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We analyze both finite and infinite systems of Riccati equations derived from stochastic differential games on infinite networks. We discuss a connection to the Catalan numbers and the convergence of the Catalan functions by Fourier transforms., Comment: 9 pages

Mean Field Control Games (MFCG), introduced in [Angiuli et al., 2022a], represent competitive games between a large number of large collaborative groups of agents in the infinite limit of number and size of groups. In this paper, we prove the convergence of a three-timescale Reinforcement Q-Learning (RL) algorithm to solve MFCG in a model-free approach from the point of view of representative agents. Our analysis uses a Q-table for finite state and action spaces updated at each discrete time-step over an infinite horizon. In [Angiuli et al., 2023], we proved convergence of two-timescale algorithms for MFG and MFC separately highlighting the need to follow multiple population distributions in the MFC case. Here, we integrate this feature for MFCG as well as three rates of update decreasing to zero in the proper ratios. Our technique of proof uses a generalization to three timescales of the two-timescale analysis in [Borkar, 1997]. We give a simple example satisfying the various hypothesis made in the proof of convergence and illustrating the performance of the algorithm., Comment: arXiv admin note: text overlap with arXiv:2312.06659

We establish the convergence of the unified two-timescale Reinforcement Learning (RL) algorithm presented in a previous work by Angiuli et al. This algorithm provides solutions to Mean Field Game (MFG) or Mean Field Control (MFC) problems depending on the ratio of two learning rates, one for the value function and the other for the mean field term. Our proof of convergence highlights the fact that in the case of MFC several mean field distributions need to be updated and for this reason we present two separate algorithms, one for MFG and one for MFC. We focus on a setting with finite state and action spaces, discrete time and infinite horizon. The proofs of convergence rely on a generalization of the two-timescale approach of Borkar. The accuracy of approximation to the true solutions depends on the smoothing of the policies. We provide a numerical example illustrating the convergence.

We present the development and analysis of a reinforcement learning (RL) algorithm designed to solve continuous-space mean field game (MFG) and mean field control (MFC) problems in a unified manner. The proposed approach pairs the actor-critic (AC) paradigm with a representation of the mean field distribution via a parameterized score function, which can be efficiently updated in an online fashion, and uses Langevin dynamics to obtain samples from the resulting distribution. The AC agent and the score function are updated iteratively to converge, either to the MFG equilibrium or the MFC optimum for a given mean field problem, depending on the choice of learning rates. A straightforward modification of the algorithm allows us to solve mixed mean field control games (MFCGs). The performance of our algorithm is evaluated using linear-quadratic benchmarks in the asymptotic infinite horizon framework., Comment: Revisions made in accordance with reviewer's wishes. The revision primarily includes a detailed statement of our contribution, a justification of our multiscale approach, further explanations of our RL problem setup, numerical experiments, and relevant references in response to reviewers' comments. 29 pages; 9 figures

We introduce the notions of Collective Arbitrage and of Collective Super-replication in a discrete-time setting where agents are investing in their markets and are allowed to cooperate through exchanges. We accordingly establish versions of the fundamental theorem of asset pricing and of the pricing-hedging duality. A reduction of the price interval of the contingent claims can be obtained by applying the collective super-replication price.

In this work we propose deep learning-based algorithms for the computation of systemic shortfall risk measures defined via multivariate utility functions. We discuss the key related theoretical aspects, with a particular focus on the fairness properties of primal optima and associated risk allocations. The algorithms we provide allow for learning primal optimizers, optima for the dual representation and corresponding fair risk allocations. We test our algorithms by comparison to a benchmark model, based on a paired exponential utility function, for which we can provide explicit formulas. We also show evidence of convergence in a case for which explicit formulas are not available., Comment: 4 figures, 5 tables

We propose a mean field control game model for the intra-and-inter-bank borrowing and lending problem. This framework allows to study the competitive game arising between groups of collaborative banks. The solution is provided in terms of an asymptotic Nash equilibrium between the groups in the infinite horizon. A three-timescale reinforcement learning algorithm is applied to learn the optimal borrowing and lending strategy in a data driven way when the model is unknown. An empirical numerical analysis shows the importance of the three-timescale, the impact of the exploration strategy when the model is unknown, and the convergence of the algorithm.

The aim of this paper is to study a new methodological framework for systemic risk measures by applying deep learning method as a tool to compute the optimal strategy of capital allocations. Under this new framework, systemic risk measures can be interpreted as the minimal amount of cash that secures the aggregated system by allocating capital to the single institutions before aggregating the individual risks. This problem has no explicit solution except in very limited situations. Deep learning is increasingly receiving attention in financial modelings and risk management and we propose our deep learning based algorithms to solve both the primal and dual problems of the risk measures, and thus to learn the fair risk allocations. In particular, our method for the dual problem involves the training philosophy inspired by the well-known Generative Adversarial Networks (GAN) approach and a newly designed direct estimation of Radon-Nikodym derivative. We close the paper with substantial numerical studies of the subject and provide interpretations of the risk allocations associated to the systemic risk measures. In the particular case of exponential preferences, numerical experiments demonstrate excellent performance of the proposed algorithm, when compared with the optimal explicit solution as a benchmark.

We present a new combined \textit{mean field control game} (MFCG) problem which can be interpreted as a competitive game between collaborating groups and its solution as a Nash equilibrium between groups. Players coordinate their strategies within each group. An example is a modification of the classical trader's problem. Groups of traders maximize their wealth. They face cost for their transactions, for their own terminal positions, and for the average holding within their group. The asset price is impacted by the trades of all agents. We propose a three-timescale reinforcement learning algorithm to approximate the solution of such MFCG problems. We test the algorithm on benchmark linear-quadratic specifications for which we provide analytic solutions.

We analyze the systemic risk for disjoint and overlapping groups (e.g., central clearing counterparties (CCP)) by proposing new models with realistic game features. Specifically, we generalize the systemic risk measure proposed in [F. Biagini, J.-P. Fouque, M. Frittelli, and T. Meyer-Brandis, Finance and Stochastics, 24(2020), 513--564] by allowing individual banks to choose their preferred groups instead of being assigned to certain groups. We introduce the concept of Nash equilibrium for these new models, and analyze the optimal solution under Gaussian distribution of the risk factor. We also provide an explicit solution for the risk allocation of the individual banks, and study the existence and uniqueness of Nash equilibrium both theoretically and numerically. The developed numerical algorithm can simulate scenarios of equilibrium, and we apply it to study the bank-CCP structure with real data and show the validity of the proposed model.

Mean field games (MFG) and mean field control problems (MFC) are frameworks to study Nash equilibria or social optima in games with a continuum of agents. These problems can be used to approximate competitive or cooperative games with a large finite number of agents and have found a broad range of applications, in particular in economics. In recent years, the question of learning in MFG and MFC has garnered interest, both as a way to compute solutions and as a way to model how large populations of learners converge to an equilibrium. Of particular interest is the setting where the agents do not know the model, which leads to the development of reinforcement learning (RL) methods. After reviewing the literature on this topic, we present a two timescale approach with RL for MFG and MFC, which relies on a unified Q-learning algorithm. The main novelty of this method is to simultaneously update an action-value function and a distribution but with different rates, in a model-free fashion. Depending on the ratio of the two learning rates, the algorithm learns either the MFG or the MFC solution. To illustrate this method, we apply it to a mean field problem of accumulated consumption in finite horizon with HARA utility function, and to a trader's optimal liquidation problem.

The problem of portfolio optimization when stochastic factors drive returns and volatilities has been studied in previous works by the authors. In particular, they proposed asymptotic approximations for value functions and optimal strategies in the regime where these factors are running on both slow and fast timescales. However, the rigorous justification of the accuracy of these approximations has been limited to power utilities and a single factor. In this paper, we provide an accurate analysis for cases with general utility functions and two timescale factors by constructing sub- and super-solutions to the fully nonlinear problem so that their difference is at the desired level of accuracy. This approach will be valuable in various related stochastic control problems.

Trading frictions are stochastic. They are, moreover, in many instances fast-mean reverting. Here, we study how to optimally trade in a market with stochastic price impact and study approximations to the resulting optimal control problem using singular perturbation methods. We prove, by constructing sub- and super-solutions, that the approximations are accurate to the specified order. Finally, we perform some numerical experiments to illustrate the effect that stochastic trading frictions have on optimal trading., Comment: 21 pages, 6 figures, 3 tables

The study of linear-quadratic stochastic differential games on directed networks was initiated in Feng, Fouque \& Ichiba \cite{fengFouqueIchiba2020linearquadratic}. In that work, the game on a directed chain with finite or infinite players was defined as well as the game on a deterministic directed tree, and their Nash equilibria were computed. The current work continues the analysis by first developing a random directed chain structure by assuming the interaction between every two neighbors is random. We solve explicitly for an open-loop Nash equilibrium for the system and we find that the dynamics under equilibrium is an infinite-dimensional Gaussian process described by a Catalan Markov chain introduced in \cite{fengFouqueIchiba2020linearquadratic}. The discussion about stochastic differential games is extended to a random two-sided directed chain and a random directed tree structure., Comment: 24 pages. arXiv admin note: text overlap with arXiv:2003.08840

We present a Reinforcement Learning (RL) algorithm to solve infinite horizon asymptotic Mean Field Game (MFG) and Mean Field Control (MFC) problems. Our approach can be described as a unified two-timescale Mean Field Q-learning: The \emph{same} algorithm can learn either the MFG or the MFC solution by simply tuning the ratio of two learning parameters. The algorithm is in discrete time and space where the agent not only provides an action to the environment but also a distribution of the state in order to take into account the mean field feature of the problem. Importantly, we assume that the agent can not observe the population's distribution and needs to estimate it in a model-free manner. The asymptotic MFG and MFC problems are also presented in continuous time and space, and compared with classical (non-asymptotic or stationary) MFG and MFC problems. They lead to explicit solutions in the linear-quadratic (LQ) case that are used as benchmarks for the results of our algorithm.

We study linear-quadratic stochastic differential games on directed chains inspired by the directed chain stochastic differential equations introduced by Detering, Fouque, and Ichiba. We solve explicitly for Nash equilibria with a finite number of players and we study more general finite-player games with a mixture of both directed chain interaction and mean field interaction. We investigate and compare the corresponding games in the limit when the number of players tends to infinity. The limit is characterized by Catalan functions and the dynamics under equilibrium is an infinite-dimensional Gaussian process described by a Catalan Markov chain, with or without the presence of mean field interaction.

The problem of portfolio allocation in the context of stocks evolving in random environments, that is with volatility and returns depending on random factors, has attracted a lot of attention. The problem of maximizing a power utility at a terminal time with only one random factor can be linearized thanks to a classical distortion transformation. In the present paper, we address the situation with several factors using a perturbation technique around the case where these factors are perfectly correlated reducing the problem to the case with a single factor. Our proposed approximation requires to solve numerically two linear equations in lower dimension instead of a fully non-linear HJB equation. A rigorous accuracy result is derived by constructing sub- and super- solutions so that their difference is at the desired order of accuracy. We illustrate our result with a particular model for which we have explicit formulas for the approximation. In order to keep the notations as explicit as possible, we treat the case with one stock and two factors and we describe an extension to the case with two stocks and two factors.

We propose a novel concept of a Systemic Optimal Risk Transfer Equilibrium (SORTE), which is inspired by the B\"uhlmann's classical notion of an Equilibrium Risk Exchange. We provide sufficient general assumptions that guarantee existence, uniqueness, and Pareto optimality of such a SORTE. In both the B\"uhlmann and the SORTE definition, each agent is behaving rationally by maximizing his/her expected utility given a budget constraint. The two approaches differ by the budget constraints. In B\"uhlmann's definition the vector that assigns the budget constraint is given a priori. On the contrary, in the SORTE approach, the vector that assigns the budget constraint is endogenously determined by solving a systemic utility maximization. SORTE gives priority to the systemic aspects of the problem, in order to optimize the overall systemic performance, rather than to individual rationality.

We consider a general class of mean field control problems described by stochastic delayed differential equations of McKean-Vlasov type. Two numerical algorithms are provided based on deep learning techniques, one is to directly parameterize the optimal control using neural networks, the other is based on numerically solving the McKean-Vlasov forward anticipated backward stochastic differential equation (MV-FABSDE) system. In addition, we establish a necessary and sufficient stochastic maximum principle for this class of mean field control problems with delay based on the differential calculus on function of measures, as well as existence and uniqueness results for the associated MV-FABSDE system.

Empirical studies indicate the presence of multi-scales in the volatility of underlying assets: a fast-scale on the order of days and a slow-scale on the order of months. In our previous works, we have studied the portfolio optimization problem in a Markovian setting under each single scale, the slow one in [Fouque and Hu, SIAM J. Control Optim., 55 (2017), 1990-2023], and the fast one in [Hu, Proceedings of IEEE CDC 2018, accepted]. This paper is dedicated to the analysis when the two scales coexist in a Markovian setting. We study the terminal wealth utility maximization problem when the volatility is driven by both fast- and slow-scale factors. We first propose a zeroth-order strategy, and rigorously establish the first order approximation of the associated problem value. This is done by analyzing the corresponding linear partial differential equation (PDE) via regular and singular perturbation techniques, as in the single-scale cases. Then, we show the asymptotic optimality of our proposed strategy within a specific family of admissible controls. Interestingly, we highlight that a pure PDE approach does not work in the multi-scale case and, instead, we use the so-called epsilon-martingale decomposition. This completes the analysis of portfolio optimization in both fast mean-reverting and slowly-varying Markovian stochastic environments.

We study a toy model of linear-quadratic mean field game with delay. We "lift" the delayed dynamic into an infinite dimensional space, and recast the mean field game system which is made of a forward Kolmogorov equation and a backward Hamilton-Jacobi-Bellman equation. We identify the corresponding master equation. A solution to this master equation is computed, and we show that it provides an approximation to a Nash equilibrium of the finite player game.

We propose a particle system of diffusion processes coupled through a chain-like network structure described by an infinite-dimensional, nonlinear stochastic differential equation of McKean-Vlasov type. It has both (i) a local chain interaction and (ii) a mean-field interaction. It can be approximated by a limit of finite particle systems, as the number of particles goes to infinity. Due to the local chain interaction, propagation of chaos does not necessarily hold. Furthermore, we exhibit a dichotomy of presence or absence of mean-field interaction, and we discuss the problem of detecting its presence from the observation of a single component process., Comment: 32 pages

Fractional stochastic volatility models have been widely used to capture the non-Markovian structure revealed from financial time series of realized volatility. On the other hand, empirical studies have identified scales in stock price volatility: both fast-time scale on the order of days and slow-scale on the order of months. So, it is natural to study the portfolio optimization problem under the effects of dependence behavior which we will model by fractional Brownian motions with Hurst index $H$, and in the fast or slow regimes characterized by small parameters $\eps$ or $\delta$. For the slowly varying volatility with $H \in (0,1)$, it was shown that the first order correction to the problem value contains two terms of order $\delta^H$, one random component and one deterministic function of state processes, while for the fast varying case with $H > \half$, the same form holds at order $\eps^{1-H}$. This paper is dedicated to the remaining case of a fast-varying rough environment ($H < \half$) which exhibits a different behavior. We show that, in the expansion, only one deterministic term of order $\sqrt{\eps}$ appears in the first order correction., Comment: arXiv admin note: text overlap with arXiv:1706.03139

In our previous paper, "A Unified Approach to Systemic Risk Measures via Acceptance Set" (\textit{Mathematical Finance, 2018}), we have introduced a general class of systemic risk measures that allow for random allocations to individual banks before aggregation of their risks. In the present paper, we prove the dual representation of a particular subclass of such systemic risk measures and the existence and uniqueness of the optimal allocation related to them. We also introduce an associated utility maximization problem which has the same optimal solution as the systemic risk measure. In addition, the optimizer in the dual formulation provides a \textit{risk allocation} which is fair from the point of view of the individual financial institutions. The case with exponential utilities which allows for explicit computation is treated in details., Comment: Keywords}: Systemic risk measures, random allocations, risk allocation, fairness

Empirical studies indicate the existence of long range dependence in the volatility of the underlying asset. This feature can be captured by modeling its return and volatility using functions of a stationary fractional Ornstein--Uhlenbeck (fOU) process with Hurst index $H \in (\frac{1}{2}, 1)$. In this paper, we analyze the nonlinear optimal portfolio allocation problem under this model and in the regime where the fOU process is fast mean-reverting. We first consider the case of power utility, and rigorously give first order approximations of the value and the optimal strategy by a martingale distortion transformation. We also establish the asymptotic optimality in all admissible controls of a zeroth order trading strategy. Then, we extend the discussions to general utility functions using the epsilon-martingale decomposition technique, and we obtain similar asymptotic optimality results within a specific family of admissible strategies.

A parsimonious generalization of the Heston model is proposed where the volatility-of-volatility is assumed to be stochastic. We follow the perturbation technique of Fouque et al (2011, CUP) to derive a first order approximation of the price of options on a stock and its volatility index. This approximation is given by Heston's quasi-closed formula and some of its Greeks. It can be very efficiently calculated since it requires to compute only Fourier integrals and the solution of simple ODE systems. We exemplify the calibration of the model with S&P 500 and VIX data.

Rough stochastic volatility models have attracted a lot of attentions recently, in particular for the linear option pricing problem. In this paper, starting with power utilities, we propose to use a martingale distortion representation of the optimal value function for the nonlinear asset allocation problem in a (non-Markovian) fractional stochastic environment (for all Hurst index $H \in (0,1)$). We rigorously establish a first order approximation of the optimal value, where the return and volatility of the underlying asset are functions of a stationary slowly varying fractional Ornstein-Uhlenbeck process. We prove that this approximation can be also generated by a fixed zeroth order trading strategy providing an explicit strategy which is asymptotically optimal in all admissible controls. Furthermore, we extend the discussion to general utility functions, and obtain the asymptotic optimality of this fixed strategy in a specific family of admissible strategies.

In this paper, we propose the uncertain volatility models with stochastic bounds. Like the regular uncertain volatility models, we know only that the true model lies in a family of progressively measurable and bounded processes, but instead of using two deterministic bounds, the uncertain volatility fluctuates between two stochastic bounds generated by its inherent stochastic volatility process. This brings better accuracy and is consistent with the observed volatility path such as for the VIX as a proxy for instance. We apply the regular perturbation analysis upon the worst case scenario price, and derive the first order approximation in the regime of slowly varying stochastic bounds. The original problem which involves solving a fully nonlinear PDE in dimension two for the worst case scenario price, is reduced to solving a nonlinear PDE in dimension one and a linear PDE with source, which gives a tremendous computational advantage. Numerical experiments show that this approximation procedure performs very well, even in the regime of moderately slow varying stochastic bounds.

We propose a model of inter-bank lending and borrowing which takes into account clearing debt obligations. The evolution of log-monetary reserves of $N$ banks is described by coupled diffusions driven by controls with delay in their drifts. Banks are minimizing their finite-horizon objective functions which take into account a quadratic cost for lending or borrowing and a linear incentive to borrow if the reserve is low or lend if the reserve is high relative to the average capitalization of the system. As such, our problem is an $N$-player linear-quadratic stochastic differential game with delay. An open-loop Nash equilibrium is obtained using a system of fully coupled forward and advanced backward stochastic differential equations. We then describe how the delay affects liquidity and systemic risk characterized by a large number of defaults. We also derive a close-loop Nash equilibrium using an HJB approach., Comment: 1 figure

In this paper, we study the portfolio optimization problem with general utility functions and when the return and volatility of underlying asset are slowly varying. An asymptotic optimal strategy is provided within a specific class of admissible controls under this problem setup. Specifically, we first establish a rigorous first order approximation of the value function associated to a fixed zeroth order suboptimal trading strategy, which is given by the heuristic argument in [J.-P. Fouque, R. Sircar and T. Zariphopoulou, {\it Mathematical Finance}, 2016]. Then, we show that this zeroth order suboptimal strategy is asymptotically optimal in a specific family of admissible trading strategies. Finally, we show that our assumptions are satisfied by a particular fully solvable model., Comment: 39pages, 3figures

The financial crisis has dramatically demonstrated that the traditional approach to apply univariate monetary risk measures to single institutions does not capture sufficiently the perilous systemic risk that is generated by the interconnectedness of the system entities and the corresponding contagion effects. This has brought awareness of the urgent need for novel approaches that capture systemic riskiness. The purpose of this paper is to specify a general methodological framework that is flexible enough to cover a wide range of possibilities to design systemic risk measures via multi-dimensional acceptance sets and aggregation functions, and to study corresponding examples. Existing systemic risk measures can usually be interpreted as the minimal capital needed to secure the system after aggregating individual risks. In contrast, our approach also includes systemic risk measures that can be interpreted as the minimal capital funds that secure the aggregated system by allocating capital to the single institutions before aggregating the individual risks. This allows for a possible ranking of the institutions in terms of systemic riskiness measured by the optimal allocations. Moreover, we also allow for the possibility of allocating the funds according to the future state of the system (random allocation). We provide conditions which ensure monotonicity, convexity, or quasi-convexity properties of our systemic risk measures.

In this paper we present a new method to compute the first-order approximation of the price of derivatives on futures in the context of multiscale stochastic volatility of Fouque \textit{et al.} (2011, CUP). It provides an alternative method to the singular perturbation technique presented in Hikspoors and Jaimungal (2008). The main features of our method are twofold: firstly, it does not rely on any additional hypothesis on the regularity of the payoff function, and secondly, it allows an effective and straightforward calibration procedure of the model to implied volatilities. These features were not achieved in previous works. Moreover, the central argument of our method could be applied to interest rate derivatives and compound derivatives. The only pre-requisite of our approach is the first-order approximation of the underlying derivative. Furthermore, the model proposed here is well-suited for commodities since it incorporates mean reversion of the spot price and multiscale stochastic volatility. Indeed, the model was validated by calibrating it to options on crude-oil futures, and it displays a very good fit of the implied volatility., Comment: 30 pages and 3 figures

We propose a simple model of inter-bank borrowing and lending where the evolution of the log-monetary reserves of $N$ banks is described by a system of diffusion processes coupled through their drifts in such a way that stability of the system depends on the rate of inter-bank borrowing and lending. Systemic risk is characterized by a large number of banks reaching a default threshold by a given time horizon. Our model incorporates a game feature where each bank controls its rate of borrowing/lending to a central bank. The optimization reflects the desire of each bank to borrow from the central bank when its monetary reserve falls below a critical level or lend if it rises above this critical level which is chosen here as the average monetary reserve. Borrowing from or lending to the central bank is also subject to a quadratic cost at a rate which can be fixed by the regulator. We solve explicitly for Nash equilibria with finitely many players, and we show that in this model the central bank acts as a clearing house, adding liquidity to the system without affecting its systemic risk. We also study the corresponding Mean Field Game in the limit of large number of banks in the presence of a common noise.

Multiscale stochastic volatility models have been developed as an efficient way to capture the principle effects on derivative pricing and portfolio optimization of randomly varying volatility. The recent book Fouque, Papanicolaou, Sircar and S{\o}lna (2011, CUP) analyzes models in which the volatility of the underlying is driven by two diffusions -- one fast mean-reverting and one slow-varying, and provides a first order approximation for European option prices and for the implied volatility surface, which is calibrated to market data. Here, we present the full second order asymptotics, which are considerably more complicated due to a terminal layer near the option expiration time. We find that, to second order, the implied volatility approximation depends quadratically on log-moneyness, capturing the convexity of the implied volatility curve seen in data. We introduce a new probabilistic approach to the terminal layer analysis needed for the derivation of the second order singular perturbation term, and calibrate to S&P 500 options data., Comment: 34 pages, 2 figures

In this paper, we study stochastic volatility models in regimes where the maturity is small, but large compared to the mean-reversion time of the stochastic volatility factor. The problem falls in the class of averaging/homogenization problems for nonlinear HJB-type equations where the "fast variable" lives in a noncompact space. We develop a general argument based on viscosity solutions which we apply to the two regimes studied in the paper. We derive a large deviation principle, and we deduce asymptotic prices for out-of-the-money call and put options, and their corresponding implied volatilities. The results of this paper generalize the ones obtained in Feng, Forde and Fouque [SIAM J. Financial Math. 1 (2010) 126-141] by a moment generating function computation in the particular case of the Heston model., Comment: Published in at http://dx.doi.org/10.1214/11-AAP801 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We propose a multi-scale stochastic volatility model in which a fast mean-reverting factor of volatility is built on top of the Heston stochastic volatility model. A singular pertubative expansion is then used to obtain an approximation for European option prices. The resulting pricing formulas are semi-analytic, in the sense that they can be expressed as integrals. Difficulties associated with the numerical evaluation of these integrals are discussed, and techniques for avoiding these difficulties are provided. Overall, it is shown that computational complexity for our model is comparable to the case of a pure Heston model, but our correction brings significant flexibility in terms of fitting to the implied volatility surface. This is illustrated numerically and with option data.

Using spectral decomposition techniques and singular perturbation theory, we develop a systematic method to approximate the prices of a variety of options in a fast mean-reverting stochastic volatility setting. Four examples are provided in order to demonstrate the versatility of our method. These include: European options, up-and-out options, double-barrier knock-out options, and options which pay a rebate upon hitting a boundary. For European options, our method is shown to produce option price approximations which are equivalent to those developed in [5]. [5] Jean-Pierre Fouque, George Papanicolaou, and Sircar Ronnie. Derivatives in Financial Markets with Stochas- tic Volatility. Cambridge University Press, 2000.

In 1999 Robert Fernholz observed an inconsistency between the normative assumption of existence of an equivalent martingale measure (EMM) and the empirical reality of diversity in equity markets. We explore a method of imposing diversity on market models by a type of antitrust regulation that is compatible with EMMs. The regulatory procedure breaks up companies that become too large, while holding the total number of companies constant by imposing a simultaneous merge of other companies. The regulatory events are assumed to have no impact on portfolio values. As an example, regulation is imposed on a market model in which diversity is maintained via a log-pole in the drift of the largest company. The result is the removal of arbitrage opportunities from this market while maintaining the market's diversity., Comment: 21 pages

Gaussian copulas are widely used in the industry to correlate two random variables when there is no prior knowledge about the co-dependence between them. The perturbed Gaussian copula approach allows introducing the skew information of both random variables into the co-dependence structure. The analytical expression of this copula is derived through an asymptotic expansion under the assumption of a common fast mean reverting stochastic volatility factor. This paper applies this new perturbed copula to the valuation of derivative products; in particular FX quanto options to a third currency. A calibration procedure to fit the skew of both underlying securities is presented. The action of the perturbed copula is interpreted compared to the Gaussian copula. A real worked example is carried out comparing both copulas and a local volatility model with constant correlation for varying maturities, correlations and skew configurations., Comment: 34 pages, 6 figures and 3 tables