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The 2008 credit crisis emphasized the importance of managing counterparty credit risk correctly. One of the ways to mitigate counterparty credit risk is by posting collateral within a derivative trade. Collateral is a borrower’s pledge of specific assets to a lender, to secure repayment of a liability.
For exchange traded derivatives, i.e., stock options, counterparty credit risk is reduced, because the two counterparties in the trade are required to post margins to the exchange. However, this does not fully eliminate counterparty credit risk. Central counterparty clearing houses (CCPs) are responsible for facilitating trades between the two counterparties. CCPs benefit both parties in a transaction, because they bear most of the credit risk; however, CCPs are not risk-free. As a result, counterparty credit risk is present in exchange traded derivatives. This paper focuses on over-the-counter (OTC) traded derivatives.
Mitigation of counterparty credit risk in OTC derivatives requires collateral, but the posting of collateral is not mandatory. The posting of collateral in a derivative trade is regulated by a credit support annex (CSA). A CSA is a contract that documents collateral agreements between counterparties in trading OTC derivative securities. The trade is documented under a standard contract, called a master agreement, developed by the International Swaps and Derivatives Association (ISDA).
Prior to the 2008 credit crisis, the London Interbank Offered Rate (LIBOR) was seen as an acceptable proxy for the risk-free rate. LIBOR is the rate at which banks could freely borrow and lend. The LIBOR rate includes a spread for the credit risk of the banks. An overnight interest rate swap (OIS) is a swap for which the overnight rate is exchanged for a fixed interest rate for a certain tenor (see Hull and White ). An overnight index swap references an overnight rate index, such as the Fed funds rate, as the underlying one for its floating leg, while the fixed leg would be set at an assumed rate.
The difference between LIBOR and the OIS rate is known as the LIBOR-OIS spread. Prior to the 2008 credit crisis, this spread was only a few basis points; it was stable and not significant (see Gregory  (p. 286)).
Hunzinger and Labuschagne  reports that the default of banks, such as Lehman Brothers and Bear Stearns, disproved the myth in the 2008 credit crisis that banks are risk-free. As a result, the LIBOR-OIS spread spiked to hundreds of basis points in the aftermath of the Lehman default in September 2008, and has remained significant ever since. The LIBOR-OIS spread reached 364.42 basis points (see Gregory ). These shifts made it apparent that LIBOR incorporates an adjustment for the credit risk of the banks; therefore, LIBOR is an imperfect proxy for the risk-free rate. The OIS rate appears to be the preferred choice as a proxy for the interest-free rate (see Hull and White ).
The 2008 credit crisis drove home the realization that banks are not risk-free, and banks became unable to borrow at preferential rates. This resulted in banks charging a funding value adjustment (FVA) on transactions. FVA is the difference between the price of the derivative and the collateral posted in the trade, multiplied by the difference between growing this amount at the funding rate and the collateral rate. This amount is then discounted back to the initial time. In essence, FVA is a correction made to the risk-free price of an OTC derivative to account for the funding cost in a financial institution.
The inclusion of FVA in pricing financial instruments is a controversial issue. Hull and White  argue against it. They argue that the funding costs and benefits realized in a trade violate the idea of risk-neutral pricing and should not be included in the pricing of the derivative. Inclusion of FVA into the price of a derivative trade violates the law of one price in the market, because the two counterparties will price a trade differently (also see Hull and White ).
Standard pricing theory ignores the intricacies of the collateralization of the market. The posting of collateral in a derivative trade changes the traditional way in which a derivative is priced. Piterbarg  notes fundamental facts regarding derivative pricing when collateral is posted. Posting collateral in an OTC trade may mitigate counterparty credit risk and funding costs; however, this depends on the collateral posted in the trade and how often this collateral is readjusted according to market movements.
Piterbarg  extends the Black-Scholes-Merton continuous-time model to include collateral in a derivative trade and also shows how the posting of collateral in a derivative trade affects the price. Piterbarg’s paper won him the Quant of the Year Award in 2011.
For more information on FVA and collateral, the reader is referred to Hull and White , Piterbarg  and Laughton and Vaisbrot .
In practice, a continuous-time model is difficult to implement and is usually discretized to facilitate implementation. The aims of this paper are to extend the Cox, Ross and Rubenstein (CRR) discrete-time model to include dividends and collateral, to discretize Piterbarg’s model, which includes collateral and dividends, and to show that the two resulting models coincide. Furthermore, we show that the difference in price of a CSA trade and a non-CSA trade is a funding value adjustment. The paper concludes with a numerical implementation of our results.
The paper is organized as follows: In Section 2, we extend the CRR model to include collateral on a dividend paying stock. In Section 3, we illustrate one of the main results in the paper, which is that the amount of collateral posted within a derivative trade will change the rate used to discount future expected cash flows. In Section 4, Piterbarg’s PDE, which represents the value of a collateralized derivative trade, is presented in a theorem. Next, in Section 5, the Feynman-Kac is applied to Piterbarg’s PDE in order to write the PDE as an expectation. A numerical implementation is provided in Section 6, followed by a concluding section.
2. The CRR Model with Collateral and Dividends
In this section, we extend the CRR model to include collateral on a dividend paying stock. The type of derivative replicated in this model is not specified, and it is assumed that the trade is between two counterparties that cannot default during the life of the trade.
The notation used is as in Piterbarg . The three different interest rates and the dividend yield required are:
the rate paid on collateral,
the rate paid on a repurchase agreement of non-dividend paying stock,
the rate paid on unsecured funding, and
the dividend yield on the stock.
The relationship between the rates is:It is further assumed that in this model, the rates are nominal annual compounded continuous (NACC) and deterministic.
The replicating portfolio, denoted by , is constructed in such a way that the portfolio perfectly replicates the derivative trade at all times and in all states of the world. The quantities constituting the replicating portfolio are:
Δ amount of stock, and
γ amount of cash.
2.1. Cash Account of the CRR Model with Collateral
The cash amount, denoted by γ, can be split into three different accounts, namely a funding account, a repo account and a collateral account:
The amount in the funding account is denoted by . This amount is the difference between two cash amounts, namely the price of the hedging portfolio and the collateral amount. The cash price of the hedging portfolio is denoted by , and the collateral amount is denoted by ; hence . This amount is the amount needed to be borrowed or lent at the unsecured funding rate, . The model makes the assumption that one can borrow or lend cash out at the funding rate without incurring counterparty credit risk.
The amount in the repo account is denoted by . This account consists of the cash invested or borrowed in order to fund the stock position, , through a repurchase agreement.
The amount in the collateral account is denoted by . This account contains the collateral posted with the counterparty. This account grows at the collateral rate . The assumptions made about the posted collateral in this model are that no hair cut is applied to the collateral value and that the currency of the collateral and the settlement of the derivative are the same.
The dividend amount, , is the value of the dividend paid on the stock position. When the stock is placed in a repurchase agreement, then the dividend paid by the stock is received by the counterparty holding the stock position in the repo. The counterparty that places the stock in the repo agreement will lose out on the dividend; however, the counterparty is compensated by only paying the repo rate less the dividend yield on the repurchase agreement. Therefore, the repo rate of a dividend paying stock is . The short rate of a particular interest rate at time is denoted by . This is the interest rate from time up to .
Figure 1 illustrates the three accounts that make up the cash account and the rates that are payable or earned on each account.
Figure 1. Illustration of the various accounts making up the cash account and the rates payable/earned on each.
Figure 1. Illustration of the various accounts making up the cash account and the rates payable/earned on each.
2.2. One Period Binomial Model with Collateral
The price of a derivative trade with collateral in a one period binomial is presented in the following theorem:
The price of a general collateralized derivative trade with the absence of counterparty credit risk in a one-period binomial model is:The conditional expectation operator, denoted by, is conditioned on the stock price at time, where n is a natural number.
If it is assumed that the collateral rate is a proxy for the risk-free rate, then the price of the derivative trade is the sum of the risk-free price and a funding value adjustment,