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19

To sum up what is discussed in the CFA curriculum, it discusses 3 types of spreads. They are used to compare a risky bond to a Treasury bond (assumed to be risk-free). Nominal spread Simply computes the difference between the YTM of the risk-free bond and the YTM of the risky bond. The major problem of this measure is that it doesn't take into account the ...


7

As with most derivatives that have early exercise, you are going to want to price this using a grid scheme. I have priced callable loans with floors using the Generalized Vasicek model at my old hedge fund, and it is fairly easy to handle. As a matter of fact my students are doing that very problem as homework this week, and my reference implementation ...


7

This really is an arbitrage. It is caused by differences in supply and demand between the interest cashflow and the principal cashflow and by differences in the financing rates on the two STRIPS. As you noted, the price difference is small, and it would take 30 years to guarantee convergence. In addition, the outstanding amount of the 30-year coupon strip ...


7

There are "perpetual" bonds and preferred shares that are traded in the corporate credit markets that exactly match your conditions above. They are recorded in the 10-K at notional value $X$. The "close-out" feature is an embedded call. You should assume your favorite stochastic interest rate (and/or credit) model and run a PDE solver, tree, or other grid ...


7

I assume that you are working in a single curve theory. While this theory used to do well, it is not adapted to today's market and — as Brian B pointed it out — you cannot get a useful information from swap rates alone. The swap rate $S(t)$ at $t$ for a given tenor $T$ and period $P$ is the fixed rate such that a swap starting at $t$ and ending at $t+T$ ...


6

In practice, I would begin with the recovery assumption. In the case of Greece, dealers are probably already quoting recovery swaps, allowing you to set this parameter directly. In general, you have to be willing to make assumptions based on history or on conversations with bankruptcy experts. Once I have the recovery assumption, I can take any ...


6

Well, that's still a very general question. A few elements of answer : Bonds pay interest on a regular basis, semiannual for US treasury and corporate bonds, annual for others such as Eurobonds, and quarterly for others. You need to distinguish between fixed coupon bonds, zero coupon bonds, bonds with an amortization schedule, floating rate notes based on ...


6

Q1 - Yes, debt load has an impact on the stock price. For instance, say you are valuing a company with a discounted cash flow model, while the interest won't affect the operational cash flows, it will increase the cost of capital. With that, the perceived value will be less than a similar company with less debt. Debt will also affect the volatility of the ...


6

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: ...


6

The Hull-White model can represents the risk free rate as a stochastic process, that is, in terms of expected return and volatility. The zero curve only gives you expected returns and you have to find a source to calibrate volatility, as FQuant told you. Common volatility sources used for this calibration are historical series of the zero curve or ...


6

I'm familiar with the library, but not with the way it is exported to R. Anyway: gearings are optional multipliers of the LIBOR fixing (some bonds might pay, for instance, 0.8 times the LIBOR) and spreads are the added spreads. In your case, the gearing is 1 and the spread is 0.0140 (that is, 140 bps; rates and spread must be expressed in decimal form). ...


6

Your observations are pretty much correct. The groupings are because of the fine print "Note how I have expanded the drift and volatility terms at $t = T$; in the above these are evaluated at $r$ and $T$." on the same page (p.528). Basically, $w$ is a function of both $r$ and $t$. Since we want to use $w(r,T)$ instead of $w(r,t)$, we taylor expand ...


6

This is called on the run/off the run arbitrage, a type of convergence trade. The basic idea is that as the liquidity premium disappears for the on-the-run issue, the price will fall and converge to the price of previous issues. Here are a couple papers - http://people.stern.nyu.edu/lpederse/courses/LAP/papers/SearchBargaining/VayanosWeill.pdf ...


6

Day-count conventions. You can't live with them, you can't live without them. The reason the prices differ is that the pricing engine can't calculate correctly the time over which the first coupon is discounted, and thus it gets slightly different discount factors to apply to the coupon amounts. Please sit down, it'll take some explaining. Ultimately, both ...


5

Yes, you are correct. Duration is additive, so your aggregate portfolio duration is the weighted average of your individual durations as you present in point 2. That holds assuming a close to flat yield curve and parallel (additive) shifts. If that's not the case, the situation gets a bit more complex. Unfortunately, right now I couldn't find any ...


5

After struggling through the Pianca paper due to its poor proofing ($F$ is never defined but appears to be face value, and $n$ is implied to be the number of periods remaining but is instead maturity), I seem to have it worked out. Using the lambertW function in gsl, I have it replicated in R: # Estimate duration using various closed-form formulae # ...


5

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 ...


5

The Macaulay duration is a measure of how sensitive a bond's price is to changes in interest rates. Duration is related to, but differs from, the slope of the plot of bond price against yield-to-maturity. The slope of the price-yield curve is $-\frac{D}{1+r}P,$ where $D$ is Macaulay duration, $P$ is bond price, and $r$ is yield. Here's how the definition ...


5

Forward interest rates are negative whenever the yield curve is negatively sloped. The US term structure was inverted most recently around 2007. Hard to find bank deposits that have negative yields (find countries experiencing deflation and you may find it), however, treasury bills during recent times of financial stress have yielded a negative rate. The ...


5

The answer to your first four questions is affirmative. Option-adjusting the spread makes an equivalence between everything theoretically possible, but the quality of results depends significantly on the quality of your interest rate model and its calibration. My personal opinion, though, is that the results need to be treated carefully because the OAS ...


5

Short answer It's complicated. A satisfactory solution is not known. Long answer A satisfactory solution is not known and research is ongoing. That doesn't mean there is nothing interesting to say about it. The phrasing in the question is not entirely correct: First off all, there's is no risk free arbitrage between bonds and stocks. Both are risky and ...


4

The risk implied by Euribor or EONIA (or their swaps) is for lending to another prime rated bank. These rate indexes represent where contributor banks are offering funds to each other in the interbank market. Contributing banks are mostly rated P-1 (Moody’s) or A-1 (S&P). You wouldn’t use these rates for govt discount curves because the risk doesn’t ...


4

One could say that a CDS price is determined by the physical default probability and the risk premium. The physical PD (PPD) is the actual probability of company defaulting within the given period of time. It is purely a theoretical concept as no one really knows what this probability is. We could estimate it using some models or credit ratings, but those ...


4

The SABR model has an overly fat right tail. If you do the CMS replication using cash-settled swaptions you find that you need ridiculously high strikes.


4

Yes, it is definitely possible to do so. With a long fixed-income portfolio, you'd typically be buying puts on treasury futures or writing calls on them (writing calls may not be feasible if you're an institutional investor due to regulatory reasons). In general, duration for long puts/short calls would be negative. However see caveats below: Typically, ...


4

If you're able to work with the results from the paper cited (Pianca, Maximum Duration of Below Par Bonds: A Closed-Form Formula), congratulations! You have the hard part done! Maximum durations for par and premium bonds are trivial. Here is a figure directly from the cited paper: Some points about the figure: the market interest rate used is ...


4

You don't say which duration, but it's generally okay to use effective duration: $$ duration (eff) = \frac{-1}{P(r)}*\frac{Price(r+b) - Price(r-b)}{2*b} $$ where $r$ = rate and $b$ = yield shock. Although, to address Brian's point, the mortgage contains an embedded call option that creates negative convexity, so the three re-pricings, $P(r)$, $P(r+b)$, ...


4

A swap does not require a model because its price can be derived from the yield curve without any assumptions about how the yield curve may move in the future. The PFE however is an indication of by how much the swap's mark-to-market may move between now and a moment in the future. It is of course influenced by how volatile rates are. The more volatile ...


4

There are two flaws in the argument. The simpler one is that expectations give information about probability distributions (premise D). I think this is what John was referring to in his comment. The fact that the expectation of a forward rate in period X is Y% tells us nothing about the implied probability distribution in that period, and certainly doesn't ...


4

Quantmod package claims to support downloading data from Federal Reserve Bank of St. Louis Economic database, which contains plenty of rates time series. It should be pretty simple to get them into R using function getSymbols, in the same manner as getSymbols("DEXJPUS",src="FRED") # FX rates from FRED



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