Bubbles, convexity and the Black–Scholes equation

Abstract

A bubble is characterized by the presence of an underlying asset whose discounted price process is a strict local martingale under the pricing measure. In such markets, many standard results from option pricing theory do not hold, and in this paper we address some of these issues. In particular, we derive existence and uniqueness results for the Black-Scholes equation, and we provide convexity theory for option pricing and derive related ordering results with respect to volatility. We show that American options are convexity preserving, whereas European options preserve concavity for general payoffs and convexity only for bounded contracts.

Can time-homogeneous diffusions produce any distribution?

“…Let u N (x, t) = E x |Y t | ∧ N , where the index indicates that Y 0 = x. By a maximum principle argument, compare [7], u N (x, t) ≤ h(x, t) independently of N . Consequently, by monotone convergence, E 0 |Y t | ≤ h(0, t) < ∞.…”

Numerical option pricing in the presence of bubbles

“…It is shown in [2] that if g is of at most linear growth, then the option price u given in (2) is a classical solution to the Black-Scholes equation (3). However, it is well-known that there are multiple solutions of at most linear growth.…”

“…To determine the stochastic solution given by (2), one may employ the following observation made in [2] and [3].…”

“…One possibility to identify this solution would be to replace the pay-off function g by g ∧M and find the unique bounded solution to the corresponding pricing equation. When letting M → ∞, the smallest solution is obtained, compare [2]. This method, however, is not straightforward to analyze numerically since, for each M , an upper bound for the state space has to be chosen, tending to infinity with M .…”

“…In these models several basic properties of option prices may fail such as the put-call parity, the no-dominance principle and the uniqueness of solutions to the corresponding Black-Scholes equation, compare [1], [3] and [4]. For a discussion on existence and uniqueness of solutions to the Black-Scholes equation in one-dimensional diffusion models with bubbles, see [2].…”