Lishang Jiang

A new well-posed algorithm to recover implied local volatility

Abstract This paper presents a new algorithm to calibrate the option pricing model, i.e. the algorithm that recovers the implied local volatility function from market option prices in the optimal control framework. A unique optimal control is shown to exist. Our algorithm is well-posed. Our numerical experiments show that, with the help of the techniques developed in the field of optimal control, the local volatility function is recovered very well.

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 parabolic variational inequality arising from the valuation of fixed rate mortgages

In this paper a one-dimensional parabolic variational inequality which typically arises in option pricing of fixed rate mortgage loan is studied. The main goal is to study the properties of the free boundary. The monotonicity and

Basket CDS pricing with interacting intensities

C^\infty

Convergence of the Binomial Tree Method for American Options in a Jump-Diffusion Model

smoothness of free boundary are proved and its behavior near expiry is considered as well.

On the rate of convergence of the binomial tree scheme for American options

We propose a factor contagion model for correlated defaults. The model covers the heterogeneous conditionally independent portfolio and the infectious default portfolio as special cases. The model assumes that the hazard rate processes are driven by external common factors as well as defaults of other names in the portfolio. The total hazard construction method is used to derive the joint distribution of default times. The basket CDS rates can be computed analytically for homogeneous contagion portfolios and recursively for general factor contagion portfolios. We extend the results to include the interacting counterparty risk and the stochastic intensity process.

Numerical analysis on binomial tree methods for a jump-diffusion model

The paper studies the binomial tree method for American options in a jump-diffusion model. We employ the theory of viscosity solution to show uniform convergence of the binomial tree method for American options. We also prove existence and convergence of the optimal exercise boundary in the binomial tree approximation. In addition, the terminal value of the optimal exercise boundary is given for American options in jump-diffusion models.

Optimal Convergence Rate of the Binomial Tree Scheme for American Options with Jump Diffusion and Their Free Boundaries

An American put option can be modelled as a variational inequality. With a penalization approximation to this variational inequality, the convergence rate

INTENSITY-BASED MODELS FOR PRICING MORTGAGE-BACKED SECURITIES WITH REPAYMENT RISK UNDER A CIR PROCESS

O\big((\Delta x)^{2/3}\big)