1

Interest Rate Derivative Pricing when Banks are Risky and Markets are Illiquid Geoffrey R. Harris1

and Tao L. Wu

Stuart School of Business, Illinois Institute of Technology Third and Current Draft: May 17, 2011

Abstract: We examine the relative values of interest rate derivative contracts and cash LIBOR, particularly during the credit crisis from 2007 until present. There are substantial deviations from the results predicted by standard arbitrage pricing theory. We analyze the historical behavior of these deviations and their relationship to market measures of credit risk and liquidity. We introduce four different stochastic models of credit risk and liquidity, which make specific predictions of how these pricing deviations depend on the maturity of the contracts and the tenor of the underlying LIBOR rates. We then compare these predictions to empirical observations of the relative values of different forward rate agreements. JEL Classification: G12, G13, G01. Keywords: Interest Rate Derivatives, Interbank Markets, Eurodollar Futures, Forward Rate Agreements, Credit Risk, Liquidity Risk, Credit Crisis, Financial Crises.

1 Corresponding Author. Department of Finance, Stuart School of Business, Illinois Institute of Technology. 565 W. Adams St., Chicago, IL 60661. Tel. (312)-906-6533. Email: gharris@stuart.iit.edu. This research was supported by funds from the Stuart School of Business, Illinois Institute of Technology. We would like to acknowledge fruitful and helpful correspondence and discussions with Tom Bielecki, John Bilson, John Dean, Jonathan Harris, Tom Jacobs, Andrew Johnson and Xuan Zhou.

2

1. Introduction: Underlying the standard methods of pricing derivative contracts is the assumption that

market participants can borrow and lend in unlimited amounts at the risk-free rate. The most frequently traded interest rate derivatives have payoffs that depend on the inter-bank borrowing rate (LIBOR). Traditionally, the pricing and risk management of these interest rate derivatives have assumed that this rate is ‘risk-free’.

Since the advent of the financial crisis that began in 2007, these assumptions have been violated to a considerable degree. In this work, we explore the implications of this for the relative valuation of simple interest-rate derivative contracts that depend on interbank lending rates.

This paper is organized as follows. We begin by providing more detail about the theoretical context of this study and its relationship to previous literature. Next, we present intuitive arguments that illustrate how credit risk and illiquidity leads to the breakdown of basic relationships among different forward rates. We document that these relationships held until the start of the credit crisis in August 2007, but were then subsequently violated, leading to relative mispricings of interest rate derivative contracts. We shall refer to these mispricings as ‘anomalies’. We perform an empirical regression analysis, illustrating how these anomalies depend on market variables. To assess whether bid-ask spreads can explain the size of the mispricings, we examine market data for interbank loans traded on an exchange. We then introduce more formal models of credit and liquidity risk that can explain these mispricings, and show how these models imply differences in the behavior of the term structure of these pricing anomalies.

Derivative agreements are priced by appealing to arguments based on the absence of arbitrage. Assume that two parties engage in a derivative transaction, i.e. a contract that depends on the value of an underlying asset, such as a stock or bond. Both parties can then hedge the impact of this derivative through a strategy that entails buying or selling other derivatives or the underlying asset. If the hedge is perfect, the market risk incurred by taking on the derivative position is eliminated. The hedged portfolio, by absence of arbitrage, should have a return equal to the risk-free rate. This constraint enforces a relationship between the values of the derivative and the instruments that are used as a hedge.

The parties must borrow or lend money to establish the hedge position. The price of the derivative then depends on the borrowing and lending rates, the price of the underlying asset, and the parameters that determine the hedging strategy (e.g. the volatility of the asset). Until recently, it has been nearly universal practice, when pricing derivatives, to determine borrowing and lending rates from yield curves based on LIBOR (London Interbank Offer Rate) for a given currency and from the prices of derivative contracts based on these LIBORs, namely Eurodollar Futures contracts and interest rate swaps. On each business day, at 11 a.m. London time, the British Bankers Association (BBA) polls a set of banks, asking them to provide their best estimate of the rates which they would pay to borrow an amount of substantial size from other peer banks for terms of 1 day, 1 week, 2 weeks, one month, two months, and monthly up to one year. The USD LIBOR panel consists of sixteen banks. The LIBOR rate for each term is then computed by discarding the highest and lowest quartile of these quotes and averaging the remaining quotes.

3

The interest rate used in derivative pricing is often referred to as the ‘risk-free’ rate. In the traditional arbitrage arguments used to price derivatives, it is implicitly assumed that no default will occur in the transactions carried out as part of the hedging strategy. If a default were to occur, the hedge could incur substantial credit losses, and its proceeds would not provide a perfect hedge for the derivative payoff. Furthermore, the hedging strategy assumes that the hedger not only can finance at the LIBOR rate today, but will be able to finance at this rate in the future. This might not be the case if their credit quality were to deteriorate significantly.

When the credit quality of major financial institutions is strong, LIBOR is at least quite close to the risk-free rate, and the risk of default on a LIBOR-based loan is very small. A substantial portion of the institutions that transact large volumes of derivatives, such as J.P. Morgan and Citigroup, can indeed finance at these rates, and it is therefore reasonable to use LIBOR as the rate used in arbitrage arguments. When the risk of default of these institutions is small, the assumption that LIBOR is risk-free should not lead to substantial mispricing of derivatives. From August 2007 on through 2009, however, banks were under severe financial distress, and their perceived risk of default was significant. During a financial crisis such as this, lending at LIBOR, in carrying out a trading strategy, does expose an arbitrageur to significant credit risk. Furthermore, there is considerable uncertainty that financial institutions that could borrow at LIBOR would be able to continue to do so in the future.

Hedging arguments used to derive arbitrage arguments will break down if the interbank lending market becomes highly illiquid, when it is difficult and to execute or unwind hedges, or if doing so entails large transaction costs. Lack of liquidity may also constrain the amount that an institution can borrow on the interbank market. This violates the assumption of the availability of unlimited borrowing implicit in arbitrage arguments.

It is difficult to disentangle the effects of illiquidity and credit risk. This is, in part, because the two factors are strongly correlated. Market illiquidity spikes up when perceptions of credit risk increase; securities issued by more credit risky entities tend to be less liquid. There is a significant body of research that addresses the relevant impact of credit risk and liquidity on bonds and default swaps; see Lesmond, Chen and Wei (2005), Longstaff, Mithal and Neis (2005) and Ericsson and Renault (2006). Most of these studies conclude that the yield premium for illiquidity is significant, and of the same order of magnitude as the premium for credit risk. Recent work by Schwarz (2009) has also demonstrated large differences in yields between instruments with essentially identical credit risk, but differences in liquidity. It therefore seems plausible that these anomalies can only be fully explained with models of both credit and liquidity risk. Note, however, that the standard models used to price and hedge credit risky instruments are developed by characterizing the impact of losses due to default and not explicitly modeling liquidity risk. They may be adequate for pricing and risk management, if the impact of credit and liquidity risk is approximately of the same form. In this case, the impact of liquidity together with default risk would simply be reflected by the values of the model parameters, which are typically obtained via calibration to market prices.

In this paper, although we study anomalies that violate standard arbitrage pricing arguments for interest rate derivatives, we still work within the framework of arbitrage pricing theory. This is valid if the additional sources of credit and liquidity risk can be hedged; i.e. if markets are complete. Such hedging could be carried out with credit derivatives and basis swaps, though it is not guaranteed that these will in practice succeed as effective hedges for the credit

4

and liquidity risk factors that affect the valuation of interest rate derivatives. Particularly in a crisis period with high illiquidity, the effectiveness of these hedges may be reduced; they may be less liquid and the assumption of market completeness might be strongly violated. In this case, arbitrage pricing models might not work well. One is confronted with a complex incomplete markets problem, which has been approached in many different ways in the literature, many of which are reviewed by Staum (2007). These approaches often lead to bounds rather than exact prices2

Financial institutions compute adjustments to the prices of their derivative contracts due to the possibility that the counterparties in these contracts may default; see Duffie and Huang (1999), Pykhtin (2005), Canabarro (2010) and Morini and Prampolini (2010). These adjustments, often known as CVA, an acronym for credit valuation adjustment, are essentially nil for derivatives traded on exchanges, such as Eurodollar futures. They can have a significant value for over-the-counter (OTC) derivatives, such as swaps

. Much of the analysis in this paper is based on market data from 2009 and early 2010, during which markets in basis swaps and default swaps were liquid, so that the assumption of market completeness seems a plausible approximation.

3. For OTC contracts, the size of these adjustments is reduced considerably through netting agreements, which stipulate that, for purposes of determining obligations in the event of counterparty bankruptcy, the value of all derivatives between a pair of counterparties is aggregated4

. The bulk of over-the-counter derivative transactions are collateralized; this significantly reduces the impact of counterparty credit risk on pricing. The impact of collateralization on derivatives pricing has been explored by Johannes and Sundaresan (2003). More recently, significant progress by Piterbarg (2010), Burgard and Kjaer (2010), Fujii and Takahashi (2010) and Bielecki (2011) and Bielecki, Cialenco and Iyigunler (2011) has been made in modeling the dependence of derivative prices on funding rates and the structure of collateralization agreements.

The analysis of this paper complements the work referenced above on derivative counterparty risk. Although counterparty credit risk and collateralization are of considerable significance in the pricing of derivative securities, particularly during the last few years, we shall argue that it has a minimal role in explaining the anomalies in interest rate derivatives that are the focus of this paper. This is primarily because the anomalies we discuss depend on relatively short maturity instruments which are struck with initial value zero (e.g. par swaps and forward rate agreements struck at the current forward rate). These anomalies are largely a consequence of the breakdown of arbitrage arguments connecting the interbank lending and derivatives market, due

2 Bergman (1995) works out bounds for prices of equity options, given that in hedging, one borrows and lends at different rates. This analysis can be extended to interest rate derivatives, and is quite simple for swaps and FRAs. The resulting bounds, however, are quite wide. 3 The aggregate amount of derivatives transacted by large banks is so large that the total CVA is considerable. For instance, JP Morgan Chase reports the mark-to-market change of their CVA annually; in 2008, this was -2.3 Billion and in 2009 this was 1.9 Billion (see the J. P. Morgan Chase Annual Report (2009)). 4 When netting is legally enforced (as is usually the case), the aggregate CVA for a set of contracts is typically much smaller than the sum of the CVAs computed if each agreement were considered to be in isolation. This is because the different agreements may have offsetting values.

5

to the uncertainty of whether an institution that can currently fund at LIBOR will be able to do so in the future. This is of relevance to derivatives with payoffs that depend on LIBOR, i.e. most interest rate derivatives, which constitute a large portion of the over-the-counter derivatives market.

This work also extends a strand of literature that addressed the relative pricing of cash LIBOR rates and Eurodollar/Euribor futures and of swaps and Eurodollar futures; see Sundaresan (1991), Grinblatt and Jegadeesh (1995), Gupta and Subrahmanyam (2000) and Chance (2006). Differences in pricing between rates inferred from related cash and derivative contracts, such as cash LIBOR rates and Eurodollar/Euribor futures, can also be explained by slight differences in the payoffs of these contracts. For instance, the non-linearity in the dependence of Eurodollar futures contracts on the price of the underlying debt, and the differences in timing of cash flows between futures and forward contracts are the source of ‘convexity corrections’ in the difference between futures and forwards rates. These convexity corrections depend on the volatility of interest rates, and have a strong increasing dependence on contract maturity. Grinblatt and Jegadeesh (1995) argued that credit risk did not explain the relative pricing of Eurodollar futures and cash forward rates. In fact, in their study, the effect of credit risk would have led to relative pricing of the sign opposite to that observed in the data. Their empirical observations of the futures-forwards basis, however, did provide evidence of the presence of convexity corrections. In our analysis, we find the opposite sign in the futures-forwards basis, as the maturities of our contracts are short, so that the convexity correction is negligible, and because our analysis is performed over a period in which credit and liquidity risk are elevated.

2. The Anomalies and their Consequences We now explain how credit and liquidity effects lead to the failure of the ‘composition’

relationship, an identity that is used frequently in fixed income calculations and particularly when bootstrapping yield curves. In later sections of this paper, we will introduce formal models that explain this behavior. In this section, we present intuitive arguments that illustrate how credit and liquidity effects lead to modifications of the traditional arbitrage arguments.

The standard practice in building yield curves based on LIBOR is to infer LIBOR yields from the prices of Eurodollar futures and swaps. The rates that underlie Eurodollar futures (and many swaps) are 3-month LIBOR. From the yield curves that are constructed, one then infers forward rates. These forward rates may span maturities other than 3 months, so that, for instance, the forward value of 6-month LIBOR can be inferred from these yield curves, and one can then price instruments that depend on 6-month LIBOR. One consequence of the pricing anomalies that we observe is that this procedure no longer works well; e.g. one cannot exactly price derivatives based on 6-month LIBOR using standard yield curve construction techniques applied to 3-month LIBOR rates.

The standard relationship between forwards of different maturities can be derived by imposing absence of arbitrage on a forward agreement plus its hedge. Consider a bank that enters into a forward agreement at time t, which requires it to pay a fixed payment R(t,S,S+T)T at time S+T, in exchange for receiving a floating rate payment equal to f(S,S, S+T)T at time S+T 5

5 This agreement is not quite identical to a standard Forward Rate Agreement (FRA), which is studied in the remainder of the paper. The interest payments in a standard FRA are discounted and exchanged in arrears, in this case, at time S rather than S + T.

.

6

f(t,S,S+T) denotes the forward rate for borrowing at the LIBOR rate from time S to S+T, as evaluated at time t. The value of this forward agreement changes as interest rates move, but this can be hedged if the bank

1. Deposits 1/(1+ f(t,t,S)S) for a term S+T at time 0, earning interest at a rate of f(t,t,S+T). 2. Borrows 1/(1+ f(t,t,S)S) for a term S, paying interest at a rate of f(t,t,S). 3. Rolls over this loan at time S for an additional term T, at a rate of f(S, S, S+T).

The net cash flows exchanged in the forward plus hedge are independent of the value of f(S, S, S+T); they are known with certainty at time t. The value of R(t,S,S+T) for which the forward agreement has value zero at time t is (by definition) the forward rate f(t,S,S+T). By absence of arbitrage, this value is determined by the requirement that the net cash flows at time S+T vanish. It follows that f(t,S,S+T) satisfies

(1) ));,,(360

1()),,(360

1))(,,(360

1( TSttfddTSStfddSttfdd tTSSTStS +−

+=+−

+−

+ ++

dx represents the number of days until time x. We refer to this expression as the composition relationship; it relates the value of forwards of different maturities and is an essential tool in yield curve bootstrapping.

The arbitrage argument assumes that the bank will be able to borrow at LIBOR at time S. However, a bank that can borrow at LIBOR at an earlier time t, may be unable to borrow at LIBOR at a future time S > t. LIBOR is a rate with refreshed credit quality, indicative of the rate paid by a high-credit quality bank, such as a bank on the LIBOR panel. If an issuer pays LIBOR now, it is more likely that their credit quality will deteriorate than improve; this is because credit risk is mean-reverting and credit quality has a tendency to revert to a mean level which is worse than that of a typical LIBOR-paying bank.

As the hedge may underperform in the event of credit deterioration, the bank should have to pay a forward rate lower than the rate computed via the idealized arbitrage argument. This suggests that

(2) )).,,(360

1()),,(360

1))(,,(360

1( TSttfdd

TSStfdd

Sttfdd tTSSTStS +

−+<+

−+

−+ ++

In previous literature, most notably the work of Collin-Dufresne and Solnik (2001), LIBOR has been modeled as a rate with refreshed credit quality and the effect of credit deterioration of LIBOR-quality counterparties on asset prices has been analyzed. Collin-Dufresne and Solnik demonstrated how these models could explain the relative values of long-dated securities, comparing swap rates and AA bond yields. In this paper, we examine a related phenomenon, applied to the relative pricing of short-dated derivative and cash securities. In the work of Collin-Dufresne and Solnik (2001), the risk of credit deterioration is treated as idiosyncratic; individual credit spreads can ‘jump’, but not in conjunction with moves in LIBOR, which simply diffuses. In the recent crisis, much of the risk has been systemic rather than idiosyncratic.

Underlying the arbitrage argument is the assumption that financial institutions can borrow unlimited amounts at their funding rate. With limited market liquidity, this may not be the case, and to borrow beyond a certain amount, they may have to pay interest rates beyond their base funding rate. If the amount that they need to borrow is extremely high, or liquidity very limited, they may simply be unable to borrow, or their marginal borrowing rates may be so

7

punitive that it may be preferable for them to raise cash by liquidating assets. Particularly in an environment in which funding is limited, they will incur a loss liquidating these assets6

This is the risk that institutions face in the advent of a liquidity crisis. It is sometimes termed rollover risk and is common to all financial institutions, as they rely on short-term funding to purchase long-maturity assets. The hedge to the forward agreement is an archetype of the aggregate set of transactions executed by financial institutions, as it entails funding a longer-term deposit with two shorter-term loans. When a liquidity crisis strikes, institutions hoard liquidity, and they incur a cost for engaging in transactions that require short-term funding

.

7

Note that if one executes the reverse of the forward agreement, then the hedge entails a deposit that is to be rolled over at time S; when hedging the bank uses long-term funding to buy shorter-dated assets. There is a benefit to having the deposit mature at the intermediate time. If a liquidity crisis in underway at that point, the maturing deposit can be used to replenish cash on hand, and this will reduce the aggregate amount of distressed assets that the institution will have to sell at a loss.

, such as the hedge to a forward agreement. To the extent that a crisis will increase funding costs for all banks, this risk will lead to an increase in forward LIBOR, pushing up the slope of the term structure. The potential for an individual institution to have marginal funding costs above LIBOR, or to incur costs in liquidating assets in order to service debt, however, will lead to violations of the composition relationship, as expressed in equation (2).

The violations of the composition identity that we have discussed are due to a combination of credit and liquidity risk that cannot be easily disentangled, as an institution’s funding rate depends both on its credit quality and the market liquidity for its debt. There are, however, further consequences of limited liquidity in the interbank market, such as greater bid-ask spreads.

In the discussion in this paper, we still assume that Eurodollar futures, swaps (and FRAs) remain liquid relative to the interbank lending market8

6 The connection between illiquidity of assets (market illiquidity) and constraints on funding and capital (funding illiquidity) has been explored, for instance, by Brunnermeier and Pedersen (2009). These authors develop a model of liquidity spirals, in which market and funding illiquidity reinforce and amplify each other. Market and funding liquidity are also referred to outside and inside liquidity; see Bolton, Santos and Scheinkman (2009).

. That is, illiquidity affects the relative prices of these instruments by virtue of its presence in the underlying interbank lending market, and illiquidity in this market is far more severe and consequential than illiquidity in the market for derivatives. Illiquidity in the interbank market has the consequence of increasing bid-ask spreads; this can have a significant impact on the validity of the composition relationship. The bank earns the bid rate on the deposit, but pays the offer rate when borrowing. When executing the hedge to the forward agreement, the bank would then have a shortfall equal to the value of the bid-ask spread (determined at t = 0), accrued over the lifetime of the transaction. Absence of arbitrage then implies that

7 Rollover risk can also accelerate or catalyze bankruptcy. He and Xiong (2010) incorporate rollover risk into a structural model of default to capture this effect. 8 The swaps and Eurodollar futures markets are highly liquid. In other markets, such as the treasury market, illiquidity has a significant impact on valuation. Liquidity effects are relevant when constructing yield curves from treasury securities, as some treasury notes (e.g. newly issued notes) trade at a premium due to high demand and limited supply (see, for instance, Brandt and Kavajecz (2004)).

8

(3)

))0((360

)),0,0(360

1()),,0(360

1))(,0,0(360

1( spreadaskbiddTSfdTSSfddSfd TSTSSTSS −+++<+−

++ +++

One can reverse the argument, so that the hedge requires borrowing over time S + T, and a deposit that is rolled over at time S. In this case, one obtains a lower bound for the LHS of the above equation, but this bound depends on the bid-ask spread at time S (which is not known at time 0). Note that the presence of a bid-ask spread is consistent with the violation of the composition identity; however unlike the prior argument appealing to credit risk, it does not explain the direction of the violation.

Beginning in 2009, several working papers proposed modifications of traditional bootstrapping techniques and pricing formulas so that they will work in the market conditions during the credit crisis9

This new framework for bootstrapping yield curves and pricing interest rate derivatives has several distinct shortcomings. First of all, the requirement that one model a set of forward curves (one per spanning tenor) rather than a single forward curve makes fixed income valuation and risk management computationally far more complex. It is more difficult to design a hedging strategy with a minimal set of hedging instruments. Furthermore, there is limited market information available to construct these curves. In the U.S. market, Eurodollar futures contracts provide a great deal of information that can be used to construct the forward curve with a spanning tenor of 3 months. Some information for other spanning tenors (e.g. 6 months) can be inferred from FRAs and swaps (with floating rate frequencies of twice per year), but there are a limited number of these instruments with sufficient liquidity.

. They introduce a separate discounting curve to present-value cash flows, as well as a series of separate forward curves, one for each tenor of LIBOR (e.g. one month, three month, six month, etc...) that loosely correspond to market FRA rates. The analytics are similar to those proposed by Kijima, Tanaka and Wong (2009) to price cross-currency swaps with basis risk, i.e. relative credit and liquidity risk between borrowing in different currencies. Mercurio (2009) provides a theoretical justification for this procedure. These authors also address the pricing of interest rate derivatives given multiple curves based on different LIBORs, using, as in Mercurio (2009), a market model approach that works with separate diffusions for LIBOR rates of each tenor. None of these papers, however, works out an underlying credit or liquidity model that ties together the different forward curves.

In this situation, it would be advantageous to develop models that relate forwards of all spanning tenors. Their relationship would be more complex than the composition relationship that follows from traditional arbitrage arguments, but has failed during the credit crisis. It would depend on model parameters that capture the credit-riskiness of the market participants that trade derivatives. One could then calibrate these model parameters using market data about FRAs and swaps with only a few different spanning tenors, along with Eurodollar futures and possibly cash LIBOR. These models would also guide one in developing strategies to hedge interest rate risk with a more limited set of hedging instruments, e.g. Eurodollar Futures and OIS-LIBOR basis swaps. Developing and analyzing underlying models of credit and liquidity that explain the

9 See Mercurio (2009), Morini (2009), Ametrano and Bianchetti (2009), Bianchetti (2010), Kenyon (2010), Fujii, Shimada and Takahashi (2010) and Pallavicini and Tarenghi (2010) and Andersen and Piterbarg (2010).

9

relative values of fixed income instruments during the credit crisis is one of the primary goals of this paper.

3. Inconsistencies in the Pricing of Forward Rate Agreements and Cash LIBOR rates. Forward Rate Agreements (FRAs) and Eurodollar Futures are derivatives that are used to

lock-in one-period lending rates. Eurodollar Futures contracts depend solely on three-month borrowing rates. FRAs, which are transacted over-the-counter, depend on underlying rates of various tenors. Therefore, using FRA rates, we will be able to analyze how violations of the composition identity (1) depend on the underlying tenors. We begin with an analysis of mispricing of FRAs.

A FRA is structured in the following way: the seller must pay, two days after time S, the amount per unit notional

(4) )),(

3601

),((360 TSSrdd

fTSSrddSTS

STS

+−

+

−+−+

+ .

f is the fixed rate specified in the FRA contract; r(S,S+T) is the LIBOR rate fixed at time S that has a maturity of T. This can be re-expressed as a forward agreement to, at time t, receive a zero-coupon bond maturing at time S+T with face value (1 + f(dS+T -dS)/360) in exchange for a fixed payment of $1. This assumes, of course, that the zero-coupon bond will have a yield of LIBOR at time S; i.e. the bond issuer will be able to fund at LIBOR then. The seller’s position can be replicated by a long position in the underlying asset – i.e. a zero-coupon bond maturing at time T, financed by borrowing until time S, or equivalently, by lending at LIBOR until time T, and borrowing at LIBOR until time S. The arbitrage argument that leads to the price of the FRA constructs a riskless portfolio from a long (short) position in the FRA, a short (long) position in a zero-coupon bond maturing at T, and a long (short) position in a zero-coupon bond maturing at S. The arbitrage argument fails if the bond issuer cannot default or fund at LIBOR at time S.

The arbitrage argument implies that the value of the FRA at time t on a rate with maturity T that is fixed at time S is then

(5) ),(

3601

3601

),(360

1

1),,,(TStr

dd

ddf

Strdd

fTSStVtTS

STS

tS +−

+

−+

−−

+=+

+

+

.

Market data for FRAs specifies the contract rate f * which ensures that the FRA has value zero. So, we expect

(5) )1),(

3601

),(360

1(

360

1),(* −−

+

+−

+

−=+

+

+ Strdd

TStrdd

ddTSSf

Ts

tTS

STS

.

We check this relationship empirically, comparing FRA contract market rates f*market(S, S+T) with rates f*(S, S +T) calculated using equation (5), with cash LIBOR prices r(t,S) and r(t,S+T), for S = 3, 6 and 9 months and T = 3 months and 6 months. Data for FRA contracts10

10 The data comes from Bloomberg, which queries various sources.

span the period from January 2000 until June 2009.

10

The results show a dramatic change in the difference between market and calculated rates beginning in August 2007. To remove the impact of asynchronicity between LIBOR and FRA data, we take twenty-day moving averages of the data. The market FRA contract dates reported are taken at the end of the NY trading day. LIBOR, however, is computed by polling banks at 11 a.m. in London. The LIBOR rate does not reflect the price of an actual transaction at a specific time, but certainly changes in the market between 11 a.m. GMT and 4 p.m. ET will lead to differences between our market quotes for FRAs and the values computed from LIBOR. On most days, one would expect these differences to be small (a few basis points), and it is plausible that they are relatively uncorrelated from day to day. In that case, the differences would mostly disappear when averaged over a number of trading days.

Figure 1: Spread between forward rates inferred from cash LIBOR and FRA contract rates, averaged over 20-days to remove most of the impact of asynchronicity.

The differences of the twenty-day moving average remain below 10 basis points from January 2000 until August 2007. It first falls below -20 basis points in August 2007. It subsequently oscillates between -10 and -40 basis points until fall of 2008, when it spikes below -100 basis points. This coincides with the market dislocations associated with the collapse of Lehman Brothers.

During the crisis period, forward rates implied by FRA prices are considerably lower than those computed from LIBOR. This is what would be expected if the risk of deterioration of bank credit was significant enough to affect the FRA-cash LIBOR relationship. This follows from the inequality in equation (2), which dictates that forward LIBOR from period S to S+T would be less than the value inferred from cash LIBOR for periods S and S+T.

A similar analysis can be applied to a comparison of 1-year par swap rates and swap rates calculated from cash LIBOR rates. For instance, in a one-year swap (evaluated, using the

11

convention above, at t = 0) in which the fixed payments are made semi-annually, standard arbitrage arguments imply that the par swap rate is:

(6) ;

360360360)

21,0(

360360)1,0(

360)1,0()

360

)21,0(

1(

126126612

126

1 tttdrtdr

rdrd

K+

−+

+=

the rates r(0, ½) and r(0,1) are 6 and 12 month LIBOR, while t6 and t12 denote the number of 30/360 days in the next 6 and 12 months. Until the summer of 2007, market quotes for 1-year swap rates were generally within a few basis points of the value calculated from this relationship11

This behavior is observed for other currencies. For instance, many one-year Euribor swaps are structured so that the fixed payment is made once at maturity, while the floating rate payments are semi-annual. Arbitrage arguments imply that the corresponding one-year Euribor swap rate should then equal 365/360 multiplied by 12 month Euribor. Again, the difference between these quantities is a few basis points prior to the credit crisis; it surges up to about 80 basis points in September 2008.

. With the advent of the crisis, the difference between market and calculated swap rates grew considerably, reaching about 100 basis points in September of 2008 and falling to about 40 basis points in early 2009.

4. The Futures-Forward Basis. Next, we turn to an analysis of the relative pricing of Eurodollar Futures contracts and

cash LIBOR rates. Eurodollar futures contracts are traded on the Chicago Mercantile Exchange (CME). Each contract matures two London business days before the 3rd Wednesday of each month; the March, June, September and December contracts are traded most heavily. A single Eurodollar futures contract of maturity T is settled at date T + 2 with a value of V(T) = $2,500*(100 - 100* 3 month LIBOR fixed at T).

The Eurodollar futures contract differs from a FRA in that it is marked-to-market daily, with the buyer or seller posting margin, while FRA payments are exchanged at maturity. In practice, however, most forward rate agreements are struck as part of a larger derivatives portfolio, which is collateralized. Daily changes in the value of the FRA will affect the amount of collateral that needs to be posted; this is effectively similar to the margin agreement associated with Eurodollar futures contracts. It will not be exactly the same, however as the collateral margin requirements depend on the specific details of the bilateral collateral agreement between counterparties, and this varies between different counterparties.

Futures and FRA contracts not only differ in the timing of their cash flows; their payoffs are not exactly the same. The primary difference is that the FRA payoff (maturing at time S on a rate that extends to time S+T) has a factor in the denominator that discounts from time S+T to S; the Eurodollar futures contract does not. Using standard arbitrage arguments, one can show that the quoted price of the Eurodollar futures contract with maturity S should satisfy P(t) = 100 (1 - Et[3 month LIBOR(S)]),

11 Much of this difference in basis points is presumably attributed to time of day differences between the LIBOR quote and the market level of interest rates at the close of the day, when the swap rate data is taken.

12

with the expectation taken in the risk-neutral measure. As LIBOR is a non-linear function of bond prices, this expectation depends on the dynamics of the interest rates, i.e. the particular interest rate model that we choose and its parameters. It will not equal the forward rate that is inferred from FRAs.

There has been a considerable amount of study devoted to the difference between the rate inferred directly from the quoted price of the futures contract (subtracting the price from 100 and dividing by 100) and the forward rate (inferred from contracts that are not collateralized). We refer to this difference as the Eurodollar futures-forward basis. Given a particular interest rate model (e.g. Vasicek or Cox-Ingersoll-Ross), one can compute the Eurodollar futures contract price exactly, and arrive at an expression for this basis. A very general result, appearing in Hunt

and Kennedy (2000), e.g., is that the futures-forward basis equals),0(

))(,cov(SB

SMrS− ; here M(S) is

the value of a money-market account at time S and B(0,S) is a S-maturity zero coupon bond. This term is often referred to as a ‘convexity correction’. From this formula, one can see that the convexity correction is of magnitude –(σrS)2f(S), with σ representing the volatility of the futures rate rS; f(S) is of order one or less for S < 1 year. The product σrS is typically of order .01. For short maturity contracts this correction is small. Assuming, for instance, a generously large value of .02 for σrS implies a futures-forward basis of less than two basis points for contracts maturing within six months.

In previous empirical work on the futures-forward basis, the forwards have been computed from cash LIBOR rates. When we measure the basis using our data, inferring futures rates directly from Eurodollar futures contract prices, and forward rates from cash LIBOR, using equation (5), the results look just like the difference between forward rates inferred from FRAs and forwards inferred from cash LIBOR. For the period of the financial crisis, the discrepancy is far larger than a few basis points, and of the sign opposite that would be explained just by the convexity correction. The twenty-day moving average of the basis reaches 130 basis points in fall of 2008; before the financial crisis, it always remained below 10 basis points.

13

Figure 2: Spread between 3 month forwards maturing in 3 months, using rates inferred from cash LIBOR to those from FRA contract rates and Eurodollar Futures (neglecting the convexity correction). The twenty-day moving average of this basis is shown.

The sign and magnitude of the basis is opposite and far larger than that observed using data from the 1990’s. In that period, it was observed by Grinblatt and Jegadeesh (1995) that the basis could not be explained by credit risk, as the sign was wrong.

From the same plot, one can infer the discrepancy between forward rates implied by FRAs and futures rates from Eurodollar futures contracts. In this case, we expect only a small difference due to the convexity correction.12

12 Earlier studies on the futures-forwards basis examined longer-dated contracts, for which the convexity correction, which is approximately proportional to the square of contract maturity, is far larger.

Indeed, we only observe very small discrepancies of a few basis points at most between implied FRA forward rates and Eurodollar futures rates. The discrepancies are larger in the earlier part of our dataset, and show no substantial increase during the financial crisis. These small differences are likely not simply due to the convexity correction. This may in part be due to asynchronicity in the futures and FRA data. Additionally, the comparison requires interpolation of future contract values, as the FRA maturities roll each day, while the future contract maturities are fixed on the third Monday of the month. We used cubic spline to interpolate futures rates. While the upcoming Eurodollar futures contracts are extremely liquid, the FRA market is somewhat less liquid and transparent, and this may

14

contribute to pricing discrepancies. Poorer data or less liquidity may have caused these discrepancies to be somewhat larger in the earlier years of our dataset.

If one enters into a FRA and then hedges the FRA with Eurodollar futures contracts, executing the hedge does not require one to borrow or lend the entire notional amount. Instead, one has to just fund the additional amount required for posting margin for the futures contracts. In contrast, the relationship between forwards inferred from FRAs was based on hedging the FRA with borrowing or lending in the interbank market. It is the additional credit and liquidity risk inherent in this market (e.g. of former LIBOR-paying counterparties no longer being able to borrow at LIBOR) that caused the standard arbitrage argument to break down. This should not significantly impact the relative pricing of FRAs and Eurodollar futures.

5. Practical Implications for Yield Curve Building and Pricing of Derivatives The data so far illustrate that during the crisis, there are extremely large differences

between forwards inferred from cash LIBOR and derivatives. The forwards implied by FRAs and Eurodollar futures, however, are nearly identical. If one is building a model to price interest-rate derivatives, the implication of this is simply that one should not use longer maturity cash LIBOR rates to determine the yield curve. Instead, Eurodollar futures or FRA prices should be used; as Eurodollar futures contracts are more liquid, it is indeed standard practice to use them in yield curve construction. There is considerable liquidity in the Eurodollar futures market extending out several years. To build the very long-end of the yield curve, swap rates are generally used. In summary, derivative prices should be used to calibrate any yield curve that is to be used for pricing derivatives.

In fact, there are further inconsistencies in relative pricing of derivatives that make interest rate derivative pricing considerably more difficult. This is a consequence of the failure of the composition identity and the resulting relationship expressed in equation (2). Eurodollar futures are based on 3-month LIBOR, and using these to calibrate a yield curve should be sufficient to price interest-rate derivatives based on 3-month LIBOR. When given a swap, cap or swaption based on LIBOR with a different tenor, e.g., 6-month LIBOR, one must then extract 6-month forward rates from the yield curve. Before the credit crisis, it was standard practice to use the composition identity to infer these 6-month forward rates. When this fails, Eurodollar futures are not sufficient to price these other interest rate derivatives.

As a simple example of this, consider FRAs based on both 3 and 6-month LIBOR. The FRAs based on 3-month LIBOR that we have examined above are 3 to 6 (3 X 6), 6 to 9 and 9 to 12 month, i.e. they are based on r(t,t+T) with T = 3 months. If the composition identity were to hold, we could use these FRA contract rates to determine the rates for 3 to 9 month and 6 to 12 month FRAs.

Our FRA dataset consists of contracts with underlying tenors of one month, three months, six months and one year. Contracts mature one through 21 months in the future. We compute anomalies by comparing sequential FRA contracts based on 1-month LIBOR with contracts based on 3-month LIBOR, sequential contracts based on 3-month LIBOR with contracts based on 6-month LIBOR and sequential contracts based on 6-month LIBOR with contracts based on 12-month LIBOR. In particular, we define

15

(7)

∑′

=

′++′−++′+−++++=′ττ

ττττττ

ττη1

))],)1(,(1log()),,(1[log(1),,,(i

iTtiTttfTtTttfTt

The quantity η represents the excess (continuously compounded) yield that one would

obtain, if at time t, one locked in interest payments (spanning the period t+T to t+T + τ) for a FRA that matured at time T on underlying LIBOR of term τ, compared to locking in a sequence of reinvested interest payments beginning at time T with shorter terms τ′ that span (t+T, t+T + τ′, … t+T + τ). With standard arbitrage pricing theory, η should be identically zero.

We expect to observe small non-zero values of η due to the effects of asynchronicity and bid-ask spread. First, note that each anomaly involves the relationship between long and short positions in 3 or 4 FRA contracts. FRA rates are quoted with bid ask spreads ranging from 2 to 5 basis, with larger bid-ask spreads typically attributed to FRAs with underlying rates of 6 and 12 month LIBOR. We use closing prices; for different FRAs, these may not represent exactly simultaneous quotes, particularly if the FRA is thinly traded. This is particularly relevant for the FRAs with underlying rates of 6 and 12 month LIBOR, which are less frequently traded. This means that on a given day, that an anomaly of 5 – 10 basis points is not particularly significant. However, a persistent anomaly of 10 basis points or so over a period of many days likely is meaningful. We observe anomalies far greater than 10 basis points.

Our dataset contained open, close, low and high FRA rates, along with an indication of both bid and ask for the closing rates. Furthermore, we were able to examine plots of intraday prices for portions of the dataset (in the latter part of 2009). The intraday plots indicated that there were significantly fewer trades in the 6 and 12 month LIBOR FRAs. They also contained occasional spikes in the FRA rates during the day; these spikes were more prevalent for less liquid FRAs. If one of the spikes occurred at the end of the day, this would lead to an outlying FRA rate which was significantly off-market.

We apply a filter to our data, rejecting FRA rates on days in which the open and high quotes or the close and high quotes differed by more than 15 basis points. This should have the effect of removing closing quotes that occur on the infrequent spikes, and as well removing data on days with particularly high volatility, when data asynchronicity is more likely to be a problem. With this filter, nearly all of the FRA rates from the period September through December of 2008 are rejected, along with many of the FRA rates spanning the period November 2007 through March 2008. We do retain most of the data for 2009, particularly for the more liquid contracts.

Figures 3 and 4 illustrate the behavior of a selected set of FRA anomalies during mid-2008 (between the demise of Bear-Stearns and before the default of Lehman brothers), as well for the calendar year 2009.

16

Figure 3: Anomalies, representing the excess yield obtained in investing at longer-term FRA rates rather than a set of shorter duration rolling contracts. This data is from the early months of the credit crisis.

17

Figure 4: As in figure 4, except the data for the anomalies spans 2009 and the first half of 2010.

The anomalies between 1-month and 3-month LIBOR FRAs, for different contract

maturity dates T, tend to move in tandem. As a function of T, the term structure is downward sloping in 2008, and through May of 2009. In the latter part of 2009, however, the term structure inverts, and becomes upwards sloping. In the plot of 2009 data, we also illustrate the behavior of anomalies between 3 and 6 month LIBOR and 6 and 12 month LIBOR FRAs. By the end of 2009, the FRA anomalies corresponding to the shortest maturity T, with underlying rates of 1 and 3-month LIBOR are small enough to be immaterial, while the longer-dated FRA anomalies with longer underlying rates are still significant. Anomalies increase again in spring 2010, concurrent with the Greek debt crisis.

The results illustrated in figure 2 show that three month forward rates inferred from FRAs and Eurodollar futures are nearly identical during the crisis. Therefore, the standard procedure of building yield curves using Eurodollar futures prices would lead to mispricings of six to twelve month FRAs by as much as 40 basis points in early 2009.

These anomalies are also present in the FRA rates based on Euribor. For instance, for Euribor-based FRAs, the anomaly η(t, ¼ , ¼ , ½ ) has an average value of 1.3 bp before the crisis (from 1/06 through 6/07). This is smaller than the FRA bid-ask spread. Between 1/09 and 8/10, the average of this anomaly equals 19 basis points, which also is the average of this anomaly over this time period when the underlying rates are USD LIBOR.

18

This anomaly greatly complicates the practice of curve building and derivative pricing. It is no longer sufficient to build a single forward curve, based on inferences of forward 3-month LIBOR. Instead, to price derivatives based on underlying LIBOR rates of differing tenors, one needs a series of forward curves, e.g. a curve based on estimates of forward 1-month LIBOR, 3-month LIBOR, 6-month LIBOR etc. as well as a riskless curve used to discount cash flows13

If one then has to price derivatives that mature in several years, more market information is needed to estimate η(t, T, …, …) for T well beyond one year. This information could come from the market price of swaps with floating legs that depend on LIBORs of tenors other than 3 months, i.e. both fixed-floating LIBOR swaps and basis swaps in which the receive and pay legs depend on LIBORs with different tenors.

. In principle, one could extract this information from FRA contract rates that span (t X (t+1)) months, or (t X (t+3)) months or (t X (t+6)) months. Liquidity rapidly diminishes in the FRA market for maturities greater than one-year or so. This procedure would therefore only work at the very short-end of the yield curve.

6. Empirical Analysis of Anomalies. Given that we have observed the breakdown of relationships between derivative prices,

we now turn to an empirical analysis of the source of these anomalies. The goal of this analysis is to assess which market factors are correlated with these anomalies. The results can then potentially be used to suggest hedges for portfolios with valuations that depend on these anomalies. They can serve as a guide in isolating the appropriate variables (e.g. measures of default or liquidity risk) that ultimately must be used in modeling the pricing anomalies.

For market factors that serve as proxies for credit and liquidity risk, it is natural to consider the spread between LIBOR and the `risk-free’ or bank funding rates as well as bank CDS spreads. To measure the former requires identification of the `risk-free’ and funding rate. It is not necessarily apparent how to choose the risk-free rate. In past academic literature, e.g. Collin-Dufresne and Solnik (2000), rates inferred from treasury bills and bonds have been chosen as risk-free. These instruments are highly desirable not only because they have minimal default risk, but because they are very liquid. For instance, for market participants who want to short bonds (e.g. through reverse repo), it is important to be able to quickly purchase these bonds on the market. The only bonds that generally have enough market liquidity to enable swift repurchase are treasury bonds. This creates excess demand for them, potentially lowering their yield below the true risk-free rate. This can be modeled as a liquidity convenience yield, as in Grinblatt (2001). The degree of liquidity, however, varies from one treasury instrument to another, usually depending on how recently it has been issued. Due to these characteristics, treasury rates are not typically used by practitioners as the risk-free rate.

The relevant rate that a market participant must consider in evaluating an arbitrage strategy is their funding rate. The return on a risky security depends on its excess yield over the funding rate; this should increase with the security’s riskiness. This suggests that an appropriate metric to measure liquidity and credit risk is the difference between the return of a risky security and a funding rate.

13 Derivative transactions are assumed to be collateralized with minimal credit risk, so that their cash flows should be discounted at a riskless rate. For more elaboration on this, see the next footnote.

19

Purchases of bonds by market participants are financed through repo operations; the funding rate for these is the repo rate. Most over-the-counter (OTC) derivative transactions between financial institutions are collateralized or secured. Fed Funds is the interest rate that is most often paid on collateral posted for OTC derivatives transactions. When a bank receives collateral, it generally has the right to rehypothecate it, i.e. post it to another counterparty as collateral. Therefore, $N of collateral will typically reduce its funding requirements by $N. If it pays Fed Funds interest on this collateral, it then effectively funds its derivatives operations at the Fed Funds rate14

A term structure of the Fed Funds rate can be inferred from Overnight-Indexed-Swap (OIS) rates. OIS swaps consist of one fixed and one floating leg, which both pay off at maturity

. Though OTC derivative operations are not immune from credit risk, because they are collateralized, they entail considerably less credit risk than (unsecured) interbank lending, transacted at LIBOR rates.

15. The floating leg payment is proportional to the interest accrued during the swap at the effective overnight federal funds rate, determined daily and compounded over the lifetime of the swap. These swaps are designed to allow financial institutions to hedge their funding costs. Liquidity in the OIS swap market is quite good extending out several years; longer maturity OIS-LIBOR basis swaps are also traded, allowing one to bootstrap an effective fed funds term structure out many years16

Figure 5 depicts three to six month forward rates inferred from 3 and 6 month OIS swaps, and compares these to forward rates inferred from FRAs, Eurodollar futures and cash LIBOR. The spread between cash LIBOR and the FRA/Eurodollar future rates is a consequence of the anomalies we observe; the spread between the implied OIS rate and LIBOR implied from FRAs/Eurodollar futures is a measure of credit and liquidity risk. Note that the two spreads are of the same order of magnitude and appear to track each other.

.

14 The effective Fed Funds rate is as well a rate paid for unsecured overnight lending. As it is an overnight rate, it is often argued that this rate is effectively risk-free, though this argument seems less convincing if applied at the height of the credit crisis. There is, as well, an overnight LIBOR rate. During the height of the credit crisis, overnight LIBOR spiked several hundred basis points over Fed Funds, although both represent the interest on overnight, unsecured loans between financial institutions. Overnight LIBOR, however is a ‘polled rate’ among large international banks operating in London, while Fed Funds represents the interest rate typically paid by large U.S. banks to smaller U.S. banks for overnight deposits. Thus, the rates operate in different markets, and cannot be arbitraged to an unlimited degree. The rationale for using ‘Fed-Funds’ as a risk-free rate rests largely on its role as the interest rate paid on secured transactions (e.g. collateralized derivatives) between banks. Consequently, over the last few years, market practitioners have moved towards using rates inferred from OIS swaps to present value cash flows in OTC derivatives, such as LIBOR-based swaps. 15 OIS swaps typically have a one-day lag between maturity and settlement. This leads to a very small amount of convexity in their valuation (as explained by Henrard (2008)), which we ignore in extracting forward fed funds rate. 16 There is, however, no analogue of the OIS market for repos with a great deal of market depth. This is one reason why it is preferable to use the spread to Fed Funds rather than the general collateral repo rate as a measure of liquidity and credit risk.

20

Figure 5: A comparison of forward rates inferred from cash LIBOR, FRAs, Eurodollar futures and OIS swaps.

Another measure of the credit and liquidity risk of financial institutions is their credit default swap (CDS) spread. Specifically, as a metric, we compute the average daily 5-year CDS spread for 11 of the 16 members of the US dollar LIBOR panel17

.

There is a large maturity mismatch between CDS spreads and the anomalies that we measure. Just as 5-year yields may not track three or six-month yields, CDS spreads and the anomalies might not move coherently. Figure 6 compares the 1-4 month FRA anomaly

)41,

121,

121,(tη , the LIBOR-OIS spread and the average bank CDS spread, over the time period

extending from 2007 until mid 2008 (the early stages of the credit crisis), and then through 2009 and much of 2010. The first plot depicts steady growth in the CDS spread from November 2007 through April 2008, while the anomaly and OIS-LIBOR spread remain relatively flat. From the plot of the spreads in 2009 and 2010, we observe that overall they all do trend downwards through the latter three-quarters of 2009, and then increase in the first half of 2010, but that there are periods of time in which large moves in CDS spreads are not accompanied by similar changes in the LIBOR-OIS spread or the anomalies. In particular, during the first two months of 2009, bank CDS spreads surged up by 50%, while the anomalies and LIBOR-OIS spreads only increased slightly. During this period of time, there were widely publicized fears that Citigroup and perhaps Bank of America would be taken over by the U.S. government, possibly creating

17These include Bank of America, Barclays, Credit Suisse, Deutsche Bank, HSBC, RBS, UBS, JP Morgan, Citigroup, Lloyds and Rabobank; the quality of CDS data for the remaining five LIBOR panel members was not as good as it was for these eleven banks.

21

losses for long-term bondholders. Investors may have believed that the government had and would presumably act in a manner to keep financial markets functioning, and would therefore want to ensure that interbank lending agreements were honored. It is therefore plausible that confidence in the security of interbank lending would not diminish even as CDS spreads widened. Formally, this is reflected by differences in the markets’ expectation of recovery on interbank loans versus long-term bonds upon default. The behavior of CDS spreads and measures of interbank lending risk may differ considerably both due to discrepancies in tenor (long vs. short-term) and discordant dynamics of their respective anticipated recovery rates.

Figure 6: A comparison of FRA anomalies, the forward OIS-LIBOR spread and a measure of average bank credit default swap spreads.

The lack of consistency between the interbank lending and CDS markets is also reflected in the behavior of individual bank LIBOR quotes and CDS spreads over time. Naively, one would expect that banks with higher CDS spreads would pay more to borrow on the interbank market, and would quote higher values of LIBOR in the daily polls. This would be expressed by a positive rank-order correlation between LIBOR quotes and CDS spreads. From August 2008 until August 2010, the Spearman and Kendall correlations between LIBOR quotes and CDS spreads vary from about -.5 to .5, and assume mean values of 0.02 and 0.04 that are insignificant. This is consistent with the suggestion that lenders believed that interbank loans to these banks

22

would be repaid in the event of default. 18

The lack of correlation between CDS spreads and the pricing anomalies has significant practical ramifications when one considers modeling and hedging the pricing anomalies. A natural approach would be to calibrate any credit model used to price the anomalies using CDS quotes. These empirical results suggest that any model that simultaneously explains CDS spreads and the anomalies would need to be multi-factor and/or would have to incorporate different and dynamic recoveries for interbank loans and the bonds referenced by credit default swaps.

These results also are consistent with the findings of Michaud and Upper (2008); these authors analyze data from 2007 and find that banks with higher CDS spreads do not quote higher LIBOR rates. They also find no significant correlation between day-to-day changes in the LIBOR-OIS spread and CDS spreads, although they argue that LIBOR-OIS and CDS spreads track each other over longer time scales.

To explore more thoroughly the statistical relationship between the anomalies and measures of credit, interest rate and liquidity risk, we also perform a regression analysis, regressing weekly changes in these anomalies with weekly changes in various market factors.

The asynchronicity in the data presents challenges for empirical analysis. Indeed, if one is not careful, one can easily be misled by spurious correlations that are a consequence of timing differences between data measurements. The impact of the discrepancies in measurement times can be largely eliminated when estimating the magnitude of the anomalies by taking moving averages over a number of days. This cannot be done when computing weekly changes in the anomaly and using them in a regression. A substantial portion of the weekly changes in the anomalies may be due to changes in the portion of the anomalies caused by lags in the data. Furthermore, the contributions to weekly changes arising from these timing discrepancies may be strongly correlated with weekly changes in the dependent regression variables. This is because some potential regression variables (e.g. the cash LIBOR – OIS spread) are also affected by the same timing discrepancies, while others, such as market volatilities, should be roughly proportional to observed timing discrepancies.

This can be addressed by only including variables that depend on measurements made at the close of the market in New York. In particular, none of the dependent and independent variables depend on cash LIBOR. With the FRAs used to compute the anomalies, we apply the filter described in the previous section, removing data on days when open, close and high rates were sufficiently different. This should help to reduce the impact of asynchronicity between different FRA rates used to compute these anomalies. The filtered data series has many gaps in 2007 and 2008; consequently, we only perform these regressions on weekly data from 1/6/09 through 8/16/10. In the regressions, we have also included as dependent variables measure of market volatility, and interest rate risk; these do not have significant explanatory value.

For the 1-4 month forward anomaly )41,

121,

121,(tη , we obtain these regression results:

18 The low or negative rank-order correlation might also be observed if some banks, particularly those under distress, when polled, provided misleading and unrealistic LIBOR quotes. This possibility was raised in the financial press by Mollenkamp (2008). Schwarz (2009) finds that during this period, LIBOR quoted by banks was only slightly below the levels of rates reported for transactions on a transparent electronic market, the eMID, implying that it was unlikely that banks were systematically greatly understating their borrowing costs on the interbank market.

23

Table 1: Regression of the anomaly )41,

121,

121,(tη against a set of explanatory variables

that are correlated with various measures of credit, liquidity and market risk. Explanatory Variable(s) (weekly changes) R-squared F stat p value (for F stat) Forward LIBOR-OIS spread 0.57 105.96 <.0001 Forward LIBOR (inferred from FRAs) 0.58 109.33 <.0001

Forward LIBOR-OIS spread and forward LIBOR 0.59 56.6 <.0001

5 year swap rate minus forward LIBOR 0.03 2.65 0.11 Average bank CDS spread 0.01 0.92 0.34

Average bank CDS spread, without 3 largest & smallest 0.01 0.53 0.47 3 month into 1 year swaption volatility squared. <0.01 0.05 0.82 VIX <0.01 0.01 0.92 SPX (S&P level) <0.01 0.40 0.53 Similar results, albeit with lower R-squareds, are found when regressing weekly changes in the 3-

6 month anomaly, )41,

121,

41,(tη , against changes in potential explanatory variables.

Table 2: Regression of the anomaly )41,

121,

41,(tη against a set of explanatory variables

that are correlated with various measures of credit, liquidity and market risk. Explanatory Variable(s) (weekly changes) R-squared F stat p value (for F stat) Forward LIBOR-OIS spread 0.33 39.94 <.0001 Forward TED spread 0.26 27.91 <.0001 Forward LIBOR (inferred from FRAs) 0.33 40.26 <.0001 Forward LIBOR-OIS spread and forward LIBOR 0.34 20.50 <.0001 Forward TED spread and forward LIBOR 0.33 19.74 <.0001 5 year swap rate minus forward LIBOR 0.01 0.49 0.49 Average bank CDS spread 0.01 1.02 0.31 Average bank CDS spread, without 3 largest & smallest

0.03 2.18 0.54

3 month into 1 year swaption volatility squared. 0.05 3.73 0.06 VIX (S&P implied volatility) 0.02 1.68 0.20 SPX (S&P level) 0.02 1.47 0.23

In the regressions, we find that the anomaly only demonstrates a significant correlation with the LIBOR – OIS forward spread, the TED spread and forward LIBOR. During this time period, however, short-term OIS rates and treasury rates were close to zero with little variation, so that changes in the spread were primarily due to changes in LIBOR. This is borne out by the multivariate regression results. The F-statistic is much large in the univariate regressions; it

24

declines considerably when the LIBOR-OIS or TED Spread are used as regressors together with forward LIBOR. The large R-squareds in the regression with the LIBOR-OIS spread are consistent with the explanation that the anomalies are a consequence of credit and liquidity risk. There are, however, alternative interpretations of these results. For instance, the regressions would yield large R-squareds if the quotes for FRAs that span one-month were often stale and did not move much over a week, even when quotes that FRAs that span three-months, which are more liquid, did change. In this case, changes in the anomaly would be correlated with changes in three-month LIBOR. Note that stale data will not explain the anomalies, which, as illustrated by the moving averages we have computed, are large and persistent over the crisis period.

Changes in the FRA anomalies will have an impact on the valuation of a fixed income portfolio; one might aspire to hedge this risk. The above results suggests, however, that if a hedge were based on taking a position in the VIX or CDS spreads, which encompass a measure of ‘market fear’, that it would be ineffective in reducing the weekly volatility of such a portfolio. A hedge based on LIBOR-OIS basis swaps, however, appears to have more potential of immunizing the portfolio against this risk.

7. Can Bid-Ask Spreads Explain the Anomalies? In assessing the role of illiquidity in explaining the anomalies, we would at least like to

make a qualitative assessment of the degree of illiquidity during the financial crisis. One basic question that we attempt to address is: are bid-ask spreads in the interbank market in themselves large enough to explain the size of the observed anomalies, even if one neglects other liquidity and credit effects? For this to be the case, the bid-ask spread would have to be of the same order as the actual anomaly.

It is difficult, however, to quantify the degree of illiquidity in the interbank market. In the popular press, there are anecdotal reports that during the fall of 2008, the interbank lending market ‘seized up’.19 Illiquidity in the interbank market would be manifested by large bid-ask spreads, so that LIBID (the London Interbank Bid Rate) would be considerably less than LIBOR. There is no data, however, gathered on LIBID. 20

Most interbank lending is conducted through brokers; however, a substantial volume of interbank loans are transacted on the “electronic Market for Interbank Deposit” (eMID) based in

The Federal Reserve does report the level of interbank lending among US banks, indicating how much lending is carried through Fed Funds and Repurchase Agreements combined, as well as ‘other’ categories. The Federal Reserve H.8 surveys show a decrease in interbank lending among large US commercial banks during the fall of 2008, but the decline is relatively modest, bringing lending down to the level reported in 2006. This data has been interpreted by Kehoe, Chari and Christiano (2008) as signifying that the claim that interbank lending was ‘virtually nonexistent’ during the fall of 2008 is a myth. Note that the Federal Reserve interbank lending data applies to a different (though not mutually exclusive) set of banks from those in the LIBOR panel, lending in a different market.

19 The New York Times, on September 30, 2008 reports "The money markets have completely broken down, with no trading taking place at all," said Christoph Rieger, a fixed- income strategist at Dresdner Kleinwort in Frankfurt. "There is no market any more. Central banks are the only providers of cash to the market, no-one else is lending." 20 By convention, LIBID is sometimes quoted as equal to LIBOR minus 12.5 basis points, reflecting the typical bid-ask spread in interbank lending in the early days of the interbank market, in the 1980’s. This does not mean, however, that the actual bid-ask spread is currently 12.5 basis points.

25

Milan. The eMID data set can shed some light on the bid-ask spread in the interbank lending market; eMID data has also been analyzed for this purpose by Michaud and Upper (2008) and Politi et al. (2010). Between January 1999 and August 2009, 138 Italian banks and 106 international banks from the EU, Switzerland and the US participated in this market, with typical daily volumes of 2x1010 Euros.

In addition, the eMID is a quote-driven market. Quotes are posted individually in an order book. Only borrower banks are identified by codes. Lender banks are anonymous. A trade occurs when a bank (the aggressor) actively chooses a living order. If the aggressor is lending (selling), the trade is immediately executed and money is giving to the quoter. If the aggressor is borrowing (buying), the quoter can first check its identity. Quotes are not recorded in the dataset, only trades are. In addition to rate and amount, for each trade, we know whether the aggressor is buying or selling. Only data on the Euro is available.

Since no quotes are available in the dataset, we match the closest pairs of buy and sell trades, restricting pairs to trades occurring on the same day. We examine the sell-buy21

1. The bid-ask spread.

spread to estimate an approximate upper bound for the bid-ask spread in the eMID market. The sell-buy spread between transactions is determined by three factors:

2. Movement in the level of interest rates during the time spanning the two transactions. 3. Differences in credit quality between the borrowers engaged in buy transactions

versus sell transactions. The impact of movements in rates within pairs can be largely eliminated by averaging

sell-buy spreads over a large number of pairs of transactions. As for the third factor, Schwarz (2009) argues that banks with poorer credit quality tend to borrow as aggressors, and that the sell-buy spread does indeed reflect counterparty credit risk as well as the bid-ask spread. This, however, implies that the sell-buy spread would tend, on average, to be higher than the bid-ask spread.

The computed spreads are averaged within each trading day. In figure 7, we plot the overnight sell-buy spread, further averaged over twenty days; this moving average is similar to the bid-ask spread presented by Politi et al. (2010).

21 The “sell” and “buy” here refer to the action of the aggressor bank. The ‘sell-buy’ spread refers to the buy price minus the sell price, consistent with the terminology for ‘bid-ask’ spread.

26

Figure 7: Overnight sell-buy spread for the eMID interbank market during the crisis In the six months prior to Lehman’s default, the moving average of the sell-buy spreads

ranges from about two to four basis points. The sell-buy (and consequently bid-ask) spreads do surge after Lehman’s default. As the credit crisis progressed, volumes traded on the eMID did decline. This may have been due both to lack of liquidity and from competition from lending facilities set up by the European Central Bank. The evidence from the eMID, however, is that the market did not entirely freeze up, even after Lehman’s default.

The vast majority of lending on the eMID is overnight. We examined the sell-buy spread for loans with terms of one and three months, as these are the terms relevant to much of the analysis of the FRA anomalies that we observed. In this case, the data was much sparser; on many trading days, there were no buy-sell pairs. With fewer data points, even with a twenty-day moving average, we cannot eliminate the impact of market fluctuations of the risk-free rate. In figure 8, we plot the daily average sell-buy spread for loans with three-month terms. In this case, we only average positive sell-buy spreads (some sell-buy spreads will be negative if rates fall sufficiently between the sell and buy transactions). This will tend to overstate the buy-sell spread. Still, we find that prior to Lehman’s default, most of the sell-buy spreads are below 10 basis points, and average to about 5 basis points. The results for loans of terms of one month are similar. During this same period, the short-dated anomaly representing the excess cost of borrowing for three months versus three rolling one-month periods ranged from about 15 to 25 basis points. If bid-ask spreads for LIBOR quoting banks were similar to those implied by the eMID data, then arbitrage arguments, relaxed to take into account bid-ask spreads (as in equation

27

(3)), are still insufficient to explain the observed anomalies

Figure 8: Three-month sell-buy spread for the eMID interbank market during the crisis.

We note that the eMID data consists of quotes for interbank lending in Euro, while we have primarily presented anomalies for USD LIBOR. However, as described at the end of section three and in section four, anomalies for EURIBOR are quite similar to those observed for USD LIBOR. It is also important to note that LIBOR panel banks on average have superior credit than those who participate in the eMID market and therefore enjoy a narrower bid-ask spread than the eMID banks. When banks are asked to quote LIBOR, however, these quotes refer to loans of ‘reasonable size’, which generally means several hundred million Euro. The eMID results we have presented are for much smaller notionals, having a median size of about 10 million Euro. It is possible that the bid-ask spread may be larger for bigger loans. Larger overnight loans (ONL) are transacted on the eMID; these have a median size of about 150 million Euro, albeit with far less frequency than smaller loans. The sell-buy spreads for these are only slightly larger (2-5 basis points rather than 2-4 basis points in the period prior to Lehman’s default) than those of the smaller overnight loans, implying that any size effect is likely not so significant.22

8. Modeling Risky LIBOR

All of these points suggest that the bid-ask spread for LIBOR banks is probably at least not much greater than the sell-buy spread observed on the eMID market.

Up to now, the viewpoint of this paper has been largely empirical, documenting anomalies in interest rate derivative pricing, and then performing regression analysis to explore how these anomalies are related to market-based measures of credit risk. One consequence is that,

22 Schwarz (2009) notes that better quality credits tend to transact large loans on the eMID market, with smaller notionals traded by lesser quality credits. Even so, these better quality credits would not tend to have credit quality surpassing the members of the LIBOR panel.

28

without additional models, one can no longer infer a forward rate that spans a tenor ‘T’ (that is, rates of the form f(0,t,t+T)) from a series of forward rates that span a shorter (or longer) ‘S’ (rates of the form f(0,t,t+S)). We now proceed to develop and analyze models of underlying credit and liquidity that may capture the relationships between forwards of different spanning tenors during the credit crisis.

In developing these models, it is convenient to work with continuously compounded interest rates, and to express the deviations from traditional arbitrage pricing using the following function ε(t, t1, t2):

(8) ∫ →=−−=2

1

).,,(lim),(;),(),,()(),,( 211221

t

tuv vutFutFduutFtttFtttttε

In the above, F(t, t1, t2) denotes the continuously compounded forward rate (spanning a tenor (t2 – t1) ; when F has only two entries, it represents the instantaneous forward rate. We infer values of F from market FRA (forward rate agreement) quotes. We then can express the various ‘anomalies’ that we have observed as a function of ε.

The models introduced in this section will incorporate the default risk inherent in LIBOR. They will be of the reduced form type (as developed in Duffie and Singleton (1999)), representing the spread between risky and risk free rates as diffusion processes, as expressed in the risk-neutral measure. This spread depends on

1. The instantaneous risk-free rate at time u, ru. 2. The instantaneous risk-adjusted rate for an institution that can borrow at LIBOR at

time u: lu. This rate equals the risk-free rate plus the instantaneous expected loss rate (loss given default multiplied by instantaneous hazard rate).

3. The instantaneous risk-adjusted rate at time u for an institution conditional on the specification that it was able to borrow at LIBOR at time t1: R(u, t1); e.g.

.),(111 tlttR =

Implicit in this formalism is the assumption that, upon default, given a contingent claim, that recoveries equal a given fraction of the market value of the claim (the RMV assumption). This contrasts with the assumption, also often used in credit risk modeling, that one recovers a given fraction of the par value of a debt instrument.

With the RMV assumption, it is natural to view default as an event in which all claims on the defaulting company are reduced in value by one minus the recovery fraction, but subsequently the claims are not retired, but continue on until their scheduled maturity. It is possible, as well, that default can then occur multiple times. In this case, we can make sense of forward agreements on the company’s debt that expire after default occurs, as the underlying bonds will still exist after default, albeit with reduced value. In contrast, if the company is liquidated and the debt fully retired after default, the meaning of a forward agreement on the company’s debt, post default, becomes ambiguous. The formalism that we have introduced is motivated to describe losses due to default risk. The prices of credit-risky instruments, however, are affected by liquidity risk and implicitly it is assumed that these effects are captured by the risk-adjusted loss rate. Although we refer to the models introduced in this section as ‘credit-risk models’, we also assume that when the models are calibrated to market prices, that they effectively incorporate the impact of liquidity.

29

We assume that the FRAs underlying the forward rate agreements are transacted by counterparties that fund at LIBOR and that they are fully collateralized. Collateralization largely mitigates the counterparty credit risk. The consequence of this is that we will discount cash flows at the risk-free rate. In practice, the risk-free rate (r) used is the secured funding rate, i.e. the rate paid on collateral, which in the US market is typically the effective Fed Funds rate23

(9)

. It then follows that the continuously compounded forward rate inferred from a FRA that settles at time t1, on an underlying rate of tenor t2 – t1, is

))][exp(*

)]]),([exp(*)[exp(*

log(),,()(1

1 2

1

1

2112

∫

∫ ∫

−

−−

−=− t

ts

t

t

t

ts

dsrE

dstsREdsrE

tttFtt ;

E* represents the expectation in the risk-neutral measure. In the numerator, the inner expectation is conditional on the value of r at t1, so we can bring the outer discount factor inside the inner expectation, and then remove one of the nested expectations.

One can show that if R(s,t) – rs does not vary with time s, then the anomaly ε vanishes. It does not vanish if R(s,t) – rs varies with s, but is independent of t and therefore equals ls – rs . This would be the case if the LIBOR credit spread were stochastic, while LIBOR was not a refreshed rate. In the second appendix, we show that the anomalies generated in this case, though non-zero, are still much smaller than what has been observed during the financial crisis. Therefore, it is essential that any model of forward rates incorporate the characteristic that at a time s > t, LIBOR is ‘refreshed’ so that it may not be the rate paid by the same cohort of banks that paid LIBOR at time t.

We can rewrite the expression for the forward FRA rate as (10)

))][exp(*

]))(exp())),((exp()[exp(*log(),,()(

1

2 2

1

2

1

1

2112

∫

∫ ∫ ∫

−

−−−−−

−=− t

ts

t

t

t

t

t

tssss

dsrE

dsrldsltsRdsrEtttFtt

In the numerator above the above expression, the second term captures the possibility that a counterparty that can finance at LIBOR may then see its credit quality deteriorate below LIBOR. The third term itself represents the effects due to fluctuations in the instantaneous credit spread of `LIBOR’ rated entities. Let us rewrite the numerator as

23 In Piterbarg (2010), this rate is referred to as rC.

30

(11)

∫ ∫∫

∫ ∫

∫ ∫ ∫

∫ ∫ ∫

−−−−−

+−−−

+−−−−−

=−−−−−

2

1

2

1

2

2 2

1

2 2

1

2

1

2 2

1

2

1

).))(exp(),)),((exp(cov()][exp(*

)))),((exp(),cov(exp(

]))([exp(*)])),(([exp(*)][exp(*

]))(exp())),((exp()[exp(*

1

1

1

1

t

t

t

tsss

t

ts

t

t

t

tss

t

t

t

t

t

tssss

t

t

t

t

t

tssss

dsrldsltsRdsrE

dsrtsRdsr

dsrlEdsltsREdsrE

dsrldsltsRdsrE

As in Collin-Dufresne and Solnik (2001), we will model the differences R(s,t) – ls, ls – rs as stochastic processes. Our goal, in this case, is to work with simple models that lead to analytic expressions for ε. This imposes substantial limitations on our choice of processes. Typically credit spreads models assume that the spreads obey Cox-Ingersoll-Ross (CIR) processes, or that the logarithm of the spread evolves normally with, in both cases, the possibility of jumps included as well. The CIR model is only analytically tractable (in that it admits analytical formulas for bond prices) if applied to a single spread, or multiple uncorrelated spreads; the lognormal model is not tractable at all.

Analytic solutions for multiple processes that are correlated can be obtained if we assume that the processes obey Ornstein-Uhlenbeck diffusions (with perhaps uncorrelated jumps). This is the approach taken in the work of Collin-Dufresne and Solnik (2001). This, however, does not capture the tendency of the variability of spreads to grow as spreads increase. As we want to apply our results to an analysis during the credit crisis, when there is a very large variation in the magnitude of spreads, this would be a considerable shortcoming.

Ignoring the correlation between rs, R(s,t) – ls and ls – rs is tantamount to neglecting two covariance terms in equation (11). During the credit crisis, we have observed anomalies of tens of basis points, up to one hundred basis points; the quantity ε should be of this order of magnitude to lead to these anomalies. It seems unlikely, however, that these covariance terms will be this large, particularly if we confine ourselves to calculating the anomalies for short tenors, e.g. t2 < 1, t2 – t1 = ¼. For instance, if we assume, conservatively, correlations of magnitude one, along with annualized variabilities for both the risk-free rates and spreads of two hundred basis points, these terms should not exceed a basis point or so. This suggests that it would be reasonable, at least for short tenors, to neglect these terms and work with uncorrelated diffusion processes24

24 Another argument for dropping the first covariance term,

.

∫ ∫ −−−2 2

1

)))),((exp(),cov(exp( 1

t

t

t

tss dsrtsRdsr , is the following: This covariance term can be re-

expressed as ∫ ∫ ∫∫ −−−1 2

1

2

1

2

1

)))exp()),((exp(,)exp()cov(exp( 1

t

t

t

t

t

ts

t

tss dsrtsRdsrdsr . Expanding this, we

see that this term is small if the following three covariances are also small:

.))exp(),),(cov(exp(,))exp(),cov(exp(,))exp(),cov(exp(2

1

2

1

2

1

2

1

1 2

1

1∫ ∫∫ ∫∫ ∫ −−−t

t

t

ts

t

t

t

tss

t

t

t

tss dsrdstsRdsrdsrdsrdsr

31

Without the covariance terms, we find that (12)

].)([*)]))([exp(*log()]))),(([exp(*log(),,(2

1

2

1

2

1

121 dsrlEdsrlEdsltsREttt s

t

ts

t

t

t

tsss −−−−−−−−= ∫∫ ∫ε

The latter two terms are effectively another sort of convexity correction. By Taylor expanding the log and exponential, it is apparent that they are proportional to the integrated variance of the instantaneous credit spread between times t1 and t2. For t2 –t1 = ¼ , this should be less than a basis point, and it should grow at most linearly with t2 –t1. This suggests that the bulk of the contribution to ε arises from the first term, which captures the expected deterioration in credit spread of a LIBOR-rated bank below LIBOR over the period t1 to t2.

One consequence of this observation is that if we had increased the discount rate in equation (9), so that it represented a risky discount rate (e.g. assuming no collateralization of the forward rate agreement), this would have led to only very small changes in the anomaly ε. These changes would be due to the `convexity correction’ terms that we have dropped in equations (11) and (12)25

Note also that (12) does not depend on the correlation between R(s,t1) – ls and ls – rs. In the models analyzed by Collin-Dufresne and Solnik (2001), this correlation is assumed to be zero. This undoubtedly would not hold during the recent credit crisis, when much of the risk was systematic rather than idiosyncratic. The anomaly ε, however, only will have a much smaller secondary dependence on this correlation.

. This supports our claim, made in the introduction, that considerations of derivative counterparty credit risk and collateralization should only have a very modest impact on the anomalies that we observe in this paper. It is also similar to the results observed by Piterbarg (2010), who expresses the differences between forward prices (on assets such as equities) inferred from collateralized and non-collateralized contracts as convexity corrections, and finds that these are large for long-maturity contracts.

If we ignore the last two terms in (12), our problem becomes particularly simple: model the difference R(s,t1) – ls as a stochastic process. In any model that admits analytic expressions for bond prices, one can then compute the first term in (10), given an analytical expression for ε. Given this, we can then infer all potential forward rates from instantaneous forwards. If we choose as well to model ls – rs as a stochastic process that admits analytical expression for bond

When LIBOR is treated as the risk-free rate, the first two of these three covariances appear in the calculation of the Eurodollar Futures – Forwards basis, which is very small for contracts maturing within a

year. In the second appendix, we also show that ∫ ∫−2

1

2

1

))),(exp(),),(cov(exp( 11

t

t

t

t

dstsRdstsR is small,

suggesting that ∫ ∫−2

1

2

1

))exp(),),(cov(exp( 1

t

t

t

tsdsrdstsR is small as well.

25 Shifting the rate of discounting, however, has a significant impact on the valuation of forward rate agreements and swaps which are off-market, i.e. which have fixed and floating legs that take on considerably different values. However, a FRA that has zero value when discounted at the risk-free rate will have value equal to the convexity correction (i.e. a small value) if discounted at a higher rate. The impact of collateralization is significant for FRAs or swaps that either are off-market or are of long maturity.

32

prices, we can arrive at analytical expressions for (10). This, however, would come at the expense of having to calibrate additional parameters for the second process. We now consider three simple models for R(s, t1) – ls: a countably infinite-state positive jump process, a two-state continuous Markov jump process and a shifted CIR process. With a jump process, the distribution of R(s, t1) – ls is skewed so that financial institutions will usually maintain a LIBOR credit spread, but with a jump event, their credit risk and funding costs may suddenly increase dramatically. With a diffusion process, a bank would find that its credit spread would tend to gradually drift up away from LIBOR. As most banks can maintain LIBOR funding for many years, but in a period of crisis, there is fear that their credit quality may deteriorate rapidly, a jump process would seem to better capture the way one would anticipate R(s, t1) – ls would behave. Let us denote the difference R(s, t1) – ls as φ(s, t1). The positive jump model we consider is (13) )(sdJd υϕ = with φ(t1, t1) = 0; J(s) is a Poisson process with intensity λ(s)26

(14)

. In this model, φ can increase in steps of size ν, assuming values 0, ν, 2v, 3v, … In its simplest guise, one can choose λ(s) to be a deterministic function, e.g. λ(s) = λsexp(-βs) + λL(1-exp(-βs)). In this model, we have

.))](exp(1)[(),,(2

1

221 dstsstttt

t∫ −−= νλε

Collin-Dufresne and Solnik (2001) model this process as a jump with time-varying but deterministic intensity. We would like our model to have the potential to explain the stochastic behavior of the anomaly over time. One possible way to do this is to use a stochastic Poisson intensity, e.g. with λ obeying an extended Vasicek diffusion (15) dWdssbad σλλ +−= ))(( or an extended Cox-Ingersoll-Ross diffusion:

(16) .))(( dWdssbad λσλλ +−= The CIR diffusion has the advantage of restricting (with sufficiently small σ) the intensity to be positive. The introduction of stochastic intensity also naturally induces a correlation between credit spreads of different banks, providing the random processes driving their jump intensities are correlated. In this framework, the anomalies we observe will also be strongly correlated with the LIBOR risk-free spread.

As shown in the appendix, when the Poisson intensity is stochastic, the anomaly can be expressed as (17)

.))(exp(1),(]);)(|))(),([exp(*log(),;,( 00 ∫ −−≡=−−=T

s

TuvTuhtduuTuhETst λλλλε

When λ(s) obeys either the Vasicek or CIR diffusion, this equation can be reduced to a closed-form (admittedly complex) analytical solution. The solution with a CIR diffusion is derived in the first appendix. 26 We have chosen to allow the intensity to vary with time deterministically, but fixed the jump size ν to be constant. The model could have as well been extended to include time-dependent jump sizes.

33

In this paper, we will consider cases in which the t2 ≤ 2, t2 – t1 ≤ 1. In these cases, the anomaly is very well approximated by the simpler equation:

(18) .))(exp(1),(;]))(|))([*),(),;,( 00 ∫ −−≡=≈T

s

TuvTuhdutuETuhTst λλλλε

In particular, given values of a and b(s), the `short-dated’ anomaly will be quite insensitive to the value of σ (and whether the CIR or Vasicek diffusion is used). This is because the `convexity’ correction (the error term when the expectation and exponential are switched) is very small27

. This raises the question: why then bother to work with a stochastic jump intensity? Indeed, the introduction of the stochastic equations for the intensity will have negligible impact on forward curve building (as long as the underlying LIBOR rates do not have excessively long tenors), as the forward curve is primarily sensitive to the average expected credit spread. However, higher moments of the distribution of the anomaly would be needed to price instruments that depend in a non-linear fashion on the anomaly, e.g. a yield curve spread option with a payoff that is a non-linear function of the spread between 6 and 3 month LIBOR. Furthermore, given a portfolio with a valuation that was sensitive to the size of these anomalies, for risk management purposes, one would need a model that captures the stochastic nature of the anomaly.

The jump model described above has the shortcoming that it permits only increases in the spread φ. One could extend the model (e.g. as it is more generally formulated in Collin-Dufresne and Solnik (2001) to incorporate (countably infinite) negative jumps, with dφ = ν+ dJ+ (s) – ν- dJ-

(s). This model, however, would lead to spreads that were substantially negative28

In the simplest such model, let us denote the state of the obligor at time t by X(t). The two states (X = 0 and 1) are characterized by values of φ equal to 0 and ν respectively. The instantaneous probability per unit time of transitioning from state 0 to state 1 is denoted by λ; the probability per unit time of jumping back from state 1 to state 0 is given by μ. The anomaly in this case is given by

. To avoid this possibility, we consider a jump model formulated as a continuous-time finite-state Markov process, as in the model of credit risk introduced by Jarrow, Lando and Turnbull (1997).

(19) ).)exp()0)(|)((log(),,( 1021

2

1

12 dsstXduuXsptttt

t

tt

occup νε −==−= ∫∫−

poccup is the ‘occupation time’ density; the occupation time measures the cumulative amount of time (between t1 and t2) for which X equals 1. The form of this density can be derived using combinatoric arguments similar to those at the beginning of the first appendix. A derivation is presented by Pedler (1971). With this density, one has explicitly

27 Similarly, the values of short-dated bond prices in the CIR and Vasicek models are relatively insensitive to the parameter σ. 28 A slightly negative spread is plausible, as a bank could fund slightly below LIBOR, but a large negative spread is not.

34

(20)

];)))(2())(2())(()(exp(

)))((log[exp()(),,(

120121120

12

121221

12

dssttsIsttsIstt

stts

ttttttttt

−−+−−−−

−−−−−

+−−−−−=

∫−

λµλλµλµµνµλ

νλνε

I0 and I1 are the modified Bessel functions of the first kind. Note that in our two-state Markov chain credit model, we have restricted the transition probabilities to be constant. This is for purposes of analytic tractability; it is not clear how to arrive at an analytic form for the occupation time density given stochastic (or just time-varying) transition probabilities.

A third approach is to model the difference R(s, t1) – ls ,φ(s, t1), as a continuous diffusion

process rather than a jump process. One can then directly use a Vasicek or shifted CIR process to model φ. The shifted CIR model admits an analytic solution, and has the advantage, over the Vasicek model, that the variability of the spread grows as it increases. In formulating the shifted CIR model, we specify a deterministic function ξ(s) (which, in the simplest case, equals a constant) and then express φ(s, t1) = -ξ(s) + x(s, t1); x(t1, t1) = ξ(t1) (so that φ(t1, t1) = 0) and

(21) .)( dWxdsxdx σθκ +−= This leads to the following solution for ε: (22)

.2;)1))()(exp((2

)1))()(exp((2

))]1))()(exp((2log()2log(2

)[(2)(),,(

22

12

121

1212

221

2

1

σκκ

ξ

κκσκθξε

+≡−−++

−−

+−−++−+−

++−= ∫

htthhh

ttht

tthhhhtthdsstttt

t

When ξ(s) is simply a constant ξ and ,22

≤κθσ the spread φ is bounded below by –ξ; it

mean reverts to the value θ – ξ. As we expect that some institutions will be able to finance slightly below LIBOR, it is reasonable to pick a value of ξ on the order of 10-20 basis points.

The CIR++ model is typically used to model interest rates (see Brigo and Mercurio (2006)); the shift function ξ(s) is often chosen to be piecewise constant, with levels adjusted so that the model bond prices are consistent with the term structure of interest rates. Note that in this case, the anomaly ε that we would like to fit has a two-dimensional term structure, so that adjustments to ξ will not be sufficient to produce a perfect fit to the term structure of anomalies. One would hope that one could reproduce the anomaly at least quite well via adjustments to ξ(s), κ, σ and θ. A similar situation exists when these interest rate models are generalized to include a term structure of volatility (σ) and the volatility function is then adjusted to produce a best fit to the two-dimensional swaption volatility matrix.

One shortcoming of the shifted CIR diffusion model is that, to generate anomalies comparable to those observed during the crisis, the model parameters must be chosen so it is very likely that the spread above LIBOR for any given bank that initially funds at LIBOR will grow

large rather quickly. We recall that the FRA anomaly )21,

41,

41,(tη , during early 2009, ranged

between about 30 and 40 basis points. If we choose model parameters sufficient to generate a FRA anomaly of this size, there is generally an 80% probability that the spread after 9 months

35

will be more than four times larger than the anomaly. This is illustrated in Figure 9, in which we plot the ratio between the spread after 9 months (with 80% confidence) and the anomaly. This ratio falls below 4 only when we choose a large (and somewhat unrealistic) volatility parameter σ and a small mean reversion strength κ. With these values (and θ less than about 6%), however, we cannot generate a FRA anomaly above 30 basis points.

Figure 9: This figure illustrates the difficulty in generating a large anomaly in the Shifted CIR Diffusion Model without also predicting a very high (risk-neutral) probability that any bank will see its funding rate increase far above LIBOR. The plot shows the ratio between the spread above LIBOR that the bank will pay in 9 months with 80% confidence, to a particular FRA anomaly.

If the shifted CIR diffusion model were used with the more reasonable parameter values we have checked, there would be an 80% risk-neutral probability that an institution paying LIBOR would find its financing costs at least 150 to 200 basis points above LIBOR nine months later. If the diffusions for different members of the LIBOR panel were uncorrelated, then in 9 months, nearly all the members that were in the LIBOR panel would finance at rates far above LIBOR29

29 The situation would be even more extreme if they were all positively correlated. There cannot be a large degree of negative correlation between all of these diffusions, or positive definiteness of the correlation matrix would be violated.

. Our model is meant to capture the possibility that a few banks may have their credit quality deteriorate sufficiently so that they might be removed from the LIBOR pool (or at least be among the 25% of pool members with quotes so high that they are not included in the average that determines LIBOR). It seems exceedingly unlikely that nearly all of the members of the

36

LIBOR pool would be financing at more than 100 basis points above LIBOR in nine months. For this to occur, nearly all of the members of the LIBOR pool would have to be replaced; in such a dire credit environment, it is likely that other financial institutions would then be similarly distressed. In this case, many of the institutions would probably remain in the pool, and their distressed state would be reflected in a higher LIBOR-risk-free spread.

The shifted CIR diffusion that determines the anomaly is in the risk-neutral measure, while our comments about the implausibility of nearly the entire LIBOR pool financing at well above LIBOR apply in the objective (real-world) measure. Indeed, one expects a risk-adjustment (from the market price of risk) that effectively would lead to a larger value of φ in the risk-neutral measure than in the objective measure. In the risk-neutral measure, θ – ξ would need to be several hundred basis points to generate the FRA anomalies observed in 2009. In the objective measure, however, θ – ξ would have to be far smaller (much below 100 basis points) if a significant proportion of banks are to still fund at LIBOR. For this to be the case, the market price of risk would have to be very large, and it would be difficult to explain the variation in the anomaly over time through the dynamics of the parameters in the objective measure, along with a static market price of risk30

In contrast, with the jump models, one can generate large anomalies while restraining the likelihood that nearly all of the LIBOR panel will finance at rates well above LIBOR. This is possible because the size of the anomalies increases approximately proportionately with the jump-size parameter ν, but this parameter has no impact on the likelihood that a jump will occur and that an institution will finance above LIBOR. One may, however, have to choose values of ν that may seem unreasonably large (e.g. thousands of basis points) to generate large anomalies if the jump intensity is limited.

.

9. Modeling the Impact of a Liquidity Crisis. In the previous section, we computed forward rate anomalies, given models that captured

the potential excess rate of funding for a financial institution beyond LIBOR. We now model the impact on forward rate anomalies of a more severe market dislocation, in which one or several financial institutions may need to liquidate assets to obtain cash. We assume that many banks continue to quote LIBOR rates during this liquidity crisis; i.e. we presume that the quotes are truthful, indicating the interbank market is still operating and that only some banks cannot borrow and need to liquidate assets to obtain cash.

To motivate this model, let us divide the assets of a financial institution into two categories: liquid (e.g. cash, treasury securities) assets with value C(t) and illiquid assets (sub-prime mortgage backed securities, real estate, a prime brokerage unit, …) with value A(t). Some of these assets are pledged as collateral, so we have C(t) = Cpledged (t) + Cunpledged(t) and A(t) = Apledged (t) + Aunpledged(t).

30 There have been proposals to incorporate stochastic market prices of risk into interest-rate models; see Ahmad and Wilmott (2007). Duffee (2002) and Cheridito, Filipovic and Kimmel (2007) explore generalized forms of the market-price of risk in affine models, in order to provide a better fit to empirical term-structure data. A time-dependent deterministic market price of risk is also incorporated into the models used by Liu, Longstaff and Mandell (2006) to evaluate the credit and liquidity components of swap spreads.

37

The total amount of unsecured debt that creditors should be willing to extend to this institution should then be approximately

(23) ).()()(

);()()('2

'1sec

21sec

tAtCtD

tAtCtD

pledgedpledgedured

unpledgedunpledgeduredun

κκ

κκ

+=

+=.

The κ’s are haircuts; κ1' should be only slightly less than one; the secured haircuts κ1' and κ2' should be larger than the unsecured haircuts κ 1 and κ2. Furthermore, κ2 and κ2' should be considerably less than κ1 and κ1'. This is a reflection of the possibility of significantly lower recovery from the illiquid assets, due to the uncertainty in their value, their price volatility and transaction costs and bid-ask spreads incurred through liquidation.

When a liquidity crisis occurs, the value of illiquid assets plummets. Furthermore, the haircuts κ2 and κ2' will also decline substantially, as volatility, uncertainty and bid-ask spreads increase. As the capital held be other financial institutions declines, they may require a greater return on their capital, leading them to charge higher rates and demand more collateral, reducing the values of κ2 and κ2' further. At the same time, the secured debt capacity also declines as the value of pledged illiquid assets and their associated haircuts fall. A consequence of this is that the debt capacity D(t) may be less than the outstanding debt. The financial institution will at first use its remaining cash or sell its highly liquid securities to service debt; it may also pledge additional securities as collateral for new and existing additional secured debt31

There is likely a substantial cost to selling these illiquid assets. During a liquidity crisis, very few institutions will have the debt capacity that allows them to purchase these assets, i.e. convert their liquid assets to illiquid assets. There will not be a competitive market in these assets, and the institution selling an amount a of these assets will incur a loss proportional to α, which we denote as αa. As long as α < κ1 - κ2 or α < κ1' - κ2' , however, the institution will be able to increase its debt capacity by liquidating assets. The institution can also attempt to raise capital, but during a liquidity crisis this will be difficult, and will effectively entail losses similar to those incurred through selling illiquid securities.

. Debt service requirements will be more severe if the institution has a great deal of short-term debt. When these measures are exhausted, the institution will have to sell illiquid assets either to obtain cash to service debt, or to obtain cash or risk-free securities that will serve as collateral for secured debt.

In modeling liquidity, we can characterize the state of a financial institution as: 1. Normal 2. Undergoing a liquidity crisis, meaning that the institution has to sell illiquid assets to

roll over and service its debt, i.e. its debt capacity is below its outstanding debt. 3. In default. Once a liquidity crisis takes place, there are several possible outcomes. The institution

may not be able to sell a sufficient amount of illiquid assets to bring its debt capacity up to the level of its outstanding debt. If enough debt becomes due at this point, it will not be able to roll it over, and it will default.

Given a sufficient base of illiquid assets, the institution may also be able to liquidate sufficient illiquid assets so that its debt capacity exceeds its outstanding debt; it can then roll-over (or repay outstanding debt). Additionally, the value of its assets and the associated haircuts may

31 Covenants may restrict which assets can be pledged as collateral for new secured debt.

38

increase as the market emerges from a liquidity crisis. In these cases, the institution will return to the normal state from the liquidity crisis state.

In the subsequent analysis, we shall only model the impact of losses arising from the liquidation state rather than default; we shall only consider the first two states. For this reason, we refer to the models introduced in this section as ‘liquidity models’, while calling the reduced-form models in the previous section ‘credit models’. We will then assess the impact of the potential occurrence of the second state on the valuation of a FRA.

To do this, we consider again the FRA plus its hedge. Without the risk of default or a liquidity crisis, this portfolio is riskless, but if the institution that executes the hedge and the FRA enters into the second liquidity crisis state, the hedge plus FRA will include additional cash flows that ultimately affect the value of the FRA.

Define X(t) to be an indicator of whether a given institution is undergoing a liquidity crisis is present at time t; X(t) = 1 if there is a crisis at time t; otherwise X(t) = 0. We consider an institution with insufficient liquidity during a crisis, so that if X(t) = 1, then at time t, the institution will need to liquidate assets to service its debt. For every dollar it receives for these assets sold into a market in crisis, it incurs a loss of α.

Now, consider a forward rate agreement, with initial value zero, struck at the forward rate R at which at time S32

(24)

, the buyer receives the amount

)),,(

3601

),,((360 TSSSfdd

RTSSSfddFRAPayoffSTS

STS

+−

+

−+−=

+

+ .

This payoff can be hedged by

1. Depositing ),,(

3601

1

Sttfdd tS −+at time t for a term S + T -t.

2. Borrowing the same amount at time t for a term S - t. 3. Rolling over this loan minus FRAPayoff at time S, so that it matures at S + T – t ; the rate

of interest paid will be the market rate f(S,S,S+T). If we neglect the impact of illiquidity and credit risk, the value of R is determined by

considering the combination of FRA payoff and hedge, which is independent of f(S,S, S+T) and therefore riskless with value zero.

Now let us consider the impact of a liquidity crisis on the FRA payoff plus hedge. At time S, the combination of FRA payoff and hedge entails a net payment of 1- FRAPayoff (one dollar for the loan, minus the payment received from settling the FRA). If X(S) = 1, the institution cannot borrow as much as it needs, and has to instead liquidate assets at a loss. The marginal impact of the FRA plus hedge is that the institutions’ cash will be depleted by 1- FRAPayoff at time S. This cash will be unavailable to service debt, and additional illiquid securities will then have to be liquidated. This effectively leads to a loss equal to α(1 – FRAPayoff) at time S.

What happens if a liquidity crisis is in progress at both times S and S + T? The FRA plus hedge will not have any marginal impact on how much the institution has borrowed from time S to S + T, as at time S, its ability to assume new debt is restricted regardless of whether the FRA

32 Technically, the payment is two days later, but we neglect these minor details.

39

and hedge were executed. At time S + T, however, the institution will receive the original deposit

back, which will now be worth Vdeposit(S+T) = ),,(

3601

),,(360

1

Sttfdd

TSttfdd

tS

tTS

−+

+−

+ +

. This cash infusion

will reduce the amount it needs to liquidate at S + T; the institution will therefore gain αVdeposit(S+T).

What if the economy is in the normal state at time S + T, after experiencing a liquidity crisis at time S? In this case, there will be no benefit from the deposit that is returned at S + T, as the institution will not need to liquidate assets then.

A final possibility is that there is no liquidity crisis at time S but that the economy is in crisis at time S + T. At that point, the institution either receives or pays a final cash flow, equal to the difference between the deposited funds and the funds to be paid back. This amount equals

(25)

,),,(

3601

),,(360

1)),,(

3601()),,((

360 Sttfdd

TSttfdd

TSSSfddRTSSSfddtS

tTS

STSSTS

−+

+−

+++

−+−−+

−+

++

which vanishes when R equals the forward rate as computed through traditional arbitrage pricing arguments. This small cash flow will reduce or increase the amount of assets needed to be liquidated at S + T; the FRA plus hedge will realize a benefit or incur a loss equal to α times this amount.

The FRA can be valued most easily in the forward measure, with numeraire equal to the zero coupon bond maturing at S+T. The cash flows occur at time S+T, and also at time S if a liquidity crisis occurs then. Summing up these cash flows, divided by the numeraire, and multiplying by the bond price, yields an FRA price of (26)

].),,(

3601

),,(360

1))(1(

)360

1)))(()()()((1([),,(

3601

1

Sttfdd

TSttfdd

TSX

ddRTSXSXTSXSXE

TSttfdd

V

tS

tTS

STST

tTSFRA

−+

+−

++++

−++−+++−

+−

+=

+

+

+

α

α

The FRA rate f(t,S,S+T) is the value of R for which the expected value of the above quantity (in the T-forward measure) vanishes. Let P(s,t) denote the risk-neutral probability that X(s) = 1 at time s given that X(t) =0 at an earlier time t ≤s . We shall work with very simple dynamical models of X, and will make the further assumption that the dynamics of X and the dynamics of interest rates are independent33

33 This assumption is necessary because the expected value of the FRA payoff is taken in the forward measure. We prefer to specify the dynamics of X(t) in the risk neutral, rather than the forward measure, so that it is not dependent on an additional time T. We therefore need to perform a change of numeraire; this will not affect our answer, however, as the expectation of the change of numeraire term just factors out if we make an assumption of independence between X(t) and interest rates.

.

40

We then have

(27) .)),()),(1)(,((1

),(1

),,(360

1

),,(360

1

3601

STSPSTSPtSPtTSP

Sttfdd

TSttfdd

ddR

tS

tTS

STS

+++−+++

−+

+−

+=

−+

+

+

αα

This leads to an anomaly ε (see equation 9) equal to

(28)

).)),()),(1)(,((1

),(1log(

),(1

|),()),(1(|),(

),,(

STSPSTSPtSPtTSP

dvtvP

uvuPtvP

utuP

TSStTS

S

vuvu

+++−+++

++

∂∂

−−∂

∂

=+ ∫+ ==

αα

ααε

We now examine the form of the anomaly with an explicit model for the dynamics of X(t). Let us assume that X(t) is a simple two-state continuous Markov chain: the probability per unit time that the economy moves into a crisis state is λ(t); the probability that it moves from a crisis state into a ‘normal’ state is μ(t). We work in the risk-neutral measure (assuming that we can complete the market and hedge the consequences of changes in X), so that λ and μ are risk-neutral probabilities34

Let P(s,t) denote the risk-neutral probability that X(s) = 1 at time s given that X(t) =0 at an earlier time t ≤s . Then P(s,t) obeys the equation

.

(29) );(),())()(( stsPssdsdP λµλ ++−=

if X(0) = 0, this has the solution

(30) ).))()((exp(),(;),(

)(),(),( dvvvtsMdutvM

vtsMtsPs

t

s

t∫∫ +−≡= µλλ

Note that the probability of undergoing a liquidity crisis should remain relatively low for LIBOR-quoting banks, if we take this model literally, assuming that any bank undergoing a liquidity crisis will be unable to borrow funds on the interbank market. The rationale is the same as discussed with the CIR diffusion credit model; it is unlikely that most of the banks that fund at LIBOR at a given time will be unable to so after a few months. This is because, though individual banks may provide outlying LIBOR quotes, or be removed from the LIBOR panel, it seems unlikely that nearly the entire LIBOR panel will be replaced; instead, in a crisis, LIBOR quotes will just skyrocket. The two-state Markov chain liquidity model, with substantial values of α is capable of generating large anomalies, while retaining relatively small transition probabilities (λ) into the illiquid state.

The specific liquidity model we have introduced resembles the second credit model introduced in the previous section; a two-state Markov process (entailing normal and risky states) underlies both models. The impact of a transition to the risky state differs in the models, as does the form of the anomaly. In the credit model, we restricted the state transition probabilities to be constant in time to retain analytic tractability; this is not necessary in the liquidity model. 34 Bielecki and Rutkowski (2002) show how to perform a change of numeraire (which would lead to a formulation in the risk-neutral measure) in asset pricing models with Markov chain dynamics.

41

In the analysis in this section, we have assumed that an institution that has entered into the second liquidation state will be unable to borrow any funds at all, as their debt capacity was significantly below their outstanding debt. If the balance sheet of the institution were somewhat stronger, or the liquidity crisis less severe, it may be the case that the institution in question can borrow some funds, but not as much as it requires, and will still have to liquidate assets. Under this assumption, the argument that led to equation (27) must be modified. The longer-dated deposit that matures at time S+T is an illiquid asset, and increases the institution’s debt capacity by a haircut κ multiplied by the value of the deposit, therefore reducing the amount of illiquid assets that need to be sold at a loss35

(31)

. With this modification, equation (27) is replaced with

).)),()),(1)(,((1

)),(1)(1((),,(

3601

),,(360

1

3601

STSPSTSPtSPtTSP

Sttfdd

TSttfddddR

tS

tTS

STS

+++−+++−

+−

+

+−

+=

−+

+

+

αακκ

10. Fitting Risky LIBOR models to the Term Structure of Anomalies We now turn to assessing whether the credit risk and liquidity models introduced in the

previous section can produce satisfactory fits to the anomalies observed earlier in the paper. Our goal is to fit the term structure of these anomalies, i.e. anomalies that relate LIBORs spanning various tenors, to be locked in at different times in the future. This endeavor is akin to the analysis of the ability of models of the short rate, such as Vasicek, CIR, Black-Karasinski, to fit the term structure of interest rates, as well as the term structure of implied swaption and capfloor volatility.

Two possible sources of market information that determine the anomalies are FRAs and interest rate swaps. In the U.S. market, the majority of interest rate swaps have floating rate legs that make quarterly payments proportional with coupons based on 3 month LIBOR. One could assess whether the credit risk models in the previous session produce consistent prices for swaps with monthly, quarterly, bi-annual and annual floating rate coupons (based on 1, 3, 6 and 12 month LIBOR). The quality and quantity of the FRA data that was available was superior to the data for swaps with non-quarterly floating rate payments; therefore our analysis in this section is based on FRA data.

There is an extensive literature on calibration of term structure models, generally in the context of modeling interest rates. Models can be compared by assessing their fit to time series of asset prices. The standard techniques used are Kalman Filtering, applied to the calibration to the Vasicek model in Babbs and Nowman (1999) and Maximum Likelihood Estimation, used in the estimation of CIR in Chen and Scott (1993). These methods require long time series of consistent, high quality data. In our case, the data set is relatively brief (at most two years), with gaps (after the demise of Lehman Brothers, for instance) containing less reliable data.

Given the limitations imposed by the data, we do not pursue a full-blown calibration of these models to a time series of FRA and swap prices. Instead, we examine a set of two ratios that characterize the relative sizes of the anomalies for FRAs with underlying rates of different

35 In the argument leading up to (26), this increase in their assets did not matter, as we assumed that the institution’s situation was so dire that the would not be able to borrow any funds at all and that with the addition of the asset in the FRA hedge, the debt capacity in (23) would still be below their outstanding debt, and they would still be unable to borrow.

42

duration. We shall see that the three models that we have introduced make considerably different predictions about the behavior of these ratios.

In section 5, we defined the quantity η, representing the excess yield obtained on a FRA maturing at time T on a longer underlying rate τ, compared to the return on multiple FRAs with shorter underlying rates of tenor τ′. With standard arbitrage pricing theory, η should be identically zero. In terms of the anomaly ε,

(32) ∑′

=

′+′−+−+=′ττ

ττετεττη1

),)1(,(),,(),,,(i

iTiTtTTtTt .

We expect that η will increase with the ratio τ / τ' (the anomaly increases as the two LIBOR rates being compared are further apart in maturity). It is less clear how, if the ratio τ / τ′ is held constant, η will behave as τ changes.

The ratios that we consider are:

(33) ),

2,

2,(),

2,,(

)2,,,(),(2

τττηττη

ττητχ′

′′++′

′′′

≡′TtTt

TtT

with τ′ = ½ and

(34) ),

3,

3,(),

3,,(

)2,,,(),(3

τττηττη

ττητχ′

′′++′

′′′

≡′TtTt

TtT

with τ′ = ¼. The first ratio measures the relative size of the anomaly between 12 month and 6 month

LIBOR rates with the anomaly between 6 month and 3 month LIBOR rates; the second ratio compares the anomaly between 3 and 6 month LIBOR with the anomaly between 1 and 3 month LIBOR.

What behavior do the credit models introduced in the previous section predict for the FRA anomalies and these ratios? The simplest model to analyze is the jump model. Within this model, consider the expression

(35) ∑ ∫−

=

+

−+

+−−+−≡1

'

1

'

')1(

)].([)exp()))(exp()'((exp(),',,(ττ

τ

τ

λττττi

iT

iT

sEvsTviTvTtK

It follows, since as in equation 18, the convexity corrections are small, that (36) ).,',,(),',,( ττττη TtKTt ≈

Given a fixed pair (τ′, τ) , such as (τ′, τ) = (¼ ,½), if the extended CIR or Vasicek models are used to model the stochastic jump intensity, then one can adjust the target rates b(s) to match the anomaly η for different values of T. This is akin to adjusting the target rate in the extended Vasicek or CIR models of the short rate to match the term structure of bond prices. Once one has done this, however, these models have both the strength and weakness that they make quite rigid predictions of how the anomaly varies with τ′ and τ.

For the jump model, we have

43

(37) ),

2,

2,(),

2,,(

)2,,,(),(2

ττττττττχ

′′′

++′′

′′≈′

TtKTtK

TtKT .

One can re-express the ratio (38)

For jump sizes significantly less than one (10,000 basis points), the second term is very

small, and we find that

(39) ).2

exp()',(2ττχ′

−≈vT

We have verified that this holds with a high degree of precision with the exact solution with CIR intensity, given in the first appendix. The ratio therefore is generally very close to one, and nearly independent of the parameters that characterize the diffusion of the jump intensity. Similarly, in this model, the ratio χ3 is close to ¾.

This behavior is illustrated in Figure 10, in which these ratios are plotted as a function of ν, using the credit model with stochastic jump intensity, in which the jump intensity obeys a CIR process. The calculations are based on the solution for the anomaly given in Appendix 1. The plot shows this behavior for three very different sets of CIR diffusion model parameters; for all three sets, the curves line up nearly exactly, illustrating the fact that these ratios only depend significantly on the jump-size parameter. Even this dependence is mild, and for very large jump

).

)]([)exp())exp()2

(exp()]([)exp())2

exp(1(

])]([)exp()]([)exp([))2

exp(1(21

1)(2

exp(

),2

,2

,(),2

,,(

)2,,,(

2'

22

2

2

∫ ∫

∫ ∫+ ′+

+

′+

+

′+

′

′

′−′

−+′−

−

−′−

−

−′

−

=′

′′++′

′′′

ττ

ττ

τ

τ

λττλτ

λλτ

τ

τττττττ

T

T

T

T

T

T

T

T

dssEvsvvdssEvsv

dssEvsdssEvsv

v

TtKTtK

TtK

44

sizes (2,200 bp), the ratios are only modestly below 1 and ¾ respectively.

Figure 10: This figure shows that the ratios between different FRA anomalies decline very slowly with the jump size in the CIR Positive Jump Credit Model; they are extremely insensitive to the values of other parameters.

The other models do not admit such simple analytic approximations for these ratios.

The behavior of the anomaly ratios for the two-state Markov chain credit model as a function of λ and μ is illustrated in figures 11 and 12. Empirical investigation shows that these ratios are quite insensitive to the value of the jump-size parameter υ. The similarity between the plots in figures 11 and 12 is a consequence of the fact that χ2(λ,μ) is very close to (typically within .02 of) χ2(μ,λ) and χ3(λ,μ) is close to χ3(μ,λ). As a result of this approximate symmetry, one can estimate the range of possible anomaly ratios by looking at just one these two plots. These anomaly ratios range from 1 (for χ2) and .75 (for χ3) when λ and μ are very small to about .25 (for χ2) and below .2 (for χ3) when λ and μ become very large.

45

Figure 11: This figure shows that the ratios between different FRA anomalies decline substantially as λ increases in the two-state Markov chain credit model.

Figure 12: This figure shows that the ratios between different FRA anomalies decline substantially as μ increases in the two-state Markov chain credit model. Note the similarities between these plots and those in Figure 11.

46

In the remaining two models, we shall see that the ratios are primarily sensitive to a

single parameter. When the financing spread above LIBOR is modeled as a shifted CIR process, the ratios depend primarily on the strength of mean reversion of this process (κ). Figure 13 shows this dependence. For small κ, the ratios are 1 and ¾ respectively, near the values predicted by the jump model. As κ increases, these values decline, ultimately asymptotically approaching ¼ and ⅛ respectively. The plot illustrates the behavior in three cases, for different shift functions ξ(t) and mean reversion targets θ and volatilities σ; in all cases, the dependence on these parameters is so slight that it is not visible on the plots.

Figure 13: This figure shows that the anomaly ratios generated by the Shifted CIR Diffusion Credit Model vary over a wide range as the mean reversion speed changes, but are insensitive to other parameters.

In the liquidity model, these ratios (based on equations 27-30 with constant λ and μ) are sensitive to the value of μ, which determines the rate at which the economy returns to a normal state after a liquidity crisis. When it seems like the economy may linger in the illiquid state for a long time, these ratios are much larger: 2 and 1.5 respectively. With increasing values of μ these ratios decline towards values of ½ and ⅓. In this case, there is visible, but very small, dependence on the parameter λ, which characterizes the probability of transition into the illiquid state. An increase of λ by a factor of 10 causes a decrease in the ratio by about 10% when μ is small.

47

Figure 14: These figures illustrate that in the liquidity model the FRA anomaly ratios vary over a wide range as the rate at which the economy relaxes to its normal state changes; this model admits ratios that are significantly above one. The ratios are relatively insensitive to other parameters.

We now examine the empirical behavior of these ratios. Our analysis is restricted by the quality of the data set. The anomalies represent the yields of offsetting positions in FRAs, so the uncertainty in the value of the anomalies will be larger than the uncertainty in these constituent FRA rates. If the anomalies are small (e.g. 10 basis points), than any significant imprecision in their values (e.g. a couple of basis points) will lead to substantial error in the estimates of their ratios. We therefore only examine ratios on dates in which our FRA data seems reliable and the anomalies are substantial, i.e. greater than 20 basis points. This limits us to data during the first half of 2009, along with a couple of data points in 2010 during the height of the sovereign debt crisis.

We note that during this period, bid-ask spreads for FRAs with 1 and 3 month underlying rates were typically 2 basis points; they were about 4 basis points for FRAs with 6 and 12 month underlying rates. If one believed that the empirical ratios were too high (or low), exploiting this would require offsetting positions in a number of FRA contracts based on LIBORs with three different underlying maturities. The cumulative amount one would pay due to the bid-ask

48

spreads on these FRAs would preclude exploiting FRA mispricings that are identified via ‘incorrect’ ratios of anomalies, unless the observed ratios were extremely far off.

The following plots show the behavior of these ratios during portions of 2009 and 2010:

Figure 15: The first FRA anomaly ratio, as observed empirically in 2009. Note that we have included data on dates in which the anomalies were greater than 20 basis points.

Figure 16: The second FRA anomaly ratio, as observed empirically in 2009 and briefly in 2010. Note that we have included data on dates in which the anomalies were greater than 20 basis points.

49

Given the imprecision in FRA rates, deviations of 0.1 or so in these ratios are not significant. It appears that these ratios, for much of 2009, were not so far from the lower limits obtained from the liquidity model for very large values of μ. In the late spring and summer of 2009, the ratios were considerably larger, somewhat close to the values implied by the positive jump credit model and the maximum allowable ratios admitted by the shifted CIR diffusion credit model and two-state Markov chain credit model. It is clear that the positive jump credit model does not generally provide a good fit to the FRA term structure. The other three models admit solutions that are compatible with the ratios, albeit with calibrations that require different parameter values (mean reversion speed κ for the shifted CIR credit model and relaxation time μ for the liquidity model) at different dates. For the shifted CIR credit model, the values of κ compatible with the ratios are generally larger than one. This model then effectively predicts that a LIBOR-based bank is likely to have its funding costs rise significantly above LIBOR within a year. As pointed out earlier, this seems implausible, and would only occur if most of the banks within the LIBOR panel were replaced within a year.

The two-state Markov chain credit model and liquidity model both are compatible with the observed term structure ratios, with plausible albeit very different parameter values. The liquidity model is the only model that permits ratios significantly greater than one; these occur for smaller values of the parameter μ, corresponding to an illiquid state that is likely to persist for more than a year. With the exception of a few outlying data points, we do not observe ratios greater than one. Generally, the empirical data is consistent with a value of μ that is much larger than one, indicating that for the liquidity model, the typical duration of the illiquid state is short – at most, a few months. On the other hand, from May through July of 2009, the ratio χ2 was typically greater than 0.8. This would only be consistent with parameter values of μ and λ in the two-state Markov chain credit model that are less than one, indicating that though a transition to the ‘bad’ credit state was unlikely, that the model predicted that once there, a financial institution was likely to remain in that state for more than a year. In contrast, earlier in 2009, closer to the peak of the credit crisis, the ratios are consistent with higher values of μ and λ in the two-state Markov chain credit model.

The liquidity model has a distinct practical advantage. It admits relatively simple analytical solutions when the Markov transition probabilities λ and μ are allowed to be piecewise constant functions of time. This provides a great deal of flexibility in fitting the model to market data, e.g. LIBOR basis swaps of varying maturities, through bootstrapping. It seems unlikely that simple analytical solutions can be found for the anomaly in the two-state Markov chain credit model if λ and μ are allowed to vary with time.

11. Conclusion

We define and observe time series of anomalies in the pricing of Forward Rate Agreements and Eurodollar Futures contracts, relative to each other and to bank LIBOR quotes. These anomalies are substantial from August 2007 until present. Their presence means that interest rate modeling has become more complex since the advent of the credit crisis; traditional methods of bootstrapping yield curves no longer work. Our regression results show that these anomalies track the LIBOR – OIS spread. The anomalies can potentially be hedged through

50

LIBOR – OIS basis swaps, but bank credit default swaps would likely not serve as effective hedges.

The anomalies can be explained through models of credit and liquidity risk. The anomalies resulting from credit risk models are primarily a consequence of the characteristic that LIBOR is a rate with refreshed credit quality. The model of liquidity risk that we develop incorporates the cost of funding long-term assets with short-term debt, which may be costly to roll over in a liquidity crisis. From simple one-factor models of credit and liquidity risk, we can predict ratios that characterize the term structure of these anomalies. We introduce a set of simple ratios that express how these anomalies expand with increasing underlying tenor. The two models that appear to be most successful in explaining these ratios and generating the observed anomalies are the two-state Markov chain credit model and the liquidity model. Both of these models incorporate jumps from a normal state to a distressed state, and allow jumps back to the normal state. This suggests that it is important that models that capture excess (over LIBOR) liquidity or credit risk of financial institutions should incorporate both up and down jumps.

There are a number of ways that this work can potentially be extended, though many of them would be far more promising if longer time series spanning many years of very high quality data were available. This would allow one to do a more rigorous job of calibrating these models, using filtering or maximum likelihood techniques. Likewise, these models could be extended to incorporate multiple factors, or to combine credit and liquidity risk. The latter could be accomplished, for instance, by incorporating a third default state into our liquidity model. These more complex models would have more parameters, which would be very difficult to estimate without a considerably richer data set. Finally, one could explore moving beyond standard arbitrage pricing theory, applying methods used to compute or bound asset prices in incomplete markets to the problem of reconciling the values of interest rates (LIBOR) and their derivatives.

References

Acharya, A., and Pederson, L. “Asset Pricing with Liquidity Risk.” Journal of Financial Economics 77 (2005), 375-410.

Ahmad, R., and Wilmott, P. "The Market Price of Interest-rate Risk: Measuring and Modelling Fear and Greed in the Fixed-income Markets." Wilmott Magazine (January 2007), 64-70.

Alfonso, G.; Kovner, A.; and Schoar, A. “Stressed, Not Frozen: The Federal Funds Market in the Financial Crisis.” Federal Reserve Bank of New York Staff Report 437 (2010). Ametrano, M. , and Bianchetti, M. “Bootstrapping the Illiquidity: Multiple Yield Curves Construction For Market Coherent Forward Rates Estimation.” In Modeling Interest Rates: Latest Advances for Derivatives Pricing, eds Risk Books: Mercurio (2009). Andersen, L., and Piterbarg, V. “Interest Rate Modeling: Volume I: Foundations and Vanilla Models.” Atlantic Financial Press (2010).

51

Babbs, S., and Nowman, B. “Kalman Filtering of Generalized Vasicek Term Structure Models.” Journal of Financial and Quantitative Analysis 34 (1999) 115-130. Bergman, Y. “Option Pricing with Differential Interest Rates.” Review of Financial Studies 8 (1995), 475-500. Bianchetti, M. “Two Curves, One Price: Pricing & Hedging Interest Rate Derivatives Decoupling Forwarding and Discounting Yield Curves.” Working Paper (2010). Bielecki, T. “Some Remarks on Valuation and Hedging of Financial Securities with View at Funding Bases, Collateralization and Counterparty Risk.” Working Paper, draft (2011). Bielecki, T.; Cialenco, I.; and Iyigunler, I. “Counterparty Risk and the Impact of Collateralization in CDS Contracts.” Forthcoming in World Scientific Volume (2011). Bielecki, T., and Rutkowski, M. “Credit Risk: Modeling, Valuation and Hedging.” Springer (2002). Bolton, P.; Santos, J.; and Scheinkman, J. “Outside and Inside Liquidity.” NBER Working Paper w14867 (1999).

Brandt, M., and Kavajecz, K. “Price Discovery in the U.S. Treasury Market: The Impact of Orderflow and Liquidity on the Yield Curve.” Journal of Finance 59 (2004) 2623-2654. Brigo, D., and Mercurio, M. “Theory and Practice of Interest Rate Models: With Smile, Inflation and Credit.” Springer (2006). Brunnermeier, M., and Pedersen, L. “Market Liquidity and Funding Liquidity.” Review of Financial Studies 22 (2009), 2201-2238. Burgard, C. and Kjaer, M. “PDE Representations of Options with Bilateral Counterparty Risk and Funding Costs.” Working Paper (2010). Cairns, A. “Interest Rate Models: An Introduction.” Princeton University Press (2004). Canabarro, E (ed). “Counterparty Credit Risk: Measurement, Pricing and Hedging.” Risk Books (2010). Chance, D. “Hedging Deficiency in Futures Markets.” Journal of Futures Markets 26 (2006), 189-207.

Chen, R., and Scott, L. “Maximum Likelihood Estimation for a Multifactor Equilibrium Model of the Term Structure of Interest Rates.” Journal of Fixed Income 3 (1993) 14-31.

52

Collin-Dufresne, P., and Solnik, B. "On the Term Structure of Default Premia in the Swap and LIBOR Markets." Journal of Finance 56 (2001), 1095-1115. Cox, J.; Ingersoll, J.; and Ross, S. "A Theory of the Term Structure of Interest Rates." Econometrica 53 (1985), 385–407. Duffie, D., and Huang, M. “Swap Rates and Credit Quality.” Journal of Finance 51 (1995), 921-949. Duffie, D., and Singleton, K. “Modeling Term Structure of Defaultable Bonds.” Review of Financial Studies 12 (1999), 687-720.

Ericsson, J., and Renault, O. “Liquidity and Credit Risk.” Journal of Finance 61 (2006), 2219-2250.

Fujii, M.; Shimada, Y.; and Takahashi, A. “Modeling of Interest Rate Term Structures Under Collateralization and its Implications.” Working Paper (2010). Fujii, M., and Takahashi, A. “Asymmetric and Imperfect Collateralization, Derivative Pricing, and CVA.” CARF Working Paper (2010). Grinblatt, M. “An analytic solution for interest rate swap spreads.” International Review of Finance 2 (2001), 113–149. Grinblatt, M., and Jegadeesh, N. “The Relative Pricing of Eurodollar Futures and Forward Contracts.” Journal of Finance 51 (1995), 1499-1522. Gupta, A., and Subrahmanyam, M. "An empirical examination of the convexity bias in the pricing of interest rate swaps," Journal of Financial Economics 55 (2000), 239-279. He, Z., and Xiong, W. “Rollover Risk and Credit Risk.” Working Paper (2010). Heider, F., and M. Hoerova. “Interbank Lending, Credit-Risk Premia, and Collateral.” International Journal of Central Banking 5 (2009), 5–43.

Heider, F.; Hoerova, M.; and Holthausen, C. “Liquidity Hoarding and Interbank Market Spreads: The Role of Counterparty Risk.” European Central Bank Working Paper 1126 (2009). Henrard, M. “The Irony in the Derivatives Discounting.” Working Paper (2007). Henrard, M. “Overnight Indexed Swaps and Floored Compounded Instrument in HJM One-Factor Models.” Working Paper (2008).

53

Henrard, M. “The Irony in the Derivatives Discounting Part II: The Crisis.” Working Paper (2009). Hunt, P. , and Kennedy, J. “Financial Derivatives in Theory and Practice.” Wiley (2000). Jarrow, R.; Lando, D.; and Turnbull, S. “A Markov Model for the Term Structure of Credit Risk Spreads.” Review of Financial Studies 10 (1997), 481-523. Johannes, M., and Sundaresan, S. “The Impact of Collateralization on Swap Rates.” Journal of Finance 62 (2007), 383-410. J.P. Morgan Chase Annual Report, 2009. Kehoe, Patrick J.; V. V. Chari; and Lawrence J. Christiano. “Facts and Myths about the Financial Crisis of 2008.” Working paper, Federal Reserve Bank of Minneapolis (2008). Kenyon, Chris. “Short-Rate Pricing after the Liquidity and Credit Shocks: Including the Basis.” Working Paper (2010). Kijima, M.; Tanaka, K.; and Wong, T. “A Multi-Quality Model of Interest Rates.” Quantitative Finance 9 (2009), 133-145. Lesmond, D.; Chen, L.; and Wei, J. “Corporate Yield Spreads and Bond Liquidity. “14th Annual Conference on Financial Economics and Accounting (2005). Liu, J.; Longstaff, F.; and Mandell, R. “The Market Price of Risk in Interest Rate Swaps: The Role of Default and Liquidity Risks.” Journal of Business 79 (2006), 2337-2359.

Longstaff, F.; Mithal, S.; and Neis, E. “Corporate Yield Spreads: Default Risk or Liquidity? New Evidence from the Credit Default Swap Market.” Journal of Finance 5 (2005), 2213-2253. Mercurio, F. “Interest Rates and the Credit Crunch: New Formulas and Market Models.” Working Paper (2009). Michaud, F., and Upper, C. “What Drives Interbank Rates? Evidence from the LIBOR Panel.” BIS Quarterly Review (March 2008), 47-58. Mollenkamp, C. “Bankers Cast Doubt on Key Rate Amidst Crisis.” Wall Street Journal (April 16, 2008). Morini, M. “Solving the Puzzle in the Interest Rate Market.” Working Paper (2009). Morini, M. and Prampolini, A. “Risky Funding: A Unified Framework for Counterparty and Liquidity Charges.” Working Paper (2010).

54

“Bank Lending Seizes Up Worldwide.” New York Times (September 30, 2008). Pallavicini, A., and Tarenghi, M. “Interest Rate Models with Multiple Yield Curves.” Working Paper (2010). Pedler, P. J. “Occupation Times for Two State Markov Chains.” Journal of Applied Probability 8 (1971), 381-390. Piterbarg, V. “Funding beyond Discounting: Collateral Agreements and Derivative Pricing.” Risk Magazine (2010), 97-102. Pykhtin, M (ed). “Counterparty Credit Modelling: Risk Management, Pricing and Regulation.” Risk Books (2005). Politi, M.; Iori, G.; Germano, G.; and Gabbi, G.. “The Overnight Interbank Market. Simple Statistics and Facts Before and During the Credit Crisis.” Working Paper (2010). Schwarz, K. “Mind the Gap: Disentangling Credit and Liquidity in Risk Spreads.” Working Paper (2009). Staum, J. “Incomplete Markets.” Handbook in Operations Research and Management Science, Financial Engineering 15 (2007), 511-563. Sundaresan, S. “Futures prices on Yields, Forward Prices, and Implied Forward Prices from Term Structure.” Journal of Financial and Quantitative Analysis 26 (1991), 409-424.

Appendix I: In this appendix, we derive the formulas for the anomaly given that the credit

deterioration process R(s,t) – ls = φ(s,t) is characterized by a jump process with jump size ν and stochastic intensity λ. Expressions for the anomaly can be derived when the stochastic intensity obeys an extended Vasicek-type diffusion: (1) dWdssbad σλλ +−= ))(( as well as an extended Cox-Ingersoll-Ross diffusion:

(2) .))(( dWdssbad λσλλ +−= The anomaly that we will calculate is given by the expression:

(3) )())])(|),([exp(*log(),;,( 010 tsTtdstsETstT

s

≥>=−−= ∫ λλϕλε .

We simplify this by noting that the probability density for the occurrence of k jumps at times t1, … tk between s and T is

55

(4) !

))(exp()(),...,( 1

1 k

duutttp

k

i

T

si

k

∏ ∫=

−=

λλ.

We can then write

(5) ∑ ∑∫ ∫ ∫∞

= =

=−−−=1

01

1210 ]))(|)(exp(),...,(..[*log(),;,(k

k

iik

T

s

T

s

T

sk ttkTvttpdtdtdtETst λλλε .

This expression simplifies to

(6) .)(exp(1),(]);)(|))(),([exp(*log(),;,( 00 ∫ −−≡=−−=T

s

TuvTuhtduuTuhETst λλλλε

We will first derive an expression for the quantity ε(λ0,t;t,T). When the diffusion for λ is given by the Vasicek or CIR model, then this expression can be computed via a change of variables and application of the Feynman-Kac theorem. In particular, we work with the CIR model in the remainder of this appendix, first defining )(),(),( uTuhTuq λ≡ ; then

(7).),(),(;)),(ln()()(),(

;),(),(

qTuhuqgqu

Tuhauhuabuqf

dWuqgduuqfdq

≡∂

∂+−+≡

+= σ

The Feynman-Kac theorem implies that the quantity

(8) ∫ −==−=T

t

TtTthPTttqtqduTuqETtqP )),),,((log(),;,(;])(|)),([exp(*),,( 0000 λλε

is the solution of the partial differential equation

(9) .1),,(;),(21),( 002

0

2

022

00 ==

∂∂

+∂∂

+∂∂ TTqPPq

qPtqg

qPtqf

tP σ

This equation is solved for P in the affine form:

(10)

.12

),()),(ln(

);,(),()(

));,(),(exp(),,(

22

00

=−∂

∂−−

∂∂

−=∂∂

−−=

ATthAt

TthatA

TtATthtabtB

TtBqTtATtqP

σ

The equation for A is a Riccati equation, which can be solved by defining u(t,T) such that

(11) );,(),(2

)(ln 2

TtATtht

tu σ−≡

∂∂ .0

2'"

2

=−− huauu σ

The solution for u can be expressed in terms of Bessel functions of the first kind. This is apparent if the following changes of variables are applied:

(12) .);()(

2);2

)(exp(

vamzuzzw

vtTvz

m ≡≡

≡−

−≡

−

σββ

The function w satisfies the Bessel equation

56

(13) .;0)('" 22222 mwzzwwz +≡=−++ βαα

Working backwards, applying the boundary condition that A(T,T) = 0 and defining τ≡T-t, we find that (14)

.)()(

22

)(2

2)(

;))

2exp(()

2exp((

))2

exp(()2

exp(()

2exp(22),(),(

1

22

22

1

11

2

22

ββσ

σ

βσ

σβ

τβτβ

τβτβτσσ

σ

αα

αα

αα

αα

−−−

+

−

−−+

+++

++−

≡Γ

−Γ+−

−Γ+−−−−

++−=

JJaa

JaaJ

vJvJ

vJvJvaaTtATth

In the above equation, Jα is the Bessel function of the first kind of order α. To derive this equation, we use the relationships between Bessel functions:

(15) )).()((

21)('

));()((2

)(

11

11

zJzJzJ

zJzJvzzJ

vvv

vvv

+−

+−

−=

+=

B(t,T) is obtained by integrating the product of ab(t) and h(t,T)A(t,T); when the function b(t) is piecewise constant B(t,T) can be expressed in terms of logarithms of linear combinations of Bessel functions . With b(t) = b, the expression for B(t,T) is

(16) ).)()(

))2

exp(())2

exp((ln(2),( 22

2

ββ

τβτβ

σστ

αα

αα

−

−

Γ+

−Γ+−+−=

JJ

vJvJabbaTtB

With A and B, we have a precise expression for ε(λ0,t;t,T). To derive formulas for the more general expression ε(λ0,t;s,T) we begin with

(17)

)).),(),(exp())(|)((),(exp(

)],(),(|)),([exp(*))(|)((

)],(),(|)),([exp(*

1011

1011

0

λλλλλλ

λλλλλλ

λ

TshTsAtspdTsB

TshTsqduTuqEtspd

TthTtqduTuqE

T

s

T

s

−==−−

==−==

==−

∫

∫ ∫

∫

The final expression depends on the Laplace transform of the Cox-Ingersoll-Ross transition density. This is well-known (see Cox, Ingersoll and Ross (1985) or Cairns (2004) and can be derived using a variant of the Feynman-Kac theorem. When the target rate, b(t) = b, then

57

(18)

).2)))(exp(1(

2ln(2),,(

,))(exp()2(

2),,(

));,,(),,(exp())exp())(|)((

22

22

01011

atsakaabstkD

tsaka

astkC

stkDstkCktspd

+−−−−=

−−+=

−−=−==∫

σσ

σσ

λλλλλλλ

The extension of the above expressions to piecewise constant forms of b(t) is straightforward. Combining the above expressions, we find the anomaly equals

(19) ).,),,(),((),(),),,(),((),(),;,( 00 stTshTsADTthstTshTsACTsBTst ++= λλε

This expression is relatively straightforward to code into any library (e.g. Matlab) that includes routines for evaluation of Bessel functions. There are two potential difficulties in doing this, however.

The first problem occurs on a parameter set of measure zero: when α is an integer. In this case, Jα(z) = (-1)αJ-α(z). It follow that Γ=(-1)α+1 and the numerator and denominators in the latter part of the expression for hA are both zero. In this case, the solution can be expressed as linear combinations of Bessel functions of the first and second kind, or equivalently, the limit as α approaches an integer value can be computed, without taking the limit for the numerator and denominator separately.

One can also run into trouble numerically evaluating limiting cases of these expressions, when the parameters α and β become very large. In these cases, one may have to use asymptotic expansions for Bessel functions to arrive at expressions for the anomaly. For the parameter values that we used in this study, this was not necessary. Appendix II:

We argue in this appendix that the observed FRA-cash anomaly cannot be explained by a simple credit model in which bank-lending is credit risky, but LIBOR is not modeled as a refreshed rate. We use the RMV framework and exploit the fact that observed discrepancies between rates inferred from Eurodollar futures and FRAs are very small. We present a plausibility argument, rather than a rigorous proof, as our demonstration depends on assumptions about the sign of certain covariance terms that are reasonable, but could possibly be violated in a model with particularly unusual and artificial dynamics.

As in the body of the paper: r denotes the risk-free rate, l denotes the instantaneous rate at which a bank that pays LIBOR can borrow and R(u, t) denotes the instantaneous rate paid at time u by a bank that paid LIBOR at time t.

We begin by deriving bounds on particular covariance terms in terms of the Eurodollar futures – FRA basis. These bounds hold regardless of whether LIBOR is a refreshed rate. They are predicated on the following assumption:

(1) .0)]),(exp(),cov[exp(2

1

1

1 ≥−− ∫∫t

t

t

ts dstsRdsr

This inequality is plausible, i.e. since lower risk-free rates at time t1 are more likely to accompany higher risky bond prices at t1.

58

The standard formula for valuation of Eurodollar contracts still holds in the case in which LIBOR is risky36

(2)

; the futures rate equals the expected value of LIBOR in the risk-neutral measure. If we express all rates using continuous compounding, then the difference between the yield inferred from a Eurodollar futures contract maturing at time t1 on a rate of tenor t2 – t1, and the corresponding FRA rate is:

))][exp(*

)]),([exp(*)]]),([exp(*)[exp(*log(1),,(

1

2

1

2

1

1

11

1221

∫

∫ ∫∫

−

−−

−=− t

ts

t

t

t

t

t

ts

FRAEDFut

dsrE

dstsREdstsREdsrE

tttttδ

.

This can be re-expressed as

(3)

))][exp(*

)]),([exp(*)]),(exp(),cov[exp(

)]),(exp(),),(cov[exp(1log(1),,(

1

2

12

1

1

2

1

2

1

1

1

1112

21

∫

∫∫∫

∫∫

−

−−

+−−−

=−

t

ts

t

tt

t

t

ts

t

t

t

tFRAEDFut

dsrE

dstsREdstsRdsr

dstsRdstsRtt

tttδ

The first covariance in the above expression is negative (this follows from Jensen’s inequality). We have assumed that the second covariance term is positive. In this case, both terms containing covariances have the same sign37

(4)

. It then follows that

),,()()][exp(*

)]),([exp(*)]),(exp(),cov[exp(

);,,()()]),(exp(),),(cov[exp(

2112

1

1

211211

1

2

12

1

1

2

1

2

1

tttttdsrE

dstsREdstsRdsr

tttttdstsRdstsR

EDFutFRAt

ts

t

tt

t

t

ts

EDFutFRA

t

t

t

t

−

−

−≤

−

−−

−≤−

∫

∫∫∫

∫∫

δ

δ

Note that the expression for δ does not vanish even if the risk-free and risky rates are the same. In this case, it reduces to the much-studied convexity correction applied to Eurodollar futures prices, relating covariance terms to the futures-forward basis, as in Grinblatt and Jegadeesh (1995).

Given this, let us examine the expression for the FRA – cash LIBOR basis; i.e. the discrepancy between forward rates inferred from FRAs and rates inferred from the current values of LIBOR. If, as above, the rates are expressed using continuous compounding, this basis is

36 We assume that the risk of default on the Eurodollar futures contract can be neglected (a reasonable assumption, given that it is traded on an exchange). 37 Even if the second covariance term is negative, it would be implausible that the covariance terms would be large (more than a few basis points), as they would have to cancel each other very precisely to leave only a small residual difference between yields inferred from FRAs and yields from Eurodollar futures.

59

(5) ).

)]),([exp(*)][exp(*

)]),([exp(*)]]),([exp(*)[exp(*log(1

),,(

21

2

1

11

1

12

21

∫∫

∫ ∫∫

−−

−−−

−

=−

t

t

t

ts

t

t

t

t

t

ts

FRACashLIBOR

dstsREdsrE

dstsREdstsREdsrE

tt

tttδ

Now let us assume that LIBOR is not modeled as a refreshed rate, so that all institutions that can borrow at LIBOR at a time t will be able to borrow at LIBOR at a later time s>t, i.e. R(s, t) = ls represents the instantaneous rate of borrowing of all ‘LIBOR-banks’ and is independent of t. In this case, the expression above can be manipulated into the form (6)

).)][exp(*

))exp(),cov(exp(

)][exp(*)][exp(*

))exp(,)cov(exp())exp(),cov(exp(

)][exp(*

))exp(,)cov(exp())exp(),cov(exp(

)][exp(*

))exp(),cov(exp()][exp(*1log(1

),,(

2

2

1

1

12

12

1

2

1

1

1

2

1

2

1

2

1

1

1

2

1

12

1

12

21

∫

∫∫

∫∫

∫∫∫∫

∫

∫∫∫∫

∫

∫∫∫

−

−−

−

−−

−−−

−

−

−−−

+

−

−−

+−

=−

t

ts

t

ts

t

ts

t

ts

t

ts

t

ts

t

ts

t

ts

t

ts

t

ts

t

ts

t

ts

t

ts

t

ts

t

ts

t

ts

t

ts

t

ts

FRACashLIBOR

dslE

dsldsl

dsrEdslE

dsldsldsldsr

dsrE

dsldsldsldsr

dsrE

dsldsrdslE

tt

tttδ

Substituting in the inequalities derived above, we have38

(7)

).)][exp(*

))exp(),cov(exp(

)][exp(*)][exp(*

))exp(,)cov(exp(),,()(

)][exp(*

),,()(),,()(

1)),,()exp((),,(

2

2

1

1

2

1

2

12

1

2

1

2112

2122

122112

12

211221

∫

∫∫

∫∫

∫∫

∫

−

−−

−

−

−−

−

+−

+−≤

−−−

≤

−

−−

−−

t

ts

t

ts

t

ts

t

ts

t

ts

t

ts

t

tsEDFutFRA

t

ts

EDFutFRAEDFutFRA

FRACashLIBORFRACashLIBOR

dslE

dsldsl

dslEdslE

dsldslttttt

dslE

tttttttttt

tttttttttt

δ

δδ

δδ

38 The first inequality assumes that the cash LIBOR – FRA spread is positive; during the credit crisis it was large and positive.

60

In our observations, the cash LIBOR – FRA spread exceeded 100 basis points and during the height of the crisis was well over a factor of 10 greater than the FRA – Eurodollar Futures spread. These results are not consistent with the above equation. With a 3 month underlying Eurodollar futures rate, the first and third terms on the right hand side are exactly or approximately equal to one-fourth of the FRA-Eurodollar Futures spread. The second term is far smaller. The covariance in the fourth term is likely to be positive, so this term should likely reduce, rather than add to the cash LIBOR – FRA spread.39

It therefore follows, at least within this modeling framework, that LIBOR must be treated as a refreshed rate to explain the FRA – cash LIBOR spread.

39 The covariance in the fourth term as well is unlikely to be an order of magnitude larger than

∫∫ −2

1

2

1

))exp(,)cov(exp(t

ts

t

ts dsldsl , which we have argued is bounded by the Eurodollar – Futures spread.