Questions tagged [forward-rate]

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9
votes
2answers
10k views

What does instantaneous forward mean?

Could you please help me to understand meaning of instantaneous forward rate? I mean economic interpretation at basic level. What is it used for? How can i derive it from zero rate/price? Thanks
8
votes
1answer
11k views

Calculating instantaneous forward rate from zero-coupon yield curve

I have a big dataset containing zero-coupon bond yields with different relative maturities. I fix a time horizon on my dataset and I want to calculate instantaneous forward rate. I'm going to write ...
8
votes
1answer
689 views

Transforming 3M volatilities into 6M volatilities in EUR forecast curves

I have implemented a stripping algorithm to extract forward volatilities from cap/floor flat volatilities for different currencies. I am however struggling a bit when implementing a method to convert ...
7
votes
2answers
1k views

Practical implementation of Libor Market Model

I am trying to implement a project about the BGM model, suggested in the book "The Concepts and Practice of mathematical finance" by Mark Joshi. My question is related to the forward volatility ...
6
votes
1answer
2k views

Why are multiple custom curves (swap) built for one desk?

Currently in a journey of learning and getting my hands a bit dirty with Interest Rate Swaps. Why there are multiple customized curves built by many even within one desk? For e.g. Short Rates desk ...
6
votes
1answer
764 views

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(...
5
votes
1answer
5k 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 ...
5
votes
1answer
566 views

meaning of discount term in FRA value

Consider a forward rate agreement on LIBOR (say), which starts 2 months from now, expires after 3 months and has strike $K$, and is based on $3M$ LIBOR -- $FRA_{2\times 5}$. Now the present value of ...
5
votes
1answer
707 views

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 ...
5
votes
1answer
535 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}{...
4
votes
1answer
222 views

U.S. Rate Hike Prediction

In a recent ft.com video an analyst mentioned that markets postponed their Fed rate hike expectation from September to around November 2015 due to the CNY devaluation, based on the "shift" of some "...
4
votes
3answers
1k views

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{...
4
votes
1answer
354 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,...
3
votes
3answers
1k views

How to prove martingality of forward rate under T-forward measure

Let $P(t,T)=\mathbb{E}_{Q_{R}}[e^{\int^{T}_{t}r(u)du}|\mathcal{F}_{t}]$ be the price of a 1-euro zero-coupon bond with maturity $T$ and $r(u)$ the interest rate process. Consider the the forward rate $...
3
votes
3answers
454 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 ...
3
votes
3answers
399 views

Why is the spot price not used as the forward price when a forward contract is created?

If the initial value of a forward contract is zero, surely the forward price used would be the spot price at the time the contract was created? However, my notes tell me that the forward price F, at $...
3
votes
2answers
834 views

Half of the bid-ask spread as transaction cost

I am currently reading "Deviations from Covered Interest Rate Parity" by Du et al. When establishing deviations from CIRP they consider transaction costs as follows. "We assume that the transaction ...
3
votes
0answers
98 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 + \...
3
votes
0answers
137 views

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 ...
3
votes
1answer
331 views

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 ...
2
votes
2answers
226 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(...
2
votes
2answers
6k views

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 ...
2
votes
1answer
795 views

Formula for the forward rates?

I'm reading a book about interest rate modelling. It states the following formula P(0,T) = exp(-sum of the forward rates) But I thought it's the average of the forward rates?
2
votes
1answer
253 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 ...
2
votes
1answer
312 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 ...
2
votes
1answer
4k views

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 ...
2
votes
2answers
431 views

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 ...
2
votes
2answers
869 views

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 ...
2
votes
2answers
493 views

forward space vs zero space in finance jargon

Would anyone know what does it mean to value an asset in "forward space" versus "zero space" ? where does one start from when trying to dig into the meaning of this? Thanks in advance.
2
votes
1answer
919 views

Why can a swap option be regarded as a type of Bond option?

Why can a swap option be regarded as a type of bond option? My idea: Suppose the swap rate of the swaption is $s$. Now consider a bond option expiring at $T$ with strike, $(P_K)_t = \dfrac{1}{1+s(T-...
2
votes
1answer
311 views

Dual discounted forward curve

I was wondering how to calculate the forward rates based on OIS discounting for the half year terms. I know how to do this for the full year terms -> just making sure that the two legs are equal to ...
2
votes
1answer
116 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 ...
2
votes
1answer
229 views

PCA on a portfolio of spot and forward contracts

I have a portfolio of spot and FX forwards on various currencies all based to AUD. I need to able to quantify how the changes in amount, tilt and curvature of the AUD curve would impact my p/l. ...
2
votes
1answer
371 views

What's the difference between instantaneous forward rates and observable forward rates?

Source: http://docs.fincad.com/support/developerFunc/mathref/LIBORMarketModel.htm "In contrast to models that evolve the instantaneous short rate (Hull-White, Black-Karasinski models) or ...
2
votes
1answer
88 views

Bond Interest Rate Swap Growth Rate [closed]

this should not be here because it shouldn't be here forever and eve
2
votes
1answer
279 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 ...
2
votes
1answer
289 views

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 ...
2
votes
1answer
70 views

Likelihood of a caplet ending in the money

with what likelihood would one expect an ATM caplet to end up in the money? Just as a very rough guess, from real world experience. When I consider N(d2) from the Black formula, for spot = strike = 4%...
2
votes
0answers
37 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 ...
2
votes
0answers
315 views

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 : $C_t(T,T+d)$ ...
2
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0answers
134 views

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 ...
2
votes
0answers
257 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)]...
2
votes
0answers
81 views

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^{-\...
2
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0answers
113 views

Are forward rates starting at observation date spot rates?

In part 3.2 of Lu and Neftci (2003) "Convexity Adjustments and Forward Libor Model: Case of Constant Maturity Swaps", the authors propose a new way of pricing CMS swaps, with Monte Carlo simulations. ...
2
votes
0answers
147 views

Interpolation of forward zeros-coupons bonds simulations for missing maturities (ESG data)

I have a set of economic scenarios simulated with Barrie and Hibbert ESG. The stochastic model for interest rates used is Libor Market Model Shifted. I am facing a problem with zeros-coupons prices. ...
2
votes
1answer
4k views

How to compute for basis adjusted forward rate?

To give you a brief background, I'm valuing a fixed-for-float Interest Rate Swap (IRS) using Bloomberg. I put in a notional amount in (USD) and a assigned 6MO USD LIBOR as the reference index for the ...
2
votes
0answers
682 views

Simple Forward Interest Rate Proof

Just trying to check my logic here: Let $Z(t,T)$ be a Zero-Coupon Bond with maturity $T$ bought at time $t$, $S_m$ be the spot interest rate for time $m$ and $S_n$ for time $n$ respectively, where $n ...
1
vote
1answer
409 views

FX Forward rate agreement valuation in quantlib

I am trying to value an FRA in quantlib Python using the below code: ...
1
vote
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} \...
1
vote
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% ...