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Questions tagged [forward-rate]

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Interest rates in a dedication model

In a discrete time dedication model (dedicated portfolio theory) is it more correct to use spot rates or forward rates as a proxy for the risk free rate for each time periode? https://en.wikipedia....
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Approximation of Forward Rates in discrete time

The forward rate from time $t$ to $T$ ($f_{t,T}$) can be approximated by: $$ f_{t,T}= \left[ \frac{(1+r_T)^T}{(1+r_t)^t} \right]^{\frac{1}{{T-t}}}-1 \sim \frac{(1+r_T)^T-(1+r_t)^t}{T-t} $$ Why is ...
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continuously compound forward rate formula

I want to derive the continuously compound forward rate formula according to FRA. fixed rate is $K$ and notional is $N$, $\delta=T_1-T_0$. $t<T_0<T_1$, the FRA holder at time $T_1$ need to pay ...
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Pricing Uneven FX Swaps

I'm trying to figure out how to price uneven FX swaps. I just started on a FX trading desk and have been told that the all-in rate for a 2-legged FX swap is equal to: 1) Quote for market side of net ...
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Difference between settlement of Eurodollars and FRA

I am going through J.C. Hull's chapter on FRA and EuroDollar Futures. Taking the case of FRA. I assume $T_0$ is the time when two parties entered into a FRA to fix interest rates they get on a ...
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Change of numeraire between T-forward and Bank Account

I follow a course, and get to the point that one bond price discounted by another one is a martingale: $$ \frac{P(t,T_0)}{P(t,T_1)} - \text{ is a } \mathbb{Q}^{T_1} \text{ martingale } $$ I can not ...
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Relation Between Yield Curve, First Order Derivative of YC and Forward Rate

I'm reading the book, "Derivatives Analytics with Python" by Yves Hilpisch. In an application of calibration of CIR85 process for the short-term interest rate. I found some codes which can be ...
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EURIBOR zero rates vs forward rates to project future income on a bank's loans

I work at an international bank within the M&A FIG team, and have seen that my associate uses the future daily EURIBOR 3M,6M,12M to estimate what the future interest income on a banks loans will ...
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Implied AUD Interest Rate from USDAUD FX Swap and USD Interest Rate

Can someone help me understand how to derive the implied interest rate or spot rate in BBG FXFA? I actually get why the Forward rate, F_Ask and F_Bid are derived using the formula in the picture. ...
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Soft Question: Why forward-rate curve models?

Why did people move away from short-rate models to forward-rate curve models (besides the HJM drift restriction)? Are there any benefits of short-rate models which forward-rate curve models do not ...
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Pricing caplet with Bachelier (normal dynamic) using forward measure

I'm trying to price caplet with Bachelier under forward measure, but I can't find any solution. Remind that Bachelier assumed rates follow a normal dynamic. So here what I was doing : $C_t(T,T+d)$ ...
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Bootstrap daily OIS forward rate

Can someone please show me how to derive the daily OIS forward rate in a OIS-fixed rate swap? For example, if price a paying fixed rate/receiving OIS swap in Bloomberg SWPM, Bloomberg will be able ...
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444 views

Dual Curve Bootstrapping - When to OIS discount?

I am a new quant and I am trying to understand some of the specifics of dual curve bootstrapping. For concreteness, suppose I want to build a Libor forward curve. From what I understand OIS ...
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146 views

FX hedging: forward rate and implied forward rate

In this paper (box 1 page 24): https://www.rbnz.govt.nz/-/media/ReserveBank/Files/Publications/Bulletins/2000/2000mar63-1brookeshargreaveslucaswhite.pdf It is argued that the forward rate that a ...
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Mid-curve swaption

I would like to know how the mid-curve swaption could inform us about forward volatility. In my understanding it is a swaption on a forward starting swap. Let us say the midcurve swaption expires ...
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Does Foward Rate on Libor change depending on which day you calculate it at?

Based on this article: https://en.wikipedia.org/wiki/Forward_rate If we were to calculate forward rate on libor 3M at times T1,T2 (ie, forward rate at T1, going forward T2), does that value change if ...
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How is the integral relationship between current yield curve and forward yield curve derived?

$$y(\tau) = \frac{1}{\tau} \int_{0}^{\tau} du \Big(f(u)\Big)$$ As far as I understand the forward rate is the future rate based on the expectation hypothesis. But it is unclear how many years into the ...
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How should I interpret a forward rate?

Let $L(t, S, T)$ denote the forward rate from time S to T observed at time t, assuming t < S < T. A lot of modelling work is centered around this rate, but how is this rate useful? How are we ...
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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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Difference between term spread and forward rate

Term spread is the difference of long term and short term yield. If the spread is positive, the difference indicates the extra yield received on long term bond. Forward rate is the future yield on a ...
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Dynamics of LIBOR foward rate under T-forward measure

Assume that under the physical measure $\mathbb{P}$ we have for the LIBOR forward rate $L(t):=L(t;S,T) = \frac{1}{T-S}\left(\frac{P(t,S)}{P(t,T)}-1\right)$ that $$ \mathrm{d}L(t) = L(t)\left(\mu(t)\...
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What does it mean that a swaption is atm?

This is a relatively short question: If a swaption with option tenor $S$ and swap tenor $T$ is said to be atm, does that mean that the strike of the swaption is equal to the currently (we ar at time $...
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Details of calibration of Hull-White model

Consider the one-factor Hull-White model $$ \mathrm{d}r(t) = (\theta(t)-\kappa r(t))\mathrm{d}t + \sigma\mathrm{d}W(t) $$ When one calibrates the model to market data one chooses $$ \theta(t) = \...
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Intra-period return on a long forward position

Just recall that return on a long forward position over the term of the forward contract from $\displaystyle t=0$ to $\displaystyle t=T$ is \begin{equation} R_{T,0}=\frac{S_{T}-F_{0}}{S_{0}} \end{...
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Can anybody help me understand this forward exchange rate data?

Bank of England provides data on forward exchange rates. I am a bit confused of what it means. Does a forward rate of 1.2883 in 26 June 17 (highlighted in red in the image) mean that in 26 June ...
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Historical Data on $/euro forward exchange rate?

Can anybody tell me where I could find historical forward exchange rate data between the dollar and euro? Thanks!
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153 views

Spot-Forward Relationship - Proof

Does anyone know of a decent proof for the spot-forward relationship of a currency? I've been looking on Google for hours and I'm not getting anywhere. My lecture notes are useless in that they don't ...
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61 views

option on bond future - any caplet representation out there ?

I'm trying to play with bond-future options. Bond future is a future contract on a basket of bonds. The short-side will deliver the so-called bond cheapest-to-deliver (CTD). A bond-future option is ...
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Overview of pricing using e.g. forward measure numeraire change

I am taking my first steps into numeraire change and change of probability measure. My overall struggle is relating numeraire change and all its aspects to e.g. the easy intuition behind risk-neutral ...
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Calculating Discount Margin on a floating rate bond using QuantLib

Going off Luigi's hint on this answer: Setting up Schedule for an amortizing floater in QuantLib I was able to cobble something together but I'm unable to verify if it's correct. TLDR: I was able to ...
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Transform a 3M FRA Rate to a 6M FRA Rate

I have a question whether it is possible to transform 3M FRA rates to 6M FRA rates without having any spreads available. Let's give an example: FRA 3M: FRA 1x4 FRA 2x5 FRA 3x6 FRA 4x7 FRA 5x8 FRA ...
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Convexity adjustment when payment if after interest natural term?

I've been working with a convexity adjustment for an interest rate payoff and the next question came to me: The usual problem that gives rise to the convexity adjustment I'm referring to is as ...
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307 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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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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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}{...
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Arbitrage-free quotations FRA

The rates for a FRA 3x9 are 0.95 %/1.08 %. However, according to the interest rates, the calculated forward rates should be 1.010%/1.170%. The question is whether it is arbitrage-free, and they ...
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Convert 3M rates to 6M rates using Basis Swaps (3M vs 6M)

How can I convert a 6M Libor rate e.g. 1Y Tenor to a 3M Libor rate using a basis swap 3M vs. 6M? I wanted to know the math and also an example would be great. Update: Example: ...
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Change of measure between T-forward and T*-forward contract?

I am trying to prove the need of a convexity adjustment to a forward rate by calculating the next expectation: \begin{align*} P(t_0, T_s)E^{T_s}\big(L(T_s, T_s, T_e) \mid \mathcal{F}_{t_0}\big). \end{...
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Deriving the Forward Rate Formula from the Expectation Hypothesis

The Expectation Hypothesis (EH) states that the current spot yield for any of the maturities is the geometric average of current and future short rates. $$\Big(1 + y(t=0, m=\mu) \Big)^{\mu} = \prod_{t=...
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How to calculate the product of forward rates with different reset times using Ito's lemma?

I am curious about a calculation I saw in this question. Specifically in this equation: \begin{align*} &\ L(T_s, T_p, T_e) L(T_s, T_s, T_e) \\ =&\ L(t_0, T_p, T_e) L(t_0, T_s, T_e) e^{-\...
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Identity for forward rates

In the context of interest models I came across the following identity for forward rates at time $m$ which, according to my book, has to always be fulfilled due to non-arbitrage: $$f_m(t, t+s) = \...
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140 views

Price series for an FX forward contract

Let's assume I am buying a NZD/USD 1Y forward for $1000000 on the 20/02/2017. The NZD/USD 1Y forward point is currently -270 and spot rate is 0.8325. (Example taken from here). Now I want to have a ...
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Compare the payoff of two investment with same initial capital

Investors A and B have the same initial capital to invest at time 0. Investor A invests all of the initial capital in a 6-month zero and keeps rolling over the investment into new 6-month zeros every ...
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Up-front settlement of forward contract

One has entered a forward contract to purchase oil at $F_{t,T} = S_{t}e^{(r_f + s - c)(T-t)}$. The contract is entered at time $t$ and executed at time $T$. Where: $S_{t}$ is the spot price at time $...
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How realistic are the scenarios outlined in my course?

I am currently taking a course in Financial Mathematics as part of my Maths degree. Many of the covered topics are quite basic, and revolve around potential arbitrage opportunities. For example, ...
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EUR Implied Forward Rate from Bloomberg

does anyone know about the EUR Forward Implied 3 Month Rate published by Bloomberg on the Bloomberg Page EURI3M ? First question: this is the rate from a forward curved, forward curve being ...
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63 views

Integrating Interest and Dividend Functions

How are interest rate and dividend functions integrated over time in practice? For example, what does it mean in practice to discount a current price by $e^{\int_{t_m}^{T}r_s ds }$ where $r_s$ is the ...
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Risk neutral measure of short rate model

As we all know, all affine term-structure models are members of HJM model. Under HJM model, there is a unique risk-neutral measure in both forward-rate process and bond evolving process. Hence, the ...
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Computing Correlation between Forward Rates

I have the feeling this question has an extremely simple answer but I'll put it out to the group anyway. Imagine I have data for 3M and 6M forward rates following a lognormal process, and that I ...
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FX Forward pricing with correlation between FX and Zero-Cupon

I would like to extend my question about about FX Forward rates in stochastic interest rate setup: FX forward with stochastic interest rates pricing We consider a FX process $X_t = X_0 \exp( \int_0^t(...