This paper considers utility indifference valuation of derivatives under model uncertainty and trading constraints, where the utility is formulated as an additive stochastic differential utility of both intertemporal consumption and terminal wealth,
and the uncertain prospects are ranked according to a multiple-priors model of Chen and Epstein (2002). The price is determined by two optimal stochastic control problems (mixed with optimal stopping time in the case of American option) of forward-backward stochastic differential equations. By means of backward stochastic differential equation and partial differential equation methods, we show that both bid and ask prices are closely related to the Black-Scholes risk-neutral price with modified dividend rates. The two prices will actually coincide with each other if there is no trading constraint or the model uncertainty disappears. Finally, two applications to European option and American option are discussed.
This paper studies the valuation and optimal strategy of convertible bonds as a Dynkin game by using the reflected backward stochastic differential equation method and the variational inequality method. We first reduce such a Dynkin game to an optima
l stopping time problem with state constraint, and then in a Markovian setting, we investigate the optimal strategy by analyzing the properties of the corresponding free boundary, including its position, asymptotics, monotonicity and regularity. We identify situations when call precedes conversion, and vice versa. Moreover, we show that the irregular payoff results in the possibly non-monotonic conversion boundary. Surprisingly, the price of the convertible bond is not necessarily monotonic in time: it may even increase when time approaches maturity.
A Dynkin game is considered for stochastic differential equations with random coefficients. We first apply Qiu and Tangs maximum principle for backward stochastic partial differential equations to generalize Krylov estimate for the distribution of a
Markov process to that of a non-Markov process, and establish a generalized It^o-Kunita-Wentzells formula allowing the test function to be a random field of It^os type which takes values in a suitable Sobolev space. We then prove the verification theorem that the Nash equilibrium point and the value of the Dynkin game are characterized by the strong solution of the associated Hamilton-Jacobi-Bellman-Isaacs equation, which is currently a backward stochastic partial differential variational inequality (BSPDVI, for short) with two obstacles. We obtain the existence and uniqueness result and a comparison theorem for strong solution of the BSPDVI. Moreover, we study the monotonicity on the strong solution of the BSPDVI by the comparison theorem for BSPDVI and define the free boundaries. Finally, we identify the counterparts for an optimal stopping time problem as a special Dynkin game.
Feedstock and byproduct diffusion in the root growth of aligned CNT arrays was discussed in this work. A non-dimensional modulus was proposed to differentiate catalyst-decay controlled growth deceleration from diffusion controlled one. It was found t
hat aligned MWNT arrays are usually free of feedstock diffusion while SWNT arrays are usually facing strong diffusion limit. The present method can also be utilized to predict the maximum length that CNT forest can grow in certain CVD process.
