Quantitative Finance

To promote economic stability, finance should be studied as a hard science, where scientific methods apply. When a trading strategy is proposed, the underlying model should be transparent and defined robustly to allow other researchers to understand and examine it thoroughly. Like any hard sciences, results must be repeatable to allow researchers to collaborate, and build upon each other's results. Large-scale collaboration, when applying the steps of scientific investigation, is an efficient way to leverage "crowd science" to accelerate research in finance. In this paper, we demonstrate how a real world problem in economics, an old problem still subject to a lot of debate, can be solved by the application of a crowd-powered, collaborative scientific computational framework, fully supporting the process of investigation dictated by the modern scientific method. This paper provides a real end-to-end example of investigation to illustrate the use of the framework. We intentionally selected an example that is self-contained, complete, simple, accessible, and of constant debate in both academia and the industry: the performance of a trading strategy used commonly in technical analysis. Claims of efficiency in technical analysis, referred derisively by some sources as "Black Magic", are of widespread use in mainstream media and usually met with a lot of controversy. In this paper we show that different researchers assess this strategy differently, and the subsequent debate is due more to the lack of method than purpose. Most results reported are not repeatable by other researchers. This is not satisfactory if we intend to approach finance as a hard science. To counterweight the status quo, we demonstrate what one could do by using collaborative and investigative features of contributions and leveraging the power of crowds.

In a market with stochastic interest rates, we consider an investor who can either (i) invest all if his money in a savings account or (ii) purchase zero-coupon bonds and invest the remainder of his wealth in a savings account. The indifference price of the bond is the price for which the investor could achieve the same expected utility under both scenarios. In an affine term structure setting, under the assumption that an investor has a utility function in either exponential or power form, we show that the indifference price of a zero-coupon bond is the root of an integral expression. As an example, we compute bond indifference prices and the corresponding indifference yield curves in the Vasicek setting and interpret the results.

Using neural networks, we compute bounds on the prices of multi-asset derivatives given information on prices of related payoffs. As a main example, we focus on European basket options and include information on the prices of other similar options, such as spread options and/or basket options on subindices. We show that, in most cases, adding further constraints gives rise to bounds that are considerably tighter and discuss the maximizing/minimizing copulas achieving such bounds. Our approach follows the literature on constrained optimal transport and, in particular, builds on a recent paper by Eckstein and Kupper (2019, Appl. Math. Optim.).

The use of sequential Monte Carlo within simulation for path-dependent option pricing is proposed and evaluated. Recently, it was shown that explicit solutions and importance sampling are valuable for efficient simulation of spot price and volatility, especially for purposes of path-dependent option pricing. The resulting simulation algorithm is an analog to the weighted particle filtering algorithm that might be improved by resampling or branching. Indeed, some branching algorithms are shown herein to improve pricing performance substantially while some resampling algorithms are shown to be less suitable in certain cases. A historical property is given and explained as the distinguishing feature between the sequential Monte Carlo algorithms that work on path-dependent option pricing and those that do not. In particular, it is recommended to use the so-called effective particle branching algorithm within importance-sampling Monte Carlo methods for path-dependent option pricing. All recommendations are based upon numeric comparison of option pricing problems in the Heston model.

We consider the classical problem of building an arbitrage-free implied volatility surface from bid-ask quotes. We design a fast numerical procedure, for which we prove the convergence, based on the Sinkhorn algorithm that has been recently used to solve efficiently (martingale) optimal transport problems.

Using a method rooted in information theory, we present results that have identified a large set of stocks for which social media can be informative regarding financial volatility. By clustering stocks based on the joint feature sets of social and financial variables, our research provides an important contribution by characterizing the conditions in which social media signals can lead financial volatility. The results indicate that social media is most informative about financial market volatility when the ratio of bullish to bearish sentiment is high, even when the number of messages is low. The robustness of these findings is verified across 500 stocks from both NYSE and NASDAQ exchanges. The reported results are reproducible via an open-source library for social-financial analysis made freely available.

In this work we want to provide a general principle to evaluate the CVA (Credit Value Adjustment) for a vulnerable option, that is an option subject to some default event, concerning the solvability of the issuer. CVA is needed to evaluate correctly the contract and it is particularly important in presence of WWR (Wrong Way Risk), when a credit deterioration determines an increase of the claim's price. In particular, we are interested in evaluating the CVA in stochastic volatility models for the underlying's price (which often fit quite well the market's prices) when admitting correlation with the default event. By cunningly using Ito's calculus, we provide a general representation formula applicable to some popular models such as SABR, Hull \& White and Heston, which explicitly shows the correction in CVA due to the processes correlation. Later, we specialize this formula and construct its approximation for the three selected models. Lastly, we run a numerical study to test the formula's accuracy, comparing our results with Monte Carlo simulations.

We consider the problem of computing the Credit Value Adjustment ({CVA}) of a European option in presence of the Wrong Way Risk ({WWR}) in a default intensity setting. Namely we model the asset price evolution as solution to a linear equation that might depend on different stochastic factors and we provide an approximate evaluation of the option's price, by exploiting a correlation expansion approach, introduced in \cite{AS}. We compare the numerical performance of such a method with that recently proposed by Brigo et al. (\cite{BR18}, \cite{BRH18}) in the case of a call option driven by a GBM correlated with the CIR default intensity. We additionally report some numerical evaluations obtained by other methods.

In this paper we use convolutional neural networks to find the Hölder exponent of simulated sample paths of the rBergomi model, a recently proposed stock price model used in mathematical finance. We contextualise this as a calibration problem, thereby providing a very practical and useful application.

American options are the reference instruments for the model calibration of a large and important class of single stocks. For this task, a fast and accurate pricing algorithm is indispensable. The literature mainly discusses pricing methods for American options that are based on Monte Carlo, tree and partial differential equation methods. We present an alternative approach that has become popular under the name de-Americanization in the financial industry. The method is easy to implement and enjoys fast run-times. Since it is based on ad hoc simplifications, however, theoretical results guaranteeing reliability are not available. To quantify the resulting methodological risk, we empirically test the performance of the de-Americanization method for calibration. We classify the scenarios in which de-Americanization performs very well. However, we also identify the cases where de-Americanization oversimplifies and can result in large errors.