An asset bubble, also referred to as a speculative mania or financial bubble, is an economic situation characterised by trading in an asset at a price that is far above its real value. This is often followed by a sharp drop which is known as bubble burst. As such, a large amount of wealth can be destroyed, and this may lead to a continuing economic crisis. Therefore, asset bubbles have become a subject of growing interest in mathematical finance. There are several approaches towards this subject which can be divided into two main groups: models that quantify price bubbles in a classical arbitrage-free setting and models that explain the mechanism of price bubbles. In this work, price bubbles are introduced as the difference between a minimal equilibrium price and an intrinsic value. The aim of this study was to find out how we can include sudden changes in the underlying or the economic situation and what could be its possible effect on an asset bubble. Our first approach was to introduce Markovian regime switching in the interest rate. We revealed that the bubble contains a component that is entirely based on the regime switching risk. The second approach was a Lévy model for the underlying asset. Our findings in this part provide a basis for further research on the current topic with numerical implementation.