## Quantitative Finance

## Itō’s Insight

Consider the stochastic process where is standard Brownian motion (or the Wiener process) and is a twice-differentiable function. Ito’s lemma states that The first term is recognizable from the chain rule in classical calculus, but why the second term? If is truly infinitesimal, it doesn’t even seem possible that . To understand Ito’s lemma intuitively, […]

## Profitability and Information

In the previous post (Optimal Market Exposure), we saw how trading requires information superior to the “outside world’s belief”, in order to be profitable. The trader may obtain this information by analyzing market fundamentals or technicals, or both. It is actually obvious that better information facilitates better trading, and more profits, but here we will […]

## Optimal Market Exposure

Having established in the previous post (Logarithmic Utility of Wealth) that the utility of a rational, small, long-term trader’s account is logarithmic in the account’s size, we are in a position to mathematically optimize how much risk (exposure) the trader should take on a given opportunity. We do this simply by maximizing expected utility with […]

## Logarithmic Wealth Utility

The previous post (Utility Function of Wealth) discussed and described the typical general shape of the utility function of wealth, and touched on some applications. Here, we will consider a special case, with an exact curve. Consider a rational trader – a trader that is emotionally detached from his account, and whose objective is to […]

## Utility Function of Wealth

It is often assumed that wealth has linear utility. For example, it is taken for granted that $2 is worth exactly double of $1. In most real-world scenarios, where the amount of wealth available is finite, this is not exactly true. Imagine, for example, three travellers in identical situations, except Traveller A has $2000 Traveller […]