Funding, repo and credit inclusive valuation as modified option pricing

We take the holistic approach of computing an OTC claim value that incorporates credit and funding liquidity risks and their interplays, instead of forcing individual price adjustments: CVA, DVA, FVA, KVA. The resulting nonlinear mathematical problem features semilinear PDEs and FBSDEs. We show that for the benchmark vulnerable claim there is an analytical solution, and we express it in terms of the Black-Scholes formula with dividends. This allows for a detailed valuation analysis, stress testing and risk analysis via sensitivities.Comment: 1 figur

Nonlinear valuation with XVAs: two converging approaches

When pricing OTC contracts in the presence of additional risk factors and costs, such as credit risk and funding and collateral costs, the starting “clean price” is modified additively by valuation adjustments (XVAs) that account for each factor or cost in isolation, while seemingly ignoring the combined effects. Instead, risk factors and costs can be jointly accounted for ab initio in the pricing mechanism at the level of cash flows, and this “adjusted cash flow" approach leads to a nonlinear valuation formula. While for practitioners this made more sense because it showed which discount factor is used for which cash flow (recall the multi-curve environment post-crisis), for academics, the focus was on checking that the resulting nonlinear valuation formula is consistent with the theoretical arbitrage-free “replication approach” that we also analyse in the paper. We formulate specific reasonable assumptions, which ensure that the valuation formulae obtained by the two approaches coincide, thus reinforcing both academics’ and practitioners’ confidence in adopting such nonlinear valuation formulae in a multi-curve setup

Analytical valuation of vulnerable derivative contracts with bilateral cash flows under credit, funding and wrong-way risks

We study the problem of valuing a vulnerable derivative with bilateral cash flows between two counterparties in the presence of funding, credit and wrong-way risks, and derive a closed-form valuation formula for an at-the-money (ATM) forward contract as well as a second order approximation for the general case. We posit a model with heterogeneous interest rates and default occurrence and infer a Cauchy problem for the pre-default valuation function of the contract, which includes ab initio any counterparty risk - as opposed to calculating valuation adjustments collectively known as XVA. Under a specific funding policy which linearises the Cauchy problem, we obtain a generic probabilistic representation for the pre-default valuation (Theorem 1). We apply this general framework to the valuation of an equity forward and establish the contract can be expressed as a continuous portfolio of European options with suitably chosen strikes and expiries under a particular probability measure (Theorem 2). Our valuation formula admits a closed-form expression when the forward contract is ATM (Corollary 2) and we derive a second order approximation in moneyness when the contract is close to ATM (Theorem 3). Numerical results of our model show that the forward is more sensitive to funding factors than credit ones, while higher stock funding costs increase sensitivity to credit spreads and wrong-way risk.Comment: 43 pages, 4 figure

CCPs, Central Clearing, CSA, Credit Collateral and Funding Costs Valuation FAQ: Re-hypothecation, CVA, Closeout, Netting, WWR, Gap-Risk, Initial and Variation Margins, Multiple Discount Curves, FVA?

We present a dialogue on Funding Costs and Counterparty Credit Risk modeling, inclusive of collateral, wrong way risk, gap risk and possible Central Clearing implementation through CCPs. This framework is important following the fact that derivatives valuation and risk analysis has moved from exotic derivatives managed on simple single asset classes to simple derivatives embedding the new or previously neglected types of complex and interconnected nonlinear risks we address here. This dialogue is the continuation of the "Counterparty Risk, Collateral and Funding FAQ" by Brigo (2011). In this dialogue we focus more on funding costs for the hedging strategy of a portfolio of trades, on the non-linearities emerging from assuming borrowing and lending rates to be different, on the resulting aggregation-dependent valuation process and its operational challenges, on the implications of the onset of central clearing, on the macro and micro effects on valuation and risk of the onset of CCPs, on initial and variation margins impact on valuation, and on multiple discount curves. Through questions and answers (Q&A) between a senior expert and a junior colleague, and by referring to the growing body of literature on the subject, we present a unified view of valuation (and risk) that takes all such aspects into account