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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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When to use continuous time math vs discrete time?

Seems that the theory books are all integrals in continuous time, yet in practice, discrete estimations works fine. As a newbie to this, when do you choose to use the continuous time finance vs ...
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1answer
231 views

Bermudan Swaptions

Can someone explain, in layman's terms, the mechanics behind Bermudan Swapttions ( without having recourse to pricing models )? Why are they popular? when are they used ? How are they hedged i.e ...
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1answer
146 views

Two papers - two different solutions of the Ornstein-Uhlenbeck process

Bernal 2016 says that the solution of $$ dr_{t}=\lambda*(\mu-r_{t})*dt+\sigma dW_{t} \qquad (eq.1) $$ equals $$ r_{t}=r_0*exp(-\lambda t)+\mu(1-exp(-\lambda t))+\sigma \int_{0}^{t} exp(-\lambda t)...
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234 views

Calibration of Cox-Ingersoll-Ross process that hits zero (Feller condition violation)

I'm considering a Cox-Ingersoll-Ross (CIR) process $$ dx_{t} = \alpha\left(\theta - x_{t}\right)dt + \sigma \sqrt{x_{t}}\,dW_{t}\,,\qquad \alpha,\beta,\sigma > 0 $$ which by assumption has $2\...
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61 views

Bond prices at future times under Vasick one-factor model

In Vasicek one-factor model (and in other affine models), the price of a zero-coupon bond at time $t$ conditional on the information at this time is $$P(t,T) = E[e^{-\int^T_tr(u)du}|F_t] = A(t,T)e^{-...
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1answer
107 views

On the buyside, when people quote a 'price' for a plain vanilla interest rate swap, does it include accrued interest?

The valuation date falls in between coupon payment days on the swap, does the 'price' of a swap understood to include the accrued interest (interest from the previous payment date to the valuation ...
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2answers
177 views

Why do par-yield shifts grow faster across the curve than spot-rate shifts when looking at key-rates?

Consider the following 10y key-rate shifts of bond par yields and its implied shift of bond spot rates: Assume we have the key-rates for 2y, 5y, 10y and 30y. The y-axis is in basis points, and the x-...
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2answers
3k views

Historical calibration of Hull-White model

I have a question concerning 1-factor Hull-White model. For my master project I need to calibrate it to compute Counterparty credit risk metrics. I know that the model might be calibrated either for ...
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2answers
216 views

Differential of stochastic term

Question 1: How does one come up with the equation in the red box below? It looks like some kind product rule, but I'm not sure how to apply Ito's lemma here. Bjork doesn't seem to explain it fully,...
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38 views

Differential Equation of Type Ricatti as part of Short Rate Model

I currently despair of the following solution of a differiental equation (Ricatti Type) as part of a short rate model: $$ B_t=\frac{1}{2}aB^2+bB-1 $$ First I am "guessing" a particular solution $$ ...
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1answer
276 views

Fixing Rate in Quantlib

While pricing Interest Rate Swap, I am providing Fixing rate for historical date using "addFixing(date, value)" function. But when I am trying to change value it is not happening and picking up old ...
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2answers
1k views

Different ways to express a 2s10s steepener?

Some off the top of my head 2s10s cash steepener, however this ages into a 1s9s over time 2s10s swap steepener, better/cleaner way? Are there other ways to express this curve strategy? Would you do ...
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1answer
50 views

For a market with a bank and risky assets $S_1, S_2$ with different volatility, what should be the short interest rate in this market?

Let there be two assets $S_1$ and $S_2$ s.t.for $\sigma_1 \neq \sigma_2$ $$dS_{1t}=\mu_1 S_{1t}dt+ \sigma_1S_{1t}dB_t \\dS_{2t}=\mu_2 S_{2t}dt+ \sigma_2 S_{2t}dB_t$$ . If there exists a bank, what ...
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169 views

Reason for choosing the T-forward measure to calculate expected value of forward curves

Setup I read that when simulating forward curves $(r_t(s_i))_i$ at some future time $t>0$, one is supposed to center them not around $F(0;t,t+s_i)$, but around $$\mathbb E^{\mathbb Q_{t}}[r_t(s_i)]...
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1answer
35 views

When exactly does the FOMC release the new calendar dates?

Please let me know if this is the appropriate place to post this. I know every year the federal reserve releases the calendar dates for the next year's meetings around May or June. Is there a ...
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1answer
37 views

Recreating / Extending Bond Time Series

I am trying to analyse historical yield curve dynamics within an across countries and step one is extending / recreating historical yields and/or prices. The challenge is this: lets say a 10 year ...
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0answers
32 views

Portfolio Immunization from Yield Perspective

Let's say we have the following situation: an asset (mortgage) with fixed payments, a prepayment & oas models to run through, and calculations for duration, convexity, and price, based on them. ...
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2answers
145 views

Bond Convexity and Maturity

What the reasoning for why bond convexity increases with maturity. Heuristic explanations are somewhat better as I would like a fundamental understanding. Also what causes a more convex bond to be ...
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1answer
684 views

CMS options, cash-settled/physically-settled swaptions

CMS options are traditionaly replicated using a theoritical "continuous" strip of swaptions (see for instance Hagan's paper "Convexity Conundrums : Pricing CMS Swaps, Caps and Floors"): In the paper,...
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1answer
57 views

How to simulate a path through its solution and conditional expectation / variance

Hi I want to simulate in Matlab the following stochastic integral: $ x(t) = x(s) e^{-a(t-s)} + \sigma \int_s^t e^{-a(t-u)} dW_1(u)$ with $E[x(t) \vert F_s] = x(s) e^{-a(t-s)}$ $Var[x(t) \vert ...
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1answer
73 views

Libor Swap Rates

In a 5 year Libor Swap, say fixed vs. 3 months Libor, what is the credit risk reflected by the fixed leg ? (I'm ignoring counterparty credit risk). Would the fixed leg reflect 3 month Libor quoting ...
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1answer
71 views

Measuring interest rate sensitivity for illiquid private investments?

There seems to be surprisingly little literature on this topic. If you had a portfolio consisting of an unlisted illiquid private asset class (eg private real estate, direct infrastructure or private ...
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0answers
527 views

Bloomberg zero rate calculation using shift

I used Bloomberg to calculate a zero rate under a parallel shift of 100 basis points, however I can not understand the results neather duplicate them. I included the +100 basis points by using the ...
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1answer
801 views

High convexity vs low convexity bond definition

Isn't high convexity always better than low convexity bond from the formula that $$\frac {ΔB} B=-D \frac {Δy} {1+y} + \frac 1 2 CΔy^2$$ Since $\frac 1 2 CΔy^2$ is positive no matter what so the price ...
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1answer
968 views

Generic bond yields

I was looking on historical sovereign bond yields for a project. I was wondering what is meant by "generic bond yields" mentioned on bloomberg. Somewhere else i found data about the same country but ...
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1answer
357 views

Calibration of Heston version of CIR

I'd like to calibrate a variant of Heston model for interest rates which is describe by this couple of SDE \begin{aligned}dr_t&=a(b-r_t)+\sqrt{r_t}\sigma_t dW_t^1 \\ d\sigma_t&=k(\theta-\...
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4answers
6k views

Which risk-free interest rate to use in Black-Scholes equation

Sorry but i'm new in quantitative finance. According to BS derivation the risk-free interest rate is the rate to wich the rate of a particular investment tends when the risk tends to zero. Suppose i ...
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1answer
149 views

Rebasing of Cap Volatilities

I recently found this article where towards the end the author describes a method to rebase cap volatilities. Their method works like this: for a fixed strike assume that you are given the implied ...
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1answer
2k views

Where can I find open swaption implied volatility data?

Anyone have a good place to find interest rate swaption implied volatility data? Does Bloomberg's python API allow access?
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116 views

Black 1976 caplet value

I've seen from two sources different formulas for the caplet value (Black 1976): $$Caplet_1 = N\cdot DiscountFactor_{0,k}\cdot yrFrcn_{k,k+1}\cdot [F_{k,k+1}\cdot N(d_1) - R_k\cdot N(d_2)]$$ $$ ...
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1answer
506 views

Use of cap volatilities

I have a cap volatility surface for the 6 months Libor. Can I use the same cap volatility for every cap's caplet to valuate the full cap? Example: Valuate a 18M cap (Libor 6M) by valuating 3 6M ...
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2answers
582 views

How did traders calculate that the expected number of rate hikes is 4 based on eurodollar futures on 15Feb2018?

https://www.bloomberg.com/news/articles/2018-02-14/bond-traders-swarm-2019-fed-hike-bets-after-inflation-surprise After a Wednesday report showed consumer prices rose in January by more than ...
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4answers
2k views

Why are stock index futures not used to forecast how much the stock market will rise, given that interest rates futures are used for this purpose?

In news articles, the reader often read interest rates forecasts calculated based on interest rate futures. An example is here; How did traders calculate that the expected number of rate hikes is 4 ...
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1answer
386 views

How to calculate a future contracts price?

I have the following question from Hull, problem 6.17: On August 1 a portfolio manager has a bond portfolio worth $10 million. The duration of the portfolio in October will be 7.1 years. The December ...
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2answers
90 views

Basic question re: Fed interest rate tightening and rising interest rates

February 2/8/2018 - context in case the question is still around beyond today: the stock market has been falling for almost a couple of weeks in the midst of fears of overheating of the economy (...
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1answer
609 views

Weights Blowing up in PCA

I'm using daily settlement data to get yield levels for a couple of products. From this data I am doing PCA on a rolling collection of the yield levels. I have been using sci-kit learn's PCA function, ...
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1answer
50 views

Withdrawing monthly from a bank for 40 years [closed]

Consider you have $\$104107.4099$ in the bank with a $.33\%$ monthly effective interest rate. You plan to withdraw a fixed amount X every month for 40 years, such that you make 480 withdrawals in ...
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1answer
24 views

Conversion of annual interest rate compounded monthly to monthly effective interest rate [closed]

I am given that the annual interest rate is $r=4\%$ and that it is compounded monthly. I have to find the monthly effective interest rate. If I wanted the annual effective interest rate, I would use ...
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1answer
17 views

Conversion of 1- month effective interest rate to 6-month effective interest rate [closed]

I am given that the monthly effective interest rate is $1\%$ and I would like to find the $6$ month effective interest rate for a problem. I used the formula $r_e=(1+r)^\frac{m}{n}-1=(1+.01)^\frac{12}...
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1answer
41 views

compute r(t) in Vasiceck model, what is $e^{at}r$

I know how to solve the exercise using the hint. But I do not understand where the hint is coming from. Is it just continous compounding? Can anybody explain $f(t,r) = e^{at}r$? What does it stand ...
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1answer
119 views

Call option prices in terms of maturity with negative interest rates

let's assume that interest rates are constant, $r$. When $r\geq 0$, we can see that if $T_1<T_2$ and $C_1$ (resp. $C_2$) is the price of a call option on a non-dividend paying stock with maturity $...
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2answers
3k views

Are 3 month t-bill rates in FRED annualized?

Are the 3 month t-bill rates documented by FRED here annualized? For example, the rate for January 1997 is 5.03%. Does that mean one would get a 5.03% return in 3 months, or is that an annualized rate?...
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1answer
401 views

Am Call = Euro Call if r is non-negative and Am Put = Euro Put if r is negative

It can be proven that under non-negative interest rates, it is never optimal to exercise an American call option, such that: We know, if R >= 0, the current price C of a Europen (and American) call ...
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1answer
227 views

Closed- solution for Convertible bond price two factor model

I am trying to find the closed- solution of convertible bond $V(s,r,t)$ under Vasicek model of two factor model of PDE shown in below link Ito lemma of Convertible Bond under Two-factor Model ...
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1answer
542 views

Fixing date, start date, end date in interest rate derivative valuation?

I was reading a technical report by Hagan, which can be downloaded here on the valuation of accrual swaps and range notes. It caught my attention that in the valuation he comments this: Consider ...
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1answer
62 views

Problems with Money Weighted Rate of Return [closed]

The market value of a small pension fund’s assets was 2.7m on 1 January 2000 and 3.1 m on 31 December 2000. During 2000 the only cash flows were: Bank interest and dividends totalling 125,000 ...
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1answer
169 views

Interpreting Units of Short Rate Parameters

I've estimated the parameters for the Vasicek model $$ dr(t) = a(b - r(t))dt + \sigma dW(t) $$ and the CIR model $$ dr(t) = a(b - r(t))dt + \sigma\sqrt{r(t)} dW(t) $$ to one-year Treasury yield data ...
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1answer
187 views

Equality under T-forward measure for convexity adjustment

I've been working with the convexity adjustment for interest rates that arises when changing from one measure $Q_{T_p}$ with a numéraire $N_p=P(t,T_p)$ to a measure $Q_{T_e}$ with a numéraire $N_e=P(t,...
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1answer
77 views

Price compounding: Swap versus Governments Bonds

There are different rates curve to compound prices. Since the crisis, regulators tends to favor price compounding with swap curves over IR curves deduced from governments bonds (EU regulators, french ...
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1answer
307 views

Martingale measure result application for interest rates under T-forward measure?

I've got a question about the way the equivalent martingale measure result is used for pricing derivatives. Hull states the result as the next equality: \begin{align*} f_o = g_0 E^{g}\big(\frac{f_T}{...