# Tag Info

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

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 ...

5

Ok, I've done some digging in the code. It's an issue with the LogLinear interpolation; while trying to find the correct rate for the 1-week node, the bootstrapper wanders unchecked into a region of negative rates and the logarithms blow up. At this time, I'm afraid the workaround is just to use some other interpolation. Or recompile the library and the ...

5

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 ...

4

Quantlib supports multi-curve framework (to the best of my knowledge). By the way, there's a "newer" version of that paper (authored by Pallavicini & Brigo). http://arxiv.org/abs/1304.1397 This paper might also be useful for you, very practical and basically answers any question you could have. Also see this discussion about multi-curve discounting ...

3

Your overall approach is correct. However to my knowledge it is formally more appealing to work with a parameterized and smoothed yield curve. Basically one assumes that the yield curve can be described by a smooth function $r(t,\alpha, \beta,\gamma)$ (mostly of three parameters) Given a set of market data $Y(t,T_1)\dots Y(t, T_n)$ one looks for ...

3

It's hard to be sure without seeing the inputs, but I'm guessing that the implied curve changes shape because the original curve does (which you can see from your output: except for the 1-year and 5-years points, the actual discounts are different). The reason the original curve changes is probably the different position of weekends or holidays (so that, ...

3

As @michipilli said, if $Z = 1+ as + bs^2 + cs^3$ (where I have substituted $T-t$ by $s$ for ease of notation and also suppressed the dependencies of $a$, $b$ and $c$) and $\log (1+\zeta) = \zeta - \frac{1}{2}\zeta^2 + \frac{1}{3}\zeta^3 + ...$ then, \begin{align*} \log Z &= (as + bs^2 + cs^3) - \frac{1}{2}(as + bs^2 + cs^3)^2 + \frac{1}{3}(as + ...

2

Suggest the worked examples in Chapter 5 (and for credit spreads, Chapter 17) in van Deventer, Imai and Mesler, Advanced Financial Risk Management, 2nd edition, 2013, John Wiley & Sons, Singapore. Good luck.

2

Is the author taking logs (and dividing by (T-t) etc) of our previous Z expansion from the previous page? He does, as you will see if you try to do the computation. What did you prevent to find this out by yourself? (I am trying to be constructive.) Mathematically, it doesn't add up to what the author provides as the answer. What am I missing here? ...

2

The book by Francis Diebold & Glenn Rudebusch "Yield Curve Modeling and Forecasting" addresses both a dynamic extension of Nelson-Siegel and an arbitrage-free version - may be helpful for what you are looking for. Link below: ...

2

Your steps 1. to 3. sound reasonable. I am not sure about industry practice (what industry?) I always do step 1. using PCA on historical correlations. If you plan to do a regulatory exercise better check with your regulator what he prefers. Most interesting to me is step 4. which - I think - is in general impossible to do. This can be achieved only in very ...

2

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 ...

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 ...

1

In this context, I believe carry refers to the sum of "pure" carry + roll down. Carry, in the most general sense, is the return of a position in a static world; i.e., assuming time is the only variable that is changing, what's your holding period return on a trade? When you buy a bond, the "total carry" is the sum of 1) "Pure" carry – you get interest ...

1

For the US Treasury market, zero coupon bonds are traded and they are called STRIPS. You can access them through "S GOVT" (coupon Strips) or "SP GOVT" (principal strips) on BBG. With regard to relative value trading, it's actually pretty rare that we fit models to zeros, because a lot of them are not liquid and trade differently from their coupon ...

1

For RV purposes, I have actually continued to use libor discounting for simplicity; otherwise, you'd have to model multiple curves, which become very difficult to work with... That being said, the curve has been trading very differently after the crises. For example, 5y typically didn't deviate that much from 2y and 10y on relative value basis historically, ...

1

How are the future interest rates determined? Two ways. 1) They are observed in the market, i.e. they are the best estimate of the market participants. One way is to use Bloomberg. 2) You can create your own discount curve and from that calculate the forward rates. Discount and forward curves for non-collateralized swaps must be consistent, otherwise ...

1

Assume we have $r(t)$ continuously compounded spot rate for maturity $t$. The price of the 2-year bond with semi-annual coupon $C$ is known to be $P$. We already have $r(0.5)$ and $r(1)$. We need $r(2)$ and $r(1.5) = f(r(1), r(2))$. Then $$P = C [e^{-0.5 \times r(0.5)} + e^{-r(1)}+e^{-1.5 \times r(1.5)}] + (1+C)e^{-2 \times r(2)}$$ Using linear ...

1

If you look at wikipedia then you find the definition that a par-yield is the coupon rate, such that bond prices are $100$. This is the definition. Consider $N$ bond with a given coupon rates $c_i$, times to maturity $T_i$ prices $P_i$,for $i=1,\ldots,N$. Then you can calculated the yield-to-maturity for each bond $y_i$. Some mathematics reveal that a bond ...

1

The Heath-Jarrow-Morton representations of short interest rate models (such as Hull-White) will give you an expression for the evolution of the entire forward curve, but it doesn't make the problem any easier. The closed form ZC formulae you mention above are probably your best bet.

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