23
I like to present to you a slightly different approach:
Historically, only one single yield curve was derived from different instruments, such as OIS, deposit rates, or swap rates. However, market practice nowadays is to derive multiple swap curves, optimally one for each rate tenor. This idea goes against the idea of one fully-consistent zero coupon curve, ...
20
You can't make any concrete statements about the monotonicity, convexity or even sign of the yield curve.
Yields are almost always positive, and in the past (2007 and earlier) you could find people who would argue that yields must be positive, typically using a no-arbitrage argument. But recent history has shown us that it is possible for even 10Y yields to ...
16
We assume that the short interest rate $r_t$ follows the Hull-White model, that is, the short rate $r$ and the stock price $S$ satisfies a system of SDEs of the form
\begin{align*}
dr_t &= (\theta_t -a\, r_t)dt + \sigma_0 dW_t^1,\\
dS_t &= S_t\Big[r_t dt + \sigma \Big(\rho dW_t^1 + \sqrt{1-\rho^2} dW_t^2\Big)\Big],
\end{align*}
where $a$, $\sigma_0$, ...
15
I guess it depends on what they're referring to... The traditional swap curve (LIBOR-based) is certainly not risk free, as evidenced by the experience of the financial crisis and the resulting migration to OIS discounting. The OIS curve (which is a kind of swap curve...) is now the standard risk-free curve.
The Treasury yield curve is not favored, because ...
15
A simple correlation/beta analysis of the Banks-relative-to-market versus interest rates or bond yields will tell you that the effect is real enough, whether in Europe, the US, or Japan... Likewise, a simple multiple regression of bank equity to the equity market and to swap rates will also suggest that the rates beta is almost as significant, sometimes more ...
14
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 (...
14
They are both price changes in response to a 1 bp change.
DV01 is valid for a single bond. It is the price change in response to a 1 bp change in yield of this instrument. It arises from the mathematical relationship between yield and price.
PV01 is a more general concept for all fixed income securities , not just bonds but swaps, futures and options, MBS, ...
12
Interest rates in general are far from independent and identically distributed. A high interest rate observation is quite likely to be followed by another high observation, and the volatility is likely to be higher as well. Interest rates are also mean reverting, as in most real-world situations (at least for developed markets) interest rates rarely rise ...
12
Recall that an interest rate swap has two legs, one fixed and one floating, each paid by one party to the transaction.
Now, assume you go to a big bank like JPM, and want to borrow $100MM at fixed rate. JPM will have to fund that position, which because it is a big bank it will do at floating interest rates.
But maybe JPM is worried about the effect such ...
12
Not saying this trade won't work, but there's certainly no guarantee that it will...
Given that QE will stop in October is well teleported at this point and has been expected since last year, you'd think this should be fully priced in.
Last year, when the "tapering" talk started, Treasuries did sell off quite a bit, but has since rallied all the way back. ...
12
Fed funds futures settle into the average daily Fed Funds effective rates over the month. The December 2015 futures contract therefore covers the current Fed funds target rate (0-25bp) for 16 days, and then the new rate range (expected to be 25-50bp) for 15 days.
To compute the exact probability of a rate hike involves some assumptions. For simplicity, let'...
11
CMS adjustments in single curve context can be roughly explained if you consider a CMS swaplet by the fact that there is a single payment at the CMS rate at a single date and not on the whole strip of the underlying CMS tenor schedule.
So if you are trying to hedge a CMS swaplet with the corresponding swap of CMS tenor length (with correct naïve nominal ...
11
There is actually a lot of art involved. The most simplistic framework is as follows:
The first step is to obtain a list of FOMC meeting dates. These are available currently for 2015 and 2016 here. If you're interested for rate expectations beyond 2016, you'd need to "guess" the meeting dates in the future based on past patterns.
The next step is to ...
10
Here's a research note devoted to pricing of CMS by means of a stochastic volatility model. The authors indicate in the Introduction that
an analysis of the coupon structure leads to the conclusion that CMS contracts are particularly sensitive to the asymptotic behavior of implied volatilities for very large strikes. Market CMS rates actually drive the ...
10
To answer a question with a question - are you assuming proportional or constant dividends? :)
The general consensus of the market is that dividends are somewhere between proportional (fixed yield) and constant (fixed dollar). The carry embedded into the forward prices at different strikes reflects that consensus, in fact you can establish the "constantness"...
10
(In addition to the answers of Freddy and Phil H):
With "modern" multi-curve setups: You have to distinguish between discount curves (which describe todays value of the a future fixed payoff (e.g. a zero coupon bond)) and forward curve, which describe the expectation (in a specific sense) of future interest rate fixings.
Swaps pay LIBOR rates and are ...
10
I think you are interpreting too much into the matter. The $-\frac12\sigma^2$ is just a correction term that comes from Jensen's inequality.
You need this when switching from supposedly symmetric returns (normal distribution) to the skewed price process (log-normal distribution).
I think there are no deeper truths to be found here.
10
How is this probability (45%) of Fed raising rates 3 times in 2017 calculated from Fed Funds market?
Using the following data from 12/18/16:
Jan 2017 Fed funds futures =9936,
Jan 2018 Fed Funds futures =9877
implies that 99.36-98.77 = 59bp of hikes are built in for 2017.
IF you assume the only two possibilities are 2 hikes or 3 hikes (meaning, 50bp or 75bp of hikes, assuming each hike would be 25bp), then by simple linear interpolation the probability of 3 ...
9
If you are trying to arbitrage the put-call parity, then use your collateral interest rate for the options side, and your cost of funds on the stock side of the equation. Yes, that's right, 2 different interest rates. Also, don't forget to incorporate bid-ask spreads.
If you are trying to turn a put into a call for your own book, you don't actually need ...
9
Why does USD based security valuation have to give a thing about what London Banks think?
Your question is based on false premises: the USD Libor is not determined by polling London based banks as you seem to believe, but banks on the London money market. The difference is important, as there are—of course—banks which are not based in London and active on ...
9
The idea of regime switching in volatility is rooted in the observation that volatility is usually fairly consistent and "mild", and occasionally very high, say during a market crash. The concept goes further, though. Not only does the volatility level differ markedly in different regimes, but the behavior of volatility does as well (degree of mean reversion,...
9
Of course making money is always the key issue. That (not completely facetious) comment aside:
On the practical side, in many firms IT is struggling with being clear, transparent, and intuitive in their handling of multiple curves and their associated risks. Stumbling over your own systems is an annoying way to lose money. These risks can be surprisingly ...
9
Here are some general directions:
Alternative Risk Premia
The ARP, or "smart beta," space has gained a lot of tractions over the past few years. These are rule-based strategies that provide systematic exposures to risk factors that have historically generated positive excess returns. Some of the best-known factors are, of course, trend, value, carry, etc. ...
8
You can use a matrix type seperability condition as well. This is similar but the equation has more flexibiliity. The rates are then markovian in some combinations of the Brownian motion. See More Mathematical Finance for details.
8
There are certainly (short-rate) models which assume bounded interest rates. I suppose I should clarify - the design of the model prohibits negative interest rates. Further, some models asymptotically reach some target, or mean rate which is considered mean reversion, the most famous perhaps the Vasicek.
Short rate models where rates cannot go negative:
Cox-...
8
The original Nelson Siegel paper describes a parsimonious model of the term structure using only four or three (if $\lambda_t$ is fixed). Filipovic (1999) proves that this model can never be used in a arbitrage free context, paraphrasing the abstract:
We introduce the class of consistent state space processes, which
have the property to provide an ...
8
You should take a look at the example from Hull's book.
Assume that the 6-month, 12-month, 18-month zero rates are 4%, 4.5%, and 4.8%, respectively.
Suppose we know that the 2-year swap rate is 5%, which implies that
a 2-year bond with a semiannual coupon of 5% per annum sells for par:
$$2.5 e^{-0.04 \bullet 0.5} + 2.5 e^{-0.045 \bullet 1.0}
+ 2.5 e^{-...
8
There are many reasons why a yield curve can be inverted. A default-free yield curve reflects a combination of -
market expectation of future short-term interest rates;
bond risk premium: usually positive, longer duration bonds are more volatile and riskier, so investors demand a compensation in the form of higher yields;
convexity.
Let's consider a case ...
8
It is a very interesting question. There is a brief explanation in the book Martingale methods in financial modelling. Basically, it says that, the interest short rate $r_t$ can be modeled in any martingale measure $Q$, however, as long as the zero-coupon bond price $P(t, T)$ is defined by
\begin{align*}
P(t, T) = E^{Q}\Big(e^{-\int_t^T r_s ds}
\mid \...
8
This has already been explained at the start of Chapter 4 in Brigo's book. Basically, for any affine model of the short rate $r_t$, the zero-coupon bond price has the form
\begin{align*}
P(t, T) = A(t, T)e^{-B(t, T) r_t},
\end{align*}
where $A(t, T)$ and $B(t, T)$ are deterministic functions. The yield, or zero rate, is given by
\begin{align*}
R(t, T) &= ...
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