Contributing Speakers
pdf Presentation from Conference (997 KB)
Title: Heterogeneous Risk Preferences in Financial Markets
Abstract: This paper builds a continuous time model of N heterogeneous agents whose CRRA preferences differ in their level of risk aversion and considers the effect of preference heterogeneity on financial markets. Preference heterogeneity allows markets for risky and risk free assets to clear at different levels, creating a separation between the risk free rate and market price of risk. This separation disconnects the marginal relative risk aversion from the marginal elasticity of inter-temporal substitution. This disconnect allows the risk-free rate to be low while maintaining a high market price of risk, providing a possible explanation of the risk-free rate puzzle, while avoiding the preference for early resolution of uncertainty inherent in Kreps-Porteus representative agent models. In addition, the model exhibits a correlation between dividend yields and the SDF, implying predictability in returns as observed in the market. Also, volatility moves in a non-linear way in response to shocks, creating a volatility smile similar to that observed in options pricing data. Finally, the model generates both pro- and counter-cyclical leverage cycles depending on the distribution of preferences and the level of the dividend.
pdf Presentation from Conference (623 KB)
Title: Mean-square stability analysis of an Extended Euler-Maruyama Method for a System of Stochastic Differential Equations
Abstract: In this work we introduce a new class of weak numerical schemes (We call it Extended Euler-Maruyama scheme) for solving system of Itô stochastic differential equations (SDEs). The proposed weak Extended Euler-Maruyama scheme has the potential to overcome some of the numerical instabilities that are often experienced when using explicit Euler method. This work also aims to determine the mean-square stability region of the weak modified Euler method for linear stochastic differential equations with multiplicative noises. We will show that the mean square stability region of an extended Euler-Maruyama method is bigger than the Euler-Maruyama method.
Title: The Learning Premium
Joint with Paolo Guasoni
Abstract: We find equilibrium stock prices and interest rates in a representative-agent model with uncertain dividends’ growth, gradually revealed by dividends themselves, where asset prices are rational - reflect current information and anticipate the impact of future knowledge on future prices. In addition to the usual premium for risk, stock returns include a learning premium, which reflects the expected change in prices from new information. In the long run, the learning premium vanishes, as prices and interest rates converge to their counterparts in the standard setting with known growth. The model explains the increase in price-dividend ratios of the past century if both relative risk aversion and elasticity of intertemporal substitution are above one.
pdf Presentation from Conference (1.22 MB)
Title: A Dynamic Eisenberg-Noe Model of Financial Contagion
Abstract: In this talk we will consider a generalized extension of the Eisenberg & Noe (2001) model of financial contagion to allow for time dynamics in both discrete and continuous time. Derivation and interpretation as a generalization of the Eisenberg & Noe model will be provided. In discrete time, special cases corresponding to Eisenberg & Noe with endogenous loans will be considered. Emphasis will be placed on the continuous time framework and its formulation as a stochastic differential equation driven by random operating cash flows. Mathematical results on existence and uniqueness of firm equity and losses under the discrete and continuous time models will be given.
Title: Optimal Control of MDP's with Unbounded Cost on Infinite Horizon
Abstract: We use Markov risk measures to formulate a risk averse version of a total cost problem on a controlled Markov process in infinite horizon. The one step costs are in L^1 but not necessarily bounded. We derive the conditions for the existence of the optimal strategies and present the robust dynamic programing equations. We illustrate our results in an optimal investment problem on infinite horizon.
pdf Presentation from Conference (447 KB)
Title: Model Uncertainty, Recalibration, and the Emergence of Delta-Vega Hedging
Joint with Johannes Muhle-Karbe.
Abstract: We study option pricing and hedging with uncertainty about a Black-Scholes reference model which is dynamically recalibrated to the market price of a liquidly traded vanilla option. For dynamic trading in the underlying asset and this vanilla option, delta-vega hedging is asymptotically optimal in the limit for small uncertainty aversion. The corresponding indifference price corrections are determined by the disparity between the vegas, gammas, vannas, and volgas of the non-traded and the liquidly traded options.
Title: Path-dependent PDEs in infinite dimensions
Joint with Erhan Bayraktar
Abstract: We propose a notion of generalized solutions for a class of fully nonlinear first-order path-dependent PDEs on Hilbert space. Under this notion, we prove uniqueness, existence, and stability for our equations. As an application, we study optimal control problems and differential games associated to nonlinear evolution equations with locally monotone and coercive operators, which were introduced by Liu (2011) and Liu and Röckner (2010). In particular, p-Laplace equations, 2D Navier-Stokes equations, tamed 3D Navier-Stokes equations, and power fluid laws can be dealt with in this framework. Besides optimal control of PDEs, another area of applications are large deviations for SPDEs. If time remains, possible future research directions for extensions to the second-order case will be pointed out. This is of relevance for optimal control of SPDEs.
Currently, our theory has in its scope no counterpart in the non-path-dependent case.
pdf Presentation from Conference (287 KB)
Title: Fluctuations of diffusions interacting through the ranks
Joint with Misha Shkolnikov
Abstract: We study the fluctuations of a system of diffusions interacting through the ranks when the number of diffusions goes to infinity. It is known that the empirical cumulative distribution function of such diffusions converges to a non-random limiting cumulative distribution function which satisfies the porous medium PDE. We show that the fluctuations of the empirical cumulative distribution function around its limit are governed by a suitable SPDE.
pdf Presentation from Conference (767 KB)
Title: Particle Systems with Singular Interaction: application in Systemic Risk modeling.
Joint with Mykhaylo Shkolnikov.
Abstract: In this talk, I will analyze a system of particles with singular interaction through hitting times. Such systems have been used in Neuroscience, but are also well suited for modeling Systemic Risk. I will discuss the latter application and will proceed to the analysis of the associated system. In particular, I will analyze the large-population limit of this system and will provide insights into the behavior of the limiting process. The main challenges in this work stem from the very singular type of interaction between the particles which requires the use of non-standard mathematical methods.
pdf Presentation from Conference (1.26 MB)
Title: Portfolio optimization near horizon
Joint work with Rohini Kumar
Abstract: We study the problem of portfolio optimization in a stochastic volatility model when the time horizon is small. Using asymptotic techniques, we obtain a closed-form formula for a portfolio which approximates the optimal portfolio near horizon. These results are achieved by constructing sub- and super-solutions to the "marginal HJB equation" and applying a comparison principle argument.
pdf Presentation from Conference (669 KB)
Title: A General Valuation Framework for SABR and Stochastic Local Volatility Models
Joint with Z. Cui and J. L. Kirkby
Abstract: In this talk, we propose a general framework for the valuation of options in stochastic local volatility (SLV) models with a general correlation structure, which includes the Stochastic Alpha Beta Rho (SABR) model as a special case. Standard stochastic volatility models, such as Heston, Hull-White, Scott, Stein-Stein, $\alpha$-Hypergeometric, 3/2, 4/2, mean reverting, and Jacobi stochastic volatility models, also fall within this general framework. We propose a novel double-layer continuous-time Markov chain (CTMC) approximation respectively for the variance process and the underlying asset price process. The resulting regime-switching continuous-time Markov chain is further reduced to a single CTMC on an enlarged state space. Closed-form matrix expressions for European options are derived. We also propose a recursive risk-neutral valuation technique for pricing discretely monitored path-dependent options, and use it to price Bermudan, and barrier options. In addition, we provide single Laplace transform formulae for arithmetic Asian options as well as occupation time derivatives. Numerical examples demonstrate the accuracy and efficiency of the method using several popular SLV models, and reference prices are provided for SABR, Heston-SABR, quadratic SLV, and the Jacobi model.
pdf Presentation from Conference (1.11 MB)
Title: Uncertain Volatility Models with Stochastic Bounds
Abstract: 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.
Title: Sensitivity Analysis of a Financial Network Model
Abstract: The financial system is increasingly interconnected. Cyclical interdependencies among corporations may cause that the default of one firm seriously affects other firms and even the whole financial network. To describe financial networks, L. Eisenberg and T. Noe introduced network models that became popular among researchers and practitioners. To describe the connections between firms, they use the liabilities between two firms to construct relative liability matrices. Based on this description, they compute the payouts of firms to their counterparties. However, in practice, there is no accurate record of the liabilities and researchers have to resort to estimation processes. Thus it is very important to understand possible errors of payouts due to the estimation errors. In our research, we describe estimation errors via sizes and directions of perturbations in the relative liability matrices. We quantify the effect of estimation errors to payouts using directional directives and derive its formula in the regular financial network. For a given relative liability matrix, we compute the effect to the payout of different estimation errors.
pdf Presentation from Conference (3.12 MB)
Title: Optimal Investment and Pricing in the Presence of Defaults.
Abstract: We consider the optimal investment problem when the traded asset may default, causing a jump in its price. For an investor with constant absolute risk aversion, we compute indifference prices for defaultable bonds, as well as a price for dynamic protection against default. For the latter problem, our work complements Sircar & Zariphopoulou (2007), where it is implicitly assumed the investor is protected against default. We consider a factor model where the asset's instantaneous return, variance, correlation and default intensity are driven by a time-homogenous diffusion X taking values in an arbitrary region E. We identify the certainty equivalent with a semi-linear degenerate parabolic partial differential equation with quadratic growth in both function and gradient. Under a minimal integrability assumption on the market price of risk, we show the certainty equivalent is a classical solution. In particular, our results cover when X is a one-dimensional affine diffusion and when returns, variances and default intensities are also affine. Numerical examples highlight the relationship between the factor process and both the indifference price and default insurance. Lastly, we show the insurance protection price is not the default intensity under the dual optimal measure.
pdf Presentation from Conference (511 KB)
Title: Developments in ODE, SDE, PDE, PIDE and Other Analytical Approaches and Their Applications to Modern Physics and Quantitative Finance
Abstract: We discuss certain latest developments in methodology and approaches to solve ordinary differential equations (ODE), stochastic differential equations (SDE), partial differential equations (PDE), partial integro-differential equations (PIDE) and related objects analytically.
These approaches are used in both Modern Physics and Quantitative Finance both theoretically and in practical applications. An additional advantage is that the approach that is developed in Physics could be often applied in Quantitative Finance and vice versa.
In our presentation, we will show that these analytical methodologies are making both research and its implementation in both Physics and Quantitative Finance much more efficient.
Abstract: In this talk, we briefly go over the formulation of optimal control problem with risk-aversion. Our focus will be the policy evaluation, which essentially is risk evaluation via BSDE. Specifically, we use dual representation of risk measure, we converted risk valuation to a stochastic control problem where the control is the Radon-Nikodym derivative process. By exploring optimality conditions, we show piecewise-constant density(control) provides a decent approximation to on a short interval. An dynamic programming algorithm extends the approximation to the a finite time horizon. Lastly, we give an application in risk management in conjunction with nested simulation.
Title: The Kumaraswamy Transmuted Pareto Distribution
Joint work with A. Akinsete, G. Aryal and H. Long
Abstract: In this modern era of statistical distribution theory, many applied researchers and experts propose several new models so that real data set can be analyzed mostly in the field of finance, medical, engineering, biology, physics, computer science and others. In this work, a new five-parameter Kumaraswamy transmuted Pareto (KwTP) distribution is introduced and studied. We discuss various mathematical and statistical properties of the distribution including obtaining expressions for the moments, quantiles, mean deviations, skewness, kurtosis, reliability and order statistics. The estimation of the model parameters is performed by the method of maximum likelihood. We compare the distribution with few other distributions to show its versatility in modeling data with heavy tail.
Title: Optimal investment with discretionary stopping-a binomial approach
Abstract: In this work, we extend the classical binomial framework in options pricing, combine it with the convex duality and dual control theory, and develop a novel (recombining) binomial framework to solve optimal investment problems with discretionary stopping, e.g. mixed optimal stochastic control problems. Our proposed binomial framework is simple to implement and allows for general forms of utility functions. We determine the early exercise boundary together with the optimal investment strategy before the early exercise. We illustrate and test our method in the cases of several general utility functions, including CRRA, SAHARA and Yaari utility functions.
Title: Recovering Linear Equations of XVA in Bilateral Contracts
Abstract: We investigate conditions to represent derivative price under XVA explicitly. As long as we consider different borrowing/lending rates, XVA problem becomes a non-linear equation and this makes finding explicit solution of XVA difficult. It is shown that the associated valuation problem is actually linear under some proper conditions so that we can have the same complexity in pricing as classical pricing theory. Moreover, the conditions mentioned above is mild in the sense that it can be obtained by choosing adequate covenants between the investor and counterparty.
Title: Martingale optimal transport with stopping
Abstract: We solve the martingale optimal transport problem for cost functionals represented by optimal stopping problems. The measure-valued martingale approach developed in ArXiv: 1507.02651 allows us to obtain an equivalent infinite-dimensional controller-stopper problem. We use the stochastic Perron's method and characterize the finite dimensional approximation as a viscosity solution to the corresponding HJB equation. It turns out that this solution is the concave envelope of the cost function with respect to the atoms of the terminal law. We demonstrate the results by finding explicit solutions for a class of cost functions.
Title: Optimal switching for balancing electric power systems
Abstract: We study the problem of offering American put and call options on electricity with immediate physical delivery for balancing a power system. We assume that the real-time electricity price is the composition of a price stack function with a stochastic process modelling physical imbalance in an electrical power system. However, as empirical analysis in the UK electricity market suggests this price stack function depends also on the current time, while the stochastic process modelling physical imbalance evolves as a mean reverting Ornstein-Uhlenbeck process. We establish a model where balancing opportunities are provided on a daily basis by a storage operator who is the owner of an electricity storage.
The storage operator optimally switches between the available states with offering balancing options for the system operator.
Using the relation with variational inequalities, we calculate the value of this problem for both parties.
This paper shows that with proper parameter choices a mutual benefit is available with the introduction of this contract, that is a financial profit is available for the storage operator and at the same time balancing cost can be cut for the electricity system operator.
Title: Optimal Control of MDP's with Unbounded Cost on Infinite Horizon
Abstract: We use Markov risk measures to formulate a risk averse version of a total cost problem on a controlled Markov process in infinite horizon. The one step costs are in L^1 but not necessarily bounded. We derive the conditions for the existence of the optimal strategies and present the robust dynamic programing equations. We illustrate our results in an optimal investment problem on infinite horizon.
