This is the first book about the emerging field of utility indifference pricing for valuing derivatives in incomplete markets. René Carmona brings together a who's who of leading experts in the field ...to provide the definitive introduction for students, scholars, and researchers. Until recently, financial mathematicians and engineers developed pricing and hedging procedures that assumed complete markets. But markets are generally incomplete, and it may be impossible to hedge against all sources of randomness.Indifference Pricingoffers cutting-edge procedures developed under more realistic market assumptions.
The book begins by introducing the concept of indifference pricing in the simplest possible models of discrete time and finite state spaces where duality theory can be exploited readily. It moves into a more technical discussion of utility indifference pricing for diffusion models, and then addresses problems of optimal design of derivatives by extending the indifference pricing paradigm beyond the realm of utility functions into the realm of dynamic risk measures. Focus then turns to the applications, including portfolio optimization, the pricing of defaultable securities, and weather and commodity derivatives. The book features original mathematical results and an extensive bibliography and indexes.
In addition to the editor, the contributors are Pauline Barrieu, Tomasz R. Bielecki, Nicole El Karoui, Robert J. Elliott, Said Hamadène, Vicky Henderson, David Hobson, Aytac Ilhan, Monique Jeanblanc, Mattias Jonsson, Anis Matoussi, Marek Musiela, Ronnie Sircar, John van der Hoek, and Thaleia Zariphopoulou.
The first book on utility indifference pricingExplains the fundamentals of indifference pricing, from simple models to the most technical onesGoes beyond utility functions to analyze optimal risk transfer and the theory of dynamic risk measuresCovers non-Markovian and partially observed models and applications to portfolio optimization, defaultable securities, static and quadratic hedging, weather derivatives, and commoditiesIncludes extensive bibliography and indexesProvides essential reading for PhD students, researchers, and professionals
Mean field games are studied by means of the weak formulation of stochastic optimal control. This approach allows the mean field interactions to enter through both state and control processes and ...take a form which is general enough to include rank and nearest-neighbor effects. Moreover, the data may depend discontinuously on the state variable, and more generally its entire history. Existence and uniqueness results are proven, along with a procedure for identifying and constructing distributed strategies which provide approximate Nash equlibria for finite-player games. Our results are applied to a new class of multi-agent price impact models and a class of flocking models for which we prove existence of equilibria.
MEAN FIELD GAMES WITH COMMON NOISE Carmona, René; Delarue, François; Lacker, Daniel
The Annals of probability,
11/2016, Volume:
44, Issue:
6
Journal Article
Peer reviewed
Open access
A theory of existence and uniqueness is developed for general stochastic differential mean field games with common noise. The concepts of strong and weak solutions are introduced in analogy with the ...theory of stochastic differential equations, and existence of weak solutions for mean field games is shown to hold under very general assumptions. Examples and counter-examples are provided to enlighten the underpinnings of the existence theory. Finally, an analog of the famous result of Yamada and Watanabe is derived, and it is used to prove existence and uniqueness of a strong solution under additional assumptions.
Full text
Available for:
BFBNIB, INZLJ, NMLJ, NUK, PNG, SAZU, UL, UM, UPUK, ZRSKP
The purpose of this paper is to provide a complete probabilistic analysis of a large class of stochastic differential games with mean field interactions. We implement the Mean-Field Game strategy ...developed analytically by Lasry and Lions in a purely probabilistic framework, relying on tailor-made forms of the stochastic maximum principle. While we assume that the state dynamics are affine in the states and the controls, and the costs are convex, our assumptions on the nature of the dependence of all the coefficients upon the statistical distribution of the states of the individual players remains of a rather general nature. Our probabilistic approach calls for the solution of systems of forward-backward stochastic differential equations of a McKean--Vlasov type for which no existence result is known, and for which we prove existence and regularity of the corresponding value function. Finally, we prove that a solution of the Mean-Field Game problem as formulated by Lasry and Lions, does indeed provide approximate Nash equilibriums for games with a large number of players, and we quantify the nature of the approximation. PUBLICATION ABSTRACT
The purpose of this paper is to provide a detailed probabilistic analysis of the optimal control of nonlinear stochastic dynamical systems of McKean–Vlasov type. Motivated by the recent interest in ...mean-field games, we highlight the connection and the differences between the two sets of problems. We prove a new version of the stochastic maximum principle and give sufficient conditions for existence of an optimal control. We also provide examples for which our sufficient conditions for existence of an optimal solution are satisfied. Finally we show that our solution to the control problem provides approximate equilibria for large stochastic controlled systems with mean-field interactions when subject to a common policy.
Full text
Available for:
BFBNIB, INZLJ, NMLJ, NUK, PNG, SAZU, UL, UM, UPUK, ZRSKP
We propose a new approach to mean field games with major and minor players. Our formulation involves a two player game where the optimization of the representative minor player is standard while the ...major player faces an optimization over conditional McKean-Vlasov stochastic differential equations. The definition of this limiting game is justified by proving that its solution provides approximate Nash equilibriums for large finite player games. This proof depends upon the generalization of standard results on the propagation of chaos to conditional dynamics. Because it is of independent interest, we prove this generalization in full detail. Using a conditional form of the Pontryagin stochastic maximum principle (proven in the Appendix), we reduce the solution of the mean field game to a forward-backward system of stochastic differential equations of the conditional McKean-Vlasov type, which we solve in the linear quadratic setting. We use this class of models to show that Nash equilibriums in our formulation can be different from those originally found in the literature.
We propose two numerical methods for the optimal control of McKean–Vlasov dynamics in finite time horizon. Both methods are based on the introduction of a suitable loss function defined over the ...parameters of a neural network. This allows the use of machine learning tools, and efficient implementations of stochastic gradient descent in order to perform the optimization. In the first method, the loss function stems directly from the optimal control problem. The second method tackles a generic forward-backward stochastic differential equation system (FBSDE) of McKean–Vlasov type, and relies on suitable reformulation as a mean field control problem. To provide a guarantee on how our numerical schemes approximate the solution of the original mean field control problem, we introduce a new optimization problem, directly amenable to numerical computation, and for which we rigorously provide an error rate. Several numerical examples are provided. Both methods can easily be applied to certain problems with common noise, which is not the case with the existing technology. Furthermore, although the first approach is designed for mean field control problems, the second is more general and can also be applied to the FBSDEs arising in the theory of mean field games.
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.
The main thrust of the article is the development of a joint stochastic model for electricity demand, and wind and solar power production in a given region. The model hinges on special statistical ...data analysis techniques including the estimation of heavy tail distributions, graphical LASSO fitting procedures, and conditional Monte Carlo simulations. Assuming the availability of point forecasts, we model the deviations from these forecasts instead of modeling the actual quantities of interest. The resolution of the model is determined by the resolution of the forecast data. For the sake of illustration, we implement our model and the corresponding simulation algorithms on data made available by NREL for the Texas region with hourly time resolution, load data at the zone level, and wind and solar power production at the generation asset level. Our numerical simulations confirm that the dependencies identified through the fitting algorithm are consistent with the relative locations of the production assets and the geographical load zones over which the data were collected.
PDF
