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16

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, ...


12

This is not a trivial question. Here's a relevant excerpt (an appetizer, really) from Hull's book (7th Edition, P. 75): It is natural to assume that the rates on Treasury bills and Treasury bonds are the correct benchmark risk-free rates for derivative traders working for financial institutions. In fact, these derivative traders usually use LIBOR rates ...


11

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. ...


10

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 ...


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.


9

If your trades are collateralized/margined, you should use the rate paid on your collateral/margin.


9

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 ...


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

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 ...


7

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 ...


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 following paper, Interpolation Schemes in the Displaced-Diffusion LIBOR Market Model and the Efficient Pricing and Greeks for Callable Range Accruals, addresses this issue: We introduce a new arbitrage-free interpolation scheme for the displaced-diffusion LIBOR market model. Using this new extension, and the Piterbarg interpolation scheme, we study ...


6

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.


6

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 ...


6

The main problem is that you cannot achieve Libor in the markets. So the old-fashioned method of discounting at Libor doesn't work any more. As an example, if you compound up the 3m Libor with today's price on a 3x6 FRA, you won't get 6m Libor. Traditionally, that would mean arbitrage, but these days it's just a fact of life. You cannot achieve 3m Libor for ...


6

(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 ...


6

Risk-neutrality isn't really a property of a model. Instead, it describes a certain calibration of a model (almost always represented by an SDE). We say a model has been calibrated to risk-neutral probabilities if model parameters can be inferred from traded security prices, and there's some anti-arbitrage assumption and hedging scheme available for ...


6

There's no class at this time to add two curves as you want, but it won't be much difficult to write it. The closest you'll get in the library is the ZeroSpreadedTermStructure class, that shows the general idea: it inherits from YieldTermStructure (by way of ZeroYieldStructure) takes a YieldTermStructure and a spread (constant, in this case) and override ...


6

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 ...


6

Quite surprised at your professor's comment, since the par yield curve is one of the most important yield curve representations! You can of course just plot the yields of coupon bonds against their time to maturity and call it the yield curve, but the curve won't be smooth because of coupon effect (e.g., when the yield curve is upward sloping, high coupon ...


6

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 ...


6

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 ...


5

I think you might use the relevant OIS-rate like EONIA or Fed Fund Rate, at least this is the current fad when discounting interest rate swaps.


5

As you said, $\mu$ is the expected return that is the expected value (mathematical expectation) of the random variable "stock return" under the objective probability measure. Assuming that returns are stationary*, the obvious way to estimate it is to compute a large number $N$ of returns $R_i$, then to average them. You also want to annualize this average ...


5

Take a look at historical short-term risk-free rate proxies such as Fed Funds, LIBOR, short Treasuries, and you will find plenty of periods where rates have been significantly above or below inflation (as measured by any CPI series) in the same period. In fact, controlling this difference, known as the real interest rate, is the primary tool of modern ...


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

The very easiest change you can make is to switch to quasirandom sampling. I favor the Niederreiter sequence, for which you can find implementations in most languages around the web. You can also get a (sometimes tremendous) speed boost by running using a control variate. Even a swap would probably reduce your variance somewhat. I don't recall the CIR ...


5

You should use the full yield curve, discounting cash flows at specific dates using the appropriate zero-coupon interest rate. As to which yield curve, that is often a matter of convention. Generally one uses the LIBOR/swaps curve for all but the most liquid products (in which case you use the treasury curve). The curve is constructed from LIBOR/Eurodollar ...


5

First step is to decide what instruments you want to include in your process for estimating the spot curve. You want to look at the following instruments for inclusion - treasury coupon strips, on-the-run treasury issues, and some off-the-run treasury issues (those not trading at liquidity discounts), or all treasury coupon securities and bills. You want ...


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 ...



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