From macro to micro: causal inference, firm valuation and trading conditions

The aim of the Ph.D. thesis is twofold. First, we investigate possible stock market mispricing to eventually build profitable investment strategies. Second, we analyze how the microscopic interactions among agents influence trading conditions thereby leading to market instabilities. As regards the study of possible mispricing, we identify via the vector autoregressive approach revenues as the primary driver process of firm growth. To do so, we employ the recent Independent Component Analysis (ICA) technique which allows us to identify contemporaneous causal relations among the considered variables. In particular, the first original contribution of the thesis is to extend the ICA methodology for singular and noisy structural vector autoregressive models; see Chapter 2. As a second original contribution, starting from the revenues, we propose a firm valuation framework incorporating the associated intrinsic uncertainty. We derive a probability distribution of fair values, we construct a market factor capturing misvaluation comovements and we propose two stock recommendation systems that hinge on the fair value distribution; see Chapters 3, 4 and 5. Finally, in the last contribution, we analyze asymptotically market stability as the number of assets and traders increase. Market instability is defined as a result of oscillating equilibrium strategies of optimal execution problems in market impact games, where the dynamical equilibrium between the activity of simultaneously trading agents generates the price dynamics. One of the main results is the connection of market instability to the market cross-impact structure when portfolios execution orders are considered; see Chapter 7, 8 and 9

A spatial measure-valued model for radiation-induced DNA damage kinetics and repair under protracted irradiation condition

In the present work, we develop a general spatial stochastic model to describe the formation and repair of radiation-induced DNA damage. The model is described mathematically as a measure-valued particle-based stochastic system and extends in several directions the model developed in Cordoni et.al. 2021, Cordoni et.al. 2022a, Cordoni et.al. 2022b. In this new spatial formulation, radiation-induced DNA damage in the cell nucleus can undergo different pathways to either repair or lead to cell inactivation. The main novelty of the work is to rigorously define a spatial model that considers the pairwise interaction of lesions and continuous protracted irradiation. The former is relevant from a biological point of view as clustered lesions are less likely to be repaired, leading thus to cell inactivation. The latter instead describes the effects of a continuous radiation field on biological tissue. We prove the existence and uniqueness of a solution to the above stochastic systems, characterizing its probabilistic properties. We further couple the model describing the biological system to a set of reaction-diffusion equations with random discontinuity that model the chemical environment. At last, we study the large system limit of the process. The developed model can be applied to different contexts, with radiotherapy and space radioprotection being the most relevant. Further, the biochemical system derived can play a crucial role in understanding an extremely promising novel radiotherapy treatment modality, named in the community FLASH radiotherapy, whose mechanism is today largely unknown

A nonlinear Kolmogorov equation for stochastic functional delay differential equations with jumps

We consider a stochastic functional delay differential equation, namely an equation whose evolution depends on its past history as well as on its present state, driven by a pure diffusive component plus a pure jump Poisson compensated measure. We lift the problem in the infinite dimensional space of square integrable Lebesgue functions in order to show that its solution is an $L^2-$valued Markov process whose uniqueness can be shown under standard assumptions of locally Lipschitzianity and linear growth for the coefficients. Coupling the aforementioned equation with a standard backward differential equation, and deriving some ad hoc results concerning the Malliavin derivative for systems with memory, we are able to derive a non--linear Feynman--Kac representation theorem under mild assumptions of differentiability

A lending scheme for a system of interconnected banks with probabilistic constraints of failure

We derive a closed form solution for an optimal control problem related to an interbank lending schemes subject to terminal probability constraints on the failure of banks which are interconnected through a financial network. The derived solution applies to a real banks network by obtaining a general solution when the aforementioned probability constraints are assumed for all the banks. We also present a direct method to compute the systemic relevance parameter for each bank within the network

Transient Impact from the Nash Equilibrium of a Permanent Market Impact Game

A large body of empirical literature has shown that market impact of financial prices is transient. However, from a theoretical standpoint, the origin of this temporary nature is still unclear. We show that an implied transient impact arises from the Nash equilibrium between a directional trader and one arbitrageur in a market impact game with fixed and permanent impact. The implied impact is the one that can be empirically inferred from the directional trader's trading profile and price reaction to order flow. Specifically, we propose two approaches to derive the functional form of the decay kernel of the Transient Impact Model, one of the most popular empirical models for transient impact, from the behaviour of the directional trader at the Nash equilibrium. The first is based on the relationship between past order flow and future price change, while in the second we solve an inverse optimal execution problem. We show that in the first approach the implied kernel is unique, while in the second case infinite solutions exist and a linear kernel can always be inferred

Action-State Dependent Dynamic Model Selection

A model among many may only be best under certain states of the world. Switching from a model to another can also be costly. Finding a procedure to dynamically choose a model in these circumstances requires to solve a complex estimation procedure and a dynamic programming problem. A Reinforcement learning algorithm is used to approximate and estimate from the data the optimal solution to this dynamic programming problem. The algorithm is shown to consistently estimate the optimal policy that may choose different models based on a set of covariates. A typical example is the one of switching between different portfolio models under rebalancing costs, using macroeconomic information. Using a set of macroeconomic variables and price data, an empirical application to the aforementioned portfolio problem shows superior performance to choosing the best portfolio model with hindsight

A stochastic approach to path-dependent nonlinear Kolmogorov equations via BSDEs with time-delayed generators and applications to finance

We prove the existence of a viscosity solution of the following path dependent nonlinear Kolmogorov equation: $$\begin{cases} \partial_{t}u(t,\phi)+\mathcal{L}u(t,\phi)+f(t,\phi,u(t,\phi),\partial_{x}u(t,\phi) \sigma(t,\phi),(u(\cdot,\phi))_{t})=0,\;t\in[0,T),\;\phi\in\mathbb{\Lambda}\, ,u(T,\phi)=h(\phi),\;\phi\in\mathbb{\Lambda}, \end{cases}$$ where $\mathbb{\Lambda}=\mathcal{C}([0,T];\mathbb{R}^{d})$, $(u(\cdot ,\phi))_{t}:=(u(t+\theta,\phi))_{\theta\in[-\delta,0]}$ and $$\mathcal{L}u(t,\phi):=\langle b(t,\phi),\partial_{x}u(t,\phi)\rangle+\dfrac {1}{2}\mathrm{Tr}\big[\sigma(t,\phi)\sigma^{\ast}(t,\phi)\partial_{xx} ^{2}u(t,\phi)\big].$$ The result is obtained by a stochastic approach. In particular we prove a new type of nonlinear Feynman-Kac representation formula associated to a backward stochastic differential equation with time-delayed generator which is of non-Markovian type. Applications to the large investor problem and risk measures via $g$-expectations are also provided.Comment: 45 page

Asymptotic expansion for some local volatility models arising in finance

In this paper we study the small noise asymptotic expansions for certain classes of local volatility models arising in finance. We provide explicit expressions for the involved coefficients as well as accurate estimates on the remainders. Moreover, we perform a detailed numerical analysis, with accuracy comparisons, of the obtained results by mean of the standard Monte Carlo technique as well as exploiting the polynomial Chaos Expansion approach