# Tag Info

7

The problem here is that your market is not arbitrage-free: JPY OIS = 10% per day, flat USD OIS = 0% per day, flat USDJPY spot = 100 USDJPY Forward for tomorrow = 100 A quick sense check is that, if you have an interest rate differential, you cannot have the FX forward equal to the spot FX. I would take advantage of the arbitrage as follows: I ...

6

A multi-curve meants that you observe the discounting instruments (such as fed funds) and projection (libor, swap curve) and solve for all of them simultaneously; as opposed to bootstrapping separately a projection curve and a discounting curve. A simple paper with examples is Numerix Model Calibration: The Multiple Curve Approach. A more detailed intro is ...

4

I've always enjoyed OpenGamma's white paper: MULTIPLE CURVE CONSTRUCTION. It's a solid starting point from an implementation perspective. Andersen & Piterbarg's "Interest Rate Modeling" (Volume 1) also has a good chapter on this topic that's pretty easy to follow.

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A plethora of instruments, a menagerie of curves Different instruments are traded in different ways, and relate to a collection of curves. Floating rate instruments depend on some index in order to calculate the cashflows, and so trading instruments which depend on different indices is implicitly trading the expectations of those indices in the future. Fed ...

3

There is no overlapping, the first instrument is tied to the LIBOR rate starting at $25/10/2019$, the second one is tied to the LIBOR Rate at $27/04/2020$. For the sake of clarity, let assume that the spot date and today's date are the same, that there is only one curve (LIBOR Curve). WE use the definition of the forward rate starting at $T$ and ending at ...

3

There are many resources describing how to build a trinomial tree for the Hull & White model (for instance http://www-2.rotman.utoronto.ca/~hull/downloadablepublications/TreeBuilding.pdf), and finite differences schemes are popular as well. These apply to the single curve case. To deal with the multi curve case while keeping everything 1 factor, the ...

3

you have a missing element in your data - you need to take into account xccy basis. when you do so, then you would get the same valuation in both methods. the key is to remember that since your payoff of 100JPY is collateralised in usd, it effectively needs to be thought of as if your payoff is cash settled in usd.

3

To expand on Marcino's correct appraisal of the matter: arbitrage was introduced with the 4 pieces of market data. i.e. JPY OIS = 10% per day, flat USD OIS = 0% per day, flat USDJPY spot = 100 USDJPY Forward for tomorrow = 100 are not consistent with no-arbitrage. Discounting is driven by how the trade is funded. e.g. if there is a collateral agreement ...

2

From $(2)$ of Piterbarg, \begin{align*} V(t) = \Delta (t) S(t) + \gamma(t), \end{align*} where $\Delta (t)= \frac{\partial V(t)}{\partial S}$, and $\gamma(t)$ is the cash account that satisfies \begin{align*} d\gamma(t) &= \big[r_C(t) C(t) + r_F(t)(V(t)-C(t))-(r_R(t)-r_D(t))\Delta(t)S(t) \big]dt\\ &=\big[r_F(t)V(t) + (r_C(t)-r_F(t)) C(t)-(r_R(t)-r_D(...

2

self financed portfolio will give you : $$dV_t = r_F(t) \underbrace{(V(t)-C(t) - \Delta S_t )}_{\text{cash position}} dt + r_C(t) \underbrace{C(t)}_{\text{posted collateral}} dt + \underbrace{\Delta dS_t}_{\text{market move}}$$ then you retrieve his equation using that under risk-neutral measure : $$\mathbb{E}[dS_t|\mathcal{F}_t]=r_F(t)S_tdt$$

2

One possible solution is to build "synthetic" short term 6M IBOR deposits by extrapolating for $T < 6\text{M}$ from the 6M IBOR deposit and 1x7, 2x8, etc. 6M IBOR FRAs as I have seen done in various places, or better by extrapolating from the known 0x6, 1x7, 2x8, etc. OIS-6M IBOR basis as suggested in section 4.4.2 of the paper you are referring to. In ...

2

Predictability - we all know what a bootstrapped curve will do when we shift a value. A minimisation, however, could jump to a new minimum at any moment. They also have unpredictable performance; sometimes a minimisation is fast, sometimes slow. Robustness - these codes have been around forever, and they work. New codes, not so much. Defendability - why is ...

2

Your rates do not overlap. You have a 6M (185/360) rate of 5%. And a forward rate agreement where the 5.2% rate starts at the end of your initial contract (4/27/20) for a period of 6M (183/360). Your first contract will earn you (1 + .05*(185/360)) = 1.025694. You will then earn (1 + .052*(183/360)) on that amount, or 1.052807 over the entire period from ...

1

There are two aspects to consider here. Aspect 1 is funding the notional of the Xccy swap and the coupons (strictly speaking this is not FVA). Aspect 2 is funding the MtM of the swap throughout the life of the trade if you hedge the non-CSA trade with an offsetting transaction against a CSA counterparty (usually the "street") (this would classify ...

1

The price of something under OIS discounting is (supposed to be) the expectation of its value under a particular measure, which specifies the measure and the interest rate of the collateral account etc. So given the price of a set of ATM Libor IRS and their relevant discount curve, the immediate thing to do is to infer the markets' expectation of forward ...

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OIS rates, and the OIS fixing, reflect unsecured lending on an overnight basis. OIS rates compounded for 1Y reflect unsecured lending for a 1Y period via rolling overnight loans for a 1 year period. If, on one any of these days, the counterparty defaulted this would result in a default of the loan. The difference between this and unsecured lending for a ...

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Generally speaking there are more inputs that are required to precisely specify the multicurve structure, and they are potentially more important. For example consider constructing a EUR interest rate curveset for 3 years, in the indexes EONIA, 3M EURIOBOR, 6M EURIBOR. The information you have available are: Some outright EONIA quotes in generic tenors; ...

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For example a Caplet with Expiry of 3year with tenor = 0.5 has to be priced (following the analytical formula) with the LIBOR rate L(0,2.5,3). Am I getting it right ? Thats right. The caplet hast a tenor of half a year and expires in 3 more years, therefore it starts at T =2.5 and ends at T = 3. (Which in this case is the forward rate)

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This is a "cheapest-to-deliver" option - in the absence of any restrictions, the rational investor would post whichever collateral class offers the best rate of return at each moment in time (of course, this could vary over the life of the trade). Thus, to price in the presence of this option, you need to model the optimal posting strategy. You mentioned ...

1

Benchmark yield curves: Make it easier for market participants to efficiently price interest rate products off such benchmark yield curves because there is a consensus and agreement on what serves as benchmark. Those could include government security yield curves, inflation adjusted/reflecting yield curves, among others. Funding curve: Is a set of rates ...

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Generally it is best to use the rates that best capture how the collateral of the instruments are priced. If the overnight collateral on the instruments is managed using offshore JPY depo, then this would be a good choice.

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The math is actually simpler than what you proposed. Z-Spread is always computed as the parallel shift in a zero curve required so as to reprice the cash flows to a bond's cash flows; i.e., you solve for the $s$ in $$P + AI = \sum_{i=1}^N c_i \cdot d(t_i) \cdot e^{-t_i \times s}$$ In the multi-curve world, you simply compute both the LIBOR OAS and OIS OAS ...

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