a bank’s weighted average cost of capital.
8. The FVA adjusted price is a Private Valuation.
Laughton and Vaisbrot (2012) countered that Hull and White’s arguments were based on complete markets where all risks can be hedged, while in practice markets are incomplete and this introduces subjectivity into valuations; hence the law of one price no longer holds. Furthermore, Laughton and Vaisbrot argue that the Black-Scholes-Merton model relies upon both no arbitrage and the ability to borrow and lend at the risk-free rate in unlimited size, while in reality there is no deep liquid two-way market in borrowing and lending cash and that apparent arbitrage opportunities are visible in the market because of the practical difficulties in conducting arbitrage. Furthermore Laughton and Vaisbrot suggest that models should be practically useful to traders and that as a result the cost of borrowing is an exogenous factor that is unaffected by a single trade and that no value should be attributed to profit or loss on own default through DVA as it is impossible to monetise.
Antonio Castagna (2012) argued against points 1, 2, 3, 4 and 5 in the list above. Discounting using the risk-free rate may not be appropriate, argues Castagna, as it does not cover the cost of the replication strategy. Hedging does not always involve buying and selling zero cost instruments as the funding rate has to be paid if money is borrowed to purchase an asset. Castagna argues that the Modigliani-Millar theorem does not apply to derivatives. The argument proposed by Hull and White (2012b) that banks invest in low yielding instruments is simply false as these assets are mostly funded through the repo market. Finally, Castagna suggests that FVA cannot be offset by gains or losses through DVA from the fair value of own debt option as any such gains cannot be realised in the event of default.
Massimo Morini (2012) argues that it makes sense for a lender to charge funding costs as it will be left with a carry loss in the event it does not default itself, even if it does not make sense for the borrower. Morini suggests that FVA might be a benefit for shareholders on the basis that the shareholders of a limited liability company are effectively holding a call option on the value of the company. Like Castagna, Morini argues against the use of the Modigliani-Millar theorem but on the basis that Hull and White assume that the market response to the choice of projects undertaken by the company is linear as expressed through funding costs. Morini demonstrates that even in the case of a simple Black-Cox model the market response is nonlinear. Finally, Morini agrees with Hull and White's assertion that risk-free discounting is appropriate in derivative pricing.
What does this debate actually mean in practice? The answer to at least some degree is that quants protest too much when theoretical arguments are challenged by the market. Ultimately mathematics is a tool used in finance to, for example, value products, risk manage them, produce economic models, etc. All models have assumptions and if the market changes and the assumption is no longer valid the models have to change to remain useful. This is not the first time that models have “failed” to some degree. Skew and smile on vanilla options have long been present showing deviations from the Black-Scholes model. This demonstrates that the market believes the distribution of the underlying asset is not log-normal as assumed by the Black-Scholes model. During the credit crisis the Gaussian Copula models used to value CDOs were unable to match market prices. However, it is clear that FVA presents a broader challenge to quantitative finance than previous issues such as volatility smile. The fundamental assumptions underlying much of quantitative finance theory since Black-Scholes are challenged by FVA. Valuation is undergoing a paradigm shift away from these standard assumptions towards models that encompass more realism and away from the simplifying assumptions of the Black-Scholes framework.
Kenyon and Green (2014c) and Kenyon and Green (2014b) demonstrated that in the presence of holding costs the assumptions underlying the Modigliani-Miller theorem are violated. Regulatory costs, through capital requirements, act as holding costs leading to the direct consequence that Modigliani-Miller does not apply to derivatives and hence to FVA. Furthermore it is clear that there is no single risk-neutral measure that spans the market and that no two market participants will agree on price. Hull and White (2012b) maintain that FVA should not be charged; however, the market as a whole has reached the opposite conclusion with numerous banks taking reserves for FVA, including a headline-grabbing figure of $1.5bn by J.P. Morgan in January 2014 (Levine, 2014).
1.4.2 Different Values for Different Purposes
The so-called law of one price argues that the same asset must trade at the same price on all markets or there is an arbitrage opportunity. For example, if gold trades at $X on market A and at $Y on market B and X < Y, in the absence of transport and other cost differentials all trades will take place on market A. The question is whether or not this argument applies to OTC unsecured derivatives markets.
Superficially it would appear that the law of one price should apply to unsecured derivatives if the term sheet for the transaction is the same across multiple banks. However, the terms under which those trades actually take place, that is the ISDA agreement, are frequently different so even in legal terms the trades are different. Add in the counterparty risk of dealing with the banks on an unsecured basis and it should be clear that each deal done with a different bank is different. If two banks offer the same derivative an arbitrager will find it very difficult to arbitrage them by taking opposite positions because of counterparty risk and capital considerations. Most unsecured derivatives are actually traded with corporate customers who use them to hedge balance sheet risks. Often the corporate will use hedge accounting rules and be focused on cash flows while the bank will use mark-to-market accounting. Many of these derivatives will be transacted one way round as corporates use the derivatives as a hedge on a natural risk. For example, many corporate fixed rate loans are structured as a floating rate loan with an interest rate swap in which the corporate receives the floating rate and pays the fixed rate. Derivatives transacted with corporates will also be frequently difficult to novate to a third party, particularly for trades with smaller corporates as they may have few banking relationships or indeed may have only one banking relationship. Other banks, particularly those from outside of the geographical region, may be reluctant to perform the credit analysis and know your client checks necessary to establish a relationship. The law of one price simply does not apply in such circumstances.
A useful analogy to consider when thinking about derivatives is that of the manufacturing industry. Consider manufacturing cars; all cars are designed to carry passengers and luggage but we clearly do not expect them all to cost the same. Cars have different designs and features and these feed into the price. The value of the car will depreciate at different rates after the purchase. The cost of the car is driven by a wide variety of factors including the cost of components, labour costs, transport costs, etc. In general the price will be determined by
Why should derivatives be any different?
What is a Derivative Price?
The price of a derivative is just the price at which the transaction is dealt. For unsecured derivatives the price achieved is the end result of a negotiation process, which lies in sharp contrast to exchange traded derivatives where prices are determined by supply and demand in a very liquid market. As a negotiation both parties will try to reach an agreed price that satisfies the needs and expectations of both parties. Figure 1.2 illustrates the negotiation process.
Figure 1.2 A diagram of a price negotiation between two parties A and B. Both parties have a most favoured price that they would ideally like to transact at and a walk way price below which they will not trade. The agreed price must lie between the most favoured price and walk away price of both parties. If these ranges do not overlap then no agreement is possible.
