Chris Kenyon

Short-Rate Pricing After the Liquidity and Credit Shocks: Including the Basis

The basis between swaps referencing funded fixings and swaps referencing overnight-collateralized fixings (e.g. 6 month Euribor vs 6 month Eonia) has increased in importance with the 2007-9 liquidity and credit crises. This basis means that new pricing models for fixed income staples like caps, floors and swaptions are required. Recently new formulae have been proposed using market models. Here we present equivalent pricing in a short-rate framework which is important for applications involving credit, like CVA, where this is often useful because default can occur at any time. Furthermore, in this new multiple-curve world, short-rate models are fundamentally altered and we describe these changes.

MVA: Initial Margin Valuation Adjustment by Replication and Regression

Initial margin requirements are becoming an increasingly common feature of derivative markets. However, while the valuation of derivatives under collateralisation (Piterbarg 2010, Piterbarg2012), under counterparty risk with unsecured funding costs (FVA) (Burgard2011, Burgard2011, Burgard2013) and in the presence of regulatory capital (KVA) (Green2014) are established through valuation adjustments, hitherto initial margin has not been considered. This paper further extends the semi-replication framework of (Burgard2013a), itself later extended by (Green2014), to cover the cost of initial margin, leading to Margin Valuation Adjustment (MVA). Initial margin requirements are typically generated through the use of VAR or CVAR models. Given the form of MVA as an integral over the expected initial margin profile this would lead to excessive computational costs if a brute force calculation were to be used. Hence we also propose a computationally efficient approach to the calculation of MVA through the use of regression techniques, Longstaff-Schwartz Augmented Compression (LSAC).

Portfolio KVA: I Theory

XVA models for the calculation of CVA, FVA (see for example (Burgard and Kjaer 2013)), KVA(Green, Kenyon, and Dennis 2014), MVA (Green and Kenyon 2014) and TVA (Kenyon and Green 2014a) have frequently been formulated at the counterparty level. However, it is clear that some elements of the Regulatory Capital Framework are calculated at entity level rather than per counterparty (BCBS-128 2006; BCBS-189 2011) and hence the corresponding KVA term includes elements that are calculated across all counterparties. The Leverage Ratio for example, is clearly a function of all bank positions. Furthermore, in the KVA model of (Green, Kenyon, and Dennis 2014) capital was assumed to be readily available in any quantity at treasury level to be allocated down to individual businesses in return for the payment of a return\gamma_K. In practice, however, the central capital pool cannot be increased or decreased on demand. Additional equity capital is normally obtained through a rights issue, a relatively rare an expensive operation for a company, while equity capital can be returned to shareholders through stock buy backs, although these operations are highly regulated. The cost of capital at portfolio level should depend on the overall capital requirement in some way. Furthermore the act of trading and hedging can no longer be done at single counterparty level. A trade and hedge as a pair will imply two counterparties and capital changes due to both transactions. The hedge attracts capital and hence to hedge to overall portfolio will require an iterative process to neutralise the risk of the whole portfolio including KVA across all counterparty positions. Hence there are entity level aspects of XVA that have hitherto been ignored. This paper rectifies this problem by extending the Burgard-Kjaer semi-replication framework to multiple counterparties and multiple underlying assets, and allowing for entity-level capital requirements and effects. The work is presented in two parts, this paper covering theoretical and algorithmic developments and paper II covering practical examples and impact analysis.

Dirac Processes and Default Risk

We introduce Dirac processes, using Dirac delta functions, for short-rate-type pricing of financial derivatives. Dirac processes add spikes to the existing building blocks of diffusions and jumps. Dirac processes are Generalized Processes, which have not been used directly before because the dollar value of non-Real numbers is meaningless. However, short-rate pricing is based on integrals so Dirac processes are natural. This integration directly implies that jumps are redundant whilst Dirac processes expand expressivity of short-rate approaches. Practically, we demonstrate that Dirac processes enable high implied volatility for CDS swaptions that has been otherwise problematic in hazard rate setups.

Regulatory-Optimal Funding

Funding is a cost to trading desks that they see as an input. Current FVA-related literature reflects this by also taking funding costs as an input, usually constant, and always risk-neutral. However, this funding curve is the output from a Treasury point of view. Treasury must consider Regulatory-required liquidity buffers, and both risk-neutral (Q) and physical measures (P). We describe the Treasury funding problem and optimize against both measures, using the Regulatory requirement as a constraint. We develop theoretically optimal strategies for Q and P, then demonstrate a combined approach in four markets (USD, JPY, EUR, GBP). Since we deal with physical measures we develop appropriate statistical tests, and demonstrate highly significant (p

CDS Pricing Under Basel III: Capital Relief and Default Protection

Basel III introduces new capital charges for CVA. These charges, and the Basel 2.5 default capital charge can be mitigated by CDS. Therefore, to price in the capital relief that CDS contracts provide, we introduce a CDS pricing model with three legs: premium; default protection; and capital relief. If markets are complete, with no CDS bond basis, then CDSs can be replicated by taking short positions in risky floating bonds issued by the reference entity and a riskless bank account. If these conditions do not hold, then it is theoretically possible that the capital relief that CDSs provide may be priced in. Thus our model provides bounds on the CDS-implied hazard rates when markets are incomplete. Under simple assumptions we show that 20% to over 50% of observed CDS spread could be due to priced in capital relief. Given that this is different for IMM and non-IMM banks will we see differential pricing?

Accounting for Derivatives with Initial Margin Under IFRS 13

Initial margin is required by central counterparties (CCPs) and under regulatory Bilateral Initial Margin (BCBS-317). This initial margin is contractually defined in Clearing Arrangements (CA) or Collateral Support Deeds (CSD) respectively. We investigate whether the cost of this initial margin over the life of the affected trades (known as Margin Valuation Adjustment, MVA) should be accounted for in Accounting Fair Value under IFRS 13. We find strong indications that MVA should be reflected in Fair Value given that to exit the affected trades any financial institution taking them over would face the same costs from the trades and additional required contracts (CA or CSD) that the first financial institution already faced. In addition the CA/CSD that give rise to MVA are contractually connected to the related trades and contribute to the overall project costs of the trades. These CA/CSD project costs are analogous to costs from cashflows due to Collateral Support Agreements (CSAs) which have a similar contractual relation to trades and CSAs are routinely included in trade pricing. Whether MVA is reported separately or as part of desk activity depends on the applicable unit of account given the internal setup of the financial institution and specific trading patterns following IFRS 9 and IFRS 13.

Warehousing Credit (CVA) Risk, Capital (KVA) and Tax (TVA) Consequences

Given the limited CDS market it is inevitable that CVA desks will partially warehouse credit risk. Thus realistic CVA pricing must include both warehoused and hedged risks. Furthermore, warehoused risks may produce profits and losses which will be taxable. Paying for capital use, will also generate potentially taxable profits with which to pay shareholder dividends. Here we extend the semi-replication approach in (Burgard and Kjaer 2013; Green and Kenyon 2014) to include partial risk warehousing and tax consequences. In doing so we introduce double-semi-replication, i.e. partial hedging of value jump on counterparty default, and TVA: Tax Valuation Adjustment. We take an expectation approach to hedging open risk and so introduce a market price of counterparty default value jump risk. We show that both risk warehousing and tax are material in a set of interest rate swap examples. “In this world nothing can be said to be certain, except death and taxes”

VAR and ES/CVAR Dependence on Data Cleaning and Data Models: Analysis and Resolution

Historical (Stressed-) Value-at-Risk ((S)VAR), and Expected Shortfall (ES), are widely used risk measures in regulatory capital and Initial Margin, i.e. funding, computations. However, whilst the definitions of VAR and ES are unambiguous, they depend on input distributions that are data-cleaning- and Data-Model-dependent. We quantify the scale of these effects from USD CDS (2004--2014), and from USD interest rates (1989--2014, single-curve setup before 2004, multi-curve setup after 2004), and make two standardisation proposals: for data; and for Data-Models. VAR and ES are required for lifetime portfolio calculations, i.e. collateral calls, which cover a wide range of market states. Hence we need standard, i.e. clean, complete, and common (i.e. identical for all banks), market data also covering this wide range of market states. This data is historically incomplete and not clean hence data standardization is required. Stressed VAR and ES require moving market movements during a past (usually not recent) window to current, and future, market states. All choices (e.g. absolute difference, relative, relative scaled by some function of market states) implicitly define a Data Model for transformation of extreme market moves (recall that 99th percentiles are typical, and the behaviour of the rest is irrelevant). Hence we propose standard Data Models. These are necessary because different banks have different stress windows. Where there is no data, or a requirement for simplicity, we propose standard lookup tables (one per window, etc.). Without this standardization of data and Data Models we demonstrate that VAR and ES are complex derivatives of subjective choices.

Regulatory-Compliant Derivatives Pricing Is Not Risk-Neutral

Regulations impose idiosyncratic capital and funding costs for holding derivatives. Capital requirements are costly because derivatives desks are risky businesses; funding is costly in part because regulations increase the minimum funding tenor. Idiosyncratic costs mean no single measure makes derivatives martingales for all market participants. Hence Regulatory-compliant pricing is not risk-neutral. This has implications for exit prices and mark-to-market.