11th European Summer School in
Financial Mathematics

Paris, 27 - 31 August 2018

Paris, 27 - 31 August 2018

## CMAP

**Main Lecture Courses**

The summer school will be structured around two main topics:

**Contract Theory: incentive policy and applications to financial regulation.**Minicourses will be given by Dylan Possamai and Stéphane Villeneuve.

Partial lecture notes of Stéphane Villeneuve are**here**.**New advances in affine process: simulations and rough models.**Minicourses will be given by Aurélien Alfonsi and Sergio Pulido.

Since the celebrated Modigliani-Miller theorem, it is well-known that in the absence of frictions, the value of a company is independent of its capital structure, i.e. the proportion of debt and equity in the liability side of the balance sheet. In other words, corporate finance decisions are irrelevant in a perfect capital market. Several types of financial frictions have been introduced to understand the corporate financial and managerial decisions observed in practice. In this course, we will focus on two majors frictions, cost of issuing new securities (chapter 1) and agency problem (chapter 2). Chapter 1 introduces the problem of optimal liquidity management where issuing costs are so high that companies use their cash reserves as precautionary savings. It has to be viewed as a general introduction. From a methodical point of view, the manager is facing challenging singular control problems. Chapter 2 introduces principal-agent frictions in a continuous-time stationary setting. The workhorse of this literature is a simple model with one principal with funds but lack of expertise makes a take-it-or-leave-it offer to a single agent with outside reservation utility to steer a project. If the contract is accepted, the agent then chooses an action which will affect an outcome and which is not observable by the principal. The main objective of the principal is to establish a contract that will incentivize the agent to align his effort with the principal'€™s interest. Although these agency frictions are different from the cost of issuance (even if the presence of these costs can be justified by agency problems), the methodology is very similar to chapter 1 and involves singular control problems.

The second part of this lecture will consists in an overview of recent progresses made in contracting theory, using the so-called dynamic programming approach. The basic situation is that of a Principal wanting to hire an Agent to do a task on his behalf, and who has to be properly incentivized. We will show how this general framework allows to treat volatility control problems arising for instance in delegated portfolio management, in electricity pricing, or in central clearing houses. If time permit, we will also analyze the situation of a Principal hiring a finite number of Agents who can interact with each other, as well as the associated mean-field problem. The theory will be mostly illustrated by examples ranging from finance and insurance applications to regulation issues.

Slides of Aurélien Alfonsi are

**here**together with a

**printed version**.

Slides of Sergio Puido are

**here**.

Affine processes are widely used in mathematical finance. The most famous examples of affine models include the Cox-Ingersoll-Ross model (1985) for interest rates and the Heston stochastic volatility model (1993) for equity.Â In this first lecture, we will start by characterizing the one-dimensional affine diffusions and presenting the main properties of affine processes. We will present several models (including Heston model) based on affine processes for different financial applications, and explain why the affine property is convenient. A particular attention will be given to Wishart processes, that are affine diffusions on the set of positive semidefinite matrices. Last, we will study in detail how to simulate and approximate affine diffusions to run Monte-Carlo methods, starting with the simulation of the CIR process.

The second part of this lecture will focus on a self-contained introduction to the class of Affine Volterra Processes. Classical affine diffusions constitute a special case, but affine Volterra processes are neither semimartingales, nor Markov processes in general. For specific state spaces, we will prove existence, uniqueness, and invariance properties of solutions of the corresponding stochastic convolution equations. We will provide exponential-affine representations of the Fourierâ€“Laplace functional in terms of the solution of an associated system of deterministic integral equations, extending well-known formulas for classical affine diffusions. Our arguments will avoid infinite-dimensional stochastic analysis as well as stochastic integration with respect to non-semimartingales, relying instead on tools from the theory of finite-dimensional deterministic convolution equations. At the end of the mini-course we will establish connections between our findings and related results from the theory of Stochastic Partial Differential Equations (SPDEs) and infinite dimensional affine processes. The motivation to study affine Volterra processes comes from applications in fi- nancial modeling. A growing body of empirical research indicates that volatility fluctuates more rapidly than Brownian motion, which is inconsistent with standard semimartingale models. Fractional volatility models have emerged as compelling alternatives, although tractability can be a challenge for these non-Markovian, nonsemimartingales models. Some of these models, such as the rough Heston model, are of the affine Volterra type involving singular kernels. Our framework subsumes and extends these examples. The mini course will be structured as follows: 1. Motivation from rough volatility modeling: The rough Heston model 2. Theory of convolution equations: Existence and invariance properties of solutions of deterministic and stochastic convolution equations 3. Definition of Affine Volterra Processes and the Fourierâ€“Laplace transform formula 4. Connections to SPDEs and infinite dimensional affine processes 5. Future lines of research: numerical and theoretical challenges