# Tag Info

## Hot answers tagged term-structure

5

The original Vasicek paper is "An equilibrium model of the term structure". If you google for it, you'll find it and you can read in his own words his motivation for developing it. In particular, what now is called the Vasicek model basically comes from applying his results to an Ornstein-Uhlenbeck model for the spot process, which he claims was proposed by ...

5

It's because of the settlement days you passed when you initialized the flat volatility curve. You're creating the spot, forward and flat volatilities as: boost::shared_ptr<BlackVarianceSurface> volatilitySurface( new BlackVarianceSurface(todaysDate, calendar, maturityArray, strikeArray, ...

5

Within the fixed income space, there's a lot of literature on PCA trading. The first 2-3 principal component factors (PCs) can typically explain 90-99% of the total variances in yield curve movement. It's also nice, because the first PC looks like a change in the overall level of the yield curve, the second PC looks like a slope change, while the third ...

5

There are two different issues at play here. One is that, of course, you want only the future cash flows to enter the calculation. This is taken care when you set the evaluation date to 6 months from today. In C++, you would say Settings::instance().evaluationDate() = today + 6*Months; I don't remember the corresponding function in QuantLibXL, but you ...

4

A few points can be noted. The CIR model is usually for a short, or instantaneous, spot rate $r_t$, which is the forward rate over an infinitesimal interval. That is, \begin{align*} r_t = \lim_{\Delta \rightarrow 0}\frac{1}{\Delta}\left(\frac{1}{P(t, t+\Delta)}-1 \right), \end{align*} where $P(t, u)$ is the price at time $t$ of a zero-coupon bond with ...

4

I will refer to "Interest Rate Models - Theory and Practice: With Smile, Inflation and Credit" by Damiano Brigo and Fabio Mercurio. In chapter 3 (One-factor short-rate models) they have a very nice table which lists some of the properties of instantaneous short rate models. In both of your models you know the distribution of $r_t$. The huge difference ...

3

VXV is a 3-month volatility index, and is currently not tradable (there are no futures on it). And since you cannot trade it, you cannot arb it.

3

Many term structure models-both single-factor and multifactor imply dynamics for the short-term riskless rate $r$ that can be nested within the following stochastic differential equation: $dr = (\alpha + \beta r)dt + \sigma r^\gamma dZ.$ These dynamics imply that the conditional mean and variance of changes in the short-term rate depend on the level of ... 2 I would look at the following metrics when quantifying "liquidity" in listed options: bid/offer spread number contracts traded and from that follows notional traded (in the option not underlying) frequency of bid/offer adjustments relative to changes in the underlying delta. frequency of liquidity added/removed on the bid and offer side even when no ... 2 Here is a general proof for all parameters in an open domain. $$dr = adt+bdW:=r\big(k(\theta-x)+\frac12\sigma^2\big)dt+\sigma rdW.$$ Let $$u(r(s),s):=e^{-\int_t^sr}B(r(s),s,T)=:\phi(s) B.$$ Then $$u(r(t),t)=\mathbf E\big[u(r(s),s)\big|r(t)\big],\, \forall t<s. \tag{1}$$ So, by Ito's Lemma, \begin{align} du(r(s),s) &= Bd\phi +\phi dB \\ &= \phi \... 1 We shall prove this by contradiction. Let\theta=0$and$\sigma=0$.$X_t=X_0e^{-kt}$and $$B(0,t)=\exp\Big(-\int_0^te^{X_0e^{-ks}}ds\Big).$$ Suppose the contrary that$B(0,t)$is affine. We should have $$B(0,t)=\exp{\left(A(0,t)-C(0,t)e^{X_0}\right)}\;\;\ \forall (t,X_0), \tag{1}$$ Differentiate the logarithm of Equation (1) with respect to$t$side,$...

1

On many occasions may the ATM volatility term structure implied from option prices exhibit non monotonicity. You could actually turn the question on its head and ask yourself why should it be monotonic? Does this reflect expectations & uncertainty about interest rates (exposure to rho?), event driven concerns about the underlying, or something else? ...

1

Hans Buehler investigated this in some detail, including in his doctoral thesis. When I tried it out some years ago, back when volatility exotics were more liquid, I found the models nearly impossible to calibrate to my satisfaction, even for the SP500 complex. I think the mathematical analogy is fair, and enjoyed Buehler's work, but in practice it won't ...

1

Conventional wisdom would have it that the system would be arbitrage free if and only if: All the implied spot and forward rates on each curve are non-negative (I.e implied discount factors are monotonic non-increasing wrt maturity) All the implied spot and forward rates on the 3M curve are greater than or equal to the corresponding rates on the OIS ...

1

Yes. The map $R(\cdot;S,T):\mathbb{R}^{2}\to\mathbb{R}$ completely describes the forward rate/spot rate term interest rate structure for each $t\geq0$. (You can think of it as the market interest rate surface for the rate $R$ at time $t$). The notation $R(t;S,T)$ is meant to remind you that $R$ is a stochastic process for $t>0$, the periods of time ...

1

I am note $100\%$ sure that I understand the question. But yes. More formally one could write $R(t,S,T)$ for the rate from $S$ to $T$ observed at $t$ and $R(t,t,T)$ for the spot.

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I think the rationale behind it is that if $r$ is the short rate, the the price of the bond is $P(t,T) = \mathbf{E}e^{- \int_t^T r_s ds }.$ As is well known by know is easy to calculate expectations of random variables of the form $e^Z$, where $Z$ is Gaussian. This model is the simplest example of a case in which the integral of the short rate as Gaussian ...

1

A PCA explains the variation in data. A slope PC is usually identified by the pattern of the signs of the loadings. If the loadings of short term contracts have the same sign which is different from the sign of the loading of longer term contracts then such a PC is identified as slope PC. It means that if this PC goes up or down it affects short term ...

1

Nelson Siegel seems to be pretty standard too

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