INTEREST RATE FUTURE - Go.factset
www.factset.comIN T ERES T R AT E F U T UREFigo Liu, Lead Financial Engineer
Interest Rate FutureFigo Liu, Lead Financial Engineerfiliu@factset.comNovember 4, 2019 Copyright2020 FactSet Research Systems Inc.All rights reserved.1FactSet Research Systems Inc. www.factset.com
Contents1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22Symbology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.1 Decoding the future ticker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2 Quoting Convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2233Valuation: non-Brazilian Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.1 Tick Size and Basis Point Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2 Relative-Price Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3344Valuation: Brazilian Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55Options on Interest Rate Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.1 non-Brazilian Money Market Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.2 Brazilian Future Option: IDI Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6676Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.1 Future Root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.2 Brazilian Future Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .888 Copyright2020 FactSet Research Systems Inc.All rights reserved.2FactSet Research Systems Inc. www.factset.com
1 IntroductionAn interest rate future is a future contract with an underlying instrument that pays interest; a contract between thebuyer and seller agreeing to the future delivery of any interest-bearing asset. The interest rate future allows thebuyer and seller to lock in the price of the interest-bearing asset for a future date. To differentiate from otherfixed-income futures such as a bond future, the underlying assets for an interest rate future must be a particularinterest rate. An interest rate future is actively used to hedge against future interest rate movement, i.e., so-calledinterest rate market risk.Due to the varying feature of the underlying interest rates, the way to calculate the price and to quote the interestrate future varies a lot. The document would mainly discuss the products that are covered by FactSet such as iBORlinked futures, Cash Rate futures, Fed Fund futures, Brazilian futures, to name a few.2 SymbologyDifferent systems may have different symbols for different futures. Also, different futures would have differentquoting conventions. FactSet symbols preserve the exchange traded ticker if possible.2.1Decoding the future tickerThe future ticker at FactSet has a convention that are typically consistent with exchange rules. It is a combination offuture root 1 , which represent the underlying of the future and expiration code (month code 2 year unit). Forexample, the future ticker EDZ18 means it is a US EuroDollar Future (on US Libor 3m and traded on CME)expiring on the third Wednesday of Dec 2018.2.2Quoting ConventionImplied Forward:Most of the Rate futures are quoted as 100 f (T, T, T n) at future expiration date. f (T, T, T n) is the forwardrate observed at time T representing a borrowing cost from time T to T n (basically it becomes a spot rate). TimeT is normally the future expiration date and n is the tenor of the forward rate, representing the length of the loan toaccrue. Before the expiration date T, the price evolves and reflected by implied forward rate from the forward curve.Depending on different future contracts on different exchanges, the rate could be US Libor 3m rate, UK Libor 3m orUS Fed Fund future, etc.To calculate the implied forward rate, a forward curve will be built and bootstrapped 3 .Yield:Some contracts such as AUD, NZD and Brazilian futures are quoted in yield convention and the valuation of futurewould be calculating a synthetic bond price based on the inputted yield.1 SeeAppendix 8.1 for full set of roots FactSet covers2 https://www.cmegroup.com/month-codes.html3 Pleaserefer to white paper Swap Curve Building at FactSet for more information Copyright2020 FactSet Research Systems Inc.All rights reserved.3FactSet Research Systems Inc. www.factset.com
3 Valuation: non-Brazilian FutureBecause futures are mark-to-market every day, the value of future is set to 0 in our valuation engine. The importantvaluation measure would be the intra-day price change due to changing rate environment. This makes the valuationengine adapt a Relative-Price approach. This approach is used by all non-Brazilian futures including iBorfutures, Fed Fund Futures, Cash Rate Futures, etc.3.1Tick Size and Basis Point ValueThese are two important concepts to value a future price changes.Tick Size: it is the minimum price fluctuation. For example, for US Libor 3m Eurodollar future, the tick size is 0.25basis point for the nearing active contract and 0.005 basis point for other contract. It means if market rate changewill round to the tick size. If the change is small, there will be no price change reflecting on futures and no PnL orrisk showing up. All tick size can be found on exchange official specs.Basis Point Value: BPV means for each basis point movement on the underlying rate, what would the future valuechange. For example, the US Libor 3m Eurodollar future BPV would be 25 and the Fed Fund Future BPV wouldbe 41.67. BPV can be found on exchange website as well or can be calculated with formula:BP V N otional 0.0001 AccrualF raction(T, T n).3.2Relative-Price ApproachAs introduced in various textbooks including [1] and [2], we get following mathematical relationships:1 P (t, S)( 1)δ P (t, T )(1)f (t, S, T ) EQT (f (S, S, T ) t)(2)f (t, S, T ) BP (t) E (P (S) t)(3)(1) is well known formula to calculate arbitrage-free forward rate f (t, S, T ) from discount factor P (t, S) assumingsimple compounding with time t observed curve.(2) is well known that forward rate is a martingale under T-forward measure. With this, the expectation of theunknown spot rate at future date is equal to implied forward rate observed today.(3) is well known that future price P (t) is a martingale under spot (bank account) measure due to its mark-to-marketfeature so that expectation of the future price at expiration date would be equal to today’s market quote.With these tools on hand, the future valuation would beP (t) EB (P (S) t) EB (100 f (S, S, T ) t) 100 EB (f (S, S, T ) t)BQTE (f (S, S, T ) t) EBQT(f (S, S, T ) t) (E (f (S, S, T ) t) E(4)(f (S, S, T ) t)) f (t, S, T ) ConvexAdj(t)(5)P (t) 100 f (t, S, T ) ConvexAdj(t)(6)As we can see, the future price can be represented as time t implied forward rate minus a ConvexityAdjustment.Convexity Adjustment: There are many ways to calculate the convexity adjustment. Model based approachincludes doing change of measure to move the forward rate from T-forward measure to spot measure. See [1] for Copyright2020 FactSet Research Systems Inc.All rights reserved.4FactSet Research Systems Inc. www.factset.com
example of using Hull-White model. However, this approach involves making assumption on volatility and modelcalibration process.At FactSet, we are using simplified model-free approach to infer the convexity adjustment from the Future marketprice where ConvexAdj(t) 100 f (t, S, T ) Pmarket (t). Notice that the convexity adjustment here contains boththe true convexity adjustment and future/cash basis spread, which is why this is a simplified approach.The Assumption: The biggest assumption behind relative-price approach is that the convexity adjustment staysthe same for different rate environment. This would enable calculating future price byP 0 (t) 100 f 0 (t, S, T ) ConvexAdj(t)(7) P (t) f (t, S, T )(8)where f 0 (t, S, T ) is the new implied forward rate given rate changes.Valuation: To calculate future quote in different scenario for scenario analysis, risk analysis and return analysis, wewill use (7). To calculate mark-to-market dollar change, by using tick size and BPV, we can value the future contractas: M T M round( f (t, S, T ), tickSize) BP V(9)4 Valuation: Brazilian FutureBrazilian Futures (DI, DDI, DCO, DAP, OCI) have different pricing mechanism as it relates pricing a bullet bondwith Brazilian conventions. It also relies on building the corresponding Brazilian curve, specifically CDI curve andSelic Curve. At this moment, the two curves are not yet built but approximated by a constant spread againstBrazilian Treasury curve. The following section would introduce the methodology by using Brazilian Treasury curve.In the appendix 8.2, the proposed new methodology is shown given CDI and Selic curve being built.DI Future: Future on CDI rateP 100000(1 i(T ) n252100 )i(T ) SpotGovt (T ) spread DIF utureQuote(T )(10)(11)where n is time to future expiration dateOC1 Future: Future on Selic rateP 100000(1 i(T ) n252100 )i(T ) SpotGovt (T ) spread OCIF utureQuote(T )(12)(13)where n is time to future expiration dateDDI Future: DI x U.S. Dollar Spread FuturesP 100000(1 Copyright2020 FactSet Research Systems Inc.All rights reserved.i(T )1005 n360 )(14)FactSet Research Systems Inc. www.factset.com
where n is time to future expiration date. And i(T) is defined in the following steps:)) nnCDI (t,T )1. Ratio (1 DDIQuote(T 360)/(1 DIQuote(T) 252 ) PPDDI100100(t,T )n)n2. i [(1 DIQuote(T) 252 ) Ratio 1]/ 360100Basically, i(T) will be same as DDI future quote if no curve is shocked. However, if curve is shocked, it will bederived from a ratio with DI future quotes which is derived from a constant spread on Brazil GOVT curve.DCO Future: OC1 x U.S. Dollar SpreadP 100000(1 i(T )100 (15)n360 )where n is time to future expiration date on act/360 day count. And i is defined in the following steps:)) nnCDI (t,T )1. Ratio (1 DCOQuote(T 360)/(1 DIQuote(T) 252 ) PPDCO100100(t,T )n)n2. i [(1 DIQuote(T) 252 ) Ratio 1]/ 360100Basically, i will be same as DCO future quote if no curve is shocked. However, if curve is shocked, it will be derivedfrom a ratio with DI future quotes which is derived from a constant spread on Brazil GOVT curve.DAP Future: DI x IPCA Spread100000P (1 i(T )100 (16)n360 )i(T ) SpotGovt (T ) spread IP CA(T )(17)where n is time to future expiration date.5 Options on Interest Rate Future5.1non-Brazilian Money Market FutureIn general, options on money market interest rate are American style and the pay off for a call at expiration date isdescribed as:V (T ) max(P (T ) K, 0) max((100 f (T )) (100 K)) max(K f (T ))(18)V (t) e r(T t) EB [V (T ) t)(19)Where P is the future price and K is the strike price and the payoff function V is convex and e r(T t) is the discountfactor. Also, the call on future price will be a put on rate.Most of the options traded on exchange has daily margin process. From formula (3) and Jensen’s inequality we canderive:EB [V (T ) t) max(EB (P (T ) t) K) max(P (t) K)(20)This shows it would never be optimal to exercise the American option. FactSet value all money market option asEuropean style.From (3), we also know that future price P(t) and derived rate f(t) is a martingale under spot measure, in which casef(t) can be modeled as:df σf k dW (t) Copyright2020 FactSet Research Systems Inc.All rights reserved.6(21)FactSet Research Systems Inc. www.factset.com
FactSet provides analytics deriving from two common models:Black Model: when k 1, the call option on rate (put option on price) valuation would be using well-known BlackModel:V (t) e r(T t) (f (t)N (d1) KN (d2))(22)2d1 ln(f (t)/K) σ (T t)/2 σ T t d2 d1 σ T t(23)(24)Bachelier Model: when k 0, the forward rate can be negative and it is well-known Bachelier model: V (t) e r(T t) ((f (t) K)N (d) σ T tφ(d)))d f (t) K σ T t(25)(26)(27)5.2Brazilian Future Option: IDI OptionBrazilian IDI Option is an option on average one-day inter-bank deposit rate. It is a tool to bet the CDI rate in thefuture. The underlying of the option is a compounded daily CDI rate: IDI index valueIDIt IDIBaseDatetY(1 CDITi )1/252(28)(1 CDITi )1/252(29)Ti BaseDateIDIT IDItTYTi tBus252K IDIt (1 RK )τt,T(P ayof fT 1 )call IDIt max(TYBus252(1 CDITi )1/252 (1 RK )τt,T, 0)(30)(31)Ti tNotice that for IDI option, the payment happens 1 business day after maturity date. People can convert the indexvalue IDI and K to rate value CDI and RK or vice versa.There are different options for modeling. One can assume IDI index follows geometric Brownian motion and useBlack Formula directly on it. However, the volatility will not be observed from market and it is not so convincing tosay a daily compounded value would follow a geometric Brownian motion without drift (trend). Another option is toassume a IDI index equivalent simple rate follows geometric Brownian motion:TY Bus252(1 CDITi )1/252 1 Rt,Tτt,T(32)Ti tBus252(1 RK )τt,T(P ayof fT 1 )call IDIt Copyright2020 FactSet Research Systems Inc.All rights reserved. Bus252 1 RKτt,TBus252 τt,Tmax(Rt,T7 RK, 0)(33)(34)FactSet Research Systems Inc. www.factset.com
T Rt,Tis not known until time T. Following same result from (2), we will get EQCDI (Rt,T t) Rt,T where Rt,T is spotrate derived from today’s discount curve. For the reason mentioned above that we currently don’t have CDI discountcurve, we will use DI quote to approximate Rt,T :Bus2521Bus252 1 Rt,T τt,T (1 F utureQuote )τt,TCDIPt,TBus252T EQCDI (Rt,T t) Rt,T (1 F utureQuote )τt,TBus252τt,T 1(35)(36) Depending on if we assume Rt,Tfollows Log-normal distribution or normal distribution, and form (36), we can applyBlack/Bachelier formula to do valuation. For call option:Bus252 CDIBlack : V (t) IDIt τt,T(Rt,T N (d1) RKN (d2))Pt,t 1,T 1 Bus252 CDIBachelier : V (t) IDIt τt,T((Rt,T RK )N (d) σ T tφ(d)))Pt,t 1,T 11CDIPt,t 1,TBus252 1 (1 F utureQuote )τt,T(37)(38)(39)Again, here since we don’t have CDI discount curve, we are using DI future quote to approximate.6 Appendix Copyright2020 FactSet Research Systems Inc.All rights reserved.8FactSet Research Systems Inc. www.factset.com
6.1Future Root6.2Brazilian Future MethodologyDI Future:P QT100000Ti t (1 CDITi100(40)1) 252where CDITi is projected daily CDI rate. This approach depends on a CDI curve built from DI futures.OC1 Future:P QT100000Ti t (1 SelicTi100(41)1) 252where SelicTi is projected daily Selic rate. This approach depends on a Selic curve built from OC1 futures.DDI Future:P 100000PU SB (t, tf x , T 1) Copyright2020 FactSet Research Systems Inc.All rights reserved.9F XtPCDI (t, T )F Xt 1 PCDI (t, tf x , T 1)(42)FactSet Research Systems Inc. www.factset.com
where PU SB is USD Discount Factor for Brazilian onshore market. This should be bootstrapped from BRLUSDXccy swaps or directly from DDI futures. The current approach approximate this formula by assumingPU SB (t,tf x ,T 1) F XtPCDI (t,tf x ,T 1) F Xt 1 is constant when the rate changes.DCO Future:P 100000PU SB (t, tf x , T 1)F XtPSelic (t, T )F Xt 1 PSelic (t, tf x , T 1)(43)where PU SB is USD Discount Factor for Brazilian onshore market. This should be bootstrapped from BRLUSDXccy swaps or directly from DCO futures. The current approach approximate this formula by assumingPU SB (t,tf x ,T 1) F XtPSelic (t,tf x ,T 1) F Xt 1 is constant when the rate changes.References[1] John C. Hull Options, Futures, and Other Derivatives(9th Edition). ISBN-13: 978-0133456318[2] Steven Shreve Stochastic Calculus for Finance II: Continuous-Time Models. Springer Finance, ISBN-13:978-1441923110[3] Marcos C. S. Carreira and Brostowicz Jr., Richard J Brazilian Derivatives and Securities: Pricing and RiskManagement of FX and Interest-Rate Portfolios for Local and Global Markets. ISBN-13: 978-1137477262 Copyright2020 FactSet Research Systems Inc.All rights reserved.10FactSet Research Systems Inc. www.factset.com
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interest rate. An interest rate future is actively used to hedge against future interest rate movement, i.e., so-called interest rate market risk. Due to the varying feature of the underlying interest rates, the way to calculate the price and to quote the interest rate future varies a lot.
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