Fundamental Theorem Of Asset Pricing Research Articles

An Elementary Approach to the Fundamental Theorems of Asset Pricing in Jump-Diffusion Markets with Credit Risks, Funding Risks, and Collateral

We present an elementary approach to fundamental theorems of asset and derivative pricing for a wide class of jump-diffusion markets in which there are credit risks, funding risks, and collateral. The approach yields new and intuitive proofs of both the first and second Fundamental Theorems of Asset Pricing. We give examples of how the framework incorporates the gamut of industry pricing techniques from simple Black-Scholes pricing to recent results known as Funding Valuation Adjustments (FVA). The main contribution is the insight and intuition into the nature of the fundamental theorems that is afforded by the simplicity of the setup and the consequent reliance on only the most basic results from linear algebra.

Some Examples of the Role of Finance in Enabling Mathematics

The abstraction of physical commodities into quantified financial contracts has played an important role in enabling the mathematisation of western society. In particular, financial ethics has provided the basis on which mathematical probability has been developed. This paper begins by summarizing recent research on the role of commercial ethics in the development of probability. This is related to the Fundamental Theorem of Asset Pricing (FTAP), the foundational theory of contemporary financial mathematics. The paper then connects the FTAP to the Dutch Book Argument (DBA). The paper’s main contribution is in describing the significance of financial practice in justifying the DBA, itself the most popular justification for subjective probability. The paper then explains how markets dynamically confirm beliefs, which is modeled by the FTAP but not adequately by the DBA. The conclusions emphasize the distinction between pure and practical reasoning and that this should be mirrored in a distinction between the mathematics of physical and practical. This point is important as mathematics is becoming more important in modelling, and directing, social systems.

Risk-Neutral Pricing and Hedging of In-Play Football Bets

ABSTRACTA risk-neutral valuation framework is developed for pricing and hedging in-play football bets based on modelling scores by independent Poisson processes with constant intensities. The Fundamental Theorems of Asset Pricing are applied to this set-up which enables us to derive novel arbitrage-free valuation formulæ for contracts currently traded in the market. We also describe how to calibrate the model to the market and how trades can be replicated and hedged.

CORRIGENDUM: “PRICING AND VALUATION UNDER THE REAL-WORLD MEASURE”

In order to prove the third fundamental theorem of asset pricing for financial markets with infinite lifetime [G. Frahm (2016) Pricing and valuation under the real-world measure, International Journal of Theoretical and Applied Finance 19, 1650006], we shall assume that the discounted price process is locally bounded. Otherwise, some principal results developed by [F. Delbaen & W. Schachermayer (1997) The Banach space of workable contingent claims in arbitrage theory, Annales de l’Institut Henri Poincaré 1, 114–144] cannot be applied.

Fatou closedness under model uncertainty

We provide a characterization in terms of Fatou closedness for weakly closed monotone convex sets in the space of $${\mathcal P}$$ -quasisure bounded random variables, where $${\mathcal P}$$ is a (possibly non-dominated) class of probability measures. Applications of our results lie within robust versions the Fundamental Theorem of Asset Pricing or dual representation of convex risk measures.

Perturbation Analysis of Sub/Super Hedging Problems

We investigate the links between various no-arbitrage conditions and the existence of pricing functionals in general markets, and prove the Fundamental Theorem of Asset Pricing therein. No-arbitrage conditions, either in this abstract setting or in the case of a market consisting of European Call options, give rise to duality properties of infinite-dimensional sub- and super-hedging problems. With a view towards applications, we show how duality is preserved when reducing these problems over finite-dimensional bases. We finally perform a rigorous perturbation analysis of those linear programming problems, and highlight, as a numerical example, the influence of smile extrapolation on the bounds of exotic options.

Local martingales in discrete time

For any discrete-time $\mathsf{P} $–local martingale $S$ there exists a probability measure $\mathsf{Q} \sim \mathsf{P} $ such that $S$ is a $\mathsf{Q} $–martingale. A new proof for this result is provided. The core idea relies on an appropriate modification of an argument by Chris Rogers, used to prove a version of the fundamental theorem of asset pricing in discrete time. This proof also yields that, for any $\varepsilon >0$, the measure $\mathsf{Q} $ can be chosen so that $\frac{\mathrm {d} \mathsf {Q}} {\mathrm{d} \mathsf{P} } \leq 1+\varepsilon $.

Remarks on the stochastic integral

In Karandikar-Rao [11], the quadratic variation [M, M] of a (local) martingale was obtained directly using only Doob’s maximal inequality and it was remarked that the stochastic integral can be defined using [M, M], avoiding using the predictable quadratic variation 〈M, M〉 (of a locally square integrable martingale) as is usually done. This is accomplished here- starting with the result proved in [11], we construct ∫ f dX where X is a semimartingale and f is predictable and prove dominated convergence theorem (DCT) for the stochastic integral. Indeed, we characterize the class of integrands f for this integral as the class L(X) of predictable processes f such that |f| serves as the dominating function in the DCT for the stochastic integral. This observation seems to be new. We then discuss the vector stochastic integral ∫ 〈f, dY〉 where f is ℝ d valued predictable process, Y is ℝ d valued semimartingale. This was defined by Jacod [6] starting from vector valued simple functions. Memin [13] proved that for (local) martingales M1, … M d : If N n are martingales such that N → N t for every t and if ∃f n such that N = ∫ 〈f n , dM〉, then ∃f such that N = ∫ 〈f, dM〉. Taking a cue from our characterization of L(X), we define the vector integral in terms of the scalar integral and then give a direct proof of the result due to Memin stated above. This completeness result is an important step in the proof of the Jacod-Yor [4] result on martingale representation property and uniqueness of equivalent martingale measure. This result is also known as the second fundamental theorem of asset pricing.

Discrete-time market models from the small investor point of view and the first fundamental-type theorem

Abstract In this paper, we discuss the no-arbitrage condition in a discrete financial market model which does not hold the same interest rate assumptions. Our research was based on, essentially, one of the most important results in mathematical finance, called the Fundamental Theorem of Asset Pricing. For the standard approach a risk-free bank account process is used as numeraire. In those models it is assumed that the interest rates for borrowing and saving money are the same. In our paper we consider the model of a market (with d risky assets), which does not hold the same interest rate assumptions. We introduce two predictable processes for modelling deposits and loans. We propose a new concept of a martingale pair for the market and prove that if there exists a martingale pair for the considered market, then there is no arbitrage opportunity. We also consider special cases in which the existence of a martingale pair is necessary and the sufficient conditions for these markets to be arbitrage free.

Convex Duality and Orlicz Spaces in Expected Utility Maximization

In this paper we report further progress towards a complete theory of state-independent expected utility maximization with semi-martingale price processes for arbitrary utility function. Without any technical assumptions we establish a surprising Fenchel duality result on conjugate Orlicz spaces, offering a new economic insight into the nature of primal optima and providing a fresh perspective on the classical papers of Kramkov and Schachermayer (1999, 2003). The analysis points to an intriguing interplay between no-arbitrage conditions and standard convex optimization and motivates study of the Fundamental Theorem of Asset Pricing (FTAP) for Orlicz tame strategies.

IMPLICIT TRANSACTION COSTS AND THE FUNDAMENTAL THEOREMS OF ASSET PRICING

This paper studies arbitrage pricing theory in financial markets with implicit transaction costs. We extend the existing theory to include the more realistic possibility that the price at which the investors trade is dependent on the traded volume. The investors in the market always buy at the ask and sell at the bid price. Implicit transaction costs are composed of two terms, one is able to capture the bid-ask spread, and the second the price impact. Moreover, a new definition of a self-financing portfolio is obtained. The self-financing condition suggests that continuous trading is possible, but is restricted to predictable trading strategies having cádlág (right-continuous with left limits) and cáglád (left-continuous with right limits) paths of bounded quadratic variation and of finitely many jumps. That is, cádlág and cáglád predictable trading strategies of infinite variation, with finitely many jumps and of finite quadratic variation are allowed in our setting. Restricting ourselves to cáglád predictable trading strategies, we show that the existence of an equivalent probability measure is equivalent to the absence of arbitrage opportunities, so that the first fundamental theorem of asset pricing (FFTAP) holds. It is also shown that the use of continuous and bounded variation trading strategies can improve the efficiency of hedging in a market with implicit transaction costs. To better understand how to apply the theory proposed we provide an example of an implicit transaction cost economy that is linear and nonlinear in the order size.

GOOD DEAL BOUNDS WITH CONVEX CONSTRAINTS

We investigate the structure of good deal bounds, which are subintervals of a no-arbitrage pricing bound, for financial market models with convex constraints as an extension of Arai & Fukasawa (2014). The upper and lower bounds of a good deal bound are naturally described by a convex risk measure. We call such a risk measure a good deal valuation; and study its properties. We also discuss superhedging cost and Fundamental Theorem of Asset Pricing for convex constrained markets.

Duality Formulas for Robust Pricing and Hedging in Discrete Time

In this paper we derive robust super- and subhedging dualities for contingent claims that can depend on several underlying assets. In addition to strict super- and subhedging, we also consider relaxed versions which, instead of eliminating the shortfall risk completely, aim to reduce it to an acceptable level. This yields robust price bounds with tighter spreads. As examples we study strict super- and subhedging with general convex transaction costs and trading constraints as well as risk-based hedging with respect to robust versions of the average value at risk and entropic risk measure. Our approach is based on representation results for increasing convex functionals and allows for general financial market structures. As a side result it yields a robust version of the fundamental theorem of asset pricing.

The Fundamental Theorems of Asset Pricing and the Closed-End Fund Puzzle

The Fundamental Theorems of Asset Pricing and the Closed-End Fund Puzzle

Viability and Arbitrage Under Knightian Uncertainty

We reconsider the microeconomic foundations of financial economics under Knightian Uncertainty. In a general framework, we discuss the absence of arbitrage, its relation to economic viability, and the existence of suitable nonlinear pricing expectations. Classical financial markets under risk and no ambiguity are contained as special cases, including various forms of the Efficient Market Hypothesis. For Knightian uncertainty, our approach unifies recent versions of the Fundamental Theorem of Asset Pricing under a common framework.

Arbitrage, hedging and utility maximization using semi-static trading strategies with American options

We consider a financial market where stocks are available for dynamic trading, and European and American options are available for static trading (semi-static trading strategies). We assume that the American options are infinitely divisible, and can only be bought but not sold. In the first part of the paper, we work within the framework without model ambiguity. We first get the fundamental theorem of asset pricing (FTAP). Using the FTAP, we get the dualities for the hedging prices of European and American options. Based on the hedging dualities, we also get the duality for the utility maximization. In the second part of the paper, we consider the market which admits nondominated model uncertainty. We first establish the hedging result, and then using the hedging duality we further get the FTAP. Due to the technical difficulty stemming from the nondominancy of the probability measure set, we use a discretization technique and apply the minimax theorem.

Noncommutative Valuation of Options

The aim of this note is to show that the classical results in finance theory for pricing of derivatives, given by making use of the replication principle , can be extended to the noncommutative world. We believe that this could be of interest in quantum probability. The main result called the First fundamental theorem of asset pricing , states that a noncommutative stock market admits no-arbitrage if and only if it admits a noncommutative equivalent martingale probability.

Fundamental Theorem of Asset Pricing Under Transaction Costs and Model Uncertainty

We prove the fundamental theorem of asset pricing for a discrete time financial market where trading is subject to proportional transaction costs and the asset price dynamic is modeled by a family of probability measures, possibly nondominated. Using a backward-forward scheme, we show that when the market consists of a money market account and a single stock, no-arbitrage in a quasi-sure sense is equivalent to the existence of a suitable family of consistent price systems. We also show that when the market consists of multiple dynamically traded assets and satisfies efficient friction, strict no-arbitrage in a quasi-sure sense is equivalent to the existence of a suitable family of strictly consistent price systems.

No-Arbitrage and Hedging with Liquid American Options

Since most of the traded options on individual stocks is of American type it is of interest to generalize the results obtained in semi-static trading to the case when one is allowed to statically trade American options. However, this problem has proved to be elusive so far because of the asymmetric nature of the positions of holding versus shorting such options.Here we provide a unified framework and generalize the fundamental theorem of asset pricing (FTAP) and hedging dualities in Bayraktar and Zhou (2015, to appear in Annals of Applied Probability, also available at http://ssrn.com/abstract=2569256) to the case where the investor can also short American options. Following this earlier work, we assume that the longed American options are divisible. As for the shorted American options, we show that the divisibility plays no role regarding arbitrage property and hedging prices. Then using the method of enlarging probability spaces proposed in Deng and Tan (2016, ArXiv e-prints), we convert the shorted American options to European options, and establish the FTAP and sub- and super-hedging dualities in the enlarged space both with and without model uncertainty.

A separating hyperplane theorem, the fundamental theorem of asset pricing, and Markov's principle

We prove constructively that every uniformly continuous convex function f:X→R+ has positive infimum, where X is the convex hull of finitely many vectors. Using this result, we prove that a separating hyperplane theorem, the fundamental theorem of asset pricing, and Markov's principle are constructively equivalent. This is the first time that important theorems are classified into Markov's principle within constructive reverse mathematics.