16
There are two parts to your question and I'd like to answer them separately.
Curve Construction
On a daily basis, you can observe prices on a large variety of instruments, whose prices are driven by news and trading flows. Based on market prices of these instruments, there are a number of ways to create discount curves/forward curves. At a very high level (...
12
I think your question can be split into two parts: (i) how to value a swap mathematically and (ii) how swaps actually work as a traded product.
Part (i):
As noob2 pointed out, "theoretically", a swap is valued with the help of two curves: one "forward" curve and one "discounting" curve. Say you want to "value" a 10-...
8
Chapter 1: Goldilocks is ousted by the bears
Once upon a time, the banks used a fixing called LIBOR as a measure of the risk-free interest rate. Then the big hairy crisis came along and ate all our assumptions, leaving just the bones of the fixing (upon which everything else still fixes) and the mantle of risk-free rate proxy was passed on to a family of ...
6
It is incorrect to use 1m euribor or O/N euribor in a 6m Euribor forward curve. You should only use instruments based on 6M euribor, such as 1x7 FRA, 6x12 FRA or swaps v 6m Euribor, as you have done in your second example. The actual 6m euribor fixing itself can be thought of as a 0x6 FRA out of spot.
Before the financial crisis basis between different ...
5
Let's step back and look at the reason for making a DV01 calculation first before answering the question;
The reason for making a DV01 calculation is to quantify what market movements has impact on the valuation of the trade.
Since the 'flat' forecast curve won't be affected by market movements the answer is (using pre-2008 methodology): The floating ...
5
When pricing a swap with at least one floating leg referencing some index, if the fixing date of the index is before evaluation date, but the floating coupon period start date, end date, and payment date are after the evaluation date, then you need to use the realized fixing rate to calculate the coupon and discount it down from its payment date.
If you ...
5
I think I understand the question, but maybe not.
In USD market, the most liquid IR swaps have floating leg reset quarterly from 3Mo LIBOR. (The fixed leg is semi-annual. Ths will change when LIBOR is discountinued; it looks like the most common SOFR floaters will have annual frequency both for fixed and floating legs). (Market conventions differ for other ...
3
I don't think they are implying that future interest rates are predictable. They may be speaking of implied forward rates as predictors of future rates or, generally, of the yield curve as an expectation of the future path of short-term interest rates.
If $P(0,T_1)=1/(1+r_1)$ and $P(0,T_2)=1/(1+r_2)$ are the prices today of two "risk-free" zero coupon bonds ...
3
Which currency are you looking at ?
Say that your 1y swap would have yearly fixed payments vs 3M floating payments.
Your 1.5y swap would probably have:
a fixed payment 6m after effective date and another fixed payment 18m
after effective date
regular quarterly floating payments
Your curve was built with 1y and 2y swaps, nothing in the middle ? Then yes, ...
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
Assume the (annualised, continuously compounded) forward rate between two nodes, say $t_{10}$ and $t_{12}$, is constant, say $ f_{10,12}$, then the discount factors of the two consecutive knots will be linked as follows:
$D_{12}=D_{10}e^{-f_{10,12} \left(t_{12}-t_{10}\right)}=D_{10}e^{-2f_{10,12}}$
From which is then easy to infer the formula for $t_{11}$,
...
2
You have
$\beta_1=\frac{1}{(1+r)^n}\frac{1}{(1+r)^\theta}$
and
$\beta_2=\frac{1}{(1+r)^n}\frac{1}{(1+\theta r)}$.
Both are equal when $\theta=1$. If you consider simple interest then go for $\beta_2$. If you would like compound interest within fraction of year then pick $\beta_1$.
However, because $\theta$ is between 0 and 1 then values $\beta$'s won't ...
2
The short answer is that there's no consensus. A popular method is to shock each input instrument by 1 bp (i.e., change the futures rate by 1bp, the swap rates by 1bp, OIS rates by 1bp, etc.), rebuild the curve, and then reprice the instrument of interest to obtain its curve sensitivity. This of course is not quite a "parallel" shift of any curve (e.g., a ...
2
An OIS interest rate swap rate with annual-annual freq is determined under one year by:
$$1 + d_i s_i = \prod_{j=1}^{n(i)}(1+ d_j r_j) \; , \quad \text{where} \quad d_i = \sum_{j=0}^{n(i)} d_j \;.$$
Each $r_j$ is a forecast overnight OIS rate which as you can see are compounded in the floating side.
Therefore a discount factor in the future, for maturity $...
2
By no arbitrage, market participants need to agree on the values of the discount factor, even if they are using different conventions (day count, compounding period) to convert the discount factor into a rate.
For example, consider two discount factors computed using continuous compounding, where one is computed using the 30/360 day count (year fraction $t_{...
2
Suppose you wanted to value a 5Y EUR IRS with a USD cash collateralised curve this is the broad process:
Get the 5Y EUR 3M / OIS basis, say this is 10bps: This establishes the discounting basis in the local (EUR) currency.
Now get the 5Y EUR/USD Cross-currency basis, say this is EUR 3M-IBOR - 40bps: This establishes your link to dollars.
Now get the 5Y ...
2
Equation 2 gives the annual zero rate for all tenors. In practice, people sometimes quote rates f less than one year using Equation 1, but in general , equation 2 is used.
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 ...
2
You should be aware that Bloomberg doesn't have a yield curve. Bloomberg has market prices of the instruments to build a yield curve and the result will depend on the particular configuration for your user. Some of these choices would be:
Instruments (Deposits, Futures, FRAS, Swaps)
Interpolation Method
Curve Side (Bid, Ask, Mid)
OIS DC Stripping
If you ...
2
The average of simulated discount factors from the Hull-White model and market discount factor are the same in theory but very similar in the simulation due to numerical error.
I draw one figure which compares two discount factors and shows their difference.
red line : mean of simulated discount factors
blue line : market discount factor
green line : ...
2
A first step would obvisouly be to check if the curve you built replicates the input instruments.
A second step might be to check the forwards to see if there is irregular behaviour around the curve nodes. Look at a plot of daily 3m or 6m forwards which should be smooth. Different interpolation methods will generate diferent results and for this might I ...
2
For simplicity, let's say that your time $0.003$ equals 1 day, and your second pillar (probably $0.083$ instead of $0.00833$) equals 1 week.
What you do: Approximate the short rate with the 1-day interest rate.
What they do: Employ additional information about the shape of the yield curve at the short end, i.e. extrapolating from the first two available ...
1
Try Howard Corb's book ["Interest Rate Swaps and Other Derivatives", Columbia Business School Publishing, 2012].
And your 2y rate looks odd which should be intuitively obvious - simple arithmetic mean of your 6m, 6mf6m and 12mf6m rates is 5.6%. What does that suggest about your 18mf6m rate for the 2y rate to be 7%?!
1
if you are asking how CME collateral is discounted, then you have two considerations:
What does the CME give you on your USD cash? That's simple, it's OIS. You don't get the interest immediately, but instead I think once a month. I'm not sure if they compound it - but I would imagine they do as it references OIS.
What's your funding situation. For ...
1
This question Setting the r in put-call parity? shows the details are subtle and nuanced, but the answer is to use the rate paid on the collateral or margin.
1
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; ...
1
It's difficult to see your screenshot. But I think you should just follow some real examples online instead of having people find out what's wrong on your side.
This is an excel example, go play with it.
webuser.bus.umich.edu/Organizations/FinanceClub/resources/BootstrappingMath.xls
1
I don't recommend linear interpolation of DFs and the swap rates you are applying this to are either against 12M libor which is illiquid or you are not accounting for Quarterly or Semi-Annual floating sides. And what I'm going to suggest uses a single curve framework which is long outdated. But that being said and given the nature of what's been asked...
...
1
Update (2018-10-09):
This solution is more correct. It's a class that solves for the DM using the class ForwardSpreadedTermStructure.
public class DMFinder : ISolver1d
{
private readonly List<Cashflow> leg_;
private readonly double dm_;
private readonly DayCounter dayCounter_;
private readonly Compounding compounding_;
private ...
1
Your valuation date is $t=$ Thu 10-Nov-11. The swaps start on the spot date which is $t + 2$ business days = Mon 14-Nov-11. The usual approach is to extrapolate between $t$ and the first curve pillar, in a manner consistent with the interpolation method that you are using for representing your discount curve. For instance if you use linear interpolation of ...
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