Quantitative Finance

Al-Foraih, Mishari; Burés, Òscar; Pospíšil, Jan; Vives, Josep

doi: 10.48550/arxiv.2312.00405pmid: N/A

Abstract:Using Malliavin calculus techniques, we obtain formulas for computing Greeks under different rough Volterra stochastic volatility models. Due to the fact that underlying prices are not always square integrable, we extend the classical integration by parts formula to integrable but not necessarily square integrable functionals. First of all, we obtain formulas for general stochastic volatility (SV) models, concretely the Greeks Delta, Gamma, Rho, Vega and we introduce the Greek with respect to the roughness parameter. Then, the particular case of rough Volterra SV models is analyzed. Finally, three examples are treated in detail: the family of alpha-RFSV models, that includes rough versions of SABR and Bergomi models, a mixed alpha-RFSV model with two different Hurst parameters representing short (roughness) and long memory, and the rough Stein-Stein model. For different models and Greeks we show a numerical convergence of our formulas in Monte Carlo simulations and depict for example a dependence of the Greeks on the roughness parameter.

Andrès, Hervé; Boumezoued, Alexandre; Jourdain, Benjamin

doi: 10.48550/arxiv.2312.15950pmid: N/A

Abstract:We propose a new model for the forecasting of both the implied volatility surfaces and the underlying asset price. In the spirit of Guyon and Lekeufack (2023) who are interested in the dependence of volatility indices (e.g. the VIX) on the paths of the associated equity indices (e.g. the S\&P 500), we first study how vanilla options implied volatility can be predicted using the past trajectory of the underlying asset price. Our empirical study reveals that a large part of the movements of the at-the-money-forward implied volatility for up to two years time-to-maturities can be explained using the past returns and their squares. Moreover, we show that this feedback effect gets weaker when the time-to-maturity increases. Building on this new stylized fact, we fit to historical data a parsimonious version of the SSVI parameterization (Gatheral and Jacquier, 2014) of the implied volatility surface relying on only four parameters and show that the two parameters ruling the at-the-money-forward implied volatility as a function of the time-to-maturity exhibit a path-dependent behavior with respect to the underlying asset price. Finally, we propose a model for the joint dynamics of the implied volatility surface and the underlying asset price. The latter is modelled using a variant of the path-dependent volatility model of Guyon and Lekeufack and the former is obtained by adding a feedback effect of the underlying asset price onto the two parameters ruling the at-the-money-forward implied volatility in the parsimonious SSVI parameterization and by specifying Ornstein-Uhlenbeck processes for the residuals of these two parameters and Jacobi processes for the two other parameters. Thanks to this model, we are able to simulate highly realistic paths of implied volatility surfaces that are free from static arbitrage.

Chen, An; Ferrari, Giorgio; Zhu, Shihao

doi: 10.48550/arxiv.2312.02943pmid: N/A

Abstract:This paper investigates the consumption and investment decisions of an individual facing uncertain lifespan and stochastic labor income within a Black-Scholes market framework. A key aspect of our study involves the agent's option to choose when to acquire life insurance for bequest purposes. We examine two scenarios: one with a fixed bequest amount and another with a controlled bequest amount. Applying duality theory and addressing free-boundary problems, we analytically solve both cases, and provide explicit expressions for value functions and optimal strategies in both cases. In the first scenario, where the bequest amount is fixed, distinct outcomes emerge based on different levels of risk aversion parameter $\gamma$: (i) the optimal time for life insurance purchase occurs when the agent's wealth surpasses a critical threshold if $\gamma \in (0,1)$, or (ii) life insurance should be acquired immediately if $\gamma>1$. In contrast, in the second scenario with a controlled bequest amount, regardless of $\gamma$ values, immediate life insurance purchase proves to be optimal. Finally, we extend the analysis to consider a scenario in which the individual earmarks part of her initial wealth for inheritance, where a critical wealth threshold consistently emerges.

Gubbels, K. B.; Ypma, J. Y.; Oosterlee, C. W.

doi: 10.48550/arxiv.2312.13195pmid: N/A

Abstract:We introduce a class of copulas that we call Principal Component Copulas (PCCs). This class combines the strong points of copula-based techniques with principal component analysis (PCA), which results in flexibility when modelling tail dependence along the most important directions in high-dimensional data. We obtain theoretical results for PCCs that are important for practical applications. In particular, we derive tractable expressions for the high-dimensional copula density, which can be represented in terms of characteristic functions. We also develop algorithms to perform Maximum Likelihood and Generalized Method of Moment estimation in high-dimensions and show very good performance in simulation experiments. Finally, we apply the copula to the international stock market to study systemic risk. We find that PCCs lead to excellent performance on measures of systemic risk due to their ability to distinguish between parallel and orthogonal movements in the global market, which have a different impact on systemic risk and diversification. As a result, we consider the PCC promising for capital models, which financial institutions use to protect themselves against systemic risk.

Xia, Weixuan

doi: 10.48550/arxiv.2312.00266pmid: N/A

Abstract:The main objective of this paper is to develop a martingale-type solution to optimal consumption--investment choice problems ([Merton, 1969] and [Merton, 1971]) under time-varying incomplete preferences driven by externalities such as patience, socialization effects, and market volatility. The market is composed of multiple risky assets and multiple consumption goods, while in addition there are multiple fluctuating preference parameters with inexact values connected to imprecise tastes. Utility maximization is a multi-criteria problem with possibly function-valued criteria. To come up with a complete characterization of the solutions, first we motivate and introduce a set-valued stochastic process for the dynamics of multi-utility indices and formulate the optimization problem in a topological vector space. Then, we modify a classical scalarization method allowing for infiniteness and randomness in dimensions and prove results of equivalence to the original problem. Illustrative examples are given to demonstrate practical interests and method applicability progressively. The link between the original problem and a dual problem is also discussed, relatively briefly. Finally, using Malliavin calculus with stochastic geometry, we find optimal investment policies to be generally set-valued, each of whose selectors admits a four-way decomposition involving an additional indecisiveness risk-hedging portfolio. Our results touch on new directions for optimal consumption--investment choices in the presence of incomparability and time inconsistency, also signaling potentially testable assumptions on the variability of asset prices. Simulation techniques for set-valued processes are studied for how solved optimal policies can be computed in practice.

Leung, Pauline; Pei, Zhuan

doi: 10.48550/arxiv.2312.17123pmid: N/A

Abstract:Evidence on the effectiveness of retraining U.S. unemployed workers primarily comes from evaluations of training programs, which represent one narrow avenue for skill acquisition. We use high-quality records from Ohio and a matching method to estimate the effects of retraining, broadly defined as enrollment in postsecondary institutions. Our simple method bridges two strands of the dynamic treatment effect literature that estimate the treatment-now-versus-later and treatment-versus-no-treatment effects. We find that enrollees experience earnings gains of six percent three to four years after enrolling, after depressed earnings during the first two years. The earnings effects are driven by industry-switchers, particularly to healthcare.