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      Subsequently, the original formulation of Black-Scholes in terms of partial differential equations has been replaced by measure-theoretic probability through the work of Harrison and Kreps (1979), Harrison and Pliska (1983) and Geman, Karoui and Rochet (1995). The Black-Scholes model itself has been steadily adapted to match market prices such as through the use of market implied volatilities that display a skew or smile relative to the single flat volatility assumption of Black-Scholes, implicitly indicating that the stock price distribution has fatter tails than those implied by the log-normal distribution. Nevertheless, the essential framework of risk-neutral valuation through replication has remained largely unchanged even though the mathematical machinery used by quantitative analysts has been enhanced significantly and the computational power available to derivatives businesses has grown exponentially.

      The survival of the model is best illustrated through belief in the law of one price. The law of one price can be stated as follows:

      This simple statement seems very persuasive at first glance. If an asset is quoted at price x on market X and at price y on market Y and x < y then asset buyers acting rationally should that is, this market permits arbitrage.1 However, in the context of derivatives it is not clear that the law of one price always applies:

      1. buy assets from market X if they need to consume the asset

      2. buy assets from market X and sell on market Y to make a riskless profit of (y − x),

      • Over-the-counter (OTC) derivatives trade under bilateral agreements brokered under ISDA rules. The considerable variations in the terms of these legal rules mean that each ISDA (possibly coupled with a CSA) is effectively unique and hence so are the derivatives contracts traded between the two parties to the ISDA agreement.

      • Counterparty risk is always present in practice because even under the strongest CSA terms there will still be delays between movements in portfolio mark-to-market valuations and calls/returns on collateral. Counterparty risk makes each derivative with a different counterparty unique with a distinct valuation. The traditional understanding of the law of one price no longer applies, as there are multiple derivatives with the same basic parameters with different prices. However, the law of one price could be preserved if we consider each pair of counterparties to be a different “market”, although this reduces the “law” to irrelevance. If both counterparties to a trade use a unilateral model of CVA, where only the risk of the counterparty defaulting is considered, neither party will agree on the value of the transaction so the value is asymmetric between the two counterparties. The introduction of bilateral models for counterparty risk and DVA allows symmetry of valuation to be restored,2 but FVA models have again broken the symmetry. The introduction of KVA and the realisation that different institutions have different capital regimes has broken the symmetry irrevocably.

      • Counterparties clearly have asymmetric access to markets.

      • Many derivatives, particularly for large corporates, are transacted on an auction basis. This means that there is one agreed price with the derivative provider that the corporate ultimately selects. However, individual derivative dealers will have different values for the same underlying transaction.

      • Once transacted many corporate derivatives are essentially illiquid. Novations of trades to third parties do occur, but infrequently. If the derivative dealer were to instigate a novation this might threaten the banking relationship. Smaller counterparties will typically have a limited number of banking relationships or perhaps only one banking relationship. Novations would likely prove impossible to do in such cases as with no established banking relationship it is unlikely other derivative providers would have sufficient information on the small counterparty to be able to provide the required credit limits. The deal could only be torn up by agreement with the relationship bank.

      • Counterparties will often transact a derivative under completely different accounting regimes. For example, a corporate may hold a derivative under IRS 39 hedge accounting rules, while the bank counterparty may include the derivative in a trading book under mark-to-market accounting rules.

      The only case where the law of one price might be said to hold is in the case of very liquid exchange-traded derivatives such as futures or exchange-traded options (ETOs). In such cases, given the derivative is entirely commoditised and that margin arrangements are equal for all market participants then the price can be said to be that of the last transaction that took place on the exchange. However, the law of one price has remained persistent in the world of quantitative finance.

      The reality of the derivatives market in the aftermath of the default of Lehman brothers is very different from the idealised one encapsulated in the assumptions underlying the Black-Scholes model and risk-neutral valuation:

      • Apparent arbitrage opportunities sometimes persist in the market. For example the Repurchase Overnight Index Average Rate (RONIA) is frequently higher than the Sterling Overnight Index Average Rate (SONIA) despite the fact RONIA relates to a secured lending market while SONIA relates to unsecured lending. In reality this reflects an inability to close the apparent arbitrage and market segmentation. The repo and OIS swap markets are separate, while constructing a trade which would close the apparent arbitrage is difficult because of the large notional of the offsetting positions that would be required.

      • Banks cannot borrow money unsecured at a hypothetical “risk-free rate” and bank funding costs are significantly higher than any rate considered to be risk free such as an overnight index swap (OIS) rate.

      • Banks cannot borrow money in any quantity. At various points following the collapse of Lehman Brothers, European banks found it difficult to fund themselves in US dollars, prompting central banks to set up large cross-currency swap positions (Lanman and Black, 2011).

      • Short selling regulations (European Parliament and the Council of the European Union, 2012a) have made short selling more difficult to do in practice.

      • The cost of trading derivatives is significant, particularly if KVA is considered.

      The assumptions underlying risk neutral valuation are very clearly at variance with market conditions after the default of Lehman Brothers.3

      This book, therefore, comprises two things. Firstly it is a practical guide to CVA, DVA, FVA and KVA, including both the mathematical models and the implementation of systems to perform the calculations. Secondly, it is a guide to the future of derivative valuation but one that is as yet incomplete as the transition away from risk-neutral valuation to a more realistic valuation framework is not yet complete. This chapter presents a picture of how derivative pricing and valuation has changed since the credit crisis and as such provides an introduction to the book as a whole but one that is naturally coupled with the last chapter on the future of derivatives.

      1.2 Prices and Values

      1.2.1 Before the Fall…

      In 1996 when I first entered the derivatives market, the standard work on derivative pricing was John Hull’s seminal Options, Futures, and Other Derivatives, then in its third edition (1997) and now in its eight edition (2011). For vanilla fixed income derivatives such as interest rate swaps the standard reference was Miron and Swanell’s Pricing and Hedging Swaps (1991). The modelling approach for the valuation of interest rate swaps in 1996 involved the discounting of the future fixed cash flows using a discount function and the floating leg could be readily replicated with two notional cash flows at the start and end of the swap, if there was no margin on the floating leg, or equivalently by discounting the projected forward rate cash flows of the floating leg. There was a single discounting and projection curve for each currency. The yield curve (sometimes known as the swap curve to distinguish it from bond curves) was constructed using a simple bootstrap approach starting with cash deposits, followed by interest rate futures contracts and completed by interest rate swaps. Once the zero-coupon bond prices for

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