Quantitative Finance

Variable Annuity (VA) products expose insurance companies to considerable risk because of the guarantees they provide to buyers of these products. Managing and hedging these risks requires insurers to find the value of key risk metrics for a large portfolio of VA products. In practice, many companies rely on nested Monte Carlo (MC) simulations to find key risk metrics. MC simulations are computationally demanding, forcing insurance companies to invest hundreds of thousands of dollars in computational infrastructure per year. Moreover, existing academic methodologies are focused on fair valuation of a single VA contract, exploiting ideas in option theory and regression. In most cases, the computational complexity of these methods surpasses the computational requirements of MC simulations. Therefore, academic methodologies cannot scale well to large portfolios of VA contracts. In this paper, we present a framework for valuing such portfolios based on spatial interpolation. We provide a comprehensive study of this framework and compare existing interpolation schemes. Our numerical results show superior performance, in terms of both computational efficiency and accuracy, for these methods compared to nested MC simulations. We also present insights into the challenge of finding an effective interpolation scheme in this framework, and suggest guidelines that help us build a fully automated scheme that is efficient and accurate.

We present a detailed analysis and implementation of a splitting strategy to identify simultaneously the local-volatility surface and the jump-size distribution from quoted European prices. The underlying model consists of a jump-diffusion driven asset with time and price dependent volatility. Our approach uses a forward Dupire-type partial-integro-differential equations for the option prices to produce a parameter-to-solution map. The ill-posed inverse problem for such map is then solved by means of a Tikhonov-type convex regularization. The proofs of convergence and stability of the algorithm are provided together with numerical examples that substantiate the robustness of the method both for synthetic and real data.

In this article we investigate a state-space representation of the Lee-Carter model which is a benchmark stochastic mortality model for forecasting age-specific death rates. Existing relevant literature focuses mainly on mortality forecasting or pricing of longevity derivatives, while the full implications and methods of using the state-space representation of the Lee-Carter model in pricing retirement income products is yet to be examined. The main contribution of this article is twofold. First, we provide a rigorous and detailed derivation of the posterior distributions of the parameters and the latent process of the Lee-Carter model via Gibbs sampling. Our assumption for priors is slightly more general than the current literature in this area. Moreover, we suggest a new form of identification constraint not yet utilised in the actuarial literature that proves to be a more convenient approach for estimating the model under the state-space framework. Second, by exploiting the posterior distribution of the latent process and parameters, we examine the pricing range of annuities, taking into account the stochastic nature of the dynamics of the mortality rates. In this way we aim to capture the impact of longevity risk on the pricing of annuities. The outcome of our study demonstrates that an annuity price can be more than 4% under-valued when different assumptions are made on determining the survival curve constructed from the distribution of the forecasted death rates. Given that a typical annuity portfolio consists of a large number of policies with maturities which span decades, we conclude that the impact of longevity risk on the accurate pricing of annuities is a significant issue to be further researched. In addition, we find that mis-pricing is increasingly more pronounced for older ages as well as for annuity policies having a longer maturity.

We pose the decumulation strategy for a Defined Contribution (DC) pension plan as a problem in optimal stochastic control. The controls are the withdrawal amounts and the asset allocation strategy. We impose maximum and minimum constraints on the withdrawal amounts, and impose no-shorting no-leverage constraints on the asset allocation strategy. Our objective function measures reward as the expected total withdrawals over the decumulation horizon, and risk is measured by Expected Shortfall (ES) at the end of the decumulation period. We solve the stochastic control problem numerically, based on a parametric model of market stochastic processes. We find that, compared to a fixed constant withdrawal strategy, with minimum withdrawal set to the constant withdrawal amount, the optimal strategy has a significantly higher expected average withdrawal, at the cost of a very small increase in ES risk. Tests on bootstrapped resampled historical market data indicate that this strategy is robust to parametric model misspecification.

Stock price forecasting is a highly complex and vitally important field of research. Recent advancements in deep neural network technology allow researchers to develop highly accurate models to predict financial trends. We propose a novel deep learning model called ST-GAN, or Stochastic Time-series Generative Adversarial Network, that analyzes both financial news texts and financial numerical data to predict stock trends. We utilize cutting-edge technology like the Generative Adversarial Network (GAN) to learn the correlations among textual and numerical data over time. We develop a new method of training a time-series GAN directly using the learned representations of Naive Bayes' sentiment analysis on financial text data alongside technical indicators from numerical data. Our experimental results show significant improvement over various existing models and prior research on deep neural networks for stock price forecasting.

We propose a novel algorithm which allows to sample paths from an underlying price process in a local volatility model and to achieve a substantial variance reduction when pricing exotic options. The new algorithm relies on the construction of a discrete multinomial tree. The crucial feature of our approach is that -- in a similar spirit to the Brownian Bridge -- each random path runs backward from a terminal fixed point to the initial spot price. We characterize the tree in two alternative ways: in terms of the optimal grids originating from the Recursive Marginal Quantization algorithm and following an approach inspired by the finite difference approximation of the diffusion's infinitesimal generator. We assess the reliability of the new methodology comparing the performance of both approaches and benchmarking them with competitor Monte Carlo methods.

Using the concept of self-decomposable subordinators introduced in Gardini et al. [11], we build a new bivariate Normal Inverse Gaussian process that can capture stochastic delays. In addition, we also develop a novel path simulation scheme that relies on the mathematical connection between self-decomposable Inverse Gaussian laws and Lévy-driven Ornstein-Uhlenbeck processes with Inverse Gaussian stationary distribution. We show that our approach provides an improvement to the existing simulation scheme detailed in Zhang and Zhang [23] because it does not rely on an acceptance-rejection method. Eventually, these results are applied to the modelling of energy markets and to the pricing of spread options using the proposed Monte Carlo scheme and Fourier techniques

The history of research in finance and economics has been widely impacted by the field of Agent-based Computational Economics (ACE). While at the same time being popular among natural science researchers for its proximity to the successful methods of physics and chemistry for example, the field of ACE has also received critics by a part of the social science community for its lack of empiricism. Yet recent trends have shifted the weights of these general arguments and potentially given ACE a whole new range of realism. At the base of these trends are found two present-day major scientific breakthroughs: the steady shift of psychology towards a hard science due to the advances of neuropsychology, and the progress of artificial intelligence and more specifically machine learning due to increasing computational power and big data. These two have also found common fields of study in the form of computational neuroscience, and human-computer interaction, among others. We outline here the main lines of a computational research study of collective economic behavior via Agent-Based Models (ABM) or Multi-Agent System (MAS), where each agent would be endowed with specific cognitive and behavioral biases known to the field of neuroeconomics, and at the same time autonomously implement rational quantitative financial strategies updated by machine learning. We postulate that such ABMs would offer a whole new range of realism.

The Polynomial Chaos Expansion (PCE) technique recovers a finite second order random variable exploiting suitable linear combinations of orthogonal polynomials which are functions of a given stochas- tic quantity {\xi}, hence acting as a kind of random basis. The PCE methodology has been developed as a mathematically rigorous Uncertainty Quantification (UQ) method which aims at providing reliable numerical estimates for some uncertain physical quantities defining the dynamic of certain engineering models and their related simulations. In the present paper we exploit the PCE approach to analyze some equity and interest rate models considering, without loss of generality, the one dimensional case. In particular we will take into account those models which are based on the Geometric Brownian Motion (gBm), e.g. the Vasicek model, the CIR model, etc. We also provide several numerical applications and results which are discussed for a set of volatility values. The latter allows us to test the PCE technique on a quite large set of different scenarios, hence providing a rather complete and detailed investigation on PCE-approximation's features and properties, such as the convergence of statistics, distribution and quantiles. Moreover we give results concerning both an efficiency and an accuracy study of our approach by comparing our outputs with the ones obtained adopting the Monte Carlo approach in its standard form as well as in its enhanced version.

Subdiffusion is a well established phenomenon in physics. In this paper we apply the subdiffusive dynamics to analyze financial markets. We focus on the financial aspect of time fractional diffusion model with moving boundary i.e. American and barrier option pricing in the subdiffusive Black-Scholes (B-S) model. Two computational methods for valuing American options in the considered model are proposed - the weighted finite difference (FD) and the Longstaff-Schwartz method. In the article it is also shown how to valuate numerically wide range of barrier options using the FD approach.