# Tagged Questions

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### Valuing derivatives under stochastic interest rates

I would like to price a European option with maturity equals to 5 years. To do this, I'm using the Black-Scholes model with stochastic interest rates. Suppose I choose the CIR model for the risk-...
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### Discount factor taking into account yield curve shape

I have always been told that the discount factor formula is just: $$DF(T) = \frac{1}{(1+L_{t_0})^T}$$ where $L_{t_0}$ is the LIBOR rate on one period (the first one I guess) and $T$ the number of ...
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### How to show that the exponential Vasicek model is not an affine term-structure model?

From the pricing formula, we know that the value at time $t\in [0,T]$ of a zero coupon bond maturing at time $T$ is $$B(t,T)=E\left(\exp{\left(-\int_{t}^{T}r_sds\right)}\bigg|\mathcal{F}_t\right).$$...
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### Why does the forward rate curve lies above the spot rate curve and the yield to maturity curve?

I saw a picture of 3 different yield curves, a spot rate curve, a forward curve, and a yield to maturity curve. The forward curve was at the top, the YTM curve at the bottom. I don't understand why.
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### Why is the term structure of the implied volatility surface non-monotonic?

Does this reflect expectations & uncertainty about interest rates (exposure to rho?), event driven concerns about the underlying, or something else?
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### What is drift in interest rate term structure model

I was studying about the interest rate term structures and i came across term structure model with (and without) drift. I am really unsure about what this drift is in this equation for term structure ...
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### Is trading mean reversion of small principal components of prices profitable?

Many have told me that it is a good idea to look at the third principal component (PC) of yield curve movements, as well as third and fourth PC of G10 currencies. They claim these PCs represent "...
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### VXV vs. VIX futures: arbitrage opportunities?

At a first glance, VXV and VIX futures should not be compared at all: VXV is an underlying index, whilst VIX futures are derivatives written on a different underlying index, that is, VIX. As instance,...
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### Spotting humps in implied volatility term structure

Sometimes implied volatility term structure performs a hump shape. How can I measure the frequency of hump appearence? Basically, I have several years of daily data on IV TS, how can I count the ...
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### affine arbitrage free class of nelson siegel yield curve

I'm studying statistics for finance at university. Last week i read the working paper on "The Affine Arbitrage-Free Class of Nelson-Siegel Term Structure Models". I would like to reproduce in R ...
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### Applying interest rate models for volaility rate

To what extent may the interest rate models be applied for modeling implied volatity? The story: I was checking different stochastic option pricing models for being able to replicate implied ...
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### State of Art - Nelson Siegel Modeling

My idea is to work with dynamic Nelson Siegel models(DNS) on my master's thesis. As I am finishing undergraduation this year I started researching on the subject. I wonder what is being discussed in ...
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### Correct form for State Space Equation for Kalman Filter for DNS

In this paper: http://www.ssc.upenn.edu/~fdiebold/papers/paper55/DRAfinal.pdf in eqns 3,5 the state eqn has the mean removed. $(z_t-\mu)=A(z_{t-1}-\mu) + \epsilon_t$ $y_t=C z_t + \delta_t$ ...
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### Forecast biasness of VIX term structure

I'm interested in the topic of VIX futures being overpriced, so I'm looking for different models to find evidence for it. Asensio 2013 uses a regression to evaluate the forecast biasness of the VIX ...
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### How to check that an interest rate curve is arbitrage free

I have 2 interest rate curves (LIBOR 3M and OIS). I want to create stress scenarios for those two curves. Is it possible that some scenarios will make my term structure arbitrageable? How can I test ...
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If I have a term structure/yield curve given by: $$f(t, T) = f(0, T) + σ^2t(T − \frac{t}{2}) + σB_t$$ and want to find the short/spot rate $r_t$, is this simply: $$f(t,t) = f(0,t) + \sigma^2t(t-\... 2answers 55 views ### Incorrect characterization of spot rate? Is the t in the red boxed R(t,T) supposed to be the same as the S in the green boxed R(S,T)? 2answers 177 views ### What is the reasoning to derive this financial model called the Vasicek Model? The model specifies that the instantaneous interest rate follows the stochastic differential equation$$\mathrm{d}r_t = a(b-r_t)\: \mathrm{d}t + \sigma \: \mathrm{d}W_t$$where W_{t} is a Wiener ... 1answer 816 views ### QuantLib: Black / BSM processes and pricing via volatility surface. Different results? I start this question with a couple of C++ functions that will be useful to show some results. So start your Visual Studio C++ Express or Ceemple or whatever you want and copy & paste this: ... 0answers 164 views ### How to de-seasonalize natural gas term structure data? I need to de-seasonalize Nat Gas futures data for a project and am hoping to get good suggestions. As we all know natural gas futures are priced higher for the winter months and to analyze/model the ... 1answer 111 views ### Inferring signals in absence of sign of principal components (PCA)? PCA seems to be very popular in dimension reduction applications and for extracting the top PCs which explain the data. One such application in futures is on the term structure to obtain the level, ... 1answer 430 views ### Are there any other standard rates term structure decomposition than PCA? PCA is sometimes used to estimate components in the rates term structure. Are there any other standard method discussed in the literature or used in practice, what are their advantages and ... 1answer 832 views ### How to price a bond at specified dates in QuantLib I am wondering what's the most efficient way (i.e. the method which involves the fewest arguments) to price a bond at a specified date, e.g. a future date (as instance, 6 months from now) in QuantLib. ... 1answer 1k views ### On short-rate-models: Black-Karasinski (with constant parameters) compared to Vasicek When modelling the term structure of interest rates, one widespread possibility is using the Black-Karasinski model, which is given by the following stochastic process$$d\ln{r}=[\theta(t)-a(t)\ln{r}]...
If an increasing process $X_t$ has a known Laplace transform $\mathbb{E} e^{-s X_t} = m_t(s)$, define its hitting time $\tau$ to some level $B$ to be $$\tau = \inf\{ u > 0 : X_u \geq B \}.$$ Can ...