英文标题：
《Nonlinear Valuation under Collateral, Credit Risk and Funding Costs: A
  Numerical Case Study Extending Black-Scholes》
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作者：
Damiano Brigo, Qing Liu, Andrea Pallavicini, David Sloth
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最新提交年份：
2014
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英文摘要：
  We develop an arbitrage-free framework for consistent valuation of derivative trades with collateralization, counterparty credit gap risk, and funding costs, following the approach first proposed by Pallavicini and co-authors in 2011. Based on the risk-neutral pricing principle, we derive a general pricing equation where Credit, Debit, Liquidity and Funding Valuation Adjustments (CVA, DVA, LVA and FVA) are introduced by simply modifying the payout cash-flows of the deal. Funding costs and specific close-out procedures at default break the bilateral nature of the deal price and render the valuation problem a non-linear and recursive one. CVA and FVA are in general not really additive adjustments, and the risk for double counting is concrete. We introduce a new adjustment, called a Non-linearity Valuation Adjustment (NVA), to address double-counting. The theoretical risk free rate disappears from our final equations. The framework can be tailored also to CCP trading under initial and variation margins, as explained in detail in Brigo and Pallavicini (2014). In particular, we allow for asymmetric collateral and funding rates, replacement close-out and re-hypothecation. The valuation equation takes the form of a backward stochastic differential equation or semi-linear partial differential equation, and can be cast as a set of iterative equations that can be solved by least-squares Monte Carlo. We propose such a simulation algorithm in a case study involving a generalization of the benchmark model of Black and Scholes for option pricing. Our numerical results confirm that funding risk has a non-trivial impact on the deal price, and that double counting matters too. We conclude the article with an analysis of large scale implications of non-linearity of the pricing equations.
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中文摘要：
我们按照帕拉维奇尼和合著者2011年首次提出的方法，开发了一个无套利框架，用于对具有抵押、交易对手信用缺口风险和融资成本的衍生品交易进行一致估值。基于风险中性定价原则，我们推导出了一个通用的定价方程，其中通过简单修改交易的支付现金流，引入了信用、借记、流动性和融资估值调整（CVA、DVA、LVA和FVA）。融资成本和违约时的具体收尾程序打破了交易价格的双边性质，使估值问题成为非线性和递归问题。CVA和FVA通常不是真正的相加调整，重复计算的风险是具体的。我们引入了一种新的调整，称为非线性估值调整（NVA），以解决重复计算问题。理论上的无风险利率从我们的最终方程中消失了。正如Brigo and Pallavicini（2014）所详细解释的那样，该框架也可以根据初始保证金和变动保证金下的CCP交易进行定制。特别是，我们考虑了不对称抵押品和融资利率、置换结清和再抵押。估值方程采用倒向随机微分方程或半线性偏微分方程的形式，可以转换为一组迭代方程，可以通过最小二乘蒙特卡罗法求解。我们在一个案例研究中提出了这样一个模拟算法，该案例研究涉及期权定价的Black和Scholes基准模型的推广。我们的数字结果证实，融资风险对交易价格有着不可忽视的影响，重复计算也很重要。最后，我们分析了定价方程非线性的大规模影响。
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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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