Questions tagged [forward-rate]

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Under HJM model where the forward rates are deterministic, how do you show that forward measures = risk neutral measure? $\mathbb{Q}^{T}=\mathbb{Q}$

I am confused about showing the forward rates = risk natural measure the HJM model and that forward prices = future prices
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1answer
60 views

How do I calculate Hull White's Theta from the discount curve?

The Question I'm currently implementing the a finite difference method for the Hull-White model, shown below: $$\mathrm{d}r(t)=\lambda[\theta(t) − r(t)]\mathrm{d}t + \sigma\mathrm{d}W(t)\tag{1}$$ This ...
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Manual Computation of Python QuantLib's NPV for Pricing of a Forward Rate Agreement

Following the question that I have asked on this link and response obtained, I have managed to price a 3x6 FRA using QuantLib Python, using the following codes: ...
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1answer
92 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
65 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
88 views

Volatility and drift of the instantaneous forward rate under risk neutral measure using the zero coupon bond

I have question about this problem. I believe I have derived $f(t,T)$ correctly using the zero-coupon bond. But I am unsure about how to go forward with the question and how to use the second part. ...
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1answer
148 views

Compounding arrear SOFR Forward rate/curve

As per ISDA protocol and supplements, they stated that the fallback rate to be used on legacy derivative contracts is the compounding in arrears SOFR rate (based on a 2-day backshift) + a fixed spread ...
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50 views

SOLVED Manually Recomputing Forward Rates from QuantLib Python

I have used QuantLib Python to compute 1-month forward rates from zero rates as at 05 December 2019. My codes can be found below: ...
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47 views

Deriving instantaneous forward rate from spot rates/ zero rates

i have the following problem. I have a set of spot rates /zero rates $\{(t,r_t)\}_{t=1}^n$ where $t$ is the time and $r_t$ is the zero rate. How can i calculate with this data the instantaneous ...
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1answer
113 views

Interpolation of Zero rate curve

I have the zero rates at certain time nodes, say 3-month, 5-month, 8-month,...,2-yr,... Now I want to interpolate the curve so that the implied one-month forward rates are piecewise linear. That is, ...
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1answer
69 views

India's FX foward market

https://www.bloomberg.com/news/articles/2021-05-04/india-asks-state-banks-to-protect-dollar-assets-on-cairn-concern Based on this article USDINR forward premium has spiked as there is abundant USD ...
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2answers
132 views

Can you perform covered interest arbitrage when the forward rate is too low?

In covered interest arbitrage, if the forward rate is too high, you can (i) exchange domestic for foreign currency today (ii) invest at the foreign deposit rate (iii) exchange back to the domestic ...
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1answer
58 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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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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2answers
305 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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3answers
186 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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44 views

how to get 3 month Forward rates from Hull white model simulation?

I implemented the Hull White one factor model in Monte Carlo simulation, and got the short rate on each node (time step =1month). my question is how to get the forward rate from the short rate? I am ...
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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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76 views

How to price a forward-rate agreement?

I don't understand how the formula on page 24 of Joshi: Concepts and Practice of MF is derived. Here is the paragraph I don't understand: A forward-rate agreement is simply an agreement to take some ...
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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
327 views

Getting a daily forward OIS rate curve with QuantLib in Python

I am trying to build a 1-day EONIA forward curve with QuantLib giving OIS yields from 1mo to 50yr as input. My current approach consists on (i) obtaining the yield curve with ...
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1answer
294 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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76 views

Expectation hypothesis, expectation under which measure?

As I understand the expectation hypothesis says that the implied forward rate, can be used to predict future spot rates? If $r_{0,2}$ is the rate for a zero coupon bond maturing in two years, and the ...
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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
466 views

Which measure is used to price a swap?

When we value the floating leg of a standard vanilla swap, we replace the expectation of the future floating rates by the forward rates known today. However my understanding is that the forward rate ...
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1answer
459 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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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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1answer
117 views

Implication of forward-rate dynamics when the short-rate follows a normal process

In the section 3.2.3 of the second edition of "Interest Rate Models - Theory and Practice" by Brigo and Mercurio, the forward-rate dynamics implied by the CIR model is derived as follow: The ...
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1answer
77 views

Hedging With Zero Coupon Bonds from The Concepts and Practice of Mathematical Finance by Mark Joshi

In section 2.5 he describes an example of arbitrage-free pricing (attached below). I have a pretty solid understanding of how we arrived at $K' = K\frac{1+d}{1+r}$, but I got a little lost when he ...
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1answer
275 views

Why is overnight more expensive than spot in an increasing forward swap values table?

I'm looking at EUR/USD fwd prices. Currently they are the following ones: These are the swap points to be added to the spot price. It seems it increases with time. Therefore, since overnight value ...
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1answer
78 views

Splitting a spot swap into a forward swap and a 3 month libor

I read the following statement: We can construct a 5 year swap using 3 month libor combined with a 3mo-4.75yr forward swap, weighted by the dv01s of each part. I am not sure I understand how this ...
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38 views

Short position in currency forward meaning

Trying to understand what being in a short position of a currency forward means. For example, given a 60-day currency forward at 0.92154 pounds/euro where euro is ...
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1answer
591 views

How to calculate the Fx Forward Points for 3M

I'm trying to find the FX Forward Points for 3M, the same as in the table. However, in the conventional way (Forward points = Spot x (USD Rate - EUR Rate) x 90/360) I get a different result. Can ...
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2answers
2k views

STIR topics: Implied FX-OIS Basis and FX Forward/Swap Pricing

if someone could provide some clarity on the below: What is meant by 'Implied FX-OIS Basis'? For example: "ON JPY trading at parity, 1W implied OIS basis moved 70BP" and "3M Implied OIS basis moved ...
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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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3answers
618 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
166 views

Link between spot and forward rates in no-arbitrage world

With reference to the forward exchange rate definition, let be: $S$: the spot rate $F$: the forward rate $r_d$ and $r_f$: respectively the domestic and foreign interest rates $DF_d$ and $DF_f$: ...
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1answer
206 views

project curve spot risk (PV01) into forward risk (PV01)

is there a (simplistic?) formula to convert spot risk PV01 into the forward risk PV01? For example, if I have a a) PV01 spot risk : 1yr = 100k/bp, 2yr = 50k/bp, 3yr = 25k/bp b) how can I project ...
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1answer
392 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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1answer
192 views

Basis Swap Dual Curve Calibration

The long end of the Libor swap curve needs to be constructed from Basis Swaps because there are no other instruments traded. Can please someone explain the concept of Dual Curve Calibration?
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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}\...
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1answer
277 views

zero-coupon bond and forward rate

My understanding, in a discrete-time setting, the relationship between a zero-coupon bond price and forward rates is: $$p(t,T)=\frac{1}{\Pi_{j=1}^{T-1}f(t,j)}.$$ where $p(t,T)$ represents the price ...
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1answer
356 views

Pricing of compounded swaps

As far as I understand, a compounded swap rolls up individual payments into one final payment which becomes: $$ V(t_n) = N \prod_{i = 0}^{n-1}(1 + d_i L_i)-N $$ where $d_i$ is the day fraction for ...
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135 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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1answer
296 views

Calculation cross-currency basis

I am trying to calculate cross-currency basis on the 3-month horizon for a certain set of currencies. The formula should be $ccb = F/S (1+y_{foreign currency}) - (1+y_{USD})$ where $y_{USD}$ is Libor ...
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2answers
240 views

Instantaneous forward rate within the HJM framework

within the HJM framework, the dynamics of the instantaneous forward rate are defined by: $$f_t(T)=f_0(T) + \int_0^t\alpha_s(T)ds+\int_0^t\sigma_s(T)dW_s$$ or in differential form: $$df_t(T)=\alpha_t(...
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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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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
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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101 views

Change of measure for BGM (LMM) Model

I've been checking the demos for BGM (LFM) forward rate model. Here's a short reminder to help you follow: Now, take the following $$\frac{dL_j(t)}{L_j(t)} = \sigma_j. dW^j(t) = \mu_{ij} dt + \...