Vulnerable options pricing under uncertain volatility model

In this paper, we consider the pricing problem of options with counterparty default risks. We study the asymptotic behavior of vulnerable option prices in the worst case scenario under an uncertain volatility model which contains both corporate assets and underlying assets. We propose a method to estimate the price of vulnerable options when the volatility of the underlying assets is within a small interval. By imposing additional conditions on the boundary condition and cutting the obtained Black–Scholes–Barenblatt equation into two Black–Scholes-like equations, we obtain an approximate method for solving the fully nonlinear partial differential equation satisfied by the price of vulnerable options under the uncertain volatility model.

Optimal risk-taking in corporate defined benefit plans under risk-shifting

Purpose The purpose of this paper is to theoretically examine the risk-taking decision of corporate defined benefits (DB) plans. The equity holders’ investment problem that is represented by the position of a vulnerable option is solved. Design/methodology/approach The simple traditional contingent claim approach is applied, which considers only the distributions of corporate cash flow, without the model expansions, such as market imperfections, needed to explain the firms’ behavior for DB plans in previous studies. Findings The authors find that the optimal solution to the equity holders’ DB investment problem is not an extreme corner solution such as 100 percent investment in equity funds as in the literature. Rather, the solution lies in the middle range, as is commonly observed in real-world economies. Originality/value The major value of this study is that it develops a clear mechanism for obtaining an internal solution for the equity holders’ DB investment problem and it provides the understanding that the base for corporate investment behavior for DB plans should incorporate the fact that in some cases the optimal solution is in the middle range. Therefore, the corporate risk-taking behavior of DB plans is harder to identify than the results of the empirical literature have predicted.

Pricing Vulnerable Option under Jump-Diffusion Model with Incomplete Information

In this paper, the closed-form pricing formula for the European vulnerable option with credit risk and jump risk under incomplete information was derived. Noise was introduced to the option writers assets while the underlying asset price and the value of corporation were assumed to follow the jump-diffusion processes. Finally the numerical experiment showed that jumps of underlying assets would increase the value of the option, but noise of corporation value was opposite.

Valuation of the Vulnerable Option Price Based on Mixed Fractional Brownian Motion

The pricing problem of a kind of European vulnerable option was studied. The mixed fractional Brownian motion and the jump process were used to characterize the evolution of stock prices. The closed-form solution to European option pricing was obtained by applying martingale measure transformation method. At the end of this paper, some numerical experiments were adopted to compare the new pricing formula introduced in this paper with the classical Black-Scholes pricing formula. The result showed that the new pricing formula conformed to the actual financial market. In fact, the option value is positively correlated with the underlying asset price and the company’s asset price and the jump process has significant influence on the value of option.

A Binomial Tree Approach to Pricing Vulnerable Option in a Vague World

The aim of this paper is pricing the vulnerable options in a vague world. Due to the vulnerability of financial markets and the economy environment in the real world, investors cannot always have precise information about firm value and default recovery rate in vulnerable option pricing. Therefore, following the framework of Klein in 1996, a fuzzy binomial tree pricing model is derived by modelling the firm value and default recovery rate as fuzzy numbers. The numerical results show that the precise information assumption about the firm value and recovery rate in Klein model may lead to underestimate the credit risk on the values of vulnerable options. This study aims to provide insights for future research on defaultable options pricing under imprecise market information.

The European Vulnerable Option Pricing with Jumps Based on a Mixed Model

In this paper, we combine the reduced-form model with the structural model to discuss the European vulnerable option pricing. We define that the default occurs when the default process jumps or the corporate goes bankrupt. Assuming that the underlying asset follows the jump-diffusion process and the default follows the Vasicek model, we can have the expression of European vulnerable option. Then we use the measure transformation and martingale method to derive the explicit solution of it.

The Pricing of Vulnerable Options in a Fractional Brownian Motion Environment

Under the assumption of the stock price, interest rate, and default intensity obeying the stochastic differential equation driven by fractional Brownian motion, the jump-diffusion model is established for the financial market in fractional Brownian motion setting. With the changes of measures, the traditional pricing method is simplified and the general pricing formula is obtained for the European vulnerable option with stochastic interest rate. At the same time, the explicit expression for it comes into being.