Mathematical Finance

Optimal Portfolio with Sustainable Attitudes under Cumulative Prospect Theory
2023-11-10 12:00 - 13:00 (America/Chicago) | via Zoom
Speaker: Prof. Massimiliano Kaucic, Department of Economics, Business, Mathematics & Statistical Sciences, University of Trieste
Abstract: In the last five years, extreme events such as the COVID-19 pandemic and the Ukrainian crisis have highlighted the importance of corporate social responsibility and sustainable principles. Consequently, the investment process is changing toward more ethical choices. In this context, we extend the classical optimization framework under the cumulative prospect theory (CPT) in two directions. We first consider an agent who maximizes a financial CPT-value function preselecting the assets to be included in the portfolio based on their environmental, social, and governance (ESG) scores. Then, we develop a bi-objective model that optimizes financial and sustainable CPT-value functions at the same time. Numerical results obtained on an investable universe from the constituents of the STOXX Europe 600 show that introducing ESG information improves the portfolio’s financial performance.

No Math Finance seminar - Thanksgiving break
2023-11-24 14:00 - 15:00

Convergence of the fixed-point iteration for the Bass Local Volatility model
2024-01-12 12:00 - 13:00 | via Zoom
Speaker Dr. Gudmund Pammer, Dept. of Mathematics, ETH Zürich
Abstract: The Bass local volatility model introduced by Backhoff-Veraguas–Beiglböck–Huesmann–Källblad is a Markov model perfectly calibrated to vanilla options at finitely many maturities, that approximates the Dupire local volatility model. Conze and Henry-Labordère show that its calibration can be achieved by solving a fixed-point nonlinear integral equation. We complement the analysis and show, under suitable assumptions, existence and uniqueness of the solution to this equation, and establish that the fixed-point iteration scheme converges at linear rate.
The talk is based on joint work with Beatrice Acciaio and Antonio Marini.

Risk budgeting portfolios: Existence and computation
2024-01-19 12:00 - 13:00 | via Zoom
Speaker: Prof. Olivier Guéant, Department of Applied Mathematics, Université Paris 1 Panthéon-Sorbonne
Abstract: Modern portfolio theory has provided for decades the main framework for optimizing portfolios. Because of its sensitivity to small changes in input parameters, especially expected returns, the mean-variance framework proposed by Markowitz (1952) has however been challenged by new construction methods that are purely based on risk. Among risk-based methods, the most popular ones are Minimum Variance, Maximum Diversification, and Risk Budgeting (especially Equal Risk Contribution) portfolios. Despite some drawbacks, Risk Budgeting is particularly attracting because of its versatility: based on Euler’s homogeneous function theorem, it can indeed be used with a wide range of risk measures. This paper presents mathematical results regarding the existence and the uniqueness of Risk Budgeting portfolios for a very wide spectrum of risk measures and shows that, for many of them, computing the weights of Risk Budgeting portfolios only requires a standard stochastic algorithm.

On subordinated generalizations of 3 classical models of option pricing
2024-01-26 12:00 - 13:00 | via Zoom
Speaker: Dr. Grzegorz Krzyżanowski, Hugo Steinhaus Center, Faculty of Pure and Applied Mathematics, Wroclaw University of Science and Technology
Abstract: We will investigate the relation between Bachelier and Black-Scholes models driven by the infinitely divisible inverse subordinators. Such models, in contrast to their classical equivalents, can be used in markets where periods of stagnation are observed. We will introduce the subordinated Cox-Ross-Rubinstein model and prove that the price of the underlying in that model converges in distribution and in Skorokhod space to the price of underlying in the subordinated Black-Scholes model. Motivated by this fact we will price the selected option contracts using the binomial trees. The comparison to other numerical methods will be provided.

Good for the Planet, Good for the Wallet: The ESG Impact on Financial Performance in India
2024-02-09 14:00 - 15:00 | Department of Mathematics & Statistics, TTU
Speaker: Prof. Tauhidul Islam Tanin, EGADE Business School, Technologico de Monterrey
Abstract: We examine the impact of ESG practices on financial performance among Nifty 50 companies in India from 2015 to 2022. Utilizing fixed-effects panel quantile regression, we observe that the relationship between ESG practices and financial profitability varies across the ROE distribution. While the environmental pillar score and the governance pillar score negatively impact ROE across almost all quantiles with high statistical significance, the social pillar score exhibits mostly an insignificant relationship. Its impact is negative, but only mildly statistically significant at the lower end of the ROE distribution. The findings and their implications are important for investors, corporate executives, and policymakers.

On the implied volatility of European and Asian call options under the stochastic volatility Bachelier model
2024-02-16 12:00 - 13:00 | via Zoom
Speaker: Makar Pravosud, Department of Economics and Business, Universitat Pompeu Fabra
Abstract: In this paper we study the short-time behavior of the at-the-money implied volatility for European and arithmetic Asian call options with fixed strike price. The asset price is assumed to follow the Bachelier model with a general stochastic volatility process. Using techniques of the Malliavin calculus such as the anticipating Itô’s formula we first compute the level of the implied volatility when the maturity converges to zero. Then, we find a short maturity asymptotic formula for the skew of the implied volatility that depends on the roughness of the volatility model. We apply our general results to the SABR and fractional Bergomi models, and provide some numerical simulations that confirm the accurateness of the asymptotic formula for the skew.

Unpacking the ESG ratings: Does one size fit all?
2024-02-23 14:00 - 15:00 | via Zoom
Speaker: Prof. Monica Billio, Department of Economics, Ca’ Foscari University of Venice
Abstract: As ESG investing goes mainstream, investors increasingly rely on ESG ratings when making investment decisions. This study aims to delve into the overall ESG ratings provided by four prominent ESG data providers, focusing on their accounting methodologies, the relevance of the three pillars (environment, social, and governance), and the key performance indicators (KPIs) that drive these ratings. By examining a sample of European and UK companies, we question the significance of the governance and social pillars in explaining the overall ESG scores. Our findings highlight a subset of indicators that exhibit the highest correlation with ESG scores, including the presence of external audits, an environmental supply chain policy, and target emissions. This letter contributes to the ongoing ESG credibility debate and emphasizes the need for further transparency of ESG ratings.

On the Bachelier implied volatility at extreme strikes
2024-03-08 14:00 - 15:00 | via Zoom
Speaker: Dr. Fabien Le Floc’h, Nasdsaq

Abstract: Roger Lee proved that the Black-Scholes implied variance can not grow faster than linearly in log-moneyness. This paper investigates what happens in the Bachelier (or Normal) implied volatility world, making sure to cover the various aspects of vanilla option arbitrages.

No Seminar - Spring Break
2024-03-15

Bayesian Optimization of ESG Financial Investments
2024-03-18 12:00 - 13:00 | via Zoom
Speaker: Prof Eduardo César Garrido Merchán, Faculty of Economics and Business Sciences, Comillas Universidad Pontificia
Abstract: Financial experts and analysts seek to predict the variability of financial markets. In particular, the correct prediction of this variability ensures investors successful investments. However, there has been a big trend in finance in the last years, which are the ESG criteria. Concretely, ESG (Economic, Social and Governance) criteria have become more significant in finance due to the growing importance of investments being socially responsible, and because of the financial impact companies suffer when not complying with them. Consequently, creating a stock portfolio should not only take into account its performance but compliance with ESG criteria. Hence, this paper combines mathematical modelling, with ESG and finance. In more detail, we use Bayesian optimization (BO), a sequential state-of-the-art design strategy to optimize black-boxes with unknown analytical and costly-to compute expressions, to maximize the performance of a stock portfolio under the presence of ESG criteria soft constraints incorporated to the objective function. In an illustrative experiment, we use the Sharpe ratio, that takes into consideration the portfolio returns and its variance, in other words, it balances the trade-off between maximizing returns and minimizing risks. In the present work, ESG criteria have been divided into fourteen independent categories used in a linear combination to estimate a firm total ESG score. Most importantly, our presented approach would scale to alternative black-box methods of estimating the performance and ESG compliance of the stock portfolio. In particular, this research has opened the door to many new research lines, as it has proved that a portfolio can be optimized using a BO that takes into consideration financial performance and the accomplishment of ESG criteria.

Supermartingale Brenier's Theorem with full-marginals constraint
2024-04-12 14:00 - 15:00 | via Zoom
Speaker: Prof. Dominykas Norgilas, Department of Mathematics, North Carolina State University
Abstract: We explicitly construct the supermartingale version of the Fréchet-Hoeffding coupling in the setting with infinitely many marginal constraints. This extends the results of Henry-Labordere et al. obtained in the martingale setting. Our construction is based on the Markovian iteration of one-period optimal supermartingale couplings. In the limit, as the number of iterations goes to infinity, we obtain a pure jump process that belongs to a family of local Lévy models introduced by Carr et al. We show that the constructed processes solve the continuous-time supermartingale optimal transport problem for a particular family of path-dependent cost functions. The explicit computations are provided in the following three cases: the uniform case, the Bachelier model and the Geometric Brownian Motion case.

Semi-analytic pricing of American options in some time-dependent jump-diffusion models
2024-04-19 12:00 - 13:00 | via Zoom
Speaker: Prof. Andrey Itkin, Department of Risk and Financial Engineering, Tandon School of Engineering, NYU
Abstract: In this paper we propose a semi-analytic approach to pricing American options for some time-dependent jump-diffusions models. The idea of the method is to further generalize our approach developed for pricing barrier, [Itkin et al., 2021], and American, [Carr and Itkin, 2021; Itkin and Muravey, 2023], options in various time-dependent one factor and even stochastic volatility models. Our approach i) allows arbitrary dependencies of the model parameters on time; ii) reduces solution of the pricing problem for American options to a simpler problem of solving an algebraic nonlinear equation for the exercise boundary and a linear Fredholm-Volterra equation for the the option price; iii) the options Greeks solve a similar Fredholm-Volterra linear equation obtained by just differentiating Eq. (25) by the required parameter.

Portfolio selection under non-gaussianity and systemic risk: A machine learning based forecasting approach
2024-04-19 14:00 | via Zoom
Speaker: Prof. Abderrahim Taamouti, Management School, University of Liverpool
Abstract: The Sharpe-ratio-maximizing portfolio becomes questionable under non-Gaussian returns, and it rules out, by construction, systemic risk, which can negatively affect its out-of-sample performance. In the present work, we develop a new performance ratio that simultaneously addresses these two problems when building optimal portfolios. To robustify the portfolio optimization and better represent extreme market scenarios, we simulate a large number of returns via a Monte Carlo method. This is done by obtaining probabilistic return forecasts through a distributional machine learning approach in a big data setting and then combining them with a fitted copula to generate return scenarios. Based on a large-scale comparative analysis conducted on the US market, the backtesting results demonstrate the superiority of our proposed portfolio selection approach against several popular benchmark strategies in terms of both profitability and minimizing systemic risk. This outperformance is robust to the inclusion of transaction costs.

Hedging with temporary price impact
2024-04-26 12:00 - 13:00 | via Zoom
Speaker: Prof. Peter Bank, Department of Mathematics, Technical University of Berlin
Abstract: We consider the problem of hedging a European contingent claim in a Bachelier model with temporary price impact as proposed by Almgren and Chriss (J Risk 3:5–39, 2001). Following the approach of Rogers and Singh (Math Financ 20:597–615, 2010) and Naujokat and Westray (Math Financ Econ 4(4):299–335, 2011), the hedging problem can be regarded as a cost optimal tracking problem of the frictionless hedging strategy. We solve this problem explicitly for general predictable target hedging strategies. It turns out that, rather than towards the current target position, the optimal policy trades towards a weighted average of expected future target positions. This generalizes an observation of Gârleanu and Pedersen (Dynamic portfolio choice with frictions. Preprint, 2013b) from their homogenous Markovian optimal investment problem to a general hedging problem. Our findings complement a number of previous studies in the literature on optimal strategies in illiquid markets as, e.g., Gârleanu and Pedersen (Dynamic portfolio choice with frictions. Preprint, 2013b), Naujokat and Westray (Math Financ Econ 4(4):299–335, 2011), Rogers and Singh (Math Financ 20:597–615, 2010), Almgren and Li (Option hedging with smooth market impact. Preprint, 2015), Moreau et al. (Math Financ. doi:10.1111/mafi.12098, 2015), Kallsen and Muhle-Karbe (High-resilience limits of block-shaped order books. Preprint, 2014), Guasoni and Weber (Mathematical Financ. doi:10.1111/mafi.12099, 2015a; Nonlinear price impact and portfolio choice. Preprint, 2015b), where the frictionless hedging strategy is confined to diffusions. The consideration of general predictable reference strategies is made possible by the use of a convex analysis approach instead of the more common dynamic programming methods.

Gold-backed cryptocurrencies: A hedging tool against categorical and regional financial stress
2024-05-01 08:00 - 17:00, 2024-08-30 14:00 - 15:00 | via Zoom
Speaker: Prof. Md. Rayfayet Alam, Dept. of Finance and Economics, University of Tennessee at Chattanooga
Abstract: This study evaluates the potential of gold-backed cryptocurrencies, such as Tether Gold and PAX Gold, as a hedge and safe haven against global, regional, and categorical financial stresses. Hedge and safe haven properties of gold-backed cryptocurrencies are also compared with those of gold and Bitcoin. For the analyses, dynamic conditional correlation (DCC) and quantile coherency techniques are applied to daily data from February 2020 to March 2023. The results show that Tether Gold and PAX Gold are strong safe havens against the US and equity-valuation-related financial stress but weak safe havens against global financial stress. Tether Gold is a weak safe haven against credit-related financial stress as well. Tether Gold is a strong hedge against US financial stress but a weak hedge against aggregate financial stress of developed economies and that of emerging economies. In our sample, gold-backed cryptocurrencies usually outperform gold and Bitcoin as a hedge and safe haven against financial stresses. The Quantile coherency analysis shows that Tether Gold is a hedge against low to moderate financial stress and a safe haven against extreme financial stresses. These findings have important implications for investors, risk-managers and policy makers.

Elementary function solutions to the Bachelier model generated by Lie point symmetries
2024-05-03 12:00 - 13:00 | via Zoom
Speaker: Dr. Evangelos Melas, Department of Mathematics, University of Thessaly
Abstract: Under the recent negative interest rate situation, the Bachelier model has been attracting attention and adopted for evaluating the price of interest rate options. In this paper we find the Lie point symmetries of the Bachelier partial differential equation (PDE) and use them in order to generate new classes of denumerably infinite elementary function solutions to the Bachelier model from elementary function solutions to it, which we derived in a previous publication.

Quanto Option Pricing on a Multivariate Lévy Process Model with Generative Artificial Intelligence
2024-08-23 14:00 - 15:00 | via Zoom
Speaker: Prof. Aaron YS Kim, College of Business, Stony Brook University
Abstract: In this study, we discuss a machine learning technique to price exotic options with two underlying assets based on a non-Gaussian Levy process model. We introduce a new multivariate Levy process model named the generalized normal tempered stable (gNTS) process, which is defined by time-changed multivariate Brownian motion. Since the gNTS process does not provide a simple analytic formula for the probability density function (PDF), we use the conditional real-valued non-volume preserving (CRealNVP) model, which is a type of flow-based generative network. Then, we discuss the no-arbitrage pricing on the gNTS model for pricing the quanto option whose underlying assets consist of a foreign index and foreign exchange rate. We present the training of the CRealNVP model to learn the PDF of the gNTS process using a training set generated by Monte Carlo simulation. Next, we estimate the parameters of the gNTS model with the trained CRealNVP model using the empirical data observed in the market. Finally, we provide a method to find an equivalent martingale measure on the gNTS model and to price the quanto option using the CRealNVP model with the risk-neutral parameters of the gNTS model.

Optimal Portfolios with Sustainable Assets - Aspects for Life Insurers
2024-09-13 14:00 - 15:00 | via Zoom
Speaker: Prof. Ralf Korn, Dept. of Mathematics, RPTU Kaiserslautern-Landau
Abstract: Since August 2022 customers have to be asked if they are interested in sustainable investment when entering a pension contract. Hence, the provider has to be prepared to offer suitable investment opportunities. Further, the provider has to manage the new risks and chances of those assets in the whole portfolio. We therefore especially look at possible consequences for optimal portfolio decisions of a life insurer and suggest modeling approaches for the evolution of the demand and the sustainability ratings for sustainable assets. We will solve various portfolio problems under sustainability constraints explicitly and suggest further research topics. As a special feature for a life insurer, we particularly look at the role of the actuarial reserve fund and the annual declaration of its return.

To hedge or not to hedge? Cryptocurrencies, gold and oil against stock market risk
2024-09-20 12:00 - 13:00 | via Zoom
Speaker: Prof. Agata Kliber, Dept of Applied Mathematics, Poznan University of Economics & Business
co-Authors: Prof. Krzysztof Echaust, Dept. of Operations Research & Mathematical Economics, Poznan University of Economics & Business
Prof. Małgorzata Just, Dept. of Finance & Accounting, Poznan University of Life Sciences
Abstract: The article aims to determine whether any hedging strategy against stock market risk, performed using instruments popular in the literature (gold, cryptocurrencies and oil), can beat index futures. As a hedging strategy, we understand a pair-wise portfolio consisting of a long position in stocks and a short position in a hedging instrument put together to minimise the portfolio variance. As a benchmark, we analyse optimal and naive hedging strategies with futures contracts. We demonstrate that, regardless of the stock market, the best hedging strategy focused on variance minimisation requires using index futures. Both strategies: the optimisation-based one and the naive one, beat the dynamic strategies utilising the remaining hedging assets. Therefore, from a risk-minimisation point of view, investors have no motivation to implement cryptocurrencies, gold or oil in hedging strategy against stock market risk. The results are robust with respect to hedging against tail risk.

ESG performance and investment efficiency: The impact of information asymmetry
2024-09-27 12:00 - 13:00 | via Zoom
Speaker: Prof. Seda Erdogan, Dept. International Trade & Finance, Kadir Has University
Abstract: This paper investigates the relationship between firms’ engagement in environmental, social, and governance (ESG) activities and corporate investment efficiency, using 1,094 firms from 21 countries in Europe, covering the years 2002–2019. We conduct our estimations using fixed effects panel data techniques and address potential endogeneity with instrumental variables (IV) estimations. We provide evidence that overall ESG engagement is positively and significantly associated with investment efficiency. Analyzing overinvestment and underinvestment scenarios shows that ESG engagement decreases only overinvestment problems. Within the underinvestment scenario, we observe that ESG engagement is beneficial only for firms with higher information asymmetries. Thus, information asymmetry matters in the underinvestment case. We next show that four firm-level channels—information asymmetry, financial constraints, cash flows, and risk—link ESG performance to investment inefficiency. Additional analysis shows that firms with extreme ESG scores (i.e., very low and very high) do not experience significant reductions in investment inefficiency. Altogether, our findings draw attention to the critical role of ESG performance and information asymmetry in determining corporate investment efficiency.

Option Pricing under a Generalized Black–Scholes Model with Stochastic Interest Rates, Stochastic Strings, and Lévy Jumps
2024-10-04 14:00 - 15:00 | via Zoom
Speaker: Prof. Steven P. Clark, Dept. of Finance, UNC Charlotte
Abstract: We introduce a novel option pricing model that features stochastic interest rates along with an underlying price process driven by stochastic string shocks combined with pure jump Lévy processes. Substituting the Brownian motion in the Black–Scholes model with a stochastic string leads to a class of option pricing models with expiration-dependent volatility. Further extending this Generalized Black–Scholes (GBS) model by adding Lévy jumps to the returns generating processes results in a new framework generalizing all exponential Lévy models. We derive four distinct versions of the model, with each case featuring a different jump process: the finite activity lognormal and double–exponential jump diffusions, as well as the infinite activity CGMY process and generalized hyperbolic Lévy motion. In each case, we obtain closed or semi-closed form expressions for European call option prices which generalize the results obtained for the original models. Empirically, we evaluate the performance of our model against the skews of S&P 500 call options, considering three distinct volatility regimes. Our findings indicate that: (a) model performance is enhanced with the inclusion of jumps; (b) the GBS plus jumps model outperform the alternative models with the same jumps; (c) the GBS-CGMY jump model offers the best fit across volatility regimes.

Seminar Cancelled
2024-10-11 14:00 - 15:00 | via Zoom
Title: Time changes, Fourier transforms and the joint calibration to the S&P500/VIX smiles
Speaker: Prof. Laura Ballotta, Bayes Business School, City University of London
Abstract: We develop a model based on time changed Lévy processes and study its ability of reproducing the joint S&P500/VIX implied volatility smiles and the VIX futures prices – a problem known in the literature as the `joint calibration problem’. The model admits semi-analytical characteristic functions for the key quantities, and therefore efficient Fourier based pricing schemes can be deployed. We focus on a specification of the proposed general setting which uses purely discontinuous processes. Results from the application to market data show satisfactory performances in solving the joint calibration problem, and therefore demonstrate that the class of affine processes can provide a workable fit.

Elicitability and identifiability of tail risk measures
2024-10-18 13:00 - 14:00 | via Zoom
Speaker: Dr. Tobias Fissler, Department of Mathematics, ETH Zurich
Abstract: Tail risk measures are fully determined by the distribution of the underlying loss beyond its quantile at a certain level, with Value-at-Risk and Expected Shortfall being prime examples. They are induced by law-based risk measures, called their generators, evaluated on the tail distribution. This talk establishes joint identifiability and elicitability results of tail risk measures together with the corresponding quantile, provided that their generators are identifiable and elicitable, respectively. As an example, we establish the joint identifiability and elicitability of the tail expectile together with the quantile. The corresponding consistent scores constitute a novel class of weighted scores, nesting the known class of scores of Fissler and Ziegel for the Expected Shortfall together with the quantile. For statistical purposes, our results pave the way to easier model fitting for tail risk measures via regression and the generalized method of moments, but also model comparison and model validation in terms of established backtesting procedures.
The talk is based on joint work with Ruodu Wang, Fangda Liu and Linxiao Wei.

Estimation and backtesting of risk measures with emphasis on distortion risk measures
2024-10-25 10:00 - 11:00 | via Zoom
Speaker: Prof. Hideatsu Tsukahara, Dept.. of Economics, Seijo University, Tokyo
Abstract: Statistical methodology has an important role to play in risk measurement. In this paper, we will review and discuss some statistical issues on risk measures. Examples we consider are value-at-risk, expected shortfall, expectiles, and distortion risk measures. Several methods of estimating these risk measures based on time series data have been proposed, and we will try to explain in some detail. Another main issue we would like to address is a problem of backtesting: the evaluation of risk measurement procedures using historical data, by comparing ex ante estimates of loss distributions or risk measures with the ex post realized losses. There have been several suggestions concerning backtestability of risk measures, which will be discuss in detail. We also examine and suggest backtesting procedures for predictive distributions, expected shortfall and distortion risk measures.

Pricing options with a new hybrid neural network model
2024-11-04 12:00 - 13:00 | via Zoom
Speaker: Dr. Yossi Shvimer, Research Associate, School of Finance and Management, SOAS University of London
Abstract: A novel hybrid option pricing model using a deep learning neural network has been developed. The hybrid model keeps the traditional option pricing model with the same input parameters while simultaneously adjusting the model with neural network methods to improve accuracy when applied to real market data, especially in OTM options. The new hybrid model demonstrates superior accuracy compared to both traditional parametric and non-parametric option pricing models for both Call and Put options across all moneyness levels. The empirical results of the hybrid model provide an explanation for the deviation from the Put-Call parity observed in real market data.

Stochastic dominance, stochastic volatility, and jump risk: new theory interprets old results
2024-11-15 14:00 - 15:00 | via Zoom
Speaker: Prof Stylianos Perrakis, John Molson School of Business, Concordia Univ, Montreal
Abstract: The stochastic dominance (SD) approach is applied to the valuation of index options in frictionless markets for a wide class of stochastic volatility (SV) processes. SD allows for the derivation of a unique, exponential option pricing kernel based on the physical underlying return and volatility dynamics. A lower bound and an upper bound on option prices are obtained, for a wide class of stochastic volatility jump (SVJ) processes that feature jumps in addition to diffusion. Using parameter estimates for the physical process from high-profile studies, the bounds are shown to be remarkably tight, especially for the empirically important class of short-term near-the-money options. The bounds are in many cases inconsistent with separate parameter estimates for the risk-neutral process that are extracted from observed option prices: for many option series, the risk-neutral value exceeds the SD upper bound. This inconsistency points at the possibility that the distributional shape of the risk-neutral process is mis-specified or that the parameters are estimated without properly taking the option bid-ask spread into account.

Inverse Problem for Forecasting Stock Options Prices
2024-11-22 14:00 - 15:00 | via Zoom
Speaker: Dr. Kirill Golubnichiy, Dept of Math & Statistics, Texas Tech University
Abstract: We present a new heuristic mathematical model for accurate forecasting of prices of stock options for 1-2 trading days ahead of the present one. This new technique uses the Black-Scholes equation supplied by new intervals for the underlying stock and new initial and boundary conditions for option prices. The Black-Scholes equation was solved in the positive direction of the time variable, This ill-posed initial boundary value problem was solved by the so-called Quasi-Reversibility Method (QRM). This approach with an added trading strategy was tested on the market data for 368 stock options and good forecasting results were demonstrated. In the current paper, we use the geometric Brownian motion to provide an explanation of that effectivity using computationally simulated data for European call options. We also provide a convergence analysis for QRM. The key tool of that analysis is a Carleman estimate.

Deep learning-based portfolio optimization with transaction costs
2025-01-17 09:00 - 10:00 | via Zoom
Speaker: Prof. Aihua (Eva) Zhang, College of Science, Math & Tech., Wenzhou-Kean University, Wenzhou China
Abstract: In order to obtain the optimal portfolio strategy maximizing the accumulated terminal wealth with transaction costs, in this paper, we propose a new prediction-based portfolio method combining with a long short-term memory (in short, LSTM) network which is an extended type of recurrent neural networks in deep learning. Our proposed method, named as LSTM-Prediction-based Portfolio (LSTM-PbP) with transaction costs, consists of two technical steps: finding the optimal portfolio strategy and predicting the future relative prices. For the price prediction, we use multi-layer LSTM; while for the optimal portfolio strategy, we solve the constraint maximization problem via relative entropy. We then update the future portfolio weights using the predicted prices and past portfolio weights. We iterate the process until the final investment period. Numerical experiments are also provided to show the accumulated wealth by following the obtained optimal portfolio strategy in comparison with the accumulated wealth under buy-and-hold strategy. The numerical results show that our model consistently outperforms the buy-and-hold strategy.
