Fahuai Yi

Finite Horizon Optimal Investment and Consumption with Transaction Costs

This paper concerns continuous-time optimal investment and the consumption decision of a constant relative risk aversion (CRRA) investor who faces proportional transaction costs and a finite time horizon. In the no-consumption case, it has been studied by Liu and Loewenstein [Review of Financial Studies, 15 (2002), pp. 805-835] and Dai and Yi [J. Differential Equations, 246 (2009), pp. 1445-1469]. Mathematically, it is a singular stochastic control problem whose value function satisfies a parabolic variational inequality with gradient constraints. The problem gives rise to two free boundaries which stand for the optimal buying and selling strategies, respectively. We present an analytical approach to analyze the behaviors of free boundaries. The regularity of the value function is studied as well. Our approach is essentially based on the connection between singular control and optimal stopping, which is first revealed in the present problem.

A Variational Inequality Arising from European Installment Call Options Pricing

In this paper we consider a parabolic variational inequality arising from European continuous installment call options pricing and prove the existence and uniqueness of the solution to the problem. Moreover, we obtain

A free boundary problem arising from pricing convertible bond

C^\infty

A Variational Inequality from Pricing Convertible Bond

regularity and the bounds of the free boundary, as well as the limit of the free boundary as

Investment, consumption and hedging with lump-sum payoff in finite horizon under incomplete market

\tau=T-t\rightarrow+\infty

Finite-Horizon Optimal Investment with Transaction Costs: A Parabolic Double Obstacle Problem

. Eventually we show its numerical result by the binomial method.

Global classical solution of Muskat free boundary problem

In this article we study the behaviours of the optimal conversion boundary (i.e. free boundary) of an American-style convertible bond with finite horizon (i.e. parabolic case). We prove the existence and the uniqueness of the strong solution of the problem and the boundedness and smoothness of the free boundary. Moreover, we characterize the free boundarys start point and present two numerical results.

A variational inequality arising from American installment call options pricing

The model of pricing American-style convertible bond is formulated as a zero-sum Dynkin game, which can be transformed into a parabolic variational inequality (PVI). The fundamental variable in this model is the stock price of the firm which issued the bond, and the differential operator in PVI is linear. The optimal call and conversion strategies correspond to the free boundaries of PVI. Some properties of the free boundaries are studied in this paper. We show that the bondholder should convert the bond if and only if the price of the stock is equal to a fixed value, and the firm should call the bond back if and only if the price is equal to a strictly decreasing function of time. Moreover, we prove that the free boundaries are smooth and bounded. Eventually we give some numerical results.

A parabolic variational inequality arising from the valuation of strike reset options

Entrepreneurs often face undiversifiable idiosyncratic risks from their business investment. This paper analyses the joint decisions of investment, consumption/savings and portfolio selection in the models with lump-sum payoffs in finite horizon under incomplete markets. It shows that the optimal exercising boundary is decreasing w.r.t. and continuous in . Moreover, the free boundary is smooth. When the agent can trade a risky asset to partially hedge against investment risk, this ability will make the agent better off relative to the self-insurance setting. The main contribution of this paper is to generalize the results with respect to infinite horizon in Miao and Wang (2007) to the finite horizon case by PDE technique.