In this paper we present a simple, but new, approximation methodology for pricing a call option in a Black \& Scholes market characterized by stochastic interest rates. The method, based on a straightforward Gaussian moment matching technique applied to a conditional Black \& Scholes formula, is quite general and it applies to various models, whether affine or not. To check its accuracy and computational time, we implement it for the CIR interest rate model correlated with the underlying, using the Monte Carlo simulations as a benchmark. The method's performance turns out to be quite remarkable, even when compared with analogous results obtained by the affine approximation technique presented in Grzelak and Oosterlee (2011) and by the expansion formula introduced in Kim and Kunimoto (1999), as we show in the last section.

This paper is focused on American option pricing in the subdiffusive Black Scholes model. Two methods for valuing American options in the considered model are proposed. The weighted scheme of the finite difference (FD) method is derived and the main properties of the method are presented. The Longstaff-Schwartz method is applied for the discussed model and is compared to the previous method. In the article it is also shown how to valuate wide range of Barrier options using the FD approach. The proposed FD method has $2-\alpha$ order of accuracy with respect to time, where $\alpha\in(0,1)$ is the subdiffusion parameter, and $2$ with respect to space.

During the COVID-19 pandemic of 2019/2020, authorities have used temporary ad-hoc policy measures, such as lockdowns and mass quarantines, to slow its transmission. However, the consequences of widespread use of these unprecedented measures are poorly understood. To contribute to the understanding of the economic and human consequences of such policy measures, we therefore construct a mathematical model of an economy under the impact of a pandemic, select parameter values to represent the global economy under the impact of COVID-19, and perform numerical experiments by simulating a large number of possible policy responses. By varying the starting date of the policy intervention in the simulated scenarios, we find that the most effective policy intervention occurs around the time when the number of active infections is growing at its highest rate. The degree of the intervention, above a certain threshold, does not appear to have a great impact on the outcomes in our simulations, due to the strongly concave relationship we assume between production shortfall and reduction in the infection rate. Our experiments further suggest that the intervention should last until after the peak determined by the reduced infection rate. The model and its implementation, along with the general insights from our policy experiments, may help policymakers design effective emergency policy responses in the face a serious pandemic, and contribute to our understanding of the relationship between the economic growth and the spread of infectious diseases.

We adopt deep learning models to directly optimise the portfolio Sharpe ratio. The framework we present circumvents the requirements for forecasting expected returns and allows us to directly optimise portfolio weights by updating model parameters. Instead of selecting individual assets, we trade Exchange-Traded Funds (ETFs) of market indices to form a portfolio. Indices of different asset classes show robust correlations and trading them substantially reduces the spectrum of available assets to choose from. We compare our method with a wide range of algorithms with results showing that our model obtains the best performance over the testing period, from 2011 to the end of April 2020, including the financial instabilities of the first quarter of 2020. A sensitivity analysis is included to understand the relevance of input features and we further study the performance of our approach under different cost rates and different risk levels via volatility scaling.

We investigate state-dependent effects of fiscal multipliers and allow for endogenous sample splitting to determine whether the US economy is in a slack state. When the endogenized slack state is estimated as the period of the unemployment rate higher than about 12 percent, the estimated cumulative multipliers are significantly larger during slack periods than non-slack periods and are above unity. We also examine the possibility of time-varying regimes of slackness and find that our empirical results are robust under a more flexible framework. Our estimation results point out the importance of the heterogenous effects of fiscal policy and shed light on the prospect of fiscal policy in response to economic shocks from the current COVID-19 pandemic.

We examine dynamic coupling and feedback effects between High Frequency Traders (HFTs) and how they can destabilize markets. We develop a general framework for modelling dynamic interaction based on recurrence relations, and use this to show how unexpected latency and feedback can trigger oscillatory instability between HFT market makers with inventory constraints. Our analysis suggests that the modelled instability is an unintentional emergent behaviour of the market that does not depend on the complexity of HFT strategies - even apparently stable strategies are vulnerable. Feedback instability can lead to substantial movements in market prices such as price spikes and crashes.

The traditional way of building a yield curve is to choose an interpolation on discount factors, implied by the market tradable instruments. Since then, constructions based on specific interpolations of the forward rates have become the trend. We show here that some popular interpolation methods on the forward rates correspond exactly to classical interpolation methods on discount factors. This paper also aims at clarifying the differences between interpolations in terms of discount factors, instantaneous forward rates, discrete forward rates, and constant period forward rates.

The outbreak of the novel coronavirus (COVID-19) has caused unprecedented disruptions to financial and economic markets around the globe, leading to one of the fastest U.S. stock market declines in history. However, in the past we have seen the market recover and we can expect the market to recover again, and on this basis we assume the Standard and Poor's 500 (S&P500) index will reach a minimum before rising again in the not-too-distant future. Here we present four forecast models of the S&P500 based on COVID-19 projections of deaths released on 02/04/2020 by the University of Washington and the 2-months consideration since the first confirmed case occured in USA. The decline and recovery in the index is estimated for the following three months. The forecast is a projection of a prediction with fluctuations described by $q$-gaussian distributions. Our forecast was made on the premise that: (a) The prediction is based on a deterministic trend that follows the data available since the initial outbreak of COVID-19, and (b) fluctuations derived from the S&P500 over the last 24 years.

The Markov-modulated Poisson process is utilised for count modelling in a variety of areas such as queueing, reliability, network and insurance claims analysis. In this paper, we extend the Markov-modulated Poisson process framework through the introduction of a flexible frequency perturbation measure. This contribution enables known information of observed event arrivals to be naturally incorporated in a tractable manner, while the hidden Markov chain captures the effect of unobservable drivers of the data. In addition to increases in accuracy and interpretability, this method supplements analysis of the latent factors. Further, this procedure naturally incorporates data features such as over-dispersion and autocorrelation. Additional insights can be generated to assist analysis, including a procedure for iterative model improvement.

Implementation difficulties are also addressed with a focus on dealing with large data sets, where latent models are especially advantageous due the large number of observations facilitating identification of hidden factors. Namely, computational issues such as numerical underflow and high processing cost arise in this context and in this paper, we produce procedures to overcome these problems.

This modelling framework is demonstrated using a large insurance data set to illustrate theoretical, practical and computational contributions and an empirical comparison to other count models highlight the advantages of the proposed approach.

This paper is an attempt to study fundamentally the valuation of insurance contracts. We start from the observation that insurance contracts are inherently linked to financial markets, be it via interest rates, or -- as in hybrid products, equity-linked life insurance and variable annuities -- directly to stocks or indices. By defining portfolio strategies on an insurance portfolio and combining them with financial trading strategies we arrive at the notion of insurance-finance arbitrage (IFA). A fundamental theorem provides two sufficient conditions for presence or absence of IFA, respectively. For the first one it utilizes the conditional law of large numbers and risk-neutral valuation. As a key result we obtain a simple valuation rule, called QP-rule, which is market consistent and excludes IFA.

Utilizing the theory of enlargements of filtrations we construct a tractable framework for general valuation results, working under weak assumptions.

The generality of the approach allows to incorporate many important aspects, like mortality risk or dependence of mortality and stock markets which is of utmost importance in the recent corona crisis. For practical applications, we provide an affine formulation which leads to explicit valuation formulas for a large class of hybrid products.

We examine an expected utility maximization problem with an uncertain time horizon, a classical example being a life insurance contract due at the time of death. Life insurance contracts usually have an option-like form leading to a non-concave optimization problem. We consider general utility functions and give necessary and sufficient optimality conditions, deriving a computationally tractable algorithm. A numerical study is done to illustrate our findings. Our analysis suggests that the possible occurrence of a premature stopping leads to a reduced performance of the optimal portfolio compared to a setting without time-horizon uncertainty.

The tick size, which is the smallest increment between two consecutive prices for a given asset, is a key parameter of market microstructure. In particular, the behavior of high frequency market makers is highly related to its value. We take the point of view of an exchange and investigate the relevance of having different tick sizes on the bid and ask sides of the order book. Using an approach based on the model with uncertainty zones, we show that when side-specific tick sizes are suitably chosen, it enables the exchange to improve the quality of liquidity provision.

Although there is a wide use of technical trading rules in stock markets, the profitability of them still remains controversial. This paper first presents and proves the upper bound of cumulative return, and then introduces many of conventional technical trading rules. Furthermore, with the help of bootstrap methodology, we investigate the profitability of technical trading rules on different international stock markets, including developed markets and emerging markets. At last, the results show that the technical trading rules are hard to beat the market, and even less profitable than the random trading strategy.

This paper explores an optimal investing problem for a retiree facing longevity risk and living standard risk. We formulate the optimal investing problem as an optimal portfolio choice problem under a time-varying risk capacity constraint. Under the specific condition on model parameters, we show that the value function is a $C^2$ solution of the HJB equation and derive the optimal investment strategy in terms of second-order ordinary differential equations. The optimal portfolio is nearly neutral to the stock market movement if the portfolio's value is at a sufficiently high level; but, if the portfolio is not worth enough to sustain the retirement spending, the retiree actively invests in the stock market for the higher expected return. In addition, we solve an optimal portfolio choice problem under a leverage constraint and show that the optimal portfolio would lose significantly in stressed markets. This paper shows that the time-varying risk capacity constraint has important implications for asset allocation in retirement.

We consider the general class of spectrally positive L\'evy risk processes, which are appropriate for businesses with continuous expenses and lump sum gains whose timing and sizes are stochastic. Motivated by the fact that dividends cannot be paid at any time in real life, we study $\textit{periodic}$ dividend strategies whereby dividend decisions are made according to a separate arrival process.

In this paper, we investigate the impact of fixed transaction costs on the optimal periodic dividend strategy, and show that a periodic $(b_u,b_l)$ strategy is optimal when decision times arrive according to an independent Poisson process. Such a strategy leads to lump sum dividends that bring the surplus back to $b_l$ as long as it is no less than $b_u$ at a dividend decision time. The expected present value of dividends (net of transaction costs) is provided explicitly with the help of scale functions. Results are illustrated.

This paper investigates the hedging performance of pegged foreign exchange market in a regime switching (RS) model introduced in a recent paper by Drapeau, Wang and Wang (2019). We compare two prices, an exact solution and first order approximation and provide the bounds for the error. We provide exact RS delta, approximated RS delta as well as mean variance hedging strategies for this specific model and compare their performance. To improve the efficiency of the pricing and calibration procedure, the Fourier approach of this regime-switching model is developed in our work. It turns out that: 1 -- the calibration of the volatility surface with this regime switching model outperforms on real data the classical SABR model; 2 -- the Fourier approach is significantly faster than the direct approach; 3 -- in terms of hedging, the approximated RS delta hedge is a viable alternative to the exact RS delta hedge while significantly faster.

The 2008 mortgage crisis is an example of an extreme event. Extreme value theory tries to estimate such tail risks. Modern finance practitioners prefer Expected Shortfall based risk metrics (which capture tail risk) over traditional approaches like volatility or even Value-at-Risk. This paper provides a quantum annealing algorithm in QUBO form for a dynamic asset allocation problem using expected shortfall constraint. It was motivated by the need to refine the current quantum algorithms for Markowitz type problems which are academically interesting but not useful for practitioners. The algorithm is dynamic and the risk target emerges naturally from the market volatility. Moreover, it avoids complicated statistics like generalized pareto distribution. It translates the problem into qubit form suitable for implementation by a quantum annealer like D-Wave. Such QUBO algorithms are expected to be solved faster using quantum annealing systems than any classical algorithm using classical computer (but yet to be demonstrated at scale).

In this essay, we have comprehensively evaluated the feasibility and suitability of adopting the Machine Learning Models on the forecast of corporation fundamentals (i.e. the earnings), where the prediction results of our method have been thoroughly compared with both analysts' consensus estimation and traditional statistical models. As a result, our model has already been proved to be capable of serving as a favorable auxiliary tool for analysts to conduct better predictions on company fundamentals. Compared with previous traditional statistical models being widely adopted in the industry like Logistic Regression, our method has already achieved satisfactory advancement on both the prediction accuracy and speed. Meanwhile, we are also confident enough that there are still vast potentialities for this model to evolve, where we do hope that in the near future, the machine learning model could generate even better performances compared with professional analysts.

Interbank contagion can theoretically exacerbate losses in a financial system and lead to additional cascade defaults during downturn. In this paper we produce default analysis using both regression and neural network models to verify whether interbank contagion offers any predictive explanatory power on default events. We predict defaults of U.S. domiciled commercial banks in the first quarter of 2010 using data from the preceding four quarters. A number of established predictors (such as Tier 1 Capital Ratio and Return on Equity) are included alongside contagion to gauge if the latter adds significance. Based on this methodology, we conclude that interbank contagion is extremely explanatory in default prediction, often outperforming more established metrics, in both regression and neural network models. These findings have sizeable implications for the future use of interbank contagion as a variable of interest for stress testing, bank issued bond valuation and wider bank default prediction.