How are bond futures prices adjusted?

Discussion in 'Financial Futures' started by Herkfsu, Apr 6, 2019.

  1. Herkfsu


    I assume its like an equity index in that it is adjusted for dividends(interest), but are there any other funding interest fees to buy?

    As I understand it the price executed on a forward contract is typically adjusted upward to count for the interest that would have been earned during that time since no money is exchanged initially. Though for futures, money is exchanged initially, though its typically a small fraction of the total notional amount. So are prices adjusted according to the margin requirements on the instrument? ie if margin requirements were 50%, the price would be adjusted upward to account for 50% of the interest that would otherwise have been earned.

    Also, do brokers charge for margin on futures in the same way they would for equities. ie if I put 1,000,000 in an account and bought 10,000,000 in bond futures, would I have to pay interest on that 9,000,000 I borrowed? Or is that built into the price via the above?

    I hope I made sense with these questions. Any help would be greatly appreciated. Thanks!
  2. Robert Morse

    Robert Morse Sponsor

    Bond Futures are forward contracts to buy/sell the bond on the expiration date. There is not current interest paid or adjusted for. You are betting on future rates/prices. And, you do not put up part of the purchase price like a margin transactions in equity. The margin represents a Performance Bond to have money available to cover losses and counter-party risk.

    When you buy any future, you just need to have the value of the initial margin, then maintenance margin available. Since you are not buying and taking delivery of anything, there is no interest on the notional value of the contract.
    ajensen likes this.
  3. Robert Morse

    Robert Morse Sponsor

    $10mm in 10-year notes, contract size $100,000, would require 100 ZN futures, margin is $1265/future or $126,500 for 100 to control $10mm in 10-year notes.

    Check this out-
  4. Herkfsu


    Thanks Robert!

    Just so I am clear... When you have 1,000,000 in a brokerage account and you buy 2,000,000 worth of AAPL, you are charged interest on that extra 1,000,000(by your broker). That is not the case when buying/shorting futures, correct?
  5. Herkfsu


    How would it not be adjusted for interest paid? It would have to otherwise one could buy the bond(collect interest along the way) and sell the forward for arbitrage. Forward price should be adjusted higher. Where am I off on this?
  6. Robert Morse

    Robert Morse Sponsor

    Correct. You just need liquidating equity of more than the SPAN margin requirement. Notional value of the delivered future is not relevant. In equities, you need to buy the stock, deliver cash for the security on the settlement date, so you need to have the money in the account or borrow it from your broker.
  7. Robert Morse

    Robert Morse Sponsor

    Futures pay no interest in the same way as options pay no interest. Futures are a derivative of the deliverable symbol, in this case a 10 year note. The idea is to get the Bond at expiration at a price or hedge the changes in price, or speculate on the change in value at the date and price you entered the trade.

    Maybe it would help to look at the future like a call options with a strike of 0 with no early assignment, but you do not have to fully pay for it and only have to have a deposit that covers the risk of the trade each day.
  8. drm7


    On the surface, this seems like a simple question, but the underlying calculations for how bond futures behave are pretty complex, and there is an entire cottage industry in the trading community called "basis trading," which attempts to exploit differences between the futures and cash bonds.

    The explanation at this link just scratches the surface. There are entire books written about this.
  9. TommyR

    TommyR yeh so note the difference is very non-trivial. in 2 dimensions your underlying-cash pair which defines the martingale signa algebra is 'a bond' and 'a future (an illegal)' not a spot rate. so v v different. worth looking at not saying oh yeh options on forwards got it got it. in my opinion. 2.2 Valuing a FRA Using FuturesDeterminingf(V0,F0) amounts to valuing a forward contract viewed as a contin-gent claim with final payoff (3). In order to do that, we shall callv0=f(V0,F0)the unknown premium to be determined.We consider an investor receiving aninitial (t= 0) amount of cash equal tov0, and engaging in a continuous tradingstrategyθ=(θt) in the futures contract,1where all cash is reinvested in thediscount bondVt.Ifwecallπtthe value of the investor’s portfolio at timet,then the processπ=(πt)isgivenbyπ0=v0, and the stochastic differentialequation:2dπt=θtdFt+πtVtdVt(6)In other words, a variationdπtin the portfolio’s value arises due to variationsdFtanddVt, and the two long positionsθtandπt/VtinFtandVtrespectively.The solution to (6), given the initial conditionπ0=v0, can be expressed as:3πt=Vt(v0V0+∫t0ˆθtdˆFt)(7)1At timet, the investor has a long positionθtin the rateFt, which actually correspondsto a short position in terms of contracts.2Note that by writing (6), we have neglected the effect of minimum margin requirements.In real life, an investor entering a futures contract could not reinvest the totality of his profitsin the discount bond, since some of his cash has to be left on his margin account.3See appendix A.2where:ˆFt=Ft/Ct(8)ˆθt=θtCt/Vt(9)and the processC=(Ct) has been defined as:Ct=exp(∫t01FsVsd〈F,V〉s)(10)In particular, our investor will have a final wealth at timeTequal to:πT=πT(v0,θ)=VT(v0V0+∫T0ˆθtdˆFt)(11)This final wealth is obviously a function of the initial premiumv0and tradingstrategyθ. Now, suppose for a moment that we could findv0andθsuch that:πT(v0,θ)=αVT(FT−K)(12)Then, an investor receiving an initial cash payment ofv0and entering the strat-egyθ, will exactly generate a final wealth equal to the final payoff of our forwardcontract. In other words, an initial investment together with adequate trading,enables the exact replication of a forward contract payoff. To avoid any pos-sibility of arbitrage, the value of this forward contract has to be the initialinvestmentv0. Hence, if we can findv0andθsatisfying (12), then we know thatv0is exactly the premium that we are looking for.Our problem of findingv0can now be rephrased in terms of the followingquestions:1. Do there existv0andθsuch that (12) holds?2. If so, how do we calculatev0?Of course, the answer to these questions will very much depend on the particularassumptions made on the processesV=(Vt)andF=(Ft). In general, it isnot true thatv0andθalways exist, and if they do, actually computingv0canbe quite tedious. However, without (for now) being more specific onVandF, we can indicate the general procedure enabling to get answers to the abovequestions: firstly, comparing (12) with (11) shows thatv0andθshould satisfythe equation:v0V0+∫T0ˆθtdˆFt=α(FT−K)(13)Now, let us assume that there exists a probability measureQ, under which theprocessˆF=(ˆFt) (as defined in (8) ) is a martingale,4and furthermore, that4See appendix C for the proof of such existence, (provided we make the right assumptions).Do not be put off by the terminology here: everything you need to know is recalled below.3themartingale representation theoremcan actually be applied:5this theoremstates the existence of a constantx0together with a processφ=(φt) such that:x0+∫T0φtdˆFt=α(FT−K)(14)Of course, we do not know explicitly whatx0andφare. But we are only inter-ested in their existence: for once we know thatx0andφdo exist, then definingv0=x0V0andθt=Vtφt/Ct, equation (14) can be rewritten as (13), whichshows the existence of a premiumv0and a strategyθsatisfying equation (12).This is the answer to the above first question.Having answered question 1, we are now left with the task of actually com-putingv0. As we shall see, there is very little to it: indeed, the nice thing aboutˆF=(ˆFt) being a martingale underQ, is that we can always write:6EQ[∫T0ˆθtdˆFt]= 0(15)and takingQ-expectation on both sides of (13), we therefore obtain:v0=αV0(EQ[FT]−K)(16)which shows that computingv0amounts to the computation of theQ-expec-tationEQ[FT]. In general, this expectation can be quite difficult to obtainexplicitly. However, if the assumptions made on the processesFandVare suchthat the processC=(Ct) as defined in (10) is actually deterministic,7then wehave the following:8EQ[FT]=EQ[ˆFTCT]=CTEQ[ˆFT]=CTF0(17)which can be substituted into (16) in order to obtain:f(V0,F0)=v0=αV0(CTF0−K)(18)This completes our task of answering questions 1 and 2. It should be remem-bered however, that before deriving anything like (18), some assumptions hadto be made. In other words, taking just any kind of diffusion for the processesFandVwill inevitably lead to the collapse of the previous developments. Whenconfronted with the task of designing our financial model, three fundamentalpoints have to be kept in mind:95See appendix D for the proof of that.6We are being slightly over optimistic here. In reality, some integrability condition has tobe met byˆθ. See appendix D.7This looks like we have an additional requirement onFandV. In fact, the assumptionofCbeing deterministic is also needed to ensure that themartingale representation theoremcan be applied. See appendix D8ˆFbeing a martingale underQ,(andF0being constant),EQ[ˆFT]=EQ[ˆF0]=F0.9As already mentioned, point 3 is in fact a prerequisite to point 2.41. We need a probability measureQ, under whichˆFis a martingale.2. Themartingale representation theoremmust be applicable.3. The processC=(Ct) should be deterministic.2.3 The Convexity AdjustmentIn the previous section, we were able to explicitly determinef(V0,F0)byequa-tion (18). Looking back at (5), it appears that the forward rateL0and futuresrateF0satisfy the equation:αV0(L0−K)=αV0(CTF0−K)(19)from which we conclude that:L0=CTF0(20)In other words, the forward rateL0is equal to the futures rateF0times aconvexity adjustmentCTgiven by:10CT=exp(∫T01FtVtd〈F,V〉t)(21)In order to give a more explicit formulation ofCT, it is now time to be morespecific about the processesF=(Ft)andV=(Vt). As detailed in appendix B,the chosen diffusion forFandVare:dFt=μ(t)Ftdt+σF(t)FtdWt(22)Vt=exp(−(T+ΔT−t)Rt)(23)dRt=γ(R∞−Rt)dt+σR(t)R∞dW′′t(24)withF0,R0>0, whereγ,R∞are strictly positive constants, and all processesμ, σF,σRare deterministic. It is of course understood thatWandW′′in (22)and (24) are standard brownian motions. Furthermore, we assume thatWandW′′have deterministic correlationρ(t).In appendix B, we show that given (22), (23) and (24), the convexity ad-justmentCTcan be expressed as:11CT=exp(−R∞∫T0(T+ΔT−t)σR(t)σF(t)ρ(t)dt)(25)10There is no particular reason to callCTaconvexity adjustment, apart from currentpractice.11Paul Doust [1] assumes log-normal diffusion for bothFandV, with deterministic corre-lationρF,V. In this case we obtain:CT=exp(∫T0σV(t)σF(t)ρF,V(t)dt)5tT+ΔTT: Continuously compounded spot rateRtFt: Futures ratesT-tΔTFigure 1:ρ(t)=e−δ(T−t)/ΔTis assumed to be the correlation between thefutures rateFtand continuously compounded spot rateRt
  10. TommyR


    orrr heat equation dirichlet with X(T)=S-K. however this requires heavy machinary from stochastic analysis which was not obvious to me a priori.
    #10     Apr 7, 2019