Rational Equilibrium Asset-Pricing Bubbles in Continuous Trading Models

Yan Theorem in L ∞ with Applications to Asset Pricing

“…However, when prices are positive and Z is only a martingale density, a straightforward application of Fatou lemma delivers S 0 ≥ P (Z ∞ S ∞ ). As remarked in [15], it is reasonable to interpret P (Z ∞ S ∞ ) as the fundamental value of the asset and, consequently, the quantity β (S) = S 0 − P (Z ∞ S ∞ ) as the "bubble" part of the asset price. In [15], however, it is assumed that the martingale density is strictly positive but it is far from clear how this relates to the no arbitrage principle.…”

Fragility of arbitrage and bubbles in local martingale diffusion models

“…Loewenstein and Willard (2000) are the first to link bubbles to local martingales. In a frictionless, complete market, based on a continuous-time diffusion, they show that bubbles may occur as a result of wealth constraints.…”

“…The above definition of bubble first appears in Heston et al (2007), for both complete and incomplete markets. In a complete market this definition is equivalent to the one used by Loewenstein and Willard (2000) and Cox and Hobson (2005), who define an asset price as a bubble if it is a strict local martingale under the risk-neutral measure 1 . Again in complete markets, Jarrow, Protter and Shimbo (2007) classify bubbles into three types.…”

Shifting martingale measures and the birth of a bubble as a submartingale

“…Suppose that at any time, the fundamental value of the asset is computed as the conditional expectation of future discounted dividends under some equivalent local martingale measure. Time consistency would require that all these conditional expectations are computed under the same martingale measure R. Denoting by S R the resulting fundamental value process, the bubble is now defined as the difference S − S R , and this will be a nonnegative local martingale under R. There is a growing literature about such bubbles and their various effects; see for instance Loewenstein and Willard [20], Cox and Hobson [4], Jarrow and Madan [12], Jarrow/Protter et al [11,13,14,16,17]. In Jarrow and Protter [15], the novel concept of a relative asset bubble is introduced, which allows the study of price bubbles for assets with bounded payoffs such as defaultable bonds.…”