Fly me: financial mathematics and low-cost airlines

Rynki instrumentów pochodnych należą obecnie do jednych z najciekawszych obszarów badań w matematyce finansowej. Są też obszarem, gdzie spotykają się teoria i praktyka, tak ważne w rzetelnie uprawianej matematyce stosowanej. Zachodzi tam ożywiona wymiana idei pomiędzy matematykami a ekonomistami, choć de facto ma ona charakter bardziej interdyscyplinarny, angażując np. informatyków i fizyków. Równolegle dynamiczny rozwój rynków i pojawianie się coraz szerszej gamy instrumentów pochodnych pozwalają na włączanie do tego zbioru coraz to nowych obiektów. W naszej pracy pokazujemy, że skoro zakładać się można „o wszystko”, a instrumenty pochodne są rodzajem giełdowego zakładu, to bilety tanich linii lotniczych można opisać jako specyficzne opcje. Uwzględniając rzeczywiste warunki rynkowe i ograniczenia narzucone w regulaminach tanich przewoźników, przedstawimy alternatywny dla już istniejących sposób podejścia do wyceny takich instrumentów oraz metody konstrukcji i optymalizacji portfela złożonego z biletów, który może mieć zastosowanie zarówno do celów spekulacyjnych, jak i korporacyjnych.

The derivatives are financial instruments whose value changes in response to the changes in underlying variables. Originally, they were based on commodities and used to reduce risk from varying prices. This strategy is often called hedging. However as financial markets have become more complex it is more and more difficult to distinguish between old fashioned hedging and speculation. At present the derivatives’ markets are one of the most interestingfie lds of research in financial mathematics. They are the point where theory and practice meet, an important factor in good applied mathematics. As a result a vivid exchange of ideas takes place between mathematicians and researchers from huge variety of fields like economics, physics and computer science. In the same time the markets are constantly evolvinga nd new financial instruments are created every day. Nowadays, derivatives are beingc onstructed not only for assets such as commodities, equities , bonds, interest rates, exchange rates, but also various indexes or even weather. In order to make a transaction a buyer and a seller are needed. In this respect a financial instrument can be understood as a bet between two parties, each assumingtha t the market will move the way that they expect. In old times derivatives were mainly used to managed risk, for instance by securing fixed price for some asset in the future. One of the types of derivatives are options that provide the right, but not the obligation, to engage in a future transaction on some fixed conditions. The transaction as such does not need to concern assets or securities, for instance the option can be exercise on certain services. In the paper we show that low-cost airlines’ tickets can be described as an option contract. While it is temptingto plug-in our description into known models for option valuation, we choose more practice oriented approach. We take into account the real life constrains and airlines’ regulations we present a realistic model for option valuation. In such a framework we show that, in principle, an arbitrage is possible. We conclude the paper with remarks on buildingan d optimizingp ortfolio of such assets, having in mind speculation/trading as well as corporate hedging.