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 Finance and StochasticsJournal Prestige (SJR): 2.997 Citation Impact (citeScore): 2Number of Followers: 19      Hybrid journal (It can contain Open Access articles) ISSN (Print) 1432-1122 - ISSN (Online) 0949-2984 Published by Springer-Verlag  [2469 journals]
• A least-squares Monte Carlo approach to the estimation of enterprise risk

Abstract: The estimation of enterprise risk for financial institutions entails a re-evaluation of the company’s economic balance sheet at a future time for a (large) number of stochastic scenarios. The current paper discusses tackling this nested valuation problem based on least-squares Monte Carlo techniques familiar from American option pricing. We formalise the algorithm in an operator setting and discuss the choice of the regressors (“basis functions”). In particular, we show that the left singular functions of the corresponding conditional expectation operator present robust basis functions. Our numerical examples demonstrate that the algorithm can produce accurate results at relatively low computational costs.
PubDate: 2022-05-13

• Log-optimal and numéraire portfolios for market models stopped at a
random time

Abstract: This paper focuses on numéraire and log-optimal portfolios when a market model $$(S,\mathbb{F},P)$$ – specified by its assets’ price $$S$$ , its flow of information $$\mathbb{F}$$ and a probability measure $$P$$ – is stopped at a random time $$\tau$$ . The flow of information that incorporates both $$\mathbb{F}$$ and $$\tau$$ , denoted by $$\mathbb{G}$$ , is the progressive enlargement of $$\mathbb{F}$$ with $$\tau$$ . For the resulting stopped model $$(S^{\tau},\mathbb{G},P)$$ , we study the two portfolios in different manners and describe their computations in terms of $$\mathbb{F}$$ -observable parameters of the pair $$(S, \tau )$$ . In particular, we single out the types of risks induced by $$\tau$$ that really affect the numéraire portfolio, and address the following questions: 1) What are the conditions on $$\tau$$ (preferably in terms of information-theoretic concepts such as entropy) that guarantee the existence of the log-optimal portfolio for $$(S^{\tau},\mathbb{G},P)$$ when that for $$(S,\mathbb{F},P)$$ already exists' 2) What are the factors that fully determine the increment in maximal expected logarithmic utility from terminal wealth for the models $$(S^{\tau},\mathbb{G},P)$$ and $$(S,\mathbb{F},P)$$ , and how can one quantify them'
PubDate: 2022-05-06

• Set-valued dynamic risk measures for processes and for vectors

Abstract: Abstract The relationship between set-valued risk measures for processes and vectors on the optional filtration is investigated. The equivalence of risk measures for processes and vectors and the equivalence of their penalty function formulations are provided. In contrast to scalar risk measures, this equivalence requires an augmentation of the set-valued risk measures for processes. We utilise this result to deduce a new dual representation for risk measures for processes in the set-valued framework. Finally, the equivalence of multi-portfolio time-consistency between set-valued risk measures for processes and vectors is provided. To accomplish this, an augmented definition for multi-portfolio time-consistency of set-valued risk measures for processes is proposed.
PubDate: 2022-04-29

• Dynamic mean–variance problem with frictions

Abstract: Abstract We study a dynamic mean–variance portfolio selection problem with return predictability and trading frictions from price impact. Applying mean-field type control theory, we provide a characterisation of an equilibrium trading strategy for an investor facing stochastic investment opportunities. An explicit equilibrium strategy is derived in terms of the solution to a generalised matrix Riccati differential equation, and a sufficient condition is also provided to ensure the latter’s well-posedness. Our solution indicates that the investor should trade gradually towards a target portfolio which accounts for return predictability, price impact and time-consistency. Moreover, an asymptotic analysis around small liquidity costs shows that the investor’s target portfolio is an equilibrium portfolio without price impact in the first-order sense, and that her first-order approximated value function does not deteriorate significantly for sufficiently small liquidity costs. Finally, our numerical results demonstrate that the target portfolio is more conservative than an equilibrium portfolio without price impact.
PubDate: 2022-03-15

• Optimal consumption with reference to past spending maximum

Abstract: Abstract This paper studies the infinite-horizon optimal consumption problem with a path-dependent reference under exponential utility. The performance is measured by the difference between the nonnegative consumption rate and a fraction of the historical consumption maximum. The consumption running maximum process is chosen as an auxiliary state process, and hence the value function depends on two state variables. The Hamilton–Jacobi–Bellman (HJB) equation can be heuristically expressed in a piecewise manner across different regions to take into account all constraints. By employing the dual transform and smooth-fit principle, some thresholds of the wealth variable are derived such that a classical solution to the HJB equation and feedback optimal investment and consumption strategies can be obtained in closed form in each region. A complete proof of the verification theorem is provided, and numerical examples are presented to illustrate some financial implications.
PubDate: 2022-03-09

• A scaling limit for utility indifference prices in the discretised
Bachelier model

Abstract: Abstract We consider the discretised Bachelier model where hedging is done on a set of equidistant times. Exponential utility indifference prices are studied for path-dependent European options, and we compute their non-trivial scaling limit for a large number of trading times $$n$$ and when risk aversion is scaled like $$n\ell$$ for some constant $$\ell >0$$ . Our analysis is purely probabilistic. We first use a duality argument to transform the problem into an optimal drift control problem with a penalty term. We further use martingale techniques and strong invariance principles and obtain that the limiting problem takes the form of a volatility control problem.
PubDate: 2022-03-01
DOI: 10.1007/s00780-022-00473-y

• Scaled insurance cash flows: representation and computation via change of
measure techniques

Abstract: Abstract We consider general multi-state life insurance payment processes and study the expected accumulated cash flows that arise when modifying the payments by scaling factors depending on the time of occurrence of specific events. Such modified payment processes arise naturally in the context of incidental policyholder behaviour. We associate to the modifications new probability measures which allow a standard representation of the expected accumulated cash flows. The measures are characterised in terms of the original measure and the scaling factors. Examples for Markov chains illuminate the relevance of our concepts and results to actuarial practice.
PubDate: 2022-02-04
DOI: 10.1007/s00780-022-00472-z

• An analytical study of participating policies with minimum rate guarantee
and surrender option

Abstract: Abstract We perform a detailed theoretical study of the value of a class of participating policies with four key features: (i) the policyholder is guaranteed a minimum interest rate on the policy reserve; (ii) the contract can be terminated by the holder at any time until maturity (surrender option); (iii) at the maturity (or upon surrender), a bonus is credited to the holder if the portfolio backing the policy outperforms the current policy reserve; (iv) due to solvency requirements, the contract ends if the value of the underlying portfolio of assets falls below the policy reserve. Our analysis is probabilistic and relies on optimal stopping and free boundary theory. We find a structure of the optimal surrender strategy which was undetected by previous (mostly numerical) studies on the topic. Optimal surrender of the contract is triggered by two ‘stop-loss’ boundaries and by a ‘too-good-to-persist’ boundary (in the language of Ekström and Vaicenavicius in Stoch. Process. Appl. 130: 806–823, 2020). Financial implications of this strategy are discussed in detail and supported by extensive numerical experiments.
PubDate: 2022-01-28
DOI: 10.1007/s00780-022-00471-0

• A time-inconsistent Dynkin game: from intra-personal to inter-personal
equilibria

Abstract: Abstract This paper studies a nonzero-sum Dynkin game in discrete time under non-exponential discounting. For both players, there are two intertwined levels of game-theoretic reasoning. First, each player looks for an intra-personal equilibrium among her current and future selves, so as to resolve time inconsistency triggered by non-exponential discounting. Next, given the other player’s chosen stopping policy, each player selects a best response among her intra-personal equilibria. A resulting inter-personal equilibrium is then a Nash equilibrium between the two players, each of whom employs her best intra-personal equilibrium with respect to the other player’s stopping policy. Under appropriate conditions, we show that an inter-personal equilibrium exists, based on concrete iterative procedures along with Zorn’s lemma. To illustrate our theoretical results, we investigate a two-player real options valuation problem where two firms negotiate a deal of cooperation to initiate a project jointly. By deriving inter-personal equilibria explicitly, we find that coercive power in negotiation depends crucially on the impatience levels of the two firms.
PubDate: 2022-01-15
DOI: 10.1007/s00780-021-00468-1

• Editorial: 25th anniversary of Finance and Stochastics

PubDate: 2022-01-01
DOI: 10.1007/s00780-021-00470-7

• Reinforcement learning and stochastic optimisation

Abstract: Abstract At the heart of financial mathematics lie stochastic optimisation problems. Traditional approaches to solving such problems, while applicable to broad classes of models, require specifying a model to complete the analysis and obtain implementable results. Even then, the curse of dimensionality challenges the viability of conventional methods to settings of practical relevance. In contrast, machine learning, and reinforcement learning (RL) particularly, promises to learn from data and overcome the curse of dimensionality simultaneously. This article touches on several approaches in the extant literature that are well positioned to merge our traditional techniques with RL.
PubDate: 2022-01-01
DOI: 10.1007/s00780-021-00467-2

• My journey through finance and stochastics

Abstract: Abstract This year, Finance and Stochastics celebrates its 25th anniversary. The journal provides a platform for the community of researchers on which they can publish their ideas and results. Publication is an outcome of research which may be conducted for a number of years before it reaches the required maturity. I find this research process to be very important. Unfortunately, it is almost impossible to decode it from reading the research publications. This special issue of Finance and Stochastics gives me an opportunity to focus on it. I am grateful I can present my personal memory of this process. Understanding why questions are asked and how the answers are found is critical.
PubDate: 2022-01-01
DOI: 10.1007/s00780-021-00453-8

• From Bachelier to Dupire via optimal transport

Abstract: Abstract Famously, mathematical finance was started by Bachelier in his 1900 PhD thesis where – among many other achievements – he also provided a formal derivation of the Kolmogorov forward equation. This also forms the basis for Dupire’s (again formal) solution to the problem of finding an arbitrage-free model calibrated to a given volatility surface. The latter result has rigorous counterparts in the theorems of Kellerer and Lowther. In this survey article, we revisit these hallmarks of stochastic finance, highlighting the role played by some optimal transport results in this context.
PubDate: 2022-01-01
DOI: 10.1007/s00780-021-00466-3

• The influence of economic research on financial mathematics: Evidence from
the last 25 years

Abstract: Abstract This is an attempt to review some of the breakthroughs in economic research as they impacted the nascent field of financial mathematics over the last 25 years. Because of the prominent role of Finance and Stochastics in the definition of this emerging field, I try to view things through the lens of its published papers, and I try to stay away from financial engineering applications.
PubDate: 2022-01-01
DOI: 10.1007/s00780-021-00469-0

• An Italian perspective on the development of financial mathematics from
1992 to 2008

Abstract: Abstract This paper is intended to be a survey of the development of financial mathematics as seen through the events that I organised, and partly co-organised, between 1992 and 2008. These events all took place in Italy between 1992 and 2003, while in 2008 I was involved in the organisation of an entire special semester in Linz (Austria); this semester is included here because it marks quite well the state-of-the-art of the period just before the so-called big financial crisis that lasted from, roughly, 2008 to 2012. Even if the survey may be affected by my personal views, it can still be seen as reflecting the actual global development since what I am going to describe here concerns major occurrences. For completeness, I also mention, although only briefly, some events that took place in Italy during the given period, but where I was not personally involved.
PubDate: 2022-01-01
DOI: 10.1007/s00780-021-00452-9

• Machine learning with kernels for portfolio valuation and risk management

Abstract: Abstract We introduce a simulation method for dynamic portfolio valuation and risk management building on machine learning with kernels. We learn the dynamic value process of a portfolio from a finite sample of its cumulative cash flow. The learned value process is given in closed form thanks to a suitable choice of the kernel. We show asymptotic consistency and derive finite-sample error bounds under conditions that are suitable for finance applications. Numerical experiments show good results in large dimensions for a moderate training sample size.
PubDate: 2021-11-22
DOI: 10.1007/s00780-021-00465-4

• Deep ReLU network expression rates for option prices in high-dimensional,
exponential Lévy models

Abstract: Abstract We study the expression rates of deep neural networks (DNNs for short) for option prices written on baskets of $$d$$ risky assets whose log-returns are modelled by a multivariate Lévy process with general correlation structure of jumps. We establish sufficient conditions on the characteristic triplet of the Lévy process $$X$$ that ensure $$\varepsilon$$ error of DNN expressed option prices with DNNs of size that grows polynomially with respect to $${\mathcal{O}}(\varepsilon ^{-1})$$ , and with constants implied in $${\mathcal{O}}(\, \cdot \, )$$ which grow polynomially in $$d$$ , thereby overcoming the curse of dimensionality (CoD) and justifying the use of DNNs in financial modelling of large baskets in markets with jumps. In addition, we exploit parabolic smoothing of Kolmogorov partial integro-differential equations for certain multivariate Lévy processes to present alternative architectures of ReLU (“rectified linear unit”) DNNs that provide $$\varepsilon$$ expression error in DNN size $${\mathcal{O}}( \log (\varepsilon ) ^{a})$$ with exponent $$a$$ proportional to $$d$$ , but with constants implied in $${\mathcal{O}}(\, \cdot \, )$$ growing exponentially with respect to $$d$$ . Under stronger, dimension-uniform non-degeneracy conditions on the Lévy symbol, we obtain algebraic expression rates of option prices in exponential Lévy models which are free from the curse of dimensionality. In this case, the ReLU DNN expression rates of prices depend on certain sparsity conditions on the characteristic Lévy triplet. We indicate several consequences and possible extensions of the presented results.
PubDate: 2021-10-01
DOI: 10.1007/s00780-021-00462-7

• Scenario-based risk evaluation

Abstract: Abstract Risk measures such as expected shortfall (ES) and value-at-risk (VaR) have been prominent in banking regulation and financial risk management. Motivated by practical considerations in the assessment and management of risks, including tractability, scenario relevance and robustness, we consider theoretical properties of scenario-based risk evaluation. We establish axiomatic characterisations of scenario-based risk measures that are comonotonic-additive or coherent, and we obtain a novel ES-based representation result. We propose several novel scenario-based risk measures, including various versions of Max-ES and Max-VaR, and study their properties. The theory is illustrated with financial data examples.
PubDate: 2021-10-01
DOI: 10.1007/s00780-021-00460-9

• Complete and competitive financial markets in a complex world

Abstract: Abstract We investigate the possibility of completing financial markets in a model with no exogenous probability measure, with market imperfections and with an arbitrary sample space. We also consider whether such an extension may be possible in a competitive environment. Our conclusions highlight the economic role of complexity.
PubDate: 2021-10-01
DOI: 10.1007/s00780-021-00463-6

• Additive logistic processes in option pricing

Abstract: Abstract In option pricing, it is customary to first specify a stochastic underlying model and then extract valuation equations from it. However, it is possible to reverse this paradigm: starting from an arbitrage-free option valuation formula, one could derive a family of risk-neutral probabilities and a corresponding risk-neutral underlying asset process. In this paper, we start from two simple arbitrage-free valuation equations, inspired by the log-sum-exponential function and an $$\ell ^{p}$$ vector norm. Such expressions lead respectively to logistic and Dagum (or “log-skew-logistic”) risk-neutral distributions for the underlying security price. We proceed to exhibit supporting martingale processes of additive type for underlying securities having as time marginals two such distributions. By construction, these processes produce closed-form valuation equations which are even simpler than those of the Bachelier and Samuelson–Black–Scholes models. Additive logistic processes provide parsimonious and simple option pricing models capturing various important stylised facts at the minimum price of a single market observable input.
PubDate: 2021-10-01
DOI: 10.1007/s00780-021-00461-8

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