Abstracts

Financial bubbles seem to exhibit a strong upward trend followed by a sharp decline when the bubble bursts. Starting from this descriptive perspective, we propose an extension of the Black–Scholes model which accommodates for this effect. In addition to the standard instantaneous expected return in the Black–Scholes model, we allow for a time-dependent positive excess return which is compensated for by a negative jump at a random time representing the bursting of the bubble. Since the excess return and the distribution of the random time may be chosen almost arbitrarily, the model is flexible enough to display different types of bubbles, including the ones in the sense of Protter (2013). Moreover, the model is tractable enough to allow for (semi-)explicit calculations and may thus be used as a toy example to study qualitative effects of financial bubbles.
We study the problem of maximising expected utility from terminal wealth for a power utility investor. Using the convex duality approach, we determine the optimal strategy and the corresponding certainty equivalent up to the solution to an integral equation (or a first-order ODE). A decomposition of the optimal strategy into the myopic and hedging demand allows to analyse the effects of the stochastic investment opportunities. On the one hand, investors with relative risk aversion below 1 speculate on an early bursting of the bubble in the sense that their optimal strategy lies below their myopic demand prior to the bursting of the bubble; the optimal strategy might even involve short-selling. On the other hand, investors with relative risk aversion above 1 hedge against a late bursting of the bubble in the sense that their optimal strategy lies above their myopic demand; the optimal strategy might even lie above the Merton proportion under extreme circumstances.
Numerical examples reveal how the optimal strategy and its myopic and hedging demand depend on the model parameters. In particular, it is shown that the optimal strategy is not fundamentally different when the stock price process is a strict local martingale (as opposed to a true martingale) under a certain class of equivalent local martingale measures including the dual minimiser of the utility maximisation problem. In addition, we illustrate the welfare loss compared the the classical Black–Scholes model and its dependence on the model parameters.