Funding Value Adjustments for Uncollateralized Derivatives

# What is meant by the funding cost of a derivative?


# What is meant by the funding cost of a derivative?












Numerous sources refer to the 'funding cost' of a derivative.

I'm confused as to exactly what cost is being referred to here. To illustrate my confusion, consider purchasing an uncollateralised OTC gold forward at market (with value of 0). I have not needed to fund anything. Now consider that forward going into the money with some positive PV. I still have not funded anything (and will not need to fund anything), and I will collect my payout at maturity. Similarly instead consider that forward going out of the money with some negative PV. I will not have to fund anything (and will not need to fund anything), and I will pay my counterpart at maturity.

Some of the aforementioned references to 'funding costs':

> We agree that in the case of a receivable, there is a funding cost C, and in the case of a payable, there is a funding benefit B https://quant.stackexchange.com/a/71965/

> FVA attempts to capture the cost of funding uncollateralised OTC derivatives. ... Similarly, a funding cost arises for the bank when a derivative has a positive market value. The purchase of an ‘in the money’ or asset position derivative requires the bank to pay cash. The incremental cost of funding this purchase can also be seen as equivalent to the cost of the bank raising funding https://www.pwc.com.au/pdf/xva-explained.pdf

(note that my hypothetical is different to this PWC scenario since the derivative was 'purchased' at zero cost, whereas the PWC scenario considers purchasing an ITM derivative).

> This article does not cover the cost of funding derivatives positions https://www.bankofengland.co.uk/-/media/boe/files/quarterly-bulletin/2014/bank-funding-costs-what-are-they-what-determines-them-and-why-do-they-matter.pdf

> Traders want to incorporate a funding value adjustment (FVA) in valuations to reflect the funding costs https://www.researchgate.net/publication/256057127_Valuing_Derivatives_Funding_Value_Adjustments_and_Fair_Value

## Answer by AKdemy (score 9)

https://quant.stackexchange.com/a/76335

The theory for pricing derivatives is based on self-financing trading strategies that replicate all the payoffs of the derivative. Hence, derivatives pricing requires funding at the risk free rate, even leaving FVA out of the picture.

WIth regards to FVA, there is considerable debate about this subject. You can look at various sources like

- The FVA debate, John Hull and Alan White, August 2012, Risk Magazine,

- In the Balance, 2013, Christoph Burgard and Mats Kjaer, 2012, Risk Magazine

- Funding, Liquidity, Credit and Counterparty Risk: Links and Implications, July 2011, Antonio Castagna, iason ltd

Burgard and Kjaer as well as Hull and White make a similar point, albeit using different arguments. They both argue that: "FVA should not be considered when determining the value of the derivatives portfolio, and it should not be considered when determining the prices the dealer should charge when buying or selling derivatives"

The central argument is that the use of a risk-free discount rate indicates the valuation is only appropriate when the bank can fund the derivative at the risk-free rate. The authors list a few reasons why the risk free rate is appropriate:

- interest rate paid on dollar cash collateral is frequently based on the effective federal funds rate (or equivalent OIS rate or SOFR rate more recently)

- Risk free is a requirement of the risk-neutral valuation principle

- trades in hedging instruments involve buying or selling assets for their market prices and are, therefore, zero net present value investments

- a well-established principle in corporate finance theory is that pricing should be kept separate from funding

Ignoring for a moment that SOFR in itself is a Secured Overnight Financing Rate (hence funding / borrowing), there are different practices and no consensus on how funding costs are handled I would say. One thing to consider though, which is strongly in favour of looking at additional funding / borrowing costs is that computing reliable Implied Vol surfaces in Equity markets requires taking borrow costs into consideration. Otherwise, you end up with a surface that does not work for both calls and puts as mentioned here. Also, when you try to match many OTC derivatives or structured products (say in Bloomberg's DLIB) you will have to add a Funding Spread (which is a direct input on the main screen of the pricing engine) to match market quotes.

This pins down to what @nbbo2 wrote, namely that dealers /banks face funding issues for various reasons, even if you as a retail trader or treasury desk may not.

## Answer by dm63 (score 7)

https://quant.stackexchange.com/a/76346

Ok so let’s say you purchased an uncollat gold forward at zero, and it went in the money (positive PV to you). I can tell you that if you are at a bank , they will charge you funding cost on your profit at a rate equal to the short term borrowing rate of the bank (say, SOFR + a spread in the US). Why? There are two possibilities : (a) did you hedge the trade using an interbank gold forward or a listed gold futures contract ? If so, that trade now has a negative PV and it is subject to daily margin, so you did actually use funding on the trade + hedge. This is typically what they assume has happened and this is why they charge funding. Then the more unlikely (b) you did not hedge the trade. Congrats, you made a pure profit which shows up as an asset on the bank balance sheet, offset by an increase in equity value of the bank. You are now using the resources of the bank to maintain your otc gold receivable, for which they need to charge you a cost of funding.

That is how the FVA charge arises if a bank writes a uncollateralized deivative. The other answers I mostly agree with , but perhaps are answering a different question. For example @nbbo2 answers why cost of funding gold is required when calculating the price of a gold forward. That is an entirely different question (specifically , the cost of funding gold is needed to calculate the gold forward price even if the contract is fully collateralized). FVA arises mostly on uncollateralized trades.

## Answer by nbbo2 (score 5)

https://quant.stackexchange.com/a/76345

I will just give a general overview.

When we talk about Funding Cost we are usually referring to the dealer or Bank that takes the other side of your derivative transaction. They have to "use their balance sheet" to hedge the trade and/or may have to set aside some capital for regulatory purposes. Either way they need to charge something for the use of capital. (In the case of a Forward they have to buy the asset and store it until delivery, for example, so they need to borrow some cash internally from some department of the Bank to do this.).

Funding cost is an important issue in Bank Management, less relevant to users of derivatives like you and me. How to do it right, without unintentionally penalizing or subsidising some department/activity, while keeping the accounting reasonably simple is a non-trivial problem IMO.

## Answer by fes (score 3)

https://quant.stackexchange.com/a/76348

Here is a simplified demonstration of the so called FVA controversy as discussed on Burgard and Kjaer 12 and Andersen et al. 18 and their proposed solution.

Assume a bank writes a derivative to a client (uncollateralized). Typically it would also purchase a (delta) hedging portfolio for this derivative. Since the financial crisis, it is not anymore realistic that this hedging portfolio could be financed at the risk free rate. Rather the funding cost includes an excess spread, which reflects the bank's default risk.

On one hand it appears this spread is a cost that should be taken into account when pricing the derivative. However, several authors have pointed that this hedging transaction is not destroying bank balance sheet value. Moreover, this would lead to issuer dependent pricing and a potential divergence in book and market prices.

Both papers argue that the way to view this controversy is that purchasing the hedging portfolio is a value transfer from equity to debt holders. This is because the funding spread reduces profits and shareholder value. At the same time, in case of default the creditors can seize additional replicating assets and are better off. They argue that an FVA or similar adjustment can be used to align incentives between debt and equity holders.

## Answer by Moomin (score 0)

https://quant.stackexchange.com/a/85624

From an accounting perspective, fair value is an exit price, meaning you must value the derivative based on what a market participant would demand to assume the position. Even if your uncollateralized forward requires no direct cash outlay, an acquiring dealer will incur real funding costs to maintain the collateralized hedges needed to risk-manage it. Therefore, FVA reflects this market-implied burden, adjusting the theoretical risk-free valuation to account for the unsecured funding rates a buyer would apply to your future cash flows.