2015/06/01 by Damiano Brigo, Brigo, Damiano, Marco Francischello +3
Economics, Econometrics and Finance · #35K58 #60H30 #91B70 #Banking stability, regulation, efficiency #Economic theories and models #FOS: Economics and business #Pricing of Securities (q-fin.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1506.00686
openalex publication_date 2015/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study conditions for existence, uniqueness and invariance of the\ncomprehensive nonlinear valuation equations first introduced in Pallavicini et\nal (2011). These equations take the form of semilinear PDEs and\nForward-Backward Stochastic Differential Equations (FBSDEs). After summarizing\nthe cash flows definitions allowing us to extend valuation to credit risk and\ndefault closeout, including collateral margining with possible\nre-hypothecation, and treasury funding costs, we show how such cash flows, when\npresent-valued in an arbitrage free setting, lead to semi-linear PDEs or more\ngenerally to FBSDEs. We provide conditions for existence and uniqueness of such\nsolutions in a viscosity and classical sense, discussing the role of the\nhedging strategy. We show an invariance theorem stating that even though we\nstart from a risk-neutral valuation approach based on a locally risk-free bank\naccount growing at a risk-free rate, our final valuation equations do not\ndepend on the risk free rate. Indeed, our final semilinear PDE or FBSDEs and\ntheir classical or viscosity solutions depend only on contractual, market or\ntreasury rates and we do not need to proxy the risk free rate with a real\nmarket rate, since it acts as an instrumental variable. The equations\nderivations, their numerical solutions, the related XVA valuation adjustments\nwith their overlap, and the invariance result had been analyzed numerically and\nextended to central clearing and multiple discount curves in a number of\nprevious works, including Pallavicini et al (2011), Pallavicini et al (2012),\nBrigo et al (2013), Brigo and Pallavicini (2014), and Brigo et al (2014).\n