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This research is devoted to studying a geometric Brownian motion with drift switching driven by a 2 χ 2 Markov chain. A discrete-time multiplicative approximation scheme was developed, and its convergence in Skorokhod topology to the continuous-time geometric Brownian motion with switching has been proved. Furthermore, in a financial market where the discounted asset price follows a geometric Brownian motion with drift switching, market incompleteness was established, and multiple equivalent martingale measures were constructed. [ABSTRACT FROM AUTHOR]

This research is devoted to studying a geometric Brownian motion with drift switching driven by a 2 × 2 Markov chain. A discrete-time multiplicative approximation scheme was developed, and its convergence in Skorokhod topology to the continuous-time geometric Brownian motion with switching has been proved. Furthermore, in a financial market where the discounted asset price follows a geometric Brownian motion with drift switching, market incompleteness was established, and multiple equivalent martingale measures were constructed.

In this paper, we study the optimization problems of minimization of shortfall risk under the jump diffusion model. Jump-diffusion asset price model is driven by nonexplosive counting process that is more general than Poisson process. We take the expected discount utility of shortfall risk as our objective function. Applying Legendre transform and stochastic analysis methods,the existence of equivalent martingale measure and the optimal portfolio is proved. Furthermore, the value function, the optimal portfolio, and the optimal terminal wealth are also proposed. [ABSTRACT FROM AUTHOR]

In a numéraire-independent framework, we study a financial market with assets which are all treated in a symmetric way. We define the fundamental value ∗ S of an asset as its super-replication price and say that the market has a strong bubble if ∗ S and deviate from each other. None of these concepts needs any mention of martingales. Our main result then shows that under a weak absence-of-arbitrage assumption (basically NUPBR), a market has a strong bubble if and only if in all numéraire s for which there is an equivalent local martingale measure (ELMM), asset prices are strict local martingales under all possible ELMMs. We show by an example that our bubble concept lies strictly between the existing notions from the literature. We also give an example where asset prices are strict local martingales under one ELMM, but true martingales under another, and we show how our approach can lead naturally to endogenous bubble birth. [ABSTRACT FROM AUTHOR]

We consider financial markets with agents exposed to external sources of risk caused, for example, by short-term climate events such as the South Pacific sea surface temperature anomalies widely known by the name El Nino. Since such risks cannot be hedged through investments on the capital market alone, we face a typical example of an incomplete financial market. In order to make this risk tradable, we use a financial market model in which an additional insurance asset provides another possibility of investment besides the usual capital market. Given one of the many possible market prices of risk, each agent can maximize his individual exponential utility from his income obtained from trading in the capital market, the additional security, and his risk-exposure function. Under the equilibrium market-clearing condition for the insurance security the market price of risk is uniquely determined by a backward stochastic differential equation. We translate these stochastic equations via the Feynman–Kac formalism into semi-linear parabolic partial differential equations. Numerical schemes are available by which these semilinear pde can be simulated. We choose two simple qualitatively interesting models to describe sea surface temperature, and with an ENSO risk exposed fisher and farmer and a climate risk neutral bank three model agents with simple risk exposure functions. By simulating the expected appreciation price of risk trading, the optimal utility of the agents as a function of temperature, and their optimal investment into the risk trading security we obtain first insight into the dynamics of such a market in simple situations. [ABSTRACT FROM AUTHOR]

We consider financial markets with agents exposed to an external source of risk which cannot be hedged through investments on the capital market alone. The sources of risk we think of may be weather and climate. Therefore we face a typical example of an incomplete financial market. We design a model of a market on which the external risk becomes tradable. In a first step we complete the market by introducing an extra security which valuates the external risk through a process parameter describing its market price. If this parameter is fixed, risk has a price and every agent can maximize the expected exponential utility with individual risk aversion obtained from his risk exposure on the one hand and his investment into the financial market consisting of an exogenous set of stocks and the insurance asset on the other hand. In the second step, the market price of risk parameter has to be determined by a partial equilibrium condition which just expresses the fact that in equilibrium the market is cleared of the second security. This choice of market price of risk is performed in the framework of nonlinear backwards stochastic differential equations. [ABSTRACT FROM AUTHOR]

The paper studies connections between risk aversion and martingale measures in a discrete-time incomplete financial market. An investor is considered whose attitude toward risk is specified in terms of the index b of constant proportional risk aversion. Then dynamic portfolios are admissible if the terminal wealth is positive. It is assumed that the return (risk) processes are bounded. Sufficient (and nearly necessary) conditions are given for the existence of an optimal dynamic portfolio which chooses portfolios from the interior of the set of admissible portfolios. This property leads to an equivalent martingale measure defined through the optimal dynamic portfolio and the index 0 < b ≤ 1. Moreover, the option pricing formula of Davis is given by this martingale measure. In the case of b = 1; that is, in the case of the log-utility, the optimal dynamic portfolio defines the numéraire portfolio. [ABSTRACT FROM AUTHOR]

We establish necessary and sufficient conditions for an H¹-martingale to be representable with respect to a collection, ..., of local martingales. M ∈ H¹(P) is representable if and only if M is a local martingale under all p.m.'s Q which are "uniformly equivalent" to P and which make all the elements of ... local martingales (Theorem 1.1). We then give necessary and sufficient conditions which are easier to verify, and only involve expectations (Theorem 1.2). We go on to apply these results to the problem of pricing claims in an incomplete financial market--establishing two conjectures of Harrison and Pliska (1981). [ABSTRACT FROM AUTHOR]