The St. Petersburg Paradox was proposed two centuries ago. In the paper, we propose a new pricing theory with several rules to solve the paradox and state that the fair pricing should be judged by the buyer and the seller independently. The seller is concerned about costs, and the buyer is concerned about the realistic prospect of returns. This pricing theory can be applied to financial market to solve the confusion with fat tail distribution.

We price European and American exchange options where the underlying asset prices are modelled using a Merton (1976) jump-diffusion with a common Heston (1993) stochastic volatility process. Pricing is performed under an equivalent martingale measure obtained by setting the second asset yield process as the numeraire asset, as suggested by Bjerskund and Stensland (1993). Such a choice for the numeraire reduces the exchange option pricing problem, a two-dimensional problem, to pricing a call option written on the ratio of the yield processes of the two assets, a one-dimensional problem. The joint transition density function of the asset yield ratio process and the instantaneous variance process is then determined from the corresponding Kolmogorov backward equation via integral transforms. We then determine integral representations for the European exchange option price and the early exercise premium and state a linked system of integral equations that characterizes the American exchange option price and the associated early exercise boundary. Properties of the early exercise boundary near maturity are also discussed.

Faced with the growing research towards crude oil price fluctuations influential factors following the accelerated development of Internet technology, accessible data such as Google search volume index are increasingly quantified and incorporated into forecasting approaches. In this paper, we apply multi-scale data that including both GSVI data and traditional economic data related to crude oil price as independent variables and propose a new hybrid approach for monthly crude oil price forecasting. This hybrid approach, based on divide and conquer strategy, consists of K-means method, kernel principal component analysis and kernel extreme learning machine , where K-means method is adopted to divide input data into certain clusters, KPCA is applied to reduce dimension, and KELM is employed for final crude oil price forecasting. The empirical result can be analyzed from data and method levels. At the data level, GSVI data perform better than economic data in level forecasting accuracy but with opposite performance in directional forecasting accuracy because of Herd Behavior, while hybrid data combined their advantages and obtain best forecasting performance in both level and directional accuracy. At the method level, the approaches with K-means perform better than those without K-means, which demonstrates that divide and conquer strategy can effectively improve the forecasting performance.

We investigate the behaviour of cryptocurrencies' return data. Using return data for bitcoin, ethereum and ripple which account for over 70% of the cyrptocurrency market, we demonstrate that $\alpha$-stable distribution models highly speculative cryptocurrencies more robustly compared to other heavy tailed distributions that are used in financial econometrics. We find that the Maximum Likelihood Method proposed by DuMouchel (1971) produces estimates that fit the cryptocurrency return data much better than the quantile based approach of McCulloch (1986) and sample characteristic method by Koutrouvelis (1980). The empirical results show that the leptokurtic feature presented in cryptocurrencies' return data can be captured by an ${\alpha}$-stable distribution. This papers covers predominant literature in cryptocurrencies and stable distributions.

We propose an algorithm which predicts each subsequent time step relative to the previous timestep of intractable short rate model (when adjusted for drift and overall distribution of previous percentile result) and show that the method achieves superior outcomes to the unbiased estimate both on the trained dataset and different validation data.

In todays global economy, accuracy in predicting macro-economic parameters such as the foreign the exchange rate or at least estimating the trend correctly is of key importance for any future investment. In recent times, the use of computational intelligence-based techniques for forecasting macroeconomic variables has been proven highly successful. This paper tries to come up with a multivariate time series approach to forecast the exchange rate (USD/INR) while parallelly comparing the performance of three multivariate prediction modelling techniques: Vector Auto Regression (a Traditional Econometric Technique), Support Vector Machine (a Contemporary Machine Learning Technique), and Recurrent Neural Networks (a Contemporary Deep Learning Technique). We have used monthly historical data for several macroeconomic variables from April 1994 to December 2018 for USA and India to predict USD-INR Foreign Exchange Rate. The results clearly depict that contemporary techniques of SVM and RNN (Long Short-Term Memory) outperform the widely used traditional method of Auto Regression. The RNN model with Long Short-Term Memory (LSTM) provides the maximum accuracy (97.83%) followed by SVM Model (97.17%) and VAR Model (96.31%). At last, we present a brief analysis of the correlation and interdependencies of the variables used for forecasting.

The present article studies geometric step options in exponential L\'evy markets. Our contribution is manifold and extends several aspects of the geometric step option pricing literature. First, we provide symmetry and parity relations and derive various characterizations for both European-type and American-type geometric double barrier step options. In particular, we are able to obtain a jump-diffusion disentanglement for the early exercise premium of American-type geometric double barrier step contracts and its maturity-randomized equivalent as well as to characterize the diffusion and jump contributions to these early exercise premiums separately by means of partial integro-differential equations and ordinary integro-differential equations. As an application of our characterizations, we derive semi-analytical pricing results for (regular) European-type and American-type geometric down-and-out step call options under hyper-exponential jump-diffusion models. Lastly, we use the latter results to discuss the early exercise structure of geometric step options once jumps are added and to subsequently provide an analysis of the impact of jumps on the price and hedging parameters of (European-type and American-type) geometric step contracts.

We introduce an adaptive Euler method for the approximate solution of the Cox-Ingersoll-Ross short rate model. An explicit discretisation is applied over an adaptive mesh to the stochastic differential equation (SDE) governing the square root of the solution, relying upon a class of path-bounded timestepping strategies which work by reducing the stepsize as solutions approach a neighbourhood of zero. The method is hybrid in the sense that a backstop method is invoked if the timestep becomes too small, or to prevent solutions from overshooting zero and becoming negative. Under parameter constraints that imply Feller's condition, we prove that such a scheme is strongly convergent, of order at least 1/2. Under Feller's condition we also prove that the probability of ever needing the backstop method to prevent a negative value can be made arbitrarily small. Numerically, we compare this adaptive method to fixed step schemes extant in the literature, both implicit and explicit, and a novel semi-implicit adaptive variant. We observe that the adaptive approach leads to methods that are competitive over the entire domain of Feller's condition.

Digital advertising markets are growing and attracting increased scrutiny. This paper explores four market inefficiencies that remain poorly understood: ad effect measurement, frictions between and within advertising channel members, ad blocking and ad fraud. These topics are not unique to digital advertising, but each manifests in new ways in markets for digital ads. We identify relevant findings in the academic literature, recent developments in practice, and promising topics for future research.

An approach to the modelling of financial return series using a class of uniformity-preserving transforms for uniform random variables is proposed. V-transforms describe the relationship between quantiles of the return distribution and quantiles of the distribution of a predictable volatility proxy variable constructed as a function of the return. V-transforms can be represented as copulas and permit the construction and estimation of models that combine arbitrary marginal distributions with linear or non-linear time series models for the dynamics of the volatility proxy. The idea is illustrated using a transformed Gaussian ARMA process for volatility, yielding the class of VT-ARMA copula models. These can replicate many of the stylized facts of financial return series and facilitate the calculation of marginal and conditional characteristics of the model including quantile measures of risk. Estimation of models is carried out by adapting the exact maximum likelihood approach to the estimation of ARMA processes.

The Levy-Levy-Solomon model (A microscopic model of the stock market: cycles, booms, and crashes, Economic Letters 45 (1))is one of the most influential agent-based economic market models. In several publications this model has been discussed and analyzed. Especially Lux and Zschischang (Some new results on the Levy, Levy and Solomon microscopic stock market model, Physica A, 291(1-4)) have shown that the model exhibits finite-size effects. In this study we extend existing work in several directions. First, we show simulations which reveal finite-size effects of the model. Secondly, we shed light on the origin of these finite-size effects. Furthermore, we demonstrate the sensitivity of the Levy-Levy-Solomon model with respect to random numbers. Especially, we can conclude that a low-quality pseudo random number generator has a huge impact on the simulation results. Finally, we study the impact of the stopping criteria in the market clearance mechanism of the Levy-Levy-Solomon model.

This paper studies the optimal dividend for a multi-line insurance group, in which each subsidiary runs a product line and is exposed to some external credit default risk. The credit default contagion is considered in the sense that one credit default event can affect the default probabilities of all surviving subsidiaries. The total dividend problem is formulated for the insurance group and we reveal for the first time that the optimal singular dividend strategy is still of the barrier type. Furthermore, we show that the optimal barrier for each subsidiary is modulated by the current default state, namely how many and which subsidiaries have defaulted will determine the dividend threshold of each surviving subsidiary. These interesting conclusions are based on the analysis of the associated recursive system of Hamilton-Jacobi-Bellman variational inequalities (HJBVIs). The existence of the classical solution is established and the proof of the verification theorem is provided. For the case of two subsidiaries, the value function and optimal barriers of dividend are given in analytical forms. Numerical examples are also presented to illustrate some economic insights.

We study optimal liquidation in the presence of linear temporary and transient price impact along with taking into account a general price predicting finite-variation signal. We formulate this problem as minimization of a cost-risk functional over a class of absolutely continuous and signal-adaptive strategies. The stochastic control problem is solved by following a probabilistic and convex analytic approach. We show that the optimal trading strategy is given by a system of four coupled forward-backward SDEs, which can be solved explicitly. Our results reveal how the induced transient price distortion provides together with the predictive signal an additional predictor about future price changes. As a consequence, the optimal signal-adaptive trading rate trades off exploiting the predictive signal against incurring the transient displacement of the execution price from its unaffected level. This answers an open question from Lehalle and Neuman [27] as we show how to derive the unique optimal signal-adaptive liquidation strategy when price impact is not only temporary but also transient.

Standard methods and theories in finance can be ill-equipped to capture highly non-linear interactions in financial prediction problems based on large-scale datasets, with deep learning offering a way to gain insights into correlations in markets as complex systems. In this paper, we apply deep learning to econometrically constructed gradients to learn and exploit lagged correlations among S&P 500 stocks to compare model behaviour in stable and volatile market environments, and under the exclusion of target stock information for predictions. In order to measure the effect of time horizons, we predict intraday and daily stock price movements in varying interval lengths and gauge the complexity of the problem at hand with a modification of our model architecture. Our findings show that accuracies, while remaining significant and demonstrating the exploitability of lagged correlations in stock markets, decrease with shorter prediction horizons. We discuss implications for modern finance theory and our work's applicability as an investigative tool for portfolio managers. Lastly, we show that our model's performance is consistent in volatile markets by exposing it to the environment of the recent financial crisis of 2007/2008.

In this article, we provide representations of European and American exchange option prices under stochastic volatility jump-diffusion (SVJD) dynamics following models by Merton (1976), Heston (1993), and Bates (1996). A Radon-Nikodym derivative process is also introduced to facilitate the shift from the objective market measure to other equivalent probability measures, including the equivalent martingale measure. Under the equivalent martingale measure, we derive the integro-partial differential equation that characterizes the exchange option prices. We also derive representations of the European exchange option price using the change-of-numeraire technique proposed by Geman et al. (1995) and the Fourier inversion formula derived by Caldana and Fusai (2013), and show that these two representations are comparable. Lastly, we show that the American exchange option price can be decomposed into the price of the European exchange option and an early exercise premium.

We introduce an arbitrage-free framework for robust valuation adjustments. An investor trades a credit default swap portfolio with a risky counterparty, and hedges credit risk by taking a position in defaultable bonds. The investor does not know the return rate of her counterparty's bond, but is confident that it lies within an uncertainty interval. We derive both upper and lower bounds for the XVA process of the portfolio, and show that these bounds may be recovered as solutions of nonlinear ordinary differential equations. The presence of collateralization and closeout payoffs leads to important differences with respect to classical credit risk valuation. The value of the super-replicating portfolio cannot be directly obtained by plugging one of the extremes of the uncertainty interval in the valuation equation, but rather depends on the relation between the XVA replicating portfolio and the close-out value throughout the life of the transaction. Our comparative statics analysis indicates that credit contagion has a nonlinear effect on the replication strategies and on the XVA.

Interest in predicting multivariate probability distributions is growing due to the increasing availability of rich datasets and computational developments. Scoring functions enable the comparison of forecast accuracy, and can potentially be used for estimation. A scoring function for multivariate distributions that has gained some popularity is the energy score. This is a generalization of the continuous ranked probability score (CRPS), which is widely used for univariate distributions. A little-known, alternative generalization is the multivariate CRPS (MCRPS). We propose a theoretical framework for scoring functions for multivariate distributions, which encompasses the energy score and MCRPS, as well as the quadratic score, which has also received little attention. We demonstrate how this framework can be used to generate new scores. For univariate distributions, it is well-established that the CRPS can be expressed as the integral over a quantile score. We show that, in a similar way, scoring functions for multivariate distributions can be "disintegrated" to obtain scoring functions for level sets. Using this, we present scoring functions for different types of level set, including those for densities and cumulative distributions. To compute the scoring functions, we propose a simple numerical algorithm. We illustrate our proposals using simulated and stock returns data.

The present article provides a novel theoretical way to evaluate tradeability in markets of ordinary exponential L\'evy type. We consider non-tradeability as a particular type of market illiquidity and investigate its impact on the price of the assets. Starting from an adaption of the continuous-time optional asset replacement problem initiated by McDonald and Siegel (1986), we derive tradeability premiums and subsequently characterize them in terms of free-boundary problems. This provides a simple way to compute non-tradeability values, e.g. by means of standard numerical techniques, and, in particular, to express the price of a non-tradeable asset as a percentage of the price of a tradeable equivalent. Our approach is illustrated via numerical examples where we discuss various properties of the tradeability premiums.