction costs prevents one from using the structure of a specific model to reduce the super-replication cost.

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Equivalent martingale measures and no-arbitrage in stochastic securities market models

Cited by 350 publications

References 8 publications

Equilibrium Pricing in Incomplete Markets Under Translation Invariant Preferences

Equilibrium Pricing in Incomplete Markets Under Translation Invariant Preferences

Problems of Mathematical Finance by Stochastic Control Methods

Convex Duality with Transaction Costs

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