We analyze the systemic risk for disjoint and overlapping groups (e.g., central clearing counterp... more 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.
We study the smoothness of the solution of the directed chain stochastic differential equations, ... more We study the smoothness of the solution of the directed chain stochastic differential equations, where each process is affected by its neighborhood process in an infinite directed chain graph, introduced by Detering et al. (2020). Because of the auxiliary process in the chain-like structure, classic methods of Malliavin derivatives are not directly applicable. Namely, we cannot make a connection between the Malliavin derivative and the first order derivative of the state process. It turns out that the partial Malliavin derivatives can be used here to fix this problem.
We discuss Cover\'s universal portfolios in the context of Stochastic Portfolio Theory. By en... more We discuss Cover\'s universal portfolios in the context of Stochastic Portfolio Theory. By enlarging the class of portfolio generating functions, we see universal portfolios are generated by functions, given excess growth rates of constant rebalanced portfolios. These generating functions and resulting universal portfolios can be represented as integrations with respect to tilted version of maximal entropy measure. In this way we may answer one of the open questions posed by Fernholz & Karatzas (2009). With analyses of concentration of measures we evaluate performance of universal portfolios. Finally, we discuss universal portfolios under large equity market models.
We propose a model to study the effects of delayed information on option pricing. We first talk a... more We propose a model to study the effects of delayed information on option pricing. We first talk about the absence of arbitrage in our model, and then discuss super replication with delayed information in a binomial model, notably, we present a closed form formula for the price of convex contingent claims. Also, we address the convergence problem as the time-step and delay length tend to zero and introduce analogous results in the continuous time framework. Finally, we explore how delayed information exaggerates the volatility smile.
Signature is an infinite graded sequence of statistics known to characterize geometric rough path... more Signature is an infinite graded sequence of statistics known to characterize geometric rough paths. While the use of the signature in machine learning is successful in low-dimensional cases, it suffers from the curse of dimensionality in high-dimensional cases, as the number of features in the truncated signature transform grows exponentially fast. With the idea of Convolutional Neural Network, we propose a novel neural network to address this problem. Our model reduces the number of features efficiently in a data-dependent way. Some empirical experiments including high-dimensional financial time series classification and natural language processing are provided to support our convolutional signature model.
average position of all the particles. When a particle i attains the boundary 0 , it is annihilat... more average position of all the particles. When a particle i attains the boundary 0 , it is annihilated (default) and a new particle (also called i) spikes immediately in the middle of particles. More precisely, let us denote by X_{t} :=(X_{t}^{1}, \ldots, X_{t}^{N}) the positions of these particles, where X_{t}^{i}(\geq 0) is the position of particle i at time t\geq 0 for i=1 , . . . , N . With the average \overline{X}_{t}:=(X_{t}^{1}+ +X_{t}^{N})/N the dynamics of the system is determined by
A Walsh diffusion on Euclidean space moves along each ray from the origin, as a solution to a sto... more A Walsh diffusion on Euclidean space moves along each ray from the origin, as a solution to a stochastic differential equation with certain drift and diffusion coefficients, as long as it stays away from the origin. As it hits the origin, it instantaneously chooses a new direction according to a given probability law, called the spinning measure. A special example is a real-valued diffusion with skew reflections at the origin. This process continuously (in the weak sense) depends on the spinning measure. We determine a stationary measure for such process, explore long-term convergence to this distribution and establish an explicit rate of exponential convergence.
We consider a mean-field model for large banking systems, which takes into account default and re... more We consider a mean-field model for large banking systems, which takes into account default and recovery of the institutions. Building on models used for groups of interacting neurons, we first study a McKean-Vlasov dynamics and its evolutionary Fokker-Planck equation in which the mean-field interactions occur through a mean-reverting term and through a hitting time corresponding to a default level. The latter feature reflects the impact of a financial institution's default on the global distribution of reserves in the banking system. The systemic risk problem of financial institutions is understood as a blow-up phenomenon of the Fokker-Planck equation. Then, we incorporate in the model an optimization component by letting the institutions control part of their dynamics in order to minimize their expected risk. Phrasing this optimization problem as a mean-field game, we provide an explicit solution in a special case and, in the general case, we report numerical experiments based ...
A Walsh diffusion on Euclidean space moves along each ray from the origin, as a solution to a sto... more A Walsh diffusion on Euclidean space moves along each ray from the origin, as a solution to a stochastic differential equation with certain drift and diffusion coefficients, as long as it stays away from the origin. As it hits the origin, it instantaneously chooses a new direction according to a given probability law, called the spinning measure. A special example is a realvalued diffusion with skew reflections at the origin. This process continuously (in the weak sense) depends on the spinning measure. We determine a stationary measure for such process, explore long-term convergence to this distribution and establish an explicit rate of exponential convergence.
This paper analyzes the market behavior and optimal investment strategies to attain relative arbi... more This paper analyzes the market behavior and optimal investment strategies to attain relative arbitrage both in the $N$ investors and mean field regimes. An investor competes with a benchmark of market and peer investors, expecting to outperform the benchmark and minimizing the %proportion of initial capital. With market price of risk processes depending on the stock market and investors respectively, the minimal initial capital required is the optimal cost in the $N$-player games and mean field games. It can be characterized as the smallest nonnegative continuous solution of a Cauchy problem. The measure flow of wealth appears in the cost, while the joint flow of wealth and strategy is in state processes. We modify the extended mean field game with common noise and its notion of the uniqueness of Nash equilibrium. There is a unique equilibrium in $N$-player games and mean field games with mild conditions on the equity market.
We address the problem of a fund manager pooling moneys from risk-averse investors and running a ... more We address the problem of a fund manager pooling moneys from risk-averse investors and running a single portfolio of riskfree and risky assets in such a way that individual investors are not penalized for participating in this mutual fund. This problem has a well-known solution when the investors maximize their terminal expected utility of wealth using a power utility function with the same risk-aversion coefficient. We solve this problem in the case of investors having different risk-aversion coefficients. It turns out that the initial contributions of investors must follow a precise formula which depends on the various risk-aversion coefficients. We also apply our approach to the case of a single investor with risk aversion depending on the assets in the portfolio.
In this short note first we review foldng and unfolding of real-valued continuous semimartingales... more In this short note first we review foldng and unfolding of real-valued continuous semimartingales discussed in ICHIBA & KARATZAS (2014). We fold the continuous semimartingale in two ways : the conventional reflection and the SKOROKHOD reflection, in order to obtain a non-negative, continuous semimartingale. We unfold this folded process in a skew manner by assigning random signs independently of the folded process. Then as a byproduct, the stochastic differential equation is obtained corresponding to skew process. We examine its solvability, some properties of such equation, and then discuss some application of this method to planar processes.
for simplicity. At the times $\{t : X_{1}(t)=X_{2}(t)\}$ of collisions of its component particles... more for simplicity. At the times $\{t : X_{1}(t)=X_{2}(t)\}$ of collisions of its component particles in $\mathbb{R}$ , they interact through their left and right local times in a skew-elastic manner. To be more precise, we shall construct and examine a probability space $(\Omega, \mathfrak{F}, \mathbb{P})$ endowed with a filtration $F=\{\mathfrak{F}(t)\}_{0\leq t<\infty}$ that satisfies the “usual conditions” of right continuity and of augmentation by -negligible sets, and on it two pairs $(B_{1}(\cdot),$ $B_{2}$ and ($X_{1}(\cdot),$ $X_{2}$ of continuous, $F$ -adapted processes, such that $(B_{1}(\cdot),$ $B_{2}$ is planar Brownian motion and $(X_{1}(\cdot),$ $X_{2}$ is a continuous planar semimartingale that starts at some given site $(X_{1}(0), X_{2}(0))=(x_{1}, x_{2})\in \mathbb{R}^{2}$ on the plane and satisfies
We study the semimartingale properties for the generalized fractional Brownian motion (GFBM) intr... more We study the semimartingale properties for the generalized fractional Brownian motion (GFBM) introduced by Pang and Taqqu (2019). We discuss the applications of the GFBM and its mixtures in financial models, including stock price models, arbitrage and rough volatility. The GFBM is self-similar and has non-stationary increments, whose Hurst parameter $H \in (0,1)$ is determined by two parameters. We identify the region of these two parameter values in which the GFBM is a semimartingale. We also establish the $p$-variation results of the GFBM, which are used to provide an alternative proof of the non-semimartingale property when $H<1/2$. We then study the semimartingale properties of the mixed process of an independent Brownian motion and a GFBM when the Hurst parameter $H \in (1/2,1)$, and derive the associated equivalent Brownian measure.
We analyze the systemic risk for disjoint and overlapping groups (e.g., central clearing counterp... more 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.
We study the smoothness of the solution of the directed chain stochastic differential equations, ... more We study the smoothness of the solution of the directed chain stochastic differential equations, where each process is affected by its neighborhood process in an infinite directed chain graph, introduced by Detering et al. (2020). Because of the auxiliary process in the chain-like structure, classic methods of Malliavin derivatives are not directly applicable. Namely, we cannot make a connection between the Malliavin derivative and the first order derivative of the state process. It turns out that the partial Malliavin derivatives can be used here to fix this problem.
We discuss Cover\'s universal portfolios in the context of Stochastic Portfolio Theory. By en... more We discuss Cover\'s universal portfolios in the context of Stochastic Portfolio Theory. By enlarging the class of portfolio generating functions, we see universal portfolios are generated by functions, given excess growth rates of constant rebalanced portfolios. These generating functions and resulting universal portfolios can be represented as integrations with respect to tilted version of maximal entropy measure. In this way we may answer one of the open questions posed by Fernholz & Karatzas (2009). With analyses of concentration of measures we evaluate performance of universal portfolios. Finally, we discuss universal portfolios under large equity market models.
We propose a model to study the effects of delayed information on option pricing. We first talk a... more We propose a model to study the effects of delayed information on option pricing. We first talk about the absence of arbitrage in our model, and then discuss super replication with delayed information in a binomial model, notably, we present a closed form formula for the price of convex contingent claims. Also, we address the convergence problem as the time-step and delay length tend to zero and introduce analogous results in the continuous time framework. Finally, we explore how delayed information exaggerates the volatility smile.
Signature is an infinite graded sequence of statistics known to characterize geometric rough path... more Signature is an infinite graded sequence of statistics known to characterize geometric rough paths. While the use of the signature in machine learning is successful in low-dimensional cases, it suffers from the curse of dimensionality in high-dimensional cases, as the number of features in the truncated signature transform grows exponentially fast. With the idea of Convolutional Neural Network, we propose a novel neural network to address this problem. Our model reduces the number of features efficiently in a data-dependent way. Some empirical experiments including high-dimensional financial time series classification and natural language processing are provided to support our convolutional signature model.
average position of all the particles. When a particle i attains the boundary 0 , it is annihilat... more average position of all the particles. When a particle i attains the boundary 0 , it is annihilated (default) and a new particle (also called i) spikes immediately in the middle of particles. More precisely, let us denote by X_{t} :=(X_{t}^{1}, \ldots, X_{t}^{N}) the positions of these particles, where X_{t}^{i}(\geq 0) is the position of particle i at time t\geq 0 for i=1 , . . . , N . With the average \overline{X}_{t}:=(X_{t}^{1}+ +X_{t}^{N})/N the dynamics of the system is determined by
A Walsh diffusion on Euclidean space moves along each ray from the origin, as a solution to a sto... more A Walsh diffusion on Euclidean space moves along each ray from the origin, as a solution to a stochastic differential equation with certain drift and diffusion coefficients, as long as it stays away from the origin. As it hits the origin, it instantaneously chooses a new direction according to a given probability law, called the spinning measure. A special example is a real-valued diffusion with skew reflections at the origin. This process continuously (in the weak sense) depends on the spinning measure. We determine a stationary measure for such process, explore long-term convergence to this distribution and establish an explicit rate of exponential convergence.
We consider a mean-field model for large banking systems, which takes into account default and re... more We consider a mean-field model for large banking systems, which takes into account default and recovery of the institutions. Building on models used for groups of interacting neurons, we first study a McKean-Vlasov dynamics and its evolutionary Fokker-Planck equation in which the mean-field interactions occur through a mean-reverting term and through a hitting time corresponding to a default level. The latter feature reflects the impact of a financial institution's default on the global distribution of reserves in the banking system. The systemic risk problem of financial institutions is understood as a blow-up phenomenon of the Fokker-Planck equation. Then, we incorporate in the model an optimization component by letting the institutions control part of their dynamics in order to minimize their expected risk. Phrasing this optimization problem as a mean-field game, we provide an explicit solution in a special case and, in the general case, we report numerical experiments based ...
A Walsh diffusion on Euclidean space moves along each ray from the origin, as a solution to a sto... more A Walsh diffusion on Euclidean space moves along each ray from the origin, as a solution to a stochastic differential equation with certain drift and diffusion coefficients, as long as it stays away from the origin. As it hits the origin, it instantaneously chooses a new direction according to a given probability law, called the spinning measure. A special example is a realvalued diffusion with skew reflections at the origin. This process continuously (in the weak sense) depends on the spinning measure. We determine a stationary measure for such process, explore long-term convergence to this distribution and establish an explicit rate of exponential convergence.
This paper analyzes the market behavior and optimal investment strategies to attain relative arbi... more This paper analyzes the market behavior and optimal investment strategies to attain relative arbitrage both in the $N$ investors and mean field regimes. An investor competes with a benchmark of market and peer investors, expecting to outperform the benchmark and minimizing the %proportion of initial capital. With market price of risk processes depending on the stock market and investors respectively, the minimal initial capital required is the optimal cost in the $N$-player games and mean field games. It can be characterized as the smallest nonnegative continuous solution of a Cauchy problem. The measure flow of wealth appears in the cost, while the joint flow of wealth and strategy is in state processes. We modify the extended mean field game with common noise and its notion of the uniqueness of Nash equilibrium. There is a unique equilibrium in $N$-player games and mean field games with mild conditions on the equity market.
We address the problem of a fund manager pooling moneys from risk-averse investors and running a ... more We address the problem of a fund manager pooling moneys from risk-averse investors and running a single portfolio of riskfree and risky assets in such a way that individual investors are not penalized for participating in this mutual fund. This problem has a well-known solution when the investors maximize their terminal expected utility of wealth using a power utility function with the same risk-aversion coefficient. We solve this problem in the case of investors having different risk-aversion coefficients. It turns out that the initial contributions of investors must follow a precise formula which depends on the various risk-aversion coefficients. We also apply our approach to the case of a single investor with risk aversion depending on the assets in the portfolio.
In this short note first we review foldng and unfolding of real-valued continuous semimartingales... more In this short note first we review foldng and unfolding of real-valued continuous semimartingales discussed in ICHIBA & KARATZAS (2014). We fold the continuous semimartingale in two ways : the conventional reflection and the SKOROKHOD reflection, in order to obtain a non-negative, continuous semimartingale. We unfold this folded process in a skew manner by assigning random signs independently of the folded process. Then as a byproduct, the stochastic differential equation is obtained corresponding to skew process. We examine its solvability, some properties of such equation, and then discuss some application of this method to planar processes.
for simplicity. At the times $\{t : X_{1}(t)=X_{2}(t)\}$ of collisions of its component particles... more for simplicity. At the times $\{t : X_{1}(t)=X_{2}(t)\}$ of collisions of its component particles in $\mathbb{R}$ , they interact through their left and right local times in a skew-elastic manner. To be more precise, we shall construct and examine a probability space $(\Omega, \mathfrak{F}, \mathbb{P})$ endowed with a filtration $F=\{\mathfrak{F}(t)\}_{0\leq t<\infty}$ that satisfies the “usual conditions” of right continuity and of augmentation by -negligible sets, and on it two pairs $(B_{1}(\cdot),$ $B_{2}$ and ($X_{1}(\cdot),$ $X_{2}$ of continuous, $F$ -adapted processes, such that $(B_{1}(\cdot),$ $B_{2}$ is planar Brownian motion and $(X_{1}(\cdot),$ $X_{2}$ is a continuous planar semimartingale that starts at some given site $(X_{1}(0), X_{2}(0))=(x_{1}, x_{2})\in \mathbb{R}^{2}$ on the plane and satisfies
We study the semimartingale properties for the generalized fractional Brownian motion (GFBM) intr... more We study the semimartingale properties for the generalized fractional Brownian motion (GFBM) introduced by Pang and Taqqu (2019). We discuss the applications of the GFBM and its mixtures in financial models, including stock price models, arbitrage and rough volatility. The GFBM is self-similar and has non-stationary increments, whose Hurst parameter $H \in (0,1)$ is determined by two parameters. We identify the region of these two parameter values in which the GFBM is a semimartingale. We also establish the $p$-variation results of the GFBM, which are used to provide an alternative proof of the non-semimartingale property when $H<1/2$. We then study the semimartingale properties of the mixed process of an independent Brownian motion and a GFBM when the Hurst parameter $H \in (1/2,1)$, and derive the associated equivalent Brownian measure.
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