英文标题：
《Invariance, existence and uniqueness of solutions of nonlinear valuation
  PDEs and FBSDEs inclusive of credit risk, collateral and funding costs》
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作者：
Damiano Brigo, Marco Francischello, Andrea Pallavicini
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最新提交年份：
2015
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英文摘要：
  We study conditions for existence, uniqueness and invariance of the comprehensive nonlinear valuation equations first introduced in Pallavicini et al (2011). These equations take the form of semilinear PDEs and Forward-Backward Stochastic Differential Equations (FBSDEs). After summarizing the cash flows definitions allowing us to extend valuation to credit risk and default closeout, including collateral margining with possible re-hypothecation, and treasury funding costs, we show how such cash flows, when present-valued in an arbitrage free setting, lead to semi-linear PDEs or more generally to FBSDEs. We provide conditions for existence and uniqueness of such solutions in a viscosity and classical sense, discussing the role of the hedging strategy. We show an invariance theorem stating that even though we start from a risk-neutral valuation approach based on a locally risk-free bank account growing at a risk-free rate, our final valuation equations do not depend on the risk free rate. Indeed, our final semilinear PDE or FBSDEs and their classical or viscosity solutions depend only on contractual, market or treasury rates and we do not need to proxy the risk free rate with a real market rate, since it acts as an instrumental variable. The equations derivations, their numerical solutions, the related XVA valuation adjustments with their overlap, and the invariance result had been analyzed numerically and extended to central clearing and multiple discount curves in a number of previous works, including Pallavicini et al (2011), Pallavicini et al (2012), Brigo et al (2013), Brigo and Pallavicini (2014), and Brigo et al (2014).
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中文摘要：
我们研究了Pallavicini等人（2011）首次提出的综合非线性估值方程的存在性、唯一性和不变性条件。这些方程采用半线性偏微分方程和正倒向随机微分方程（FBSDE）的形式。在总结了现金流定义之后，我们可以将估值扩展到信用风险和违约了结，包括可能再抵押的抵押品保证金和财政部融资成本，我们展示了在无套利环境下进行估值时，这些现金流如何导致半线性偏微分方程，或者更一般地导致FBSDE。我们提供了粘性和经典意义下此类解的存在和唯一性的条件，并讨论了套期保值策略的作用。我们展示了一个不变性定理，表明即使我们从基于本地无风险银行账户以无风险利率增长的风险中性估值方法开始，我们的最终估值方程并不依赖于无风险利率。事实上，我们的最终半线性PDE或FBSDE及其经典或粘性解决方案仅取决于合同利率、市场利率或国债利率，我们不需要将无风险利率与实际市场利率进行代理，因为它充当工具变量。方程推导、数值解、相关的XVA估值调整及其重叠，以及不变性结果已经进行了数值分析，并在许多以前的工作中扩展到中央结算和多个贴现曲线，包括Pallavicini等人（2011年）、Pallavicini等人（2012年）、Brigo等人（2013年）、Brigo和Pallavicini（2014年）以及Brigo等人（2014年）。
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Pricing of Securities        证券定价
分类描述：Valuation and hedging of financial securities, their derivatives, and structured products
金融证券及其衍生产品和结构化产品的估值和套期保值
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