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As far as I understand, a T-forward measure is associated with a situation when a zero-coupon bond with the same maturity, i.e. $P(t,t+T)$, is used as a numeraire. However, given that the yield curves, LIBOR, used to derive this bods have in the markets maturities only upto $12M$, what do we do when $T > 12M$? In such cases $P(t,t+T)$ is no longer a tradable (or even observable) asset and hence can't be used as a numerarie.

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The current LIBORs (say O/N, 1wk, 1m, 2m, 3m, 6m, and 1y) refer to the borrowing cost for borrowing starting today (technically this is the spot date, which differ by currency,e.g. T+1, but we can call it today!). Each of these rates will have its own discounting curve. Let's focus on the 3 months rate.

The current 3 month LIBOR refers to borrowing today for 3 months. To construct the discount curve, as you correctly pointed out, we need the borrowing costs for future periods, so we need additional contracts/intruments. Luckily you have the 3 months FRAs (forward rate agreements) which give you the cost of borrowing for 3 months, starting in the future. You have the future contracts, which are similar to FRAs, but there are some subtle differences between FRAs and futures as you would know. You then have the swaps referencing 3 months LIBORs, and these swaps can go to very long maturities, say 50 years. So you can construct the discount curve for 3 months LIBOR by combining these contracts (e.g.,bootstrapping). You can do the same for the other LIBORs, so each LIBOR (3m, 6m etc) can have its own discount curve.

Normally you would use the longest maturity zero coupon as the nuemraire, but you don't have to. If the nuemraire asset is such that it stays alive over the horizon of your interest, then you are obviously fine. But if not then you can use a hybrid numeriare- e.g., hybrid of zero coupon and bank account.

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  • $\begingroup$ Thank you for your answer. However, isn't the curve constructed in this way, say for 3M LIBOR, tells us what the borrowing rate is for a given maturity, say 10 years, PROVIDED that the interest is payed every 3 months? In this case, we can;t use it to derive zero-coupon bond since that bond does not pay any coupons and only pays at maturity. Thanks $\endgroup$ – Confounded Nov 25 '19 at 21:49
  • $\begingroup$ You normally see only FRA futures and swaps get used in the curve contradiction. Non govies Bonds usually get modelled in the z spread sense ( also they carry idiosyncratic risk), and liquidity is also an issue . $\endgroup$ – Magic is in the chain Nov 25 '19 at 22:08
  • $\begingroup$ It seems I wasn't clear enough in what I was trying to convey. To construct a ZC bond with maturity, say, 10Y to be used as a numeraire, we need a yield curve that gives a borrowing rate for 10 years without any repayments between now and 10 years, but there is no such curve since the longest tenor is 12 months, so a 10-year rate from this curve assumes that there will be 10 payments between now and 10 years. So, how do we get the 10Y-forward measure if there is no 10Y ZC bond? $\endgroup$ – Confounded Nov 25 '19 at 22:39
  • $\begingroup$ Ah I see. There is a simple relationship between the price of the zero coupon and the simple forward rates, please look it up - it comes up a lot. Libor is like simple forward, and you are essentially ‘stringing’ together these instruments so that there is only a payment of 1 at the maturity. $\endgroup$ – Magic is in the chain Nov 25 '19 at 22:51
  • $\begingroup$ So, how do we discount a cash flow that tskes place in, say, 13 months when the long LIBOR rate is for 12 months? $\endgroup$ – Confounded Nov 25 '19 at 23:12

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