Quantitative Finance

We perform a detailed theoretical study of the value of a class of participating policies with four key features: (i) the policyholder is guaranteed a minimum interest rate on the policy reserve; (ii) the contract can be terminated by the holder at any time until maturity (surrender option); (iii) at the maturity (or upon surrender) a bonus can be credited to the holder if the portfolio backing the policy outperforms the current policy reserve; (iv) due to solvency requirements the contract ends if the value of the underlying portfolio of assets falls below the policy reserve. Our analysis is probabilistic and it relies on optimal stopping and free boundary theory. We find a peculiar structure of the optimal surrender strategy, which was undetected by previous (mostly numerical) studies on the same topic. For that we develop new methods in order to study the regularity of the corresponding optimal stopping boundaries.

Conic martingales refer to Brownian martingales evolving between bounds. Among other potential applications, they have been suggested for the sake of modeling conditional survival probabilities under partial information, as usual in reduced-form models. Yet, conic martingale default models have a special feature; in contrast to the class of Cox models, they fail to satisfy the so-called \emph{immersion property}. Hence, it is not clear whether this setup is arbitrage-free or not. In this paper, we study the relevance of conic martingales-driven default models for practical applications in credit risk modeling. We first introduce an arbitrage-free conic martingale, namely the Φ -martingale, by showing that it fits in the class of Dynamized Gaussian copula model of Crépey et al., thereby providing an explicit construction scheme for the default time. In particular, the Φ -martingale features interesting properties inherent on its construction easing the practical implementation. Eventually, we apply this model to CVA pricing under wrong-way risk and CDS options, and compare our results with the JCIR++ (a.k.a. SSRJD) and TC-JCIR recently introduced as an alternative.

In this article we consider the infinite-horizon Merton investment-consumption problem in a constant-parameter Black - Scholes - Merton market for an agent with constant relative risk aversion R. The classical primal approach is to write down a candidate value function and to use a verification argument to prove that this is the solution to the problem. However, features of the problem take it outside the standard settings of stochastic control, and the existing primal verification proofs rely on parameter restrictions (especially, but not only, R<1), restrictions on the space of admissible strategies, or intricate approximation arguments. The purpose of this paper is to show that these complications can be overcome using a simple and elegant argument involving a stochastic perturbation of the utility function.

We prove the global existence of an incomplete, continuous-time finite-agent Radner equilibrium in which exponential agents optimize their expected utility over both running consumption and terminal wealth. The market consists of a traded annuity, and, along with unspanned income, the market is incomplete. Set in a Brownian framework, the income is driven by a multidimensional diffusion, and, in particular, includes mean-reverting dynamics. The equilibrium is characterized by a system of fully coupled quadratic backward stochastic differential equations, a solution to which is proved to exist under Markovian assumptions.

We use the superposition of the Levy processes to optimize the classic BN-S model. Considering the frequent fluctuations of price parameters difficult to accurately estimate in the model, we preprocess the price data based on fuzzy theory. The price of S&P500 stock index options in the past ten years are analyzed, and the deterministic fluctuations are captured by machine learning methods. The results show that the new model in a fuzzy environment solves the long-term dependence problem of the classic model with fewer parameter changes, and effectively analyzes the random dynamic characteristics of stock index option price time series.

In this work we present an analytical model, based on the path-integral formalism of Statistical Mechanics, for pricing options using first-passage time problems involving both fixed and deterministically moving absorbing barriers under possible non-gaussian distributions of the underlying object. We adapt to our problem a model originally proposed to describe the formation of galaxies in the universe of De Simone et al (2011), which uses cumulant expansions in terms of the Gaussian distribution, and we generalize it to take into acount drift and cumulants of orders higher than three. From the probability density function, we obtain an analytical pricing model, not only for vanilla options (thus removing the need of volatility smile inherent to the Black-Scholes model), but also for fixed or deterministically moving barrier options. Market prices of vanilla options are used to calibrate the model, and barrier option pricing arising from the model is compared to the price resulted from the relative entropy model.

In a discrete-time setting, we study arbitrage concepts in the presence of convex trading constraints. We show that solvability of portfolio optimization problems is equivalent to absence of arbitrage of the first kind, a condition weaker than classical absence of arbitrage opportunities. We center our analysis on this characterization of market viability and derive versions of the fundamental theorems of asset pricing based on portfolio optimization arguments. By considering specifically a discrete-time setup, we simplify existing results and proofs that rely on semimartingale theory, thus allowing for a clear understanding of the foundational economic concepts involved. We exemplify these concepts, as well as some unexpected situations, in the context of one-period factor models with arbitrage opportunities under borrowing constraints.

Models which postulate lognormal dynamics for interest rates which are compounded according to market conventions, such as forward LIBOR or forward swap rates, can be constructed initially in a discrete tenor framework. Interpolating interest rates between maturities in the discrete tenor structure is equivalent to extending the model to continuous tenor. The present paper sets forth an alternative way of performing this extension; one which preserves the Markovian properties of the discrete tenor models and guarantees the positivity of all interpolated rates.

The goal is to re-examine and extend the findings from the recent paper by Dumitrescu, Quenez and Sulem (2017) who studied game options within the nonlinear arbitrage-free pricing approach developed in El Karoui and Quenez (1997). We consider the setup introduced in Kim, Nie and Rutkowski (2018) where contracts of an American style were examined. We give a detailed study of unilateral pricing, hedging and exercising problems for the counterparties within a general nonlinear setup. We also present a BSDE approach, which is used to obtain more explicit results under suitable assumptions about solutions to doubly reflected BSDEs.

We study the Fundamental Theorem of Asset Pricing for a general financial market under Knightian Uncertainty. We adopt a functional analytic approach which require neither specific assumptions on the class of priors P nor on the structure of the state space. Several aspects of modeling under Knightian Uncertainty are considered and analyzed. We show the need for a suitable adaptation of the notion of No Free Lunch with Vanishing Risk and discuss its relation to the choice of an appropriate filtration. In an abstract setup, we show that absence of arbitrage is equivalent to the existence of \emph{approximate} martingale measures sharing the same polar set of P . We then specialize the results to a discrete-time framework in order to obtain true martingale measures.