Interest Rate Future

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INTEREST RATE FUTURE

Figo Liu, Lead Financial Engineer Interest Rate Future Figo Liu, Lead Financial Engineer ﬁ[email protected] November 4, 2019

1 Introduction...... 2

2 Symbology...... 2 2.1 Decoding the future ticker...... 2 2.2 Quoting Convention ...... 3

3 Valuation: non-Brazilian Future...... 3 3.1 Tick Size and Basis Point Value...... 3 3.2 Relative-Price Approach...... 4

4 Valuation: Brazilian Future...... 5

5 Options on Interest Rate Future...... 6 5.1 non-Brazilian Money Market Future ...... 6 5.2 Brazilian Future Option: IDI Option...... 7

6 Appendix...... 8 6.1 Future Root...... 8 6.2 Brazilian Future Methodology...... 8

An interest rate future is a future contract with an underlying instrument that pays interest; a contract between the buyer and seller agreeing to the future delivery of any interest-bearing asset. The interest rate future allows the buyer and seller to lock in the price of the interest-bearing asset for a future date. To diﬀerentiate from other ﬁxed-income futures such as a bond future, the underlying assets for an interest rate future must be a particular 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. The document would mainly discuss the products that are covered by FactSet such as iBOR linked futures, Cash Rate futures, Fed Fund futures, Brazilian futures, to name a few.

2 Symbology

Different systems may have different symbols for different futures. Also, different futures would have different quoting conventions. FactSet symbols preserve the exchange traded ticker if possible.

2.1 Decoding the future ticker The future ticker at FactSet has a convention that are typically consistent with exchange rules. It is a combination of future root 1, which represent the underlying of the future and expiration code (month code 2 + year unit). For example, 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.

2.2 Quoting Convention Implied 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 forward rate observed at time T representing a borrowing cost from time T to T+n (basically it becomes a spot rate). Time T is normally the future expiration date and n is the tenor of the forward rate, representing the length of the loan to accrue. 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 or US 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 future would be calculating a synthetic bond price based on the inputted yield.

1See Appendix 8.1 for full set of roots FactSet covers 2https://www.cmegroup.com/month-codes.html 3Please refer to white paper Swap Curve Building at FactSet for more information

Because futures are mark-to-market every day, the value of future is set to 0 in our valuation engine. The important valuation measure would be the intra-day price change due to changing rate environment. This makes the valuation engine adapt a Relative-Price approach. This approach is used by all non-Brazilian futures including iBor futures, Fed Fund Futures, Cash Rate Futures, etc.

3.1 Tick Size and Basis Point Value These 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 0.25 basis point for the nearing active contract and 0.005 basis point for other contract. It means if market rate change will round to the tick size. If the change is small, there will be no price change reflecting on futures and no PnL or risk 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 value change. For example, the US Libor 3m Eurodollar future BPV would be $25 and the Fed Fund Future BPV would be $41.67. BPV can be found on exchange website as well or can be calculated with formula: BPV = Notional ∗ 0.0001 ∗ AccrualF raction(T,T + n).

3.2 Relative-Price Approach As introduced in various textbooks including [1] and [2], we get following mathematical relationships: 1 P (t, S) f(t, S, T ) = ( − 1) (1) δ P (t, T ) Q f(t, S, T ) = E T (f(S, S, T )|t) (2) B P (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) assuming simple 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 the unknown 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-market feature 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 be

As we can see, the future price can be represented as time t implied forward rate minus a Convexity Adjustment. Convexity Adjustment: There are many ways to calculate the convexity adjustment. Model based approach includes doing change of measure to move the forward rate from T-forward measure to spot measure. See [1] for

Copyright © 2020 FactSet Research Systems Inc. 4 FactSet Research Systems Inc. | www.factset.com All rights reserved. example of using Hull-White model. However, this approach involves making assumption on volatility and model calibration process. At FactSet, we are using simplified model-free approach to infer the convexity adjustment from the Future market price where ConvexAdj(t) = 100 − f(t, S, T ) − Pmarket(t). Notice that the convexity adjustment here contains both the 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 stays the same for different rate environment. This would enable calculating future price by

P 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, we will use (7). To calculate mark-to-market dollar change, by using tick size and BPV, we can value the future contract as:

∆MTM = round(−∆f(t, S, T ), tickSize) ∗ BPV (9)

4 Valuation: Brazilian Future

Brazilian Futures (DI, DDI, DCO, DAP, OCI) have diﬀerent pricing mechanism as it relates pricing a bullet bond with Brazilian conventions. It also relies on building the corresponding Brazilian curve, speciﬁcally CDI curve and Selic Curve. At this moment, the two curves are not yet built but approximated by a constant spread against Brazilian 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 rate 100000 P = (10) i(T ) n 252 (1 + 100 ) i(T ) = SpotGovt(T ) + spread = DIF utureQuote(T ) (11) where n is time to future expiration date OC1 Future: Future on Selic rate 100000 P = (12) i(T ) n 252 (1 + 100 ) i(T ) = SpotGovt(T ) + spread = OCIF utureQuote(T ) (13) where n is time to future expiration date DDI Future: DI x U.S. Dollar Spread Futures 100000 P = (14) i(T ) n (1 + 100 ∗ 360 )

Copyright © 2020 FactSet Research Systems Inc. 5 FactSet Research Systems Inc. | www.factset.com All rights reserved. where n is time to future expiration date. And i(T) is deﬁned in the following steps: DDIQuote(T ) n DIQuote(T ) n PCDI (t,T ) 1. Ratio = (1 + ∗ )/(1 + ) 252 ) = 100 360 100 PDDI (t,T ) DIQuote(T ) n 252 n 2. i = [(1 + 100 ) ) ∗ Ratio − 1]/ 360 Basically, i(T) will be same as DDI future quote if no curve is shocked. However, if curve is shocked, it will be derived 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 Spread 100000 P = (15) i(T ) n (1 + 100 ∗ 360 ) where n is time to future expiration date on act/360 day count. And i is deﬁned in the following steps: DCOQuote(T ) n DIQuote(T ) n PCDI (t,T ) 1. Ratio = (1 + ∗ )/(1 + ) 252 ) = 100 360 100 PDCO (t,T ) DIQuote(T ) n 252 n 2. i = [(1 + 100 ) ) ∗ Ratio − 1]/ 360 Basically, i will be same as DCO future quote if no curve is shocked. However, if curve is shocked, it will be derived from a ratio with DI future quotes which is derived from a constant spread on Brazil GOVT curve. DAP Future: DI x IPCA Spread 100000 P = (16) i(T ) n (1 + 100 ∗ 360 ) i(T ) = SpotGovt(T ) + spread − IPCA(T ) (17) where n is time to future expiration date.

5 Options on Interest Rate Future

5.1 non-Brazilian Money Market Future In general, options on money market interest rate are American style and the pay oﬀ for a call at expiration date is described as: V (T ) = max(P (T ) − K, 0) = max((100 − f(T )) − (100 − K)) = max(K − f(T )) (18) −r(T −t) B V (t) = e E [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 discount factor. 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 can derive: B B E [V (T )|t) > max(E (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 as European style. From (3), we also know that future price P(t) and derived rate f(t) is a martingale under spot measure, in which case f(t) can be modeled as: df = σf kdW (t) (21)

Copyright © 2020 FactSet Research Systems Inc. 6 FactSet Research Systems Inc. | www.factset.com All rights reserved. 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 Black Model:

V (t) = e−r(T −t)(f(t)N(d1) − KN(d2)) (22) ln(f(t)/K) + σ2(T − t)/2 d1 = √ (23) σ T − t √ d2 = d1 − σ T − t (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))) (25) f(t) − K d = √ (26) σ T − t (27)

5.2 Brazilian Future Option: IDI Option Brazilian IDI Option is an option on average one-day inter-bank deposit rate. It is a tool to bet the CDI rate in the future. The underlying of the option is a compounded daily CDI rate: IDI index value

t Y 1/252 IDIt = IDIBaseDate (1 + CDITi ) (28)

Ti=BaseDate T Y 1/252 IDIT = IDIt (1 + CDITi ) (29)

Ti=t τ Bus252 K = IDIt(1 + RK ) t,T (30) T Bus252 Y 1/252 τt,T (P ayoffT +1)call = IDIt ∗ max( (1 + CDITi ) − (1 + RK ) , 0) (31)

Ti=t

Notice that for IDI option, the payment happens 1 business day after maturity date. People can convert the index value 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 use Black Formula directly on it. However, the volatility will not be observed from market and it is not so convincing to say a daily compounded value would follow a geometric Brownian motion without drift (trend). Another option is to assume a IDI index equivalent simple rate follows geometric Brownian motion:

T Y 1/252 ∗ Bus252 (1 + CDITi ) = 1 + Rt,T τt,T (32)

Ti=t Bus252 τt,T ∗ Bus252 (1 + RK ) = 1 + RK τt,T (33) Bus252 ∗ ∗ (P ayoffT +1)call = IDIt ∗ τt,T max(Rt,T − RK , 0) (34)

Copyright © 2020 FactSet Research Systems Inc. 7 FactSet Research Systems Inc. | www.factset.com All rights reserved. T ∗ QCDI ∗ Rt,T is not known until time T. Following same result from (2), we will get E (Rt,T |t) = Rt,T where Rt,T is spot rate derived from today’s discount curve. For the reason mentioned above that we currently don’t have CDI discount curve, we will use DI quote to approximate Rt,T :

1 Bus252 Bus252 τt,T CDI = 1 + Rt,T τt,T = (1 + F utureQuote) (35) Pt,T τ Bus252 T (1 + F utureQuote) t,T − 1 QCDI ∗ E (Rt,T |t) = Rt,T = Bus252 (36) τt,T

∗ Depending on if we assume Rt,T follows Log-normal distribution or normal distribution, and form (36), we can apply Black/Bachelier formula to do valuation. For call option:

Bus252 ∗ CDI Black : V (t) = IDIt ∗ τt,T (Rt,T N(d1) − RK N(d2))Pt,t+1,T +1 (37) √ Bus252 ∗ CDI Bachelier : V (t) = IDIt ∗ τt,T ((Rt,T − RK )N(d) + σ T − tφ(d)))Pt,t+1,T +1 (38) 1 P CDI ≈ (39) t,t+1,T +1 τ Bus252 (1 + F utureQuote) t,T Again, here since we don’t have CDI discount curve, we are using DI future quote to approximate.

6 Appendix

6.2 Brazilian Future Methodology DI Future: 100000 P = (40) T CDIT 1 Q (1 + i ) 252 Ti=t 100

where CDITi is projected daily CDI rate. This approach depends on a CDI curve built from DI futures. OC1 Future: 100000 P = (41) T SelicT 1 Q (1 + i ) 252 Ti=t 100

where SelicTi is projected daily Selic rate. This approach depends on a Selic curve built from OC1 futures. DDI Future:

FXt PCDI (t, T ) P = 100000PUSB(t, tfx,T − 1) (42) FXt−1 PCDI (t, tfx,T − 1)

Copyright © 2020 FactSet Research Systems Inc. 9 FactSet Research Systems Inc. | www.factset.com All rights reserved. where PUSB is USD Discount Factor for Brazilian onshore market. This should be bootstrapped from BRLUSD Xccy swaps or directly from DDI futures. The current approach approximate this formula by assuming P (t,t ,T −1) USB fx FXt is constant when the rate changes. PCDI (t,tfx,T −1) FXt−1 DCO Future:

FXt PSelic(t, T ) P = 100000PUSB(t, tfx,T − 1) (43) FXt−1 PSelic(t, tfx,T − 1) where PUSB is USD Discount Factor for Brazilian onshore market. This should be bootstrapped from BRLUSD Xccy swaps or directly from DCO futures. The current approach approximate this formula by assuming P (t,t ,T −1) USB fx FXt is constant when the rate changes. PSelic(t,tfx,T −1) FXt−1

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 Risk Management of FX and Interest-Rate Portfolios for Local and Global Markets. ISBN-13: 978-1137477262

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