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Volume 1, 2022





Frontiers of Mathematical Finance

June 2022 , Volume 1 , Issue 2

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Asset price bubbles: Invariance theorems
Robert Jarrow, Philip Protter and Jaime San Martin
2022, 1(2): 161-188 doi: 10.3934/fmf.2021006 +[Abstract](843) +[HTML](428) +[PDF](477.47KB)
Abstract:

This paper provides invariance theorems that facilitate testing for the existence of an asset price bubble in a market where the price evolves as a Markov diffusion process. The test involves only the properties of the price process' quadratic variation under the statistical probability. It does not require an estimate of either the equivalent local martingale measure or the asset's drift. To augment its use, a new family of stochastic volatility price processes is also provided where the processes' strict local martingale behavior can be characterized.

Quadratic variation, models, applications and lessons
Dilip B. Madan and King Wang
2022, 1(2): 189-217 doi: 10.3934/fmf.2021007 +[Abstract](585) +[HTML](277) +[PDF](1569.01KB)
Abstract:

Time changes of Brownian motion impose restrictive jump structures in the motion of asset prices. Quadratic variations also depart from time changes. Quadratic variation options are observed to have a nonlinear exposure to risk neutral skewness. Joint Laplace Fourier transforms for quadratic variation and the stock are developed. They are used to study the multiple of the cap strike over the variance swap quote attaining a given percentage price reduction for the capped variance swap. Market prices for out-of-the-money options on variance are observed to be above those delivered by the calibrated models. Bootstrapped data and simulated paths spaces are used to study the multiple of the dynamic hedge return desired by a quadratic variation contract. It is observed that the optimized hedge multiple in the bootstrapped data is near unity. Furthermore, more generally, it is exposures to negative cubic variations in path spaces that raise variance swap prices, lower hedge multiples, increase residual risk charges and gaps to the log contract hedge. A case can be made for both, the physical process being closer to a continuous time change of Brownian motion, while simultaneously risk neutrally this may not be the case. It is recognized that in the context of discrete time there are no issues related to equivalence of probabilities.

Acceptability maximization
Gabriela Kováčová, Birgit Rudloff and Igor Cialenco
2022, 1(2): 219-248 doi: 10.3934/fmf.2021009 +[Abstract](151) +[HTML](66) +[PDF](864.51KB)
Abstract:

The aim of this paper is to study the optimal investment problem by using coherent acceptability indices (CAIs) as a tool to measure the portfolio performance. We call this problem the acceptability maximization. First, we study the one-period (static) case, and propose a numerical algorithm that approximates the original problem by a sequence of risk minimization problems. The results are applied to several important CAIs, such as the gain-to-loss ratio, the risk-adjusted return on capital and the tail-value-at-risk based CAI. In the second part of the paper we investigate the acceptability maximization in a discrete time dynamic setup. Using robust representations of CAIs in terms of a family of dynamic coherent risk measures (DCRMs), we establish an intriguing dichotomy: if the corresponding family of DCRMs is recursive (i.e. strongly time consistent) and assuming some recursive structure of the market model, then the acceptability maximization problem reduces to just a one period problem and the maximal acceptability is constant across all states and times. On the other hand, if the family of DCRMs is not recursive, which is often the case, then the acceptability maximization problem ordinarily is a time-inconsistent stochastic control problem, similar to the classical mean-variance criteria. To overcome this form of time-inconsistency, we adapt to our setup the set-valued Bellman's principle recently proposed in [23] applied to two particular dynamic CAIs - the dynamic risk-adjusted return on capital and the dynamic gain-to-loss ratio. The obtained theoretical results are illustrated via numerical examples that include, in particular, the computation of the intermediate mean-risk efficient frontiers.

Making no-arbitrage discounting-invariant: A new FTAP version beyond NFLVR and NUPBR
Dániel Ágoston Bálint and Martin Schweizer
2022, 1(2): 249-286 doi: 10.3934/fmf.2021010 +[Abstract](149) +[HTML](64) +[PDF](615.8KB)
Abstract:

What is absence of arbitrage for non-discounted prices? How can one define this so that it does not change meaning if one decides to discount after all?

The answer to both questions is a new discounting-invariant no-arbitrage concept. As in earlier work, we define absence of arbitrage as the zero strategy or some basic strategies being maximal. The key novelty is that maximality of a strategy is defined in terms of share holdings instead of value. This allows us to generalise both NFLVR, by dynamic share efficienc, and NUPBR, by dynamic share viability. These new concepts are the same for discounted or undiscounted prices, and they can be used in general models under minimal assumptions on asset prices. We establish corresponding versions of the FTAP, i.e., dual characterisations in terms of martingale properties. As one expects, "properly anticipated prices fluctuate randomly", but with an endogenous discounting process which cannot be chosen a priori. An example with N geometric Brownian motions illustrates our results.

Convergence of deep fictitious play for stochastic differential games
Jiequn Han, Ruimeng Hu and Jihao Long
2022, 1(2): 287-319 doi: 10.3934/fmf.2021011 +[Abstract](171) +[HTML](60) +[PDF](811.39KB)
Abstract:

Stochastic differential games have been used extensively to model agents' competitions in finance, for instance, in P2P lending platforms from the Fintech industry, the banking system for systemic risk, and insurance markets. The recently proposed machine learning algorithm, deep fictitious play, provides a novel and efficient tool for finding Markovian Nash equilibrium of large \begin{document}$ N $\end{document}-player asymmetric stochastic differential games [J. Han and R. Hu, Mathematical and Scientific Machine Learning Conference, pages 221-245, PMLR, 2020]. By incorporating the idea of fictitious play, the algorithm decouples the game into \begin{document}$ N $\end{document} sub-optimization problems, and identifies each player's optimal strategy with the deep backward stochastic differential equation (BSDE) method parallelly and repeatedly. In this paper, we prove the convergence of deep fictitious play (DFP) to the true Nash equilibrium. We can also show that the strategy based on DFP forms an \begin{document}$ \epsilon $\end{document}-Nash equilibrium. We generalize the algorithm by proposing a new approach to decouple the games, and present numerical results of large population games showing the empirical convergence of the algorithm beyond the technical assumptions in the theorems.

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