Journal Of Investment Management, Vol. 13, No. 1, (2015), pp. 64–83 © JOIM 2015 JOIM

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OIS DISCOUNTING, INTEREST RATE DERIVATIVES, AND THE MODELING OF STOCHASTIC INTEREST RATE SPREADS John Hull a,∗ and Alan White a,†

Before 2007, derivatives practitioners used a zero curve that was bootstrapped from rates to provide “risk-free” rates when pricing derivatives. In the last few years, when pricing fully collateralized transactions, practitioners have switched to using a zero curve bootstrapped from (OIS) rates for discounting. This paper explains the calculations underlying the use of OIS rates and investigates the impact of the switch on the pricing of plain vanilla caps and swap options. It also explores how more complex derivatives providing payoffs dependent on LIBOR, or any other , can be valued. It presents new results showing that they can be handled by constructing a single tree for the evolution of the OIS rate.

1 Introduction The use of LIBOR as the risk-free rate was called into question by the credit crisis that started in Before 2007, derivatives dealers used LIBOR, the mid-2007. became increasingly reluctant short-term borrowing rate of AA-rated financial to lend to each other because of credit concerns. institutions, as a proxy for the risk-free rate. The As a result, LIBOR quotes started to rise relative zero- was boot-strapped from to other rates that involved very little . LIBOR swap rates. One of the attractions of this The TED spread, which is the spread between was that many interest rate derivatives use the three-month U.S. dollar LIBOR and the three- LIBOR rate as the reference interest rate, so these month U.S. Treasury rate, is less than 50 basis instruments could be valued using a single zero points in normal market conditions. Between curve. October 2007 and May 2009, it was rarely lower than 100 basis points and peaked at over 450 basis aJoseph L. Rotman School of Management, University of points in October 2008. Toronto, 105 St George Street, Toronto M5S 3E6, Canada. ∗Tel.: 416 978 8615; E-mail: [email protected]. These developments led the market to look for †Tel.: 416 978 3689; E-mail: [email protected]. an alternative proxy for the risk-free rate.1 The

64 First Quarter 2015 OIS Discounting,Interest Rate Derivatives, and the Modeling of Stochastic Interest Rate Spreads 65 standard practice in the market now is to deter- the pricing of interest rate swaps. We go one step mine discount rates from overnight indexed swap further by quantifying the impact of OIS discount- (OIS) rates when valuing all fully collateralized ing on several different interest rate derivatives in derivatives transactions.2 Both LIBOR and OIS different situations rates are based on interbank borrowing. However, the LIBOR zero curve is based on borrowing rates It might be thought that the switch from LIBOR for periods of one or more months, whereas the to OIS discounting simply results in a change to OIS zero curve is based on overnight borrowing the discount rate while expected payoffs from an rates.As discussed by Hull andWhite (2013a), the interest rate remain unchanged. This is for overnight interbank borrowing is not the case. Forward LIBOR rates and forward less than that for longer term interbank borrowing. swap rates also change. One of the contributions As a result the credit spread that gets impounded of this paper is to examine the relative impor- in OIS swap rates is smaller than that in LIBOR tance of discount rate changes and forward-rate swap rates. changes to the valuation of interest rate deriva- tives in different circumstances. We first discuss LIBOR incorporates a credit spread that reflects how the LIBOR zero curve is bootstrapped when the possibility that the borrowing may LIBOR discounting is used and when OIS dis- default, but this does not create credit risk in a counting is used. This allows us to show how LIBOR swap. The reason for this is that the swap the transition from LIBOR discounting to OIS participants are not lending to the banks that are discounting affects the forward rates and the val- providing the quotes from which LIBOR is deter- uation of LIBOR swaps. We then move on to mined. Therefore, they are not exposed to a loss consider how the standard market models used from default by one of these banks. LIBOR is to price interest rate caps and swap options are 3 merely an index that determines the size of the affected. payments in the swap. The credit risk, if any, This paper then considers how nonstandard in the swap is related to the possibility that the instruments which provide payoffs dependent on counterparty may fail to make swap payments LIBOR can be valued. It presents new results when due. The LIBOR swaps traded in the inter- showing that it is possible to accommodate OIS dealer market are now cleared through central discounting with a single interest rate tree describ- counterparties, which require both initial ing the OIS zero curve and its evolution. It and variation margin. The counterparty credit risk examines the impact of correlation between the in the swaps that are traded today can therefore OIS rate and the LIBOR–OIS spread on the reasonably be assumed to be zero. pricing of Bermudan swap options.

Changing the risk-free discount curve changes the values of all derivatives. In the case of derivatives 2 Background other than interest rate derivatives, implement- ing the change is usually straightforward. This In this section, we review the procedures for paper focuses on how the switch from LIBOR a riskless zero curve from LIBOR to OIS discounting affects the pricing of inter- swap rates. We start by examining how bonds est rate derivatives. Other papers such as Smith and swaps are priced. This will help to introduce (2013) have examined the nature of the calcula- our notation and provide the basis for our later tions underlying the use of OIS discounting and discussion of the impact of OIS discounting.

First Quarter 2015 Journal Of Investment Management 66 John Hull and Alan White

2.1 Interest rates and discount prices LIBOR interest rate for the period between Ti and Ti+ . Let z(T ) be the continuously compounded risk- 1 free zero-coupon interest rate observed today for The key to valuing one leg of an maturity T . The price of a risk-free discount bond is the result that, when the numeraire asset is the that pays $1 at time T is P(T ) = exp[−z(T )T ].A risk-free discount bond maturing at time Ti+1, the common industry practice is for a expected future value of any interest rate (not nec- yield to be used for discount bonds with a maturity essarily a risk-free interest rate) between Ti and 4 of one year or less. This means that Ti+1 equals the current forward interest rate (i.e. 1 the interest rate that would apply in a forward P(T ) = (1) rate agreement). This means that in a world where 1 + R(T )T interest rates are stochastic, we can use P(Ti+1) as where R(T ) is the money market yield for matu- the discount factor, providing we also assume that T rity so that the expected value of Ri equals the forward inter- − P(T ) est rate, F(Ti,Ti+ ). The value, Si(K), of the leg R(T ) = 1 1 P(T )T (2) of the swap that we are considering is therefore

Consider a in which we agree to Si(K) = L[F(Ti,Ti+1) − K] buy or sell at time Ti a discount bond maturing at × (Ti+1 − Ti)P(Ti+1) time Ti+1. Simple arguments show that the for this contract, the contract where L is the notional principal. delivery price at which the forward contract has A standard interest-rate swap is constructed of P(T )/P(T ) zero value, is i+1 i . many of these legs in which the end date for one 5 If Ti+1 −Ti is less than or equal to one year we can leg is the start date for the next leg. We define T T i also define a money market on 0 as the start date of the swap and i as the th the forward bond. This is the forward interest rate. payment date (1 ≤ i ≤ M). The total swap value It is the rate of interest that must apply between is the sum of the values for all the individual legs. times Ti and Ti+1 in order for the price at Ti of a If the start date for the first leg of the swap is time T = discount bond maturing at Ti+1 to be equal to the zero ( 0 0), the swap is a spot start swap. If forward price. The forward rate is the start date for the first leg of the swap is in the T > P(T ) − P(T ) future ( 0 0), the swap is a forward start swap. i i+1 M F(Ti,Ti+1) = (3) The total value of the legs of the swap is (Ti+1 − Ti)P(Ti+1) M−1 S(K, M) = L [F(Ti,Ti+1) − K] 2.2 Interest-rate swaps i=0

Consider one leg of an interest rate swap in which × (Ti+1 − Ti)P(Ti+1) (4) a floating rate of interest is exchanged for a spec- Using Equation (3) to replace the forward rates6 ified fixed rate of interest, K. The start date is this is Ti and the end date is Ti+1. On the start date we M−1 observe the rate that applies between Ti and Ti+ .  1 S(K, M) = L [P(T ) − P(T )]− For a swap where LIBOR is received and the fixed i i+1 LKA rate is paid, there is a payment on the end date i=0 equal to (Ri − K)L(Ti+1 − Ti) where Ri is the (5)

Journal Of Investment Management First Quarter 2015 OIS Discounting,Interest Rate Derivatives, and the Modeling of Stochastic Interest Rate Spreads 67

where T0 and ends at TM is M−1 M−1 A = P(Ti+1)(Ti+1 − Ti) SLD(K, M) = L [PLD(Ti) − PLD(Ti+1)] i=0 i=0 is the annuity factor used to determine the present − LKALD (7) value of the fixed payments on the swap. where The breakeven swap rate for a particular swap is M−1 the value of K such that the value of the swap is ALD = PLD(Ti+1)(Ti+1 − Ti) (8) zero. Using Equations (4) and (5) this is i=0 M− K = K P (T ) 1 [P(T ) − P(T )] The swap value is zero when M.If LD i i=0 i i+1 i M − A or is known for all from 0 to 1 it follows that  (6) when the swaps considered start at time zero M−1 F(Ti,Ti+ )(Ti+ − Ti)P(Ti+ ) i=0 1 1 1 PLD(TM−1) + SLD(KM,M− 1)/L A PLD(TM) = 1 + KM(TM − TM−1) For a forward start swap, the breakeven swap rate (9) is called the forward swap rate. For a spot start T = swap, the breakeven swap rate is simply known When 0 0 Equation (9) allows the discount P (T ), P (T ),... as the swap rate. We denote the swap rate for a bond prices LD 1 LD 2 to be deter- swap whose final payment date is TM as KM. mined inductively. The discount bond prices can then be turned into zero rates. The zero rate for any date that is not a swap payment date is deter- 2.3 Notation mined by interpolating between adjacent known zero rates. Forward rates can be determined from In what follows we will use the subscript LD Equation (3). to denote quantities calculated using LIBOR discounting and the subscript OD to denote quan- tities calculated using OIS discounting. For exam- 2.5 Bootstrapping LIBOR with ple, PLD(T ) and POD(T ) are the prices of risk-free OIS discounting discount bonds providing a payoff of $1 at time If OIS swap rates are assumed to be riskless, T when LIBOR discounting and OIS discounting the riskless zero curve is bootstrapped from OIS are used, respectively. Also, F (Ti,Ti+ ) and LD 1 swap rates. The procedure is similar to that just F (Ti,Ti+ ) are forward LIBOR rates between OD 1 given for bootstrapping LIBOR zero rates where times Ti and Ti+ when LIBOR discounting and 1 LIBOR rates are assumed to be riskless. One point OIS discounting are used, respectively. to note is that OIS swaps of up to one-year’s matu- rity have only a single leg resulting in a single 2.4 Bootstrapping LIBOR with LIBOR payment at maturity. discounting If the zero curve is required for maturities longer When LIBOR is assumed to define riskless rates, than the maturity of the longest OIS swap, a natu- swap rates can be used to bootstrap the LIBOR ral approach is to assume that the spread between zero curve. From Equation (5), when LIBOR dis- the OIS swap rates and the LIBOR swap rates is counting is used, the value of a swap that starts at the same for all maturities after the longest OIS

First Quarter 2015 Journal Of Investment Management 68 John Hull and Alan White maturity for which there is reliable data. An alter- just a convenient tool for calculating the expected native approach for extending the OIS zero curve values of future LIBOR when OIS discounting is is to use basis swaps where three-month LIBOR is done. exchanged for the average plus Equation (6) shows that the forward swap rate a spread. These swaps have maturities as long as for a swap with start date T and payment dates 30 years in the U.S.7 0 T1,T2,...,TM is  Under OIS discounting, in order to determine M−1 F (Ti,Ti+ )(Ti+ − Ti)P (t, Ti+ ) the value of swaps and other derivatives whose i=0 OD 1 1 OD 1 A payoffs are based on LIBOR it is necessary to OD determine the expected future LIBOR rate for the (12) period between Ti and Ti+1 when the numeraire where asset is a zero-coupon OIS bond maturing at M−1 T time i+1. These are the forward LIBOR rates, A = P (Ti+ )(Ti+ − Ti) (13) F (T ,T ) OD OD 1 1 OD i i+1 (i.e., the mid market rates that i=0 would apply in forward rate agreements when the is the annuity factor used to determine the present market uses OIS discounting). value of the fixed payments on the swap. The bootstrapping process to determine the FOD’s is straightforward. The value of a swap in which 3 OIS discounting and LIBOR rates floating is received and a fixed rate of K is paid, with start date T0 and payment dates To explore how LIBOR zero rates change when T1,T2,...,TM is OIS discounting is used, we consider the case M−1 in which the zero curve is bootstrapped from swap market data for semi-annual pay swaps with SOD(K, M) = L [FOD(Ti,Ti+1) − K] i=0 maturities from six months to 30 years using the procedures outlined in the previous section. × (Ti+ − Ti)P (Ti+ ) (10) 1 OD 1 Three different sets of LIBOR swap rates are This is zero when T0 = 0 and K = KM. It follows considered: that when the swaps considered start at time zero (1) 4 to 6: The swap rate is 4% + 2% × Swap F (T ,T ) OD M−1 M Life/30. S (K ,M− )/L (2) 5 Flat: The swap rate is 5% for all maturities. = K − OD M 1 M (11) (3) 6 to 4: The swap rate is 6% − 2% × Swap (TM − TM−1)POD(TM) Life/30. Since SOD(K1, 0) = 0 the forward rates are determined sequentially starting with M = 1. Once all forward LIBOR rates are determined, We assume that the OIS swap rates equal the LIBOR swap rates less 100 basis points. (We the LIBOR discount bond prices for maturity Tj can be calculated as tried other spreads between the OIS and LIBOR swap rates and found that results are roughly j−1 −1 proportional to the spread.) {[1 + FOD(Ti,Ti+1)](Ti+1 − Ti)} i=0 Table 1 shows how much the LIBOR zero curve is The zero rates can be determined from the dis- shifted when we switch from LIBOR discounting count bond prices. The zero rates so calculated are to OIS discounting for the three term structures

Journal Of Investment Management First Quarter 2015 OIS Discounting,Interest Rate Derivatives, and the Modeling of Stochastic Interest Rate Spreads 69

Table 1 The basis point increase in LIBOR zeros Table 2 The basis point increase in three-month caused by switching from LIBOR discounting to OIS LIBOR forward rates caused by switching from discounting. LIBOR discounting to OIS discounting. LIBOR rate Term Term Term Forward rate Term Term Term maturity structure structure structure maturity structure structure structure (Yrs) 4 to 6 flat 6 to 4 (Yrs) 4 to 6 5 flat 6 to 4 0.5 0.0 0.0 0.0 0.5 0.0 0.0 0.0 1 0.0 0.0 0.0 1 0.0 0.0 0.0 3 –0.1 0.0 0.1 3 –0.4 0.0 0.4 5 –0.3 0.0 0.3 5 –1.0 0.0 1.0 7 –0.6 0.0 0.6 7 –2.0 0.0 2.1 10 –1.3 0.0 1.3 10 –4.6 0.0 4.4 15 –3.5 0.0 3.2 15 –13.0 0.0 10.7 20 –7.6 0.0 6.1 20 –31.7 0.0 19.5 25 –15.7 0.0 9.7 25 –76.8 0.0 29.7 30 –33.5 0.0 13.8 30 –217.3 0.0 39.9

4 OIS discounting and swap pricing of swap rates. In an upward sloping term struc- ture, changing to OIS discounting lowers the Changing from LIBOR discounting to OIS dis- value of the calculated LIBOR zero rates. In a counting changes the values of LIBOR swaps. In downward sloping term structure, the reverse is the case of spot start swaps the value change can true. For a flat term structure the discount rate be expressed directly in terms of changes in the used has no effect on the zero rates calculated discount factors. First, consider the swap value from swap rates. This last point can be seen under LIBOR discounting. Since the value of the by inspecting Equation (4) which determines the floating side of an at-the-money spot start swap swap value. In a flat term structure, all the for- equals the value of the fixed side, the value of ward rates are the same so that we obtain a zero a spot start swap in which fixed is paid can be swap value if K equals the common forward rate. written as This is true regardless of the level of interest SLD(K, M) = LALD(KM − K) rates. Similarly, when OIS discounting, is used the For short maturities the determination of the value becomes LIBOR zero curve is not very sensitive to the dis- S (K, M) = LA (K − K) count rate used. However, for maturities beyond OD OD M 10 years, the impact of the switch from LIBOR Note that, because the calibration process ensures discounting to OIS discounting can be quite large. that at-the-money spot start swaps are correctly The impact on forward rates is even larger. Table 2 priced, KM is the same in both equations. As a shows three-month LIBOR forward rates cal- result the price difference is culated using OIS discounting minus the same S (K, M) − S (K, M) three-month forward rate calculated using OIS OD LD discounting. = L(KM − K)(AOD − ALD)

First Quarter 2015 Journal Of Investment Management 70 John Hull and Alan White

Because OIS discount rates are lower than LIBOR curves ensures that VLD = VOD, so that the two discount rates the last term in this expression is effects are equal in magnitude and opposite in positive so that, if the fixed rate being paid is sign. Second, if the term structure is flat (swaps below the market rate, the price change when of all maturities have the same market rate) then OIS discounting is used is positive. If the fixed the forward rates based on both the LIBOR dis- rate being paid is above the market rate, the price counting and the OIS discounting are the same. change is negative. For swaps where fixed is In this case, VOD − VLO = 0 and all of the value received, the reverse is true. change is a pure discounting effect. In all other cases both shifts in forward rates and changes in In general, this sort of simple decomposition of discount factors play a role in the value change the value change is not possible. Two factors con- that occurs when we switch from LIBOR to OIS tribute to the value change: changes in the forward discounting. rates and changes in the discount factors used in the swap valuation. For all the interest rate deriva- Table 3 shows the value differences for pay-fixed tives we will consider in the rest of this paper, we spot start swaps with a fixed rate that is 1% higher define than the market rate for the same maturity. The VLD: Value of derivative assuming LIBOR dis- discounting effect, VLO − VLD, makes the mag- counting nitude of the present value of every payment VOD: Valueof derivative assuming OIS discount- larger (farther from zero). When the term struc- ing ture is not flat some of the expected payments VLO: Value of derivative when forward rates are positive and some negative so the overall are based on LIBOR discounting and the effect of changing the discount rates may be pos- expected cash flows calculated from the for- itive or negative. For all three term structures ward rates are then discounted at OIS zero the overall effect of a switch to OIS discount- rates8 ing is over 35 basis points for 10-year swaps, over 100 basis points for 20 year swaps, and Calculating these three values allows us to sep- about 200 basis points for 30 year swaps. The arate the impact of switching from LIBOR dis- table shows that for non-flat term structures, the counting to OIS discounting into forward-rate effect can be quite large and calcu- lating forward rates using the pre-crisis approach (1) a pure discounting effect, V − V ; and LO LD can therefore lead to large errors. Indeed, for the (2) a forward-rate effect, V − V OD LO upward sloping term structure we consider, the forward rate effect is bigger than the pure dis- As pointed out by Bianchetti (2010), a common counting effect when the swap lasts for 20 or 30 mistake made by practitioners is that they calcu- years. late VLO when they should be calculating VOD. The forward-rate effect measures the error that Table 4 shows the value differences for forward this mistake gives rise to. start swaps where the swap life is 10 years. The There are two special cases in which the discount- fixed rate is paid and LIBOR is received. The ing effect and the forward-rate effect can be easily fixed rate is 1% higher than the corresponding determined. First, for at-the-money LIBOR spot forward swap rate based on LIBOR discounting. start swaps (swap rate equals market swap rate), Again, for upward and downward sloping term the calibration process used to generate the zero structures and long maturities, the impact of using

Journal Of Investment Management First Quarter 2015 OIS Discounting,Interest Rate Derivatives, and the Modeling of Stochastic Interest Rate Spreads 71 LD LD V V 0 5 5 6 7 5 0 8 7 1 2 6 7 ...... − − 4 0 39 40 38 33 21 11 36 20 − − OD 110 204 OD − − − − − − − − V V − − LO LO V V 7 4 8 8 9 539 1 6 7 4 3 3 0 − ...... − ...... OD OD V V LD LD V V 712 920 329 440 659 0 137 1 207 469 39 53 41 80 70 ...... − − 4 0 51 60 68 74 81 98 46 23 12 − − LO 411 180 − − − − − − LO − − − V V − − LD LD V V 9 9 8 0 5 0 ...... 6 6 1 8 5 7 7 − ...... − 4 0 43 53 61 68 74 80 11 38 20 OD − − 112 − − − − − − 192 OD − − − V V − − LO LO V V − − OD OD V V LD LD V V 9 0.0 9 0.0 8 0.0 0 0.0 5 0.0 0 0.0 ...... 6 0.0 6 0.0 1 0.0 8 0.0 5 0.0 7 0.0 7 0.0 − ...... − 4 0 43 53 61 68 74 80 11 38 20 LO − − − − − − − − 112 192 LO V − − − V − − LD LD V V 1 9 7 7 8 1 ...... 0 5 6 5 9 8 7 − ...... − 4 0 49 67 86 11 39 21 105 134 262 OD − − − − − 114 182 OD V − − − − − − V − − LO LO V 7 2 2 5 3 1 V ...... − 7 7 3 6 3 3 0 ...... − 13 22 33 46 71 3 1 0 218 81 10 OD − − − − − − − − 299 OD V − − − V − Term structure 4 to 6 Term structure 5 flat Term structure 6 to 4 Term structure 4 to 6 Term structure 5 flat Term structure 6 to 4 LD LD V 4 7 6 2 5 0 V ...... − 7 9 3 9 6 5 70 ...... − 35 45 53 59 63 44 4 0 LO 32 29 17 10 − − − − − − − − LO effectV effect Total effect effect Total effect effect Total − − − − The increase in spot-start swap value caused by switching from LIBOR discounting to OIS discounting. The increase in forward-start swap value, where the swap life is 10 years, caused by switching from LIBOR discounting to OIS V 0 T 7 5 3 1 1 3 5 7 30 117 20 10 Swaplife Discount(Yrs) effect Fwd. rate effect Total Discount Fwd. rate effect effect Total Discount Fwd. rate effect effect Total The swaps pay a fixed rate equal to the market swap rate plus 1%. Price differences are measured in basis points. Table 3 (Yrs) discounting. Starttime Discount Fwd. rate Discount Fwd. rate Discount Fwd. rate 10 20 The swaps pay a fixed rate equal to the forward swap rate based on LIBOR discounting plus 1%. Price differences are measured in basis points. Table 4

First Quarter 2015 Journal Of Investment Management 72 John Hull and Alan White the wrong forward rates in conjunction with OIS discounting is used, discounting can lead to large errors.  ln[KOD/K] d1,OD = √ + 0.5σOD T0 σOD T0 5 OIS discounting and swap  ln[KOD/K] pricing d2,OD = √ − 0.5σOD T0 σOD T0 An interest rate swap option is an option that gives where σ is the of the forward swap the option holder the right to enter into a swap OD rate.10 When LIBOR will be paid and fixed with a specified fixed rate, K, at some future date received, the OIS-discounting valuation becomes T0 (T0 > 0). The swap lasts from T0 to TM. In the case of LIBOR discounting, for a swap VOD = AODL[KN(−d2,OD) − KODN(−d1,OD)] where LIBOR will be received and fixed paid, the standard market model gives9 This version of the standard market model, like the original version given above, can be used to VLD = ALDL[KLDN(d1,LD) − KN(d2,LD)] imply volatilities and calculate volatility surfaces from option prices. where ALD is given by Equation (8) and KLD is the current forward swap rate for the swap underlying To examine the effect of the switch to OIS the option (given by Equation (6)) when LIBOR discounting on the prices of swap options we con- discounting is used, sider options with lives between 1 and 20 years on a 5-year swap when the volatility for all matu- [K /K]  ln LD rities is set equal to 20% for both LIBOR and OIS d1,LD = √ + 0.5σLD T0 σLD T0 discounting. In every case the option ,  K, is set equal to KLD. For the LIBOR discount- ln[KLD/K] d , = √ − 0.5σ T ing case, the options are exactly at-the-money 2 LD σ T LD 0 LD 0 while in the OIS discounting case the options are where σLD is the volatility of the forward swap approximately at-the-money. N rate and is the cumulative normal distribution Table 5 shows the value differences for options to function. In practice, this model is used to imply enter into swaps in which floating is paid for the volatilities and calculate volatility surfaces from three-term structures of swap rates. Table 6 shows option prices. similar results for options to enter into swaps in When LIBOR will be paid and fixed received the which fixed is paid. The effect of changing the dis- valuation becomes count factors is about the same for both pay fixed and pay floating swaps. However, changing for- VLD = ALDL[KN(−d2,LD) − KLDN(−d1,LD)] ward rates has a much more pronounced effect on the value of options on swaps in which the option In the case of OIS discounting the standard market holder pays the fixed rate. This is because pay- model becomes fixed (pay-floating) swap options provide payoffs when swap rates are high (low). The forward swap VOD = AODL[KODN(d1,OD) − KN(d2,OD)] rate is assumed to follow geometric Brownian where AOD is given by Equation (13) and KOD is motion and so a change in the starting forward the current forward swap rate for the swap under- swap rate has a greater effect on high swap rates lying the option given by Equation (6) when OIS than on low swap rates.

Journal Of Investment Management First Quarter 2015 OIS Discounting,Interest Rate Derivatives, and the Modeling of Stochastic Interest Rate Spreads 73 LD LD V V − − red in basis OD OD fferences are V V LO LO V V − − OD OD V V LD LD V V − − LO LO V V LD LD V V − − OD OD V V LO LO V V − − OD OD V V LD LD V V − − LO LO V V LD LD V V − − OD OD V V LO LO V V − − OD OD V V Term structure 4 to 6 Term structure 5 flat Term structure 6 to 4 Term structure 4 to 6 Term structure 5 flat Term structure 6 to 4 LD LD V V − − LO LO effectV effect Total effect effect Totaleffect effectV effect effect Total Total effect effect Total effect effect Total The increase in the price of an option to enter into a 5-year pay-fixed swap caused by switching from LIBOR discounting to OIS The increase in the price of an option to enter into a 5-year pay-floating swap caused by switching from LIBOR discounting to OIS 0 0 T T 1357 5.0 14.1 24.5 35.7 1.0 2.2 3.7 5.6 6.0 16.3 28.2 41.3 14.8 6.0 23.5 32.0 0.0 0.01 0.03 0.057 6.3 14.8 15.4 6.0 23.5 25.7 32.0 37.0 –1.2 15.3 –2.9 7.0 22.6 –5.3 28.5 –8.5 5.1 12.5 –2.1 20.5 –1.0 –3.4 28.5 –5.1 13.3 14.8 6.0 19.1 23.5 6.0 23.4 32.0 0.0 0.0 0.0 0.0 14.8 6.0 23.5 32.0 14.1 5.8 21.4 27.4 2.7 1.2 5.0 7.9 16.8 26.4 6.9 35.3 discounting. discounting. Optionlife, Discount Fwd. rate Discount Fwd. rateOption Discountlife, Discount Fwd. rate Fwd. rate Discount Fwd. rate Discount Fwd. rate points. measured in basis points. 1020 53.4 111.0 9.2 27.4 138.4 62.6 69.8 43.7 0.0 0.01020 69.8 43.7 54.8 113.2 34.1 34.7 –15.0 –53.7 39.8 –17.9 59.4 –7.9 16.2 43.7 26.8 69.8 0.0 0.0 43.7 69.8 33.7 33.4 13.4 40.4 47.2 73.9 (Yrs) (Yrs) The swaps underlying the option pay the option holder a fixed rate equal to the forward swap rate based on LIBOR discounting. Price differences are measu The swaps underlying the option require the option holder to pay a fixed rate equal to the forward swap rate calculated using LIBOR discounting. Price di Table 5 Table 6

First Quarter 2015 Journal Of Investment Management 74 John Hull and Alan White

In practice option pricing models are calibrated the OIS discounting swap option pricing model to market prices by implying volatilities. This to find the that set VOD = VLD. has implications for the implied volatility sur- The results depend on the slope of the term struc- face calculated by a dealer for whom discount- ture and the terms of the options. Figure 1 shows ing practices are not in alignment with market the volatility term structure that Dealer X would practice. Consider the case of Dealer X who calculate for options to enter into a swap that pays is the first dealer to consider switching from fixed that when the term structure of interest rates LIBOR to OIS discounting for option pricing pur- is upward sloping. The implied volatilities are poses. Since all other dealers are using LIBOR always less than the market volatility used in the discounting observed market prices for swap LIBOR discounting option pricing model. The options are given by V . When Dealer X cal- LD volatility term structure is generally downward ibrates his swap option pricing model to the sloping except for in-the-money options which observed market prices, the price differences exhibit a steeply upward sloping term structure caused by the different approaches to discount- for relatively short-term options. ing translate into differences in the implied volatility. Figure 2 shows the shape of the volatility smile for the same set of options. The implied volatil- To illustrate this, we calculate option prices for ities are highest for out-of-the-money options options with lives from 1 to 20 years on a five- and lowest for in-the-money options. The smile year swap based on LIBOR discounting, VLD. is most pronounced for short maturity options The fixed rate on the swap is the forward swap and becomes increasingly flatter as the option rate based on LIBOR discounting plus an off- maturity lengthens. For at- and out-of-the-money set of between –2% and +2%. The volatility of options longer term options have lower implied the forward swap rate is 20% for all strikes and volatilities. The reverse is true for deep in-the- maturities. These prices are then backed through money options.

Figure 1 Implied volatilities calculated using OIS discounting when the market uses LIBOR discounting. The volatility used by the market is assumed to be 20% for all strike prices and maturities. Options with lives from 1 to 20 years on a five-year swap in which fixed is paid are considered. The fixed rate on the swap is the forward swap rate based on LIBOR discounting plus offsets of –2%, –1%, 0%, 1%, and 2%.

Journal Of Investment Management First Quarter 2015 OIS Discounting,Interest Rate Derivatives, and the Modeling of Stochastic Interest Rate Spreads 75

Figure 2 Implied volatilities calculated using OIS discounting when the market uses LIBOR discounting. The volatility used by the market is assumed to be 20% for all strike prices and maturities. Options with maturities of 1, 3, 5, 10, and 20 years on a five-year pay-fixed swap are considered. The fixed rate on the swap is the forward swap rate based on LIBOR discounting plus an offset between –2%, and +2%.

A similar situation applies to a Dealer Y who uses date we observe the LIBOR rate Ri that applies LIBOR discounting when the observed market between Ti and Ti+1. Then on the end date, Ti+1, prices for swap options are given by VOD. When there is a payoff of max(Ri − K, 0)L(Ti+1 − Ti). Dealer Y calibrates his swap option pricing model to market prices he will in a similar way observe With LIBOR discounting the standard market a volatility surface resulting from the different model gives the value of the payment as pricing characteristics of the two option pricing V =[F (T ,T )N(d ) − KN(d )] models. LD LD i i+1 1,LD 2,LD × L(T − T )P (T ) Because both OIS discounting and LIBOR dis- i+1 i LD i+1 counting models are calibrated to market data the where main effect of the differences in the models will in practice be captured by the implied volatili- [F (T ,T )/K]  d = ln LD i √ i+1 + . σ T ties. As a result the models will agree perfectly 1,LD 0 5 LD i σLD Ti for all options in the calibration set and the price [F (T ,T )/K]  differences for other options will be small. ln LD i i+1 d2,LD = √ − 0.5σLD Ti σLD Ti 6 OIS discounting and cap/floor pricing where σLD is the volatility of FLD(Ti,Ti+1). Consider a caplet, one leg of an interest rate cap, With OIS discounting, the standard market model in which the floating rate of interest is capped becomes at a specified fixed rate of interest, K. We use a similar notation to that for an interest rate swap. VOD =[FOD(Ti,Ti+1)N(d1,OD) The notional principal is L, the start date is Ti, and the end or payment date is Ti+1. On the start − KN(d2,OD)]L(Ti+1 − Ti)POD(Ti+1)

First Quarter 2015 Journal Of Investment Management 76 John Hull and Alan White where Table 7 shows the value differences for caps for [F (T ,T )/K]  the three term structures of swap rates. Table 8 d = ln OD i √ i+1 + . σ T 1,OD 0 5 OD i does the same for floors. The results are similar σOD Ti to those in Tables 5 and 6. The impact of switch- [F (T ,T )/K]  ln OD i i+1 ing from LIBOR discounting to OIS discounting d2,OD = √ − 0.5σOD Ti σOD Ti increases as cap maturity increases and the dis- counting effect is greater than the effect of the where σOD is the volatility applied to FOD(Ti, change in the forward rate. Ti+1). Astandard interest-rate cap is constructed of many As discussed in Section 4, in practice the cap of these legs in which the payment date for one leg and floor models would be calibrated to observed is the rate reset date for the next leg. The total cap market prices. As a result, any dealer whose dis- value is the sum of all the individual caplet values. counting differs from market practice will induce Sometimes the same implied volatility is used for a volatility surface that differs from the market all caplets and sometimes a different volatility is volatility surface. used for each caplet. A floorlet is an instrument that provides a pay- 7 Building trees to value nonstandard off of max(K − Ri, 0)L(Ti+ − Ti) at time Ti+ . 1 1 transactions The value of the floorlet under LIBOR and OIS discounting are given by So far this paper has focused on transactions which are handled by the market using standard VLD =[KN(−d2,LD) − FLD(Ti,Ti+1) market models based on Black (1976). As we × N(−d1,LD)]L(Ti+1 − Ti)PLD(Ti+1) have shown, it is fortunate that, once forward and LIBOR has been correctly calculated, only a small modification to the standard market models is 11 VOD =[KN(−d2,OD) − FOD(Ti,Ti+1) necessary. We now move on to consider instru- ments which cannot be handled using standard × N(−d1,OD)]L(Ti+1 − Ti)POD(Ti+1) market models. A single interest rate tree describ- Similar to caps, a standard interest-rate floor is ing the evolution of the LIBOR rates is commonly constructed of many of these legs in which the used to value these instruments when LIBOR dis- payment date for one leg is the rate reset date for counting is used. In this section, we explain that the next leg. it is still possible to obtain a value using a single tree when OIS discounting is used. To examine the effect of the switch to OIS dis- counting on the prices of caps and floors, we For the sake of definiteness, consider a deriva- consider quarterly reset caps and floors with dif- tive providing payoffs dependent on three-month ferent lives. In all cases, in accordance with LIBOR.12 The general procedure is as follows: market practice, the first leg of the cap or floor is omitted so the cap starts at time 0.25. The (1) Calculate the OIS zero curve from swap rates cap or floor strike price is set equal to the cor- as discussed earlier in this paper. responding swap rate under LIBOR discounting. (2) Calculate a LIBOR zero curve when OIS dis- The forward rate volatility is 20% and the same counting is used as discussed earlier in this three term structures as before are used. paper.

Journal Of Investment Management First Quarter 2015 OIS Discounting,Interest Rate Derivatives, and the Modeling of Stochastic Interest Rate Spreads 77 LD LD V V − − OD OD V V LO LO V V − − OD OD V V LD LD V V − − LO LO V V LD LD V V − − OD OD V V LO LO V V − − OD OD V V LD LD V V − − LO LO V V LD LD V V − − OD OD V V LO LO V V − − OD OD V V Term structure 4 to 6 Term structure 5 flat Term structure 6 to 4 Term structure 4 to 6 Term structure 5 flat Term structure 6 to 4 LD LD V V − − LO LO The increase in the price of an approximately at-the-money floor caused by switching from LIBOR discounting to OIS discounting. The increase in the price of an approximately at-the-money cap caused by switching from LIBOR discounting to OIS discounting. V V 135 0.17 1.9 6.4 14.2 0.0 0.1 0.5 1.3 0.1 2.0 6.9 15.5 0.2 2.2 16.1 7.5 0.0 0.0 0.0 0.0 0.2 2.2 16.1 7.5 0.2 2.6 18.3 8.5 0.0 –0.1 –1.5 –0.6 0.2 16.8 2.5 8.0 135 0.17 2.0 7.3 17.0 0.0 –0.2 –0.8 –2.2 0.1 1.9 6.6 14.8 0.2 2.2 16.1 7.5 0.0 0.0 0.0 0.0 0.2 2.2 16.1 7.5 0.2 2.4 15.6 7.6 0.0 0.1 1.9 0.7 0.2 17.4 2.6 8.3 1020 32.3 141.6 3.6 22.4 164.0 35.8 146.6 35.6 0.0 0.0 146.6 35.6 162.4 39.8 –32.3 –4.3 130.1 35.6 Floorlife Discount(Yrs) effect Fwd. rate effect Total DiscountPrice differences are measured in basis points. Fwd. rate effect effect Total Discount Fwd. rate effect effect Total Table 8 1020 41.1 217.8 –6.7 –58.5 159.3 34.5 146.6 35.6 0.0 0.0 146.6 35.6 97.7 31.5 36.9 5.3 134.6 36.9 Caplife Discount(Yrs) effect Fwd. rate effect TotalPrice differences Discount are measured in basis points. effect Fwd. rate effect Total Discount Fwd. effect rate effect Total Table 7

First Quarter 2015 Journal Of Investment Management 78 John Hull and Alan White

(3) Assume a one-factor interest rate model for storing theArrow–Debreu prices at each node the evolution of the OIS zero curve and con- when the OIS tree is constructed in step 3.) An struct a tree for the t-period OIS rate. The iterative procedure is used to determine the drift of the short rate will typically include a unconditional expected spread at time t that function of time so that the initial OIS term matches the tree-based price with the analytic structure is matched. Procedures for building price. the tree are described in, for example, Hull and White (1994, 2014). The result of this procedure is a tree that has the (4) Roll back through the tree to calculate the OIS rate and the expected LIBOR rate at each value of a three-month bond at each node so node. The tree matches both the OIS swap rates that the three-month OIS rate is known at each and LIBOR swap rates. It can be used to value node. derivatives whose payoffs depend on LIBOR (5) Assume a one-factor model for the spread when OIS discounting is used. between the three-month LIBOR rate and the The potentially difficult part of the procedure is three-month OIS rate. Similar to the model calculating the expected spread at time t at each for the OIS rate, this will include a function node of the OIS tree for a trial value of the of time so that the LIBOR zero curve can be unconditional expected spread at time t. For some matched. models analytic results are available. In other (6) Assume a correlation between the three- cases numerical procedures can be developed.13 month OIS rate and the spread in step 5. (7) Advance through the OIS tree one step at a The appendix shows that analytic results are avail- time matching the tree-based value of a for- able when (a) a mean-reverting Gaussian model ward rate agreement based on three-month is used for both the instantaneous OIS rate and LIBOR rates with the analytic value cal- the spread and (b) a mean-reverting Gaussian culated using the procedures described in model is used for both the logarithm of the Section 3. The procedure is as follows: At OIS rate and the logarithm of the spread. The the time t step choose a trial value for the first model can be viewed as a natural extension unconditional expected spread at time t. (As of Hull and White (1990) interest rate model. mentioned, the process for the spread incor- The second can be viewed as a natural exten- porates a function of time in the drift.) Calcu- sion of Black and Karasinski (1991) interest rate late the conditional expected spread at each model. node of the OIS tree at time t based on the When valuing American-style options, there is unconditional expected spread. The expected the complication that the decision to at a three-month LIBOR rate at each node of the node may depend on the value of the spread, s,at OIS tree is then the three-month OIS rate at the node. This can be accommodated as follows. the node plus the expected spread. The value First we calculate for the node the critical value of of the at each node s above which early exercise is optimal. We will at time t can be calculated from the three- refer to this as s∗. Define q = Pr(s>s∗ | R). The month LIBOR and OIS rates at the node. value of the option at the node is Discounting these values back through the (1 − q)U + qU tree produces the tree-based estimate of the 1 2 value of the forward rate agreement. (These where U1 is the value of the option (calculated present value calculations are simplified by from the value at subsequent nodes) if there is no

Journal Of Investment Management First Quarter 2015 OIS Discounting,Interest Rate Derivatives, and the Modeling of Stochastic Interest Rate Spreads 79

exercise and U2 is the intrinsic value of the option nodes that can be reached at time (i + 1)t plus at the node. The calculation of q is explained in the present value of any payment based on the the Appendix for the two models considered. expected value of LIBOR at the node. We will illustrate the model using the extension The value of the option for various values of the of Black and Karasinski (1991). Specifically, if r LIBOR−OIS spread volatility, σy, and the cor- is the short-term OIS rate and x = ln(r), then relation, ρ, between the spread and the OIS rate dx =[θ(t) − axx]dt + σxdzx are shown in Table 9. The value of the option σ = where ax and σx are positive constants, θ(t) is a when the spread is constant, y 0, is 2.92% function of time chosen to match the OIS zero of the swap notional. If the correlation is positive curve that is calculated from OIS swap rates, and the option value increases as the spread volatil- dzx is a Wiener process. If s is the three-month ity increases while if the correlation is negative LIBOR−OIS spread and y = ln(s) the option value decreases as the spread volatil- ity increases. The reason for this is that when the dy =[ψ(t) − a y]dt + σ dz y y y OIS rate is high and the correlation is positive the where ay and σy are positive constants, ψ(t) is a LIBOR−OIS spread is greater than average. Sim- function of time, and dzy is a Wiener process. The ilarly, when the OIS rate is low and the correlation correlation between dzx and dzy is assumed to be is positive the LIBOR−OIS spread is smaller than a constant, ρ. average. As a result, the LIBOR rate (i.e., the OIS rate plus the spread) exhibits more variability than We consider a ten-year Bermudan swap option the OIS rate. This increases the value of options where early exercise can take place on each swap on the LIBOR rate. If the correlation is negative payment date on or after year two. The holder the reverse is true and the LIBOR rate exhibits has an option to enter into the swap. The swap less variability than the OIS rate resulting in lower underlying the option is a quarterly reset swap option prices. in which three-month LIBOR is received and a fixed rate is paid. The fixed rate is set equal to the The results show that a correlation between the market rate for a swap that starts in two years and spread and OIS can have an appreciable effect matures in ten years. This ensures that at the first on option prices. For example, when the spread possible exercise date, two years from now, the volatility is 15% and the correlation is 0.25 the option is approximately at-the-money. The OIS value of the option increases by about 5%. This and LIBOR term structure data is the same as that result is predicated on the assumption that we for the first of the three cases defined in Section 2. know the volatility of OIS and the spread. How- The instantaneous OIS rate volatility, σx, is 20%, ever, in practice all model parameters are typically the reversion rate, ax, is 5%, and the LIBOR– implied from the observed market prices for OIS spread reversion rate, ay, is 5%. We consider options with different strike prices and maturities. a number of different values of σy and ρ. The model is in effect a tool for pricing non- A necessarily prerequisite to valuing the Bermu- standard interest rate derivatives consistently with dan swap option is valuing the underlying swap standard interest rate derivatives. As a result, the at each node of the tree. This can be calculated calibrated OIS and spread volatility parameters by rolling back through the tree in the usual way. will be sensitive to the assumed correlation but The value of the swap at a node at time it is option prices calculated using the calibrated mod- the present value of the expected value at the els will be much less sensitive to this correlation.

First Quarter 2015 Journal Of Investment Management 80 John Hull and Alan White

Table 9 The value of a Bermudan option to enter into a 10-year swap as a percentage of the swap notional for various values of the LIBOR–OIS spread volatility, σy, and the correlations between the spread and the OIS rate, ρ.

LIBOR−OIS spread volatility, σy

ρ 0.01 (%) 0.05 (%) 0.10 (%) 0.15 (%) 0.20 (%) 0.25 (%) 0.99 2.96 3.12 3.33 3.54 3.74 3.94 0.75 2.95 3.07 3.23 3.39 3.54 3.70 0.50 2.94 3.02 3.12 3.23 3.33 3.44 0.25 2.93 2.97 3.02 3.07 3.12 3.18 0.00 2.92 2.92 2.92 2.92 2.92 2.92 –0.25 2.91 2.87 2.82 2.76 2.71 2.66 –0.50 2.90 2.82 2.71 2.61 2.52 2.42 –0.75 2.89 2.76 2.61 2.47 2.32 2.18 –0.99 2.88 2.72 2.52 2.33 2.15 1.97 The option can be exercised on any payment date on or after year 2. The instantaneous OIS rate volatility is 20% and the reversion rate is 5% (σx = 0.20 and ax = 0.05). The LIBOR–OIS spread has a reversion rate of 5% (ay = 0.05). The 4 to 6 term structure applies.

8 Conclusions This paper has shown how the standard mar- ket models for caps and swap options can be When interest rate derivatives are valued, moving adjusted to accommodate OIS discounting. For from LIBOR to OIS discounting has two effects. more complex instruments we have presented a First, forward interest rates and forward swap new procedure for adjusting the way interest rate rates change. Second, the discount rates applied trees are used to accommodate OIS discounting. to the cash flows calculated from those forward Both the OIS rate and the expected LIBOR rate rates change. The impact of the switch is small can be modeled using a single tree. A stochas- for short-dated instruments, but becomes progres- tic spread makes LIBOR more (less) volatile than sively larger as the life of the instrument becomes OIS if the correlation is positive (negative). This longer. results in higher (lower) option prices than arise One mistake that is sometimes made is to (a) when the spread is deterministic. calculate LIBOR zero rates and LIBOR forward rates and by bootstrapping LIBOR forward swap Appendix: Analytic results for two one-factor rates in the traditional way (i.e., in a world where term structure models LIBOR discounting is employed) and (b) apply OIS discount rates to the cash flows calculated In this appendix we explain how the spread by assuming that the LIBOR forward rates will conditional on the OIS rate can be calculated ana- be realized. Our results show the errors caused lytically for two cases of the model outlined in by making this mistake. The error is zero for Section 6. flat term structures, but can be as much as 200 The Gaussian model basis points for long-dated transactions and the upward or downward sloping term structures we First consider the case where both the instanta- have considered. neous OIS rate and the three-month spread follow

Journal Of Investment Management First Quarter 2015 OIS Discounting,Interest Rate Derivatives, and the Modeling of Stochastic Interest Rate Spreads 81 mean-reverting Gaussian processes. Define r as The value of E(s | r) is α(r). As indicated in the instantaneous OIS rate and s as the spread Section 6 we choose the µs iteratively so that for- between three-month LIBOR and the three-month ward LIBOR rates are matched. For this model OIS rate. The processes for r and s are Pr(s>s∗ | R) is   dr =[θ(t) − a r]dt + σ dz α(r) − s∗ r r r N √ β and where, as before, N is the cumulative normal ds =[ψ(t) − ass]dt + σsdzs distribution function. These results provide the where ar, as, σr, and σs are positive parameters, necessary ammunition to value American-style θ(t) is a function of time chosen so that the pro- options. cess for the instantaneous OIS rate is consistent with the current OIS zero curve, ψ(t) is a func- tion of time chosen so that the spread is consistent The lognormal model with the LIBOR zero curve, and dzs and dzr are Consider next the situation where both the log- Wiener processes. We assume that the correlation arithm of the instantaneous OIS rate and the ρ 14 between the two Wiener processes is . logarithm of the three-month spread follow mean- x = (r) It follows that the unconditional distributions of reverting Gaussian processes. Defining ln y = (s) r and s at time t are and ln we assume   σ2 dx =[θ(t) − a x]dt + σ dz r −2art x x x r(t) ∼ ϕ µr(t), (1 − e ) 2ar   and σ2 s −2ast s(t) ∼ ϕ µs(t), (1 − e ) dy =[ψ(t) − ayy]dt + σydzy 2as where ϕ(µ, v) denotes a normal distribution with where ax,ay,σx, and σy are positive constants, mean µ and variance v and θ(t) and ψ(t) are a function of time chosen to  t match the OIS zero curve and the LIBOR zero −art arτ x y µr(t) = r0 + e θ(τ)e dτ curve, and dz and dz are Wiener process. In this 0 case ρ is the correlation between dzx and dzy.  t −ast asτ Similar to the Gaussian model case, the distribu- µs(t) = s0 + e ψ(τ)e dτ 0 tion of y conditional on x is The distribution of s conditional on r is ϕ(α(x), β) ϕ(α(r), β) where  where − a t  σy (1 − e 2 y )/ay σ ( − e−2ast)/a s 1 s α(x) = µy + ρ  (x − µx) α(r) = µs + ρ  (r − µr) − a t − a t σx (1 − e 2 x )/ax σr (1 − e 2 r )/ar and and − a t − e−2ast 1 − e 2 y 2 1 2 β = σ2 ( − ρ2) β = σs (1 − ρ ) y 1 2as 2ay

First Quarter 2015 Journal Of Investment Management 82 John Hull and Alan White

y 8 For the lognormal model, E(s | R) = E(e | x = Depending on one’s point of view both VLD and VOD V ln R). This is may be legitimate value calculations. However, LO is not a legitimate value calculation. The forward rate exp{α(ln R) + β/2} FOD(T1,T2) is a martingale when the numeraire is POD(T2), but the forward rate FLD(T1,T2) is not a We choose the µy iteratively so that for- martingale for this numeraire. ward LIBOR rates are matched. In this case 9 See for example Hull (2015), Chapter 29, for a discus- Pr(s>s∗ | R) is sion of the standard market models used for interest rate  ∗  derivatives. α(ln R) − ln s 10 N √ . Note that σOD and σLD are in general not equal. β 11 As an alternative to using standard market models, one can value caplets or swap options by integrating over the Notes joint distribution of OIS and the expected spread. This approach is discussed by Mercurio and Xie (2012). 1 Johannes and Sundaresan (2007) argued pre-crisis that 12 The same approach can be used for derivatives depen- the prevalence of collateralization in the interest rate dent on any other reference rate. If the tenor of the swap market means that discounting at LIBOR rates is reference is τ, a single tree is used to model the OIS t- no longer appropriate. period rate and the expected spread between the τ-period 2 The reason usually given for this is that these transac- OIS rate and the τ-period reference rate. tions are funded by the collateral and cash collateral 13 A general numerical procedure that can be used is as often earns the effective federal funds rate. The OIS follows. Construct a tree for the t-OIS rate, r. Con- rate is a continually refreshed federal funds rate. Hull struct a tree step-by-step for the spread, s. At the each and White (2012, 2013a, 2013b) argue that OIS is the time step, determine the unconditional distributions for best proxy for the risk-free rate and that it should be used r and s. (In the case of the s-tree a trial value for the when valuing both collateralized and non-collateralized center of the tree is being assumed.) Assume a Gaussian transactions. copula (or some other convenient copula) to determine 3 Mercurio (2009) may have been the first researcher to E(s | r) at each node of the r-tree. The advantage of investigate how the standard market models for caps the analytic results given in the Appendix for the nor- and swap options can be adapted to accommodate OIS mal and the lognormal models is that calculations are discounting. much faster because it is not necessary to build the 4 Our objective is to keep the notation as simple as possi- s-tree. ble so we do not consider the various day count practices 14 The parameter ρ is defined as the correlation between T − T that apply in practice. However, if i+1 i and other the Wiener processes. It can be shown that it is also the time parameters are interpreted as the accrual fraction correlation between the values of r and s at time t. that applies under the appropriate , our results are in agreement with industry practice. 5 This is a small simplification. In practice, the interest rate is fixed two days before the start of the period to References which it applies. 6 This assumes that the day count convention used for Bianchetti, M. (2010). “Two Curves, One Price,” Risk 23(8) the floating rates in the swap is the same as that used to (August), 66–72. calculate the forward rates. This is the case for a standard Black, F. (1976). “The Pricing of Commodity Contracts,” swap. Journal of 3 (March), 167–179. 7 A swap of the federal funds rate for LIBOR involves the Black, F. and Karasinski, P. (1991). “Bond and Option arithmetic average of effective federal funds rate for the Pricing When Short Rates Are Lognormal,” Financial period being considered, whereas payments in an OIS Analysts Journal 47(4) (July/August), 52–59. are calculated from a geometric average of effective fed- Hull, J. (2015). Options, Futures, and Other Derivatives, eral funds rates. A “convexity adjustment” is in theory 9th edition, Englewood Cliffs: Pearson. necessary to adjust for this. See, for example, Takada Hull, J. and White, A. (1990). “Pricing Interest Rate (2011). Derivatives,” Review of Financial Studies 3(4), 573–592.

Journal Of Investment Management First Quarter 2015 OIS Discounting,Interest Rate Derivatives, and the Modeling of Stochastic Interest Rate Spreads 83

Hull, J. and White, A. (1994). “Numerical Procedures for Johannes, M. and Sundaresan, S. (2007). “The Impact of Implementing Term Structure Models I: Single Factor Collateralization on Swap Rates,” Journal of Finance Models,” Journal of Derivatives 2, 7–16. 62(1) (February), 383–410. Hull, J. and White, A. (2012). “Collateral and Credit Issues Mercurio, F. (2009). “Interest Rates and the : in Derivatives Pricing,” Forthcoming, Journal of Credit New Formulas and Market Models,” ssrn 1332205. Risk. Mercurio, F. and Xie, Z. (2012). “The Basis Goes Stochas- Hull, J. and White, A. (2013a). “LIBOR vs. OIS: The tic,” Risk 25(12) (December), 78–83. Derivatives Discounting Dilemma,” Journal of Invest- Smith, D. (2013). “Valuing Interest Rate Swaps Using OIS ment Management 11(3), 14–27. Discounting,” Journal of Derivatives 20(4), Summer, Hull, J. and White, A. (2013b). “Valuing Derivatives: 49–59. Funding Value Adjustments and Fair Value,” Financial Takada, K. (2011). “Valuation ofArithmeticAverage of Fed Analysts Journal 70(3) (May/June 2014), 46–56. Funds Rate and the Construction of the US Dollar Swap Hull, J. and White, A. (2014). “AGeneralized Procedure for Yield Curve,” ssrn-id981668. Building Trees for the Short Rate and Its Application to Determining Market Implied Volatilities,” Forthcoming, Keywords: OIS; LIBOR; swaps; ; caps; Quantitative Finance. interest rate trees

First Quarter 2015 Journal Of Investment Management