KakkoKari （仮） Another (data) science blog. By Alessandro Morita
The Carr-Madan decomposition of arbitrary payoff functions
The Carr-Madan decomposition is used in quant finance to break any payoff into a (continuous) combination of calls and puts, plus a forward. Namely, for any twice differentiable function: \[\boxed{ f(x) = f(y) + f'(y)(x-y) + \int_{-\infty}^y f''(z) (z-x)^+ dz + \int_{y}^\infty f''(z) (x-z)^+ dz }\] where $(x)^+ \equiv \max(0, x)$ is the positi... Read more 20 Sep 2022 - 1 minute read
A vector space structure for probabilities
This post is based on this article. Does it even make sense to discuss about adding two probabilities? Probabilities definitely look like vectors: they are arrays of numbers. For example, it could make sense that a coin toss would be described by an array with two numbers, something like $(0.5, 0.5)$. However, it is not obvious how they would... Read more 18 Sep 2022 - 5 minute read