Cutting Edge introduction: Law-abiding FVA

HSBC quant develops an FVA model that preserves the law of one price

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For the past three years, critics have used the law of one price to argue funding valuation adjustment (FVA) – which breaks the cherished rule – should not exist (Risk25, July 2012).

Banks currently estimate FVA, which reflects funding costs arising from hedging of uncollateralised trades, using their own funding spread, or a blend of that and the average funding spreads of other dealers (Risk April 2014 and here).

This results in banks offering firm-specific quotes, demolishing the law of one price, which states that all parties in the market will value a given derivative at the same price. Purists have argued this means there is something wrong with the way FVA is being applied; pragmatists retort that the law of one price should be discarded (Risk September 2012 and here).

In this month's first technical, CVA and FVA with liability-side pricing, Wujiang Lou, a director in the global fixed-income trading team at HSBC in New York, proposes an FVA pricing method that not only retains FVA as a required adjustment to derivatives prices, but also satisfies the law of one price, eliminating what some call the ‘funding arbitrage'. The trick, Lou says, is to use the liability-side funding rate instead of one's own funding rate in calculating FVA.

"The biggest problem with current FVA methods is that banks do not know who is going to step into a trade. The funding cost is supposed to be what you observe in the market, but for derivatives, there is no market for that, so you have all sorts of assumptions on what to use. It is really about taking the derivative's funding cost to be the liability-side cost rather than the formal assumption of using a bank's own funding cost or some average broker-dealer cost," says Lou.

The liability of a derivative is no different from other senior unsecured liabilities as long as your market risk is hedged out

For example, if JP Morgan writes an option for Morgan Stanley, assuming the former's cost of funding is Libor plus 50 basis points and the latter's spread is Libor plus 100bp, according to Lou, Morgan Stanley should use JP Morgan's rate to discount rather than its own cost. This way, in the event that a replacement trade is needed, other banks stepping in would also use JP Morgan's funding rate and would all see the same price for the trade.

"The rationale is you are holding someone's liability, which is of the same rank as holding their corporate bond and that corporate bond is priced at that firm's funding cost. The liability of a derivative is no different from other senior unsecured liabilities as long as your market risk is hedged out," says Lou.

The view has supporters. "It is theoretically appealing that if you have uncollateralised receivables from derivatives and pari passu receivables from cash instruments, such as bonds or loans from the same counterparty, you would price them the same way to the first order," argues Yi Tang, a managing director and head quant at XVA and contingent credit trading team at Wells Fargo Securities in New York.

Under these assumptions, Lou develops a Black-Scholes partial differential equation that prices in both credit valuation adjustment (CVA) and FVA within one framework. This also eliminates the dependency on credit default swap spreads when pricing CVA – an added benefit, says Lou, because liquidity in this market has been shrinking (Risk January 2015).

"Conceptually, CVA and FVA can be combined. Bond yields can be decomposed into two components; credit default risk which is related to CVA, and funding or liquidity basis. These can be combined, because in the end what matters is the final price discounted by a bond curve – so why don't you treat it as one big adjustment rather than taking two separate adjustments?" says HSBC's Lou.

Not everyone is convinced. One head of the quantitative research at a US bank points to some of the potential drawbacks in using another bank's funding curve to price your own trades.

"I'm sure the asset holder would not be happy about paying an increasingly higher rate on the collateral if the liability holder starts getting in trouble. The proposed margin mechanism could exist in theory, but does not in practice. And that is exactly the point of what FVA is about - the deviation from theoretical ideals due to treasury policies and actual real-life funding strategies," he argues.

In our second technical, Efficient XVA management: pricing, hedging and allocation, Andrew Green, head of the CVA/FVA quantitative research team at Lloyds Banking Group in London, and Chris Kenyon a director in the team, propose a comprehensive framework for optimally managing XVAs, including the functions of pricing, hedging and trade-level assignment, using faster calculations and aggregation methodologies.

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