10
votes
Accepted
Do we use the Nelson-Siegel model to calculate the yield curve?
In the beginning, we had a plot of yields of individual bonds against time to maturity, the crudest form of "yield curve."
Years later, people began hand-drawing a smoothed line through these yields ...
- 11.4k
8
votes
Is trading mean reversion of small principal components of prices profitable?
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....
- 11.4k
5
votes
How to de-seasonalize natural gas term structure data?
As a starting point to this, determining seasonality for a given market is as follows:
i) Take several years of historical spot price time series, e.g. TTF spot prices. For year $i$ work out a yearly ...
- 1,651
5
votes
Accepted
SVI model and Greeks calculation
The SVI is simply a function (empirically fit to the data) which given a maturity and a strike price K, computes a BS implied volatility $\sigma$. Once you have that implied volatility you can plug it ...
- 9,332
5
votes
What is the state of the art govie bond term structure recently
I interpret your question to be asking about curve fitting techniques (for constructing fitted par/zero curves), since a term structure model (HW, LMM, etc.) can always be constructed to fit a given ...
- 11.4k
5
votes
Accepted
What is actually going on in Monte-Carlo simulation for Mortgage backed securities?
In my understanding, the mortgage prepayment option, at any point in time, is a function of the value of the mortgage from that point in time forward. This value, in turn, is a function of the future ...
- 6,425
4
votes
Contango and backwardation in VIX futures
Since contracts on physical goods have associated costs, it makes
sense that the term structure curve would be upward sloping. Since
there is no cost associated with delivery for the VIX and contango ...
- 4,738
3
votes
Accepted
Fitting the Term structure of Discount Bonds with Ho-Lee
What they are referring to is a very simplified version of the Ho-Lee model, i.e. on that assumes $$r(t)=r(0)+{\sigma}W(t)$$ where ${\sigma}$ is a constant (annualized StDev).
For the sake of ...
- 735
3
votes
Accepted
Deriving interest rate term structure in a short rate model
This is indeed a standard result. You can convince yourself by noticing
The bank account grows from 1 at $t=\tau$ to $E\left[\exp(\int_\tau^T r(u)du)|\mathscr{F}_\tau\right]$ at time $T$
The price of ...
- 1,973
3
votes
Accepted
Valuing derivatives under stochastic interest rates
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_{\...
- 21k
3
votes
VXV vs. VIX futures: arbitrage opportunities?
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.
- 2,543
3
votes
Why should future short rates tend towards the current term structure of interest rates?
It really depends for what purpose you are using the model. Let’s say you are using it for valuation of some instrument. If you want the fair market value, then a) is irrelevant and you would instead ...
- 16.6k
2
votes
Accepted
Applying interest rate models for volaility rate
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 ...
- 14.7k
2
votes
How to check that an interest rate curve is arbitrage free
To say a curve is arbitrage-free, you need to pick an arbitrage path; a series of trades which, when followed, yield a net profit without creating exposure. We neglect counterparty exposure here, ...
- 3,669
2
votes
Accepted
How to show that the exponential Vasicek model is not an affine term-structure model?
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
$$u(...
- 2,746
2
votes
How to apply PD term structure?
By the looks of it your table is cumulative PD.
You can use the argument:
$$P(\text{Default by end year X}) = P(\text{Def. by end year X-1}) + P(\text{Not def. by end year X-1})P(\text{Def. in year ...
- 9,215
2
votes
Bond Convexity and Maturity
You can understand convexity by working out a simple example numerically yourself.
Consider two bond portfolios: P1= consists of a 6 year zero coupon bond. P2= half in a 2 year ZCB, half in a 10 year ...
- 9,332
2
votes
Accepted
Bond Convexity and Maturity
Think of a zero coupon bond - the pv_zero (t years) $= \frac{\rm{pmt}}{(1+r)^t}$
As t increases the compounding effect of that discount increases (the larger the price change)
As for rate vol - ...
- 36
2
votes
Term structure used in Geometric Brownian Motions under Risk Neutral Measure?
It should be time dependent and set to the spot forward rate $= -\frac{\partial}{\partial t} \ln(\text{discount}(t))$ when simulating in continuous time. When discretizing the simulation use the ...
- 5,622
2
votes
What is drift in interest rate term structure model
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 = (\...
- 8,022
2
votes
Why do increasing spot rates have to be equal to or larger than the corresponding par rates?
To answer this question, we must fix a bit of the vocabulary, first. I will try to stick as close as possible to your conventions:
Spot rate: (also called zero rate) is the annualised rate of return ...
- 6,425
2
votes
What is "implied skew" and "spot/vol beta"?
For what date is the chart derived? One definition for implied volatility skew is: (25 delta put implied volatility - 25 delta call implied volatility) / 50 delta. Can you test to see if this ...
- 5,662
2
votes
OIS rate to build Term structure
By definition, the overnight rate is the rate at which banks lend to each other overnight. Overnight index swaps (OIS) allow banks to 'lock in' the cost of funding overnight for a specific term. They ...
- 2,137
2
votes
Accepted
ICVS 133 Bloomberg Curve
ICVS 133 on BBG as a zero coupon curve for EUR OIS (so you wouldn't need to bootstrap). If you export to Excel, the discount factors are already in the exported sheet.
- 2,209
2
votes
Use Discrete ARMA(1,q) Process to Model Short Rate for Term Structure Fitting
You are describing something called Geometric Brownian Motion, and in the realm of short rates, you are describing the discretization short rates. For the Vasicek model, $R_t = aR_{t-1} + b +\epsilon$ ...
- 845
2
votes
Accepted
Smile Dynamics - forward variance
Ok, so
$$
d\xi_t^T = \omega e^{-k(T-t)} \xi_t^T dW_t
$$
where $W$ is standard Brownian.
Then, just by applying Ito I hope you can see that
$$
\log \xi_t^T / \xi_0^T = \omega \int_0^t e^{-k(T-u)} dW_u -...
- 1,739
2
votes
Forward interest rate curve family parametrization
You've correctly identified that the forward curve indeed has two time indices -- one for when we observe it, and one for the future date at which the forward rate applies.
I would personally take the ...
- 350
1
vote
Hazard rate and Term structure model
The paper on Jun Pan's page.
The only quotes readily observable in the market are quotes for a few tenors of the standard CDS contract. Please recall that the standard credit default swap essentially ...
- 11.9k
1
vote
backtest VIX term structure strategy
This is not an answer, but instead advice:
Since you're new to quant and volatility then you should start with something other than a volatility or a rates product because those are going to be some ...
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