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As defaultable options are subject to default risks in over-the-counter (OTC) markets, vulnerable options are one of the financial securities closely linked with the risk caused by the situation that the option issuer may be unable to execute their contractual obligations. In this study, taking account of the changes in stock prices show different degrees of long-term correlation and autocorrelation at different times in the real-world situations of the financial market, contrary to the assumption of Black-Scholes settings, we propose the closed-form pricing formulas for the vulnerable options based on a fractional Brownian environment. First of all, from the model dynamics for the underlying asset under the fractional Brownian motions, we derive partial differential equations (PDEs) for vulnerable option prices, and subsequently derive the closed-form solutions for the options based on the issuer’s credit risk via double Mellin transforms. In addition, we evaluate the accuracy of vulnerable option pricing by comparing the solutions obtained using the aforementioned closed formulas with those obtained via Monte Carlo simulations. Finally, we examine the price sensitivities of vulnerable options in fractional Brownian markets with respect to the model parameters.

In this article, we establish a viscosity method for random homogenization of an obstacle problem with nondivergence structure. We study the asymptotic behavior of the viscosity solution uε{u}_{\varepsilon }of fully nonlinear equations in a perforated domain with the stationary ergodic condition. By capturing the behavior of the homogeneous solution, analyzing the characters of the corresponding obstacle problem, and finding the capacity-like quantity through the construction of appropriate barriers, we prove that the limit profile uuof uε{u}_{\varepsilon }satisfies a homogenized equation without obstacles.

In this article, we consider the parabolic system (ui)t=∇⋅(mUm−1A(∇ui,ui,x,t)+ℬ(ui,x,t)),(1≤i≤k){({u}^{i})}_{t}=\nabla \cdot (m{U}^{m-1}{\mathcal{A}}(\nabla {u}^{i},{u}^{i},x,t)+{\mathcal{ {\mathcal B} }}({u}^{i},x,t)),\hspace{1.0em}(1\le i\le k)in the range of exponents m>n−2nm\gt \frac{n-2}{n}where the diffusion coefficient UUdepends on the components of the solution u=(u1,…,uk){\bf{u}}=({u}^{1},\ldots ,{u}^{k}). Under suitable structure conditions on the vector fields A{\mathcal{A}}and ℬ{\mathcal{ {\mathcal B} }}, we first showed the uniform L∞{L}^{\infty }boundedness of the function UUfor t≥τ>0t\ge \tau \gt 0. We also proved the law of L1{L}^{1}mass conservation and the local continuity of solution u{\bf{u}}. In the last result, all components of the solution u{\bf{u}}have the same modulus of continuity if the ratio between UUand ui{u}^{i}, (1≤i≤k1\le i\le k), is uniformly bounded above and below.

We consider the following one-phase free boundary problem: Find (u, Ω) such that Ω = {u > 0} and $$\cases{u_t=\Delta u & in $\Omega \cap Q_T$ \cr u=0,|\nabla u|=1, & at $\partial \Omega \cap Q_T$\cr u(x,0)=u_o(x) & on $\Omega_o$.}$$ $ u_t=\La u & \mbox{in $\Omega \cap Q_T$}\\ u=0, \ |\D u|=1, & \mbox{at $\partial \Omega \cap Q_T$}\\ u(x,0)=u_o(x) & \text{on $\Omega_o$}.$--> with QT = ℝⁿ × (0, T). Under the condition that Ωo is convex and log uo is concave, we show that the convexity of Ω(t) and the concavity of log u(·, t) are preserved under the flow for 0 ≤ t ≤ T as long as ∂Ω(t) and u on Ω(t) are smooth. As a consequence, we show the existence of a smooth-up-to-the-interface solution, on 0 < t < Tc, with Tc denoting the extinction time of Ω(t). We also provide a new proof of a short-time existence with C^2,α initial data on the general domain. © 2002 John Wiley & Sons, Inc.

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