Questions tagged [interest-rates]
An interest rate is the rate at which interest is paid by a borrower (debtor) for the use of money that they borrow from a lender (creditor).
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Proof verification : risk free rate [closed]
I want to prove that
$$r_t = \theta + (r_0 -\theta)e^{-kt}$$
satisfies $$dr_t = k(\theta-r_t)dt, \ r(0) = r_0$$
I have
\begin{split}\frac{1}{\theta - r_t} dr_t = kdt \Rightarrow & \int_0^t \frac{1}...
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Building a Nelson-Siegel curve
I originally posted this on Mathematics, but was told my question is better suited here.
I want to graph a yield curve with an extended version of the Nelson-Siegel-Svensson. I have the issue date, ...
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SOFR Discount Curve Construction in Nov 2021
On July 29, 2021, the Alternative Reference Rates Committee (ARRC) formally recommended the forward-looking term rates based on SOFR published by the CME Group.
CME currently publishes Term SOFR for ...
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What does M mean in DI Deposit futures contract?
I am trying to understand the forumla for DI1 Brazilian deposit future contract. I am able to figure out everything except M in the following formula:
Xt=N×M×(Pt−Pt−1Ft)
Lets say if we want to ...
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Where is the Quadratic Variation Coming from in this One-Factor Cheyette Model?
I am having difficulty switching from a general interest rate model (the quasi-gaussian or cheyette model) and a specific version of this model. In particular, I assume the following instantaneous ...
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Why do we get a higher yield when we pay the interest at the end?(bonds)
I have an example where I show that if you pay the tax at the end of the bond period, the yield after tax is higher, but I am wondering if it is possible to give an explanation as to why it is like ...
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Joke of borrows at 6, lends at 9, and leaves at 4
I’m reading J. Michael Steele’s “Stochastic Calculus and Financial Applications”, in Ch 10 he mentioned
We are all familiar with asymmetry in interest rates — witness the
worn-out joke about the ...
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Is it true that interest rates options with different maturities are free of calendar arbitrage because of the different underlying rates dynamics?
The title says it all - is it true that European style interest rates options (lets say on LIBOR 3M for the sake of simplicity) with different maturities are free of calendar arbitrage because ...
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Why is the price of any asset divided by a reference asset(numeraire) is a martingale under the measure associated with that numeraire?
why is the price of any asset divided by a reference asset(numeraire) is a martingale under the measure associated with that numeraire?
For example, if I have the price of a forward price $f_t$ and a ...
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DI futures contract value on bloomberg
I am trying to understand the Contract value for DI1 futures on bloomberg. I assume the Price of 4.630 below is the CDI one day interest rate. Where does the Tick value of 9.6169 come from and how ...
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Real rates expectations
What are the drivers of real rates?
Nominal = real rates + breakevens
breakeven = inflation expectations and what about
real rates = ?
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Intuition behind reasoning around interests-in-advance
I quote Life Insurance Mathematics (Gerber, 1997).
Let $i$ be an annual effective interest rate and $d$ an annual effective discount rate.
In case of interests-in-advance, a person investing an ...
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Replicating Bloomberg's zero rates bootstrapping
I'm interested in manually replicating the bootstrapping procedure that Bloomberg uses to built ICVS179 (RUB vs MosPrime 3M) curve up to a two years tenor as of October 12th 2021.
These are the market ...
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How to measure Steepener/Flattener/Butterfly sensitivity? (in 01)
This seems like a simple concept but I'm a bit lost. How can I calculate the dollar value sensitivity for a yield curve slope or butterfly position? I understand how DV01 can be calculated, but it ...
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Bermudan Swaption Pricing via Least-Square Monte Carlo
I have some confusion regarding pricing a Bermudan Swaption using LSMC.
Let's say the underlying swap has payment dates $T_0 < T_1 < \ldots < T_n$ and for simplicity, assuming the exercise ...
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Day Count Convention
I am currently reading the book Term-Structure Model, chapter 3, by Damir Filipovic and I have the following problem:
The Libor rate for the maturities: Over night, 1 week, 1 month, 2 months and 3 ...
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Why are T-bills yielding lower than the reverse repo rate?
With another US debt ceiling debacle looming, I just realized something that runs against my intuition:
Many of the Treasury bills maturing in 6 months (with the exception of the ones maturing near ...
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Should the Libor Market Model using spot measure as numeraire simulate an arbitrage free forward curve?
I have been looking at the following resource:
Reference Paper
Using equation [4] for the discretized version of the forward libor rate:
$\tilde{L}^i_{T_{j+1}} = \tilde{L}^i_{T_{j}} exp[\sigma^i(\sum^...
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what's the difference between instantaneous short rate and instantaneous forward rate?
In the short rate models, sometimes it models the instantaneous short rate and sometimes it models the instantaneous forward rate. Does instantaneous short rate = F(0, t + tau) and instantaneous ...
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How to resampling the risk of a specific tenor on a interest rate curve without replace the instrument?
For example, suppose we have an interest rate curve bootstrapped from multiple instruments, at the short end, we used eurodollar future (up to 2Y), at the longer end, we used interest rate swap (3Y to ...
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Are there any equity derivative instruments offering exposure to borrow rate optionality?
Just what the question says. I understand lots of equity derivatives have secondary exposure to stochastic rates, but I would like to understand if there is a payoff that has borrow rate as one of its ...
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Euro-Bund Futures fair value
I am trying to calculate the fair value via discounting the cashflows of the (synthetic underlying) for the euro bund futures (https://www.eurex.com/ex-en/markets/int/fix/government-bonds/Euro-Bund-...
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Using converted lognormal volatilities for negative rates in a lognormal Libor Market Model (LMM)
There exist formulas to convert between normal and lognormal interest rate volatilities. In the most simple form the approximation for ATM volatilities would be $\sigma_{LogNorm}=\frac{\sigma_{Norm}}{\...
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HJM drift condition problem: Show that the HJM drift condition implies $b(t) \equiv b, \rho^{2}(t) \equiv a$
I need your help with understanding and solving the HJM framework. I am hoping I can get some help as I feel so lost with HJM and learning online because of the pandemic is adding more stress. Anyway ...
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What's the difference between short rate and the bootstrapped interest rate?
This thing confused me for a long time, since we can a have a curve (e.g. LIBOR 3M) bootstrapped from the market quotes of instruments (e.g. FRA, SWAP), and we can get the spot rates and also the ...
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Calibrate 1-factor Gaussian HJM model on forward rates and ATM caps prices
I'm trying to solve the following problem as a part of the Interest Rate Models course
The algorithm that I'm following is
derive simple rates from the given forward rates via $L(0, T_i) = \frac{(1+\...
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Forward/futures contracts that satisfy $F=S\exp(rT)$
I am interested in estimating riskless rates from forward/futures data. The standard forward pricing formula is given by
$$F=S\exp(rT).$$
From this we can solve the interest rate used in pricing as a ...
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Why should future short rates tend towards the current term structure of interest rates?
I'm currently looking at the Hull-White model reproduced below:
$$\mathrm{d}r = \lambda(\theta(t)-r)\mathrm{d}t + \sigma\mathrm{d}W(t)\text{.}\tag{1}$$
I have a simplistic understanding of the model. ...
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Why does the LMM in Hull seem so different from the LMM in Brigo and Mercurio?
When I look at Hull's "Options Futures and Other Derivatives" the process for $F_k(t)$ in the rolling forward risk neutral world is specified as
$\frac{dF_k(t)}{F_k(t)} = \sum^k_{i=m(t)}\...
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Why is an FRN price equal to par on every reset date?
In the book "The money markets handbook" by Moorad Choudhry, it says that "on the coupon reset date an FRN will be priced precisely at par".
Why is this? Would it not discount the ...
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Interest Rate sensitivity due to floating rate liabilites
I am new to finance, so please bear with me
I am supposed to find the change in 'interest expense' when the borrowing rate goes up by 50bps. The liabilities include both fixed and floating rate ...
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A list of the 01's in the corporate bonds
I have frequently heard terms like DV01, CV01, PV01. Where can I get a list of these glossaries to study? I am not looking for a detailed explanation, just really a list.. Once I have the list, I can ...
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How to simulate interest rate index fixings?
When calculating the PV of an interest rate derivative (IRD) that is linked to a rate index $-$ e.g. an interest rate swap $-$ we usually require the actual, or projected, index fixings in order to ...
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Why is dividend discounted stock equal to cash discounted strike?
Doing some BS problem solving and came across this beauty:
$$S \cdot \text{exp}\Big(-D(T-t)\Big) \text{exp}\bigg(-\frac{d_{1}^{2}}{2}\bigg) = E \cdot \text{exp}\Big(-r(T-t)\Big) \text{exp}\bigg(-\frac{...
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Implied cross currency curve
For EM countries without a liquid xccy curve, how I can imply it from local government bonds or swaps?
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Difference between 5Y breakeven inflation and 5Y5Y inflation forward?
I cannot figure out the difference between the two data series found here:
https://fred.stlouisfed.org/series/T5YIE/
https://fred.stlouisfed.org/series/T5YIFR/
The 5Y breakeven inflation, to my ...
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Should you hedge theta?
Consider an arbitraty interest rate derivative $C$. Assume that in order to hedge it, you are allowed to construct portfolio $H$ of linear combination of simple instruments such as Interest Rate Swaps ...
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Option Adjusted Spread - Monte Carlo
It's my understanding that in order to calculate the option adjusted spread on a mortgage backed-security, the following steps are required:
Run a Monte Carlo simulation of interest rates
Project ...
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Option pricing under Vasicek, CIR, H-L and BDT model
I have implemented and calibrated recombining trees on Excel for the Vasicek, the Cox-Ingersoll-Ross, the Ho-Lee and the Black-Derman-Toy model. I now would like to price some options with these ...
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Determining bond price based on diffusion process for the short rate model
Suppose the diffusion process for the short rate $r_t$ under the risk-neutral measure $Q$ is given by:
$$
dr_t = \theta(t)dt+\sigma dZ_t
$$
where $Z_t$ is a Brownian motion.
I am trying to show that ...
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Free Arbitrage conditions in ATM swaption surfaces
I'm wondering how can we check free arbitrage conditions in ATM swaptions surfaces since we only have access to Expiry, Tenor and volatility?
Can someone help me please, i didn't find any article ...
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Hedging Options assuming a non-constant Yield Curve
I have read most of Shreve's Stochastic Calculus for Finance II. In it, the author prices various option types assuming an interest rate that is constant with respect to time.
We can expand this model ...
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Generate FX Carry Return Index
I would like to generate a daily carry return index for a given currency pair - lets take USDEUR as an example.
I presume this involves something like taking the spot rate and the appropriate funding/...
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Why does the rate of inflation vary over time?
Interest rates have varied significantly over the last 50+ years (source: https://www.macrotrends.net/2016/10-year-treasury-bond-rate-yield-chart ).
Is it possible to comprehensively and succinctly ...
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How to backtest with fixed-income instruments
I'm running a backtest with the 5-yr and the 30-yr treasury bills going back to 1990, both with a weighting of 25%. How do I use their daily yields to adjust the portfolio through time?
I've thought ...
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How to interpret negative fixed rate in Vanilla IRS [closed]
Currently a vanilla 4Y EUR vs. 6M Euribor IRS has a negative price e.g. -0.35%.
I do not understand how to interpret the swap when the fixed rate is negative. If I am the fixed rate payer in this swap ...
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Book that considers stochastic interest rate models in discrete time?
Are there any books that covers interest rate swaps, futures, forwards etc. but have a discrete time model? I would like to go deeper into this without having to worry about the stochastic calculus.
...
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Why do bank stock returns increase from increased credit risk?
As part of my bachelor's thesis, I am running the following regression on daily bank excess returns:
(r-rf)=Beta * Market excess return + Beta * Level(5Y)+ Beta * Credit Risk + error
Level(5Y) is the ...
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Calibration of Heston model with stochastic short rate
I have following Heston model with stochastic short rate:
\begin{eqnarray*}dS\left(t\right)&=&r\left(t\right)S\left(t\right)dt+\nu\left(t\right)S\left(t\right)dW^{S}\left(t\right)\\dr\left(t\...
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Derivation of the distribution for a CIR process
Where is it possible to find a complete derivation of the distribution of a CIR process?
There is a number of papers that claim that it is a noncentrical chi-squared distribution.
However, I cannot ...
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