An overview of financial mathematics with Python codes

Halidias Nikolaos ()
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Halidias Nikolaos: Department of Statistics and Actuarial-Financial Mathematics, University of the Aegean, Karlovassi 83200, Samos, Greece

Monte Carlo Methods and Applications, 2025, vol. 31, issue 4, 279-309

Abstract: In this note we will discuss and review some recent results concerning well-known financial mathematical problems such as portfolio construction, dynamic trading, option pricing, etc. We will use some Python codes concerning the above and we will compare the results with the existing methods and techniques. We propose also a new type of multi-asset options; the options on correlation. Using this kind of options one can refine more effectively the profit function of his/her portfolio when this contain two or more assets. It is mathematically certain that, in practice, someone will eventually apply the techniques described in this paper. This is because, regarding the portfolio construction problem, we allow the investor to employ any forecasting technique – e.g., statistical methods, machine learning, behavioral finance, intuition, etc. Subsequently, the investor can enhance both the return and safety of their portfolio by incorporating call and put options. In contrast, for the derivative pricing problem, it is evident that there is no room for forecasts (see volatility for example), as pricing involves two counterparties – the seller and the buyer – making it, metaphorically, a dance for two. For this reason, the pricing methodology proposed in this paper is model-free, ensuring that the resulting prices are fully consistent with the market values of available contracts. Moreover, the investor decides at which price to buy or sell a contract based on practical, statically implementable hedging strategies that we propose in this work. That is, any pricing method should justify why an investor ought to buy or sell a derivative at the proposed price. In other words, the method must provide a clear, economically sound rationale-typically grounded in no-arbitrage principles, replication arguments, or explicit hedging strategies – that links the quoted price to actionable, implementable decisions for market participants. What remains to be explored are advanced forecasting techniques that account for events affecting the stocks of interest to the investor, as well as the documentation of hedging strategies for path-dependent options.

Keywords: Prediction; portfolio construction; option pricing and hedging; liquidity; options on correlation (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1515/mcma-2025-2020

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