Options and Bubbles

Numerical option pricing in the presence of bubbles

“…On the other hand, interior Schauder estimates yield that u solves the equation (3). Moreover, since u M ≤ u ≤ u for all M , and since u M and u are continuous at the boundary, we find that also u is continuous at the boundary, with u(x, T ) = g(x) and u(0, t) = g(0).…”

“…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].…”

“…The authors of [1] and [3] propose that a discounted underlying asset which is a strict local martingale (i.e. a local martingale but not a martingale) under the pricing measure can be used to model a financial bubble.…”

“…a local martingale but not a martingale) under the pricing measure can be used to model a financial bubble. 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].…”