Exploring Pathwise Finance: A Unique Method Without Probabilities

Are you tired of the usual way of looking at continuous-time finance? It's time to think outside the box with a new approach called causal functional calculus. Unlike traditional methods, this one doesn't rely on probabilities or integration theories. Instead, it focuses on a concept called self-financing, which doesn't need any complex math to understand. What's interesting is that any self-financing strategy can be seen as a sort of path (or gradient), and the value of your portfolio is just a sum of these paths. And guess what? This new method is naturally free from arbitrage, which means it's fair and doesn't give any advantage to someone just because of luck or information.

So, what about hedging a certain payoff that depends on the path you take? This new approach uses something called the transition principle of Isaacs in differential games. It even has a verification theorem to help us find the best solution, and that solution is described by a special kind of equation that depends on the path you take. For something specific like an Asian option, this new method even gives us an exact answer. How cool is that?

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