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.

, Probab. Statist. Group Manchester (21 pp) Let X = (X t ) t≥0 be a strong Markov process, and let G 1 , G 2 and G 3 be continuous functions satisfying G 1 ≤ G 3 ≤ G 2 and E x sup t |G i (X t )| < ∞ for i = 1, 2, 3 . Consider the optimal stopping game where the sup-player chooses a stopping time τ to maximise, and the inf-player chooses a stopping time σ to minimise, the expected payoffwhere X 0 = x under P x . Define the upper value and the lower value of the game bywhere the horizon T (the upper bound for τ and σ above) may be either finite or infinite (it is assumed thatIf X is right-continuous, then the Stackelberg equilibrium holds, in the sense that V * (x) = V * (x) for all x with V := V * = V * defining a measurable function. If X is right-continuous and left-continuous over stopping times (quasi-left-continuous), then the Nash equilibrium holds, in the sense that there exist stopping times τ * and σ * such thatfor all stopping times τ and σ , implying also that V (x) = M x (τ * , σ * ) for all x . Further properties of the value function V and the optimal stopping times τ * and σ * are exhibited in the proof.

We show, under weaker assumptions than in the previous literature, that a perpetual optimal stopping game always has a value. We also show that there exists an optimal stopping time for the seller, but not necessarily for the buyer. Moreover, conditions are provided under which the existence of an optimal stopping time for the buyer is guaranteed. The results are illustrated explicitly in two examples.Comment: Published at http://dx.doi.org/10.1214/105051606000000204 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

We investigate some properties of American option prices in the setting of time-and leveldependent volatility. The properties under consideration are convexity in the underlying stock price, monotonicity and continuity in the volatility and time decay. Some properties are direct consequences of the corresponding properties of European option prices that are already known, and some follow by writing solutions of di erent stochastic di erential equations as time changes of the same Brownian motion.

Abstract. We characterise the value function of the optimal dividend problem with a finite time horizon as the unique classical solution of a suitable Hamilton-Jacobi-Bellman equation. The optimal dividend strategy is realised by a Skorokhod reflection of the fund's value at a time-dependent optimal boundary. Our results are obtained by establishing for the first time a new connection between singular control problems with an absorbing boundary and optimal stopping problems on a diffusion reflected at 0 and created at a rate proportional to its local time.