This paper studies asset price bubbles in a continuous time model using the local martingale framework. Providing careful definitions of the asset’s market and fundamental price, we characterize all possible price bubbles in an incomplete market satisfying the ”no free lunch with vanishing risk” and ”no dominance” assumptions. We propose a new theory for bubble birth which involves a nontrivial modification of the classical framework. We show that the two leading models for bubbles as either charges or as strict local martingales, respectively, are equivalent. Finally, we investigate the pricing of derivative securities in the presence of asset price bubbles, and we show that: (i) European put options can have no bubbles, (ii) European call options and discounted forward prices can have bubbles, but the magnitude of their bubbles must equal the magnitude of the asset’s price bubble, (iii) with no dividends, American call prices must always equal an otherwise identical European call’s price, regardless of bubbles, (iv) European put-call parity in market prices must always hold, regardless of bubbles, and (v) futures price bubbles can exist and they are independent of bubbles in the underlying asset’s price. These results imply that in a market satisfying NFLVR and no dominance, in the presence of an asset price bubble, risk neutral valuation can not be used to match call option prices. We propose, but do not implement, some new tests for the existence of asset price bubbles using derivative securities.

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. 1. Introduction. Recently

We show pathological behavior of asset price processes modeled by continuous strict local martingales under a risk-neutral measure. The inspiration comes from recent results on financial bubbles. We analyze, in particular, the effect of the strict nature of the local martingale on the usual formula for the price of a European call option, especially a strong anomaly when call prices decay monotonically with maturity. A complete and detailed analysis for the archetypical strict local martingale, the reciprocal of a three dimensional Bessel process, has been provided. Our main tool is based on a general h-transform technique (due to Delbaen and Schachermayer) to generate positive strict local martingales. This gives the basis for a statistical test to verify a suspected bubble is indeed one (or not).

It is shown that delta hedging provides the optimal trading strategy in terms of minimal required initial capital to replicate a given terminal payoff in a continuous-time Markovian context. This holds true in market models where no equivalent local martingale measure exists but only a square-integrable market price of risk. A new probability measure is constructed, which takes the place of an equivalent local martingale measure. In order to ensure the existence of the delta hedge, sufficient conditions are derived for the necessary differentiability of expectations indexed over the initial market configuration. The recently often discussed phenomenon of “bubbles ” is a special case of the setting in this paper. Several examples at the end illustrate the techniques described in this work. 1

The subsequent theorem is one of the pillars supporting the modern theory of Mathematical Finance. Fundamental Theorem of Asset Pricing: The following two statements are essentially equivalent for a model S of a financial market: (i) S does not allow for arbitrage (NA) (ii) There exists a probability measure Q on the underlying probability space (Ω, F, P), which is equivalent to P and under which the process is a martingale (EMM). We have formulated this theorem in vague terms which will be made precise in the sequel: we shall formulate versions of this theorem below which use precise definitions and avoid the use of the word essentially above. In fact, the challenge is precisely to turn this vague “meta-theorem ” into sharp mathematical results. The story of this theorem started- like most of modern Mathematical Finance- with the work of F. Black, M. Scholes [3] and R. Merton [25]. These authors consider a model S =(St)0≤t≤T of geometric Brownian motion proposed by P. Samuelson [30], which today is widely known under the name of Black–Scholes model. Presumably every reader of this article is familiar with

This paper studies a rational price bubble in a productive asset and its e¤ect on the real economy in an overlapping generations model of a small open economy with an analysis on a collateralized credit constraint. As a consequence, the small open economy is vunerable to the bubble emergence. Crucially for the policy-making, the credit constraint plays a double-bladed role in the bubble emergence. That is, with the natural credit limit which is evaluated at the fundamental value of the collateral, a bubble cannot exist; while, if the …nancial intermediary sets the credit limit at the expected value of the collateral, the credit constraint instead helps support the bubble. Hence, the tight …nancial regulation and supervision over the credit constraint are recommended for policymakers to prevent or terminate the bubble. 1

Analysis of continuous strict local martingales via h-transforms

This paper extends and re…nes the Jarrow, Protter, Shimbo [12], [13] arbitrage free pricing theory for bubbles to characterize forward and futures prices. Some new insights are obtained in this regard. In particular, we: (i) provide a canonical process for asset price bubbles suitable for empirical estimation, (ii) discuss new methods to test empirically for asset price bubbles using both spot prices and call/put option prices on the spot commodity, (iii) show that futures prices always equal their fundamental values, (iv) relate forward and futures prices under bubbles, and (v) price options on futures with asset price bubbles.

Abstract. The stochastic exponential Zt = exp{Mt − M0 − (1/2)〈M, M〉t} of a continuous local martingale M is itself a continuous local martingale. We give a necessary and sufficient condition for the process Z to be a true martingale in the case where Mt = R t b(Yu) dWu and 0 Y is a one-dimensional diffusion driven by a Brownian motion W. Furthermore, we provide a necessary and sufficient condition for Z to be a uniformly integrable martingale in the same setting. These conditions are deterministic and expressed only in terms of the function b and the drift and diffusion coefficients of Y. We also classify, via deterministic necessary and sufficient conditions, when the process Z is a.s. strictly positive, when its limit Z ∞ is a.s. strictly positive, and when Z∞ is a.s. zero. This allows us to obtain a deterministic necessary and sufficient condition in the one-dimensional setting for a discounted stock price to be a true martingale under the risk-neutral measure, and for it to be a uniformly integrable martingale. These results enable us to ascertain the existence of financial bubbles in diffusion-based models. Finally, we