Questions tagged [forward-rate]

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1answer
458 views

FX Forward rate agreement valuation in quantlib

I am trying to value an FRA in quantlib Python using the below code: ...
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1answer
136 views

why $f(t,u) \neq E_t^Q [r(u)]$ when $r$ is random?

If I suppose the short rate $r$ deterministic, and the risk neutral measure $Q$, I can write the following : $$f(t,u) = -\frac{d}{du}\ln P(t,u) = -\frac{d}{du} E_t^Q \left[ e^{-\int_t^{u}r_sds} \...
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1answer
91 views

Pricing a Forward Rate Agreement using QuantLib Python

Can someone please help with the pricing of the following forward rate agreement using QuantLib Python? A 3x6 forward rate agreement, with a notional of $100,000, the FRA rate being 6%, The FRA ...
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1answer
1k views

How to calculate one-year forward one-year rate? [closed]

I'm just a little lost on how to calculate forward rates. I know this is an easy question, but, if we are given a one-year and two-year zero rate (let's say, for the sake of the argument, 2% and 3% ...
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1answer
216 views

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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1answer
2k views

forward vs spot simply-compounded spot interest rate

Question about forward vs spot simply-compounded spot interest rate.Some definitions $P(a,b)$ a zero coupond price at time $a$ and maturity $b$ $L(a,b)$ simply compounded spot interest rate set at ...
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1answer
390 views

Constant continuous forward rate interpolation

Assume that the continuously compounded forward rate is constant between two node points. What is the interpolated discount factor between these two points? So you have the two discount factors $D_{...
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2answers
512 views

Libor Forwards from Swaps

I am trying to understand how to interpret a few forward curves that I grabbed from Bloomberg. In Bloomberg, you use ICSV command and choose the USD to Libor swap curve. I did this and grabbed the 1mo,...
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1answer
286 views

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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1answer
2k views

Derive instantaneous forward rate

Given that $P(0,T)=e^{-RT}$, how does one get the formula for the instantaneous forward rate below? Specifically, how does one get to the partial derivative in the formula? I'm sure the answer is ...
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1answer
75 views

Black (1976) model growth rate input for futures price

When using the Black 76 model for pricing European index options I've often seen people use 2 different rates: the typical risk free rate used to get the discount factor, and a growth rate used to get ...
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1answer
291 views

Forward Swap Rate calculation using Quantlib

Here, we have an example for the calculation of Forward Swap Rate - How to compute forward swap rates? Below is my Forward Swap - ...
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2answers
106 views

How would you use FRAs to find out how much Central Banks would cut/hike by?

Let's say you have The FRAs 1x4F, 2x5F, 3x6F, 4x7F, 5x8F... (meaning 3 months rate today, 3 months rate in 1 month, etc..) at 5.5, 5.5, 5.6, 5.55, 5.55. Assume today's date is 20/09/2019. You also ...
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2answers
3k views

Forward swap rate calculation from the market

Following my question Swaption valuation across time using vcub where I wanted to know how to value a swaption across time using bloomberg's vcub, I remark that I have to calculate myself the ...
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2answers
2k views

Calculating spot rates from forward rates

I am working on a problem where I am trying to calculate the forward rates from two different spot rates. I have the following: ...
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1answer
262 views

Eurodollar Future Key Rate Duration

I am having trouble understanding the Key Rate (partial) Duration profile of Eurodollar Future contracts. Using market rates and pricing date as of 11/14/2018 I have calculated the partial duration ...
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1answer
1k views

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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1answer
2k views

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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1answer
6k views

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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3answers
2k views

Calculating FRA rates

Let's assume I constructed usd libor 3M curve setting 1M rate=3M rate (so the curve is flat between 1M-3M). Will 1x4 FRA rates be good if calculated from such curve?
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1answer
63 views

construct portfolio offering risk free profit

Have trouble understanding this question, seems quite open ended. Assume that $S(0)$ is the current rate of exchange for foreign currency. Assume that and $K_n$ and $K_f$ are rates of return on home ...
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1answer
1k views

Long Forward Rate Agreement, short Eurodollar futures

For this question If you are long a FRA and short a ED future with the same fixing dates, do you have positive convexity or negative convexity? The answer is positive convexity, because a ...
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2answers
225 views

Using IRR to calculate future value of cashflow

Discounting a cashflow using given forward rates will result in the following present value: PV = 102.875 = ${5\over (1+3\%)}$ + ${5\over (1+3\%)(1+4\%)}$ + ${105\over (1+3\%)(1+4\%)(1+5\%)}$ where ...
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1answer
158 views

Forward interest rate

I have some confusion regarding forward interest rate. It seems that there are two notions of forward interest rate: The interest rate $f$ agreed at time $t$ for investment over a future time period ...
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1answer
108 views

Deriving forward rate

I want to price a 1 year future under the condition of no arbitrage and based on LOOP. At time T, I sell currency Z and buy currency L. At time $t$, we define the exchange rate as $ZL_t$. The 1 year ...
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3answers
617 views

Are forward rates for an IRS computed between reset dates or between start dates?

In order to price the floating leg of an IRS I am computing forward rates for future coupons, but I'm not sure whether I have to compute such rates between reset dates or between start dates. My ...
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1answer
73 views

How to derive the expression for the forward rate?

The following RN dynamics of a ZCB maturing at time is given: $$\frac{dZ(t,T)}{Z(t,T)} = r_tdt + \sigma_Z(t,T)dX_t$$ and the forward rate is given: $$f(t,T,T+\delta) = \frac{ln(Z(t,T)) - ln(Z(t,T,...
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1answer
338 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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1answer
169 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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1answer
2k views

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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1answer
2k 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
384 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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1answer
559 views

Forward Curves and Par Yield Curves

I'm recently reading a research paper on the yield curve by Salomon brothers and in it it states that when the forward curve is above the par yield curve, it is seen as cheaper. If for example, the ...
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0answers
83 views

How to calculate the Net Present Value (Market-to-Market Value) of a plain vanilla FX forward?

I would like to understand the math/economics behind the calculation of plain vanilla FX forwards. Let's assume the following example: ...
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0answers
60 views

Intuition behind local volatility curve shapes in interest rate environments

I have some questions regarding the intuition behind shapes for the local volatility (LV) curve as seen in quite popular models. Let's say we have the following generalized stochastic-local volatility ...
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0answers
64 views

90 day SOFR market rate to equivalent 3M Libor rate conversion

I am trying to convert a 90 day SOFR market rate into a fair equivalent 3 month Libor rate. I understand since SOFR is by nature an in-arrears backwards looking rate, so for SOFR instruments such as a ...
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0answers
134 views

Calibrate an HJM model in a multicurve setup

I am a mathematician and I'm working on my thesis on Financial Mathematics. I studied this model HJM in a multicurve setup: $$ \begin{cases} df(t,T)=a(t,T)dt+\sigma(t,T)dW_t & (\mbox{rik-free})\...
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0answers
40 views

Parametric Simulation of FX forwards

I need to simulate FX forwards for risk management (VaR) purposes. The problem is that the FX forwards are derived from : 1) Spot 2) int rates 3) and the basis. So the question is how do you ...
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1answer
2k views

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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0answers
40 views

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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0answers
68 views

Forward Exchange Rate Data: Germany x US

Would anyone know where I can find historical forward exchange rate data between germany and US, yen and US to download? In Bank of England website i already found. Thanks
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0answers
74 views

How should I use central banks rates in order to compute 2-day forward exchange rate on EURUSD

Good morning, I would like to compute a 2-day forward exchange rate on EURUSD. For that, I have historical data on EURUSD spot price, and I know that theoretically, when maturity is T we have : $F_0$...
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2answers
255 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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1answer
119 views

Differential equation involving bond price and forward rate

Given forward rate f(t,T) and bond price P(t,T) where $f(t,T) = - \frac{\partial}{\partial T} \ln P(t,T)$, $P(T,T) = 1 = P(t,t)$, T>0 and $t \in [0,T]$ Does it follow that $P(t,T) = exp(-\int_{t}^...
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2answers
303 views

Getting quarterly forward rates with QuantLib

I am trying to build a quarterly forward curve with 3 month USD Libor swap rates from 1Y to 50Y as inputs. From other posts I have looked at, I have managed to come up with this code so far: ...
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4answers
3k views

Calculating bond forward rate/price

What's the difference these two methods on calculating the bond forward rate/price. First of all I'm assuming forward rate is the same as forward price in this context, if this assumption is false, ...
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1answer
64 views

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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1answer
57 views

Money market yield question

Assuming the 92-day and 274 day interest rate is 8% (act/360, money market yield) compute the 182- day forward rate starting in 92 days (act/360, money market yield). 1)7.80% 2)8.00% 3)8.20% 4)8.40% ...
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3answers
184 views

How does the term premium of the 10y20y Treasury forward rate relate to the 30y rate?

I'm reading recent research on Treasuries and to paraphrase, it says that long term 10y20y Treasury forward rates now have a positive term premium over the long run nominal funds rate (neutral rate). ...
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2answers
129 views

Forward Rates are martingal under Forwar Measure detailled proof

So I read an other post about this : How to prove martingality of forward rate under T-forward measure But I can't see how to get from there to there : $F \left(t,T_n \right)P \left(t,T_{n+1}\...