19 votes

What does instantaneous forward mean?

1. Observable instruments, spot rates, and forward rates First remember that something observable means that you can observe/find the rate in the market by looking at traded rate instruments or ...
Pontus Hultkrantz's user avatar
13 votes
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What does instantaneous forward mean?

Given a forward rate, for example: $ F(t, T, T+\delta)$ The instantaneous forward rate $f(t,T)$ fixed in $t$ is the limit when $\delta \rightarrow 0$ of your forward rate. If the relation between ...
Yassine Q.'s user avatar
6 votes
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Half of the bid-ask spread as transaction cost

The idea of assuming that the transaction cost is one half of the bid-offer spread comes from several assumptions: the positions are marked-to-market at mid; you can actually execute at bid or ask (...
Dimitri Vulis's user avatar
6 votes
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FX Forward rate agreement valuation in quantlib

You are not giving the constructor a discountCurve. The constructor is: ...
David Duarte's user avatar
  • 5,795
5 votes

Mid-curve swaption

You can only infer forward vol by pairing a mid-curve option with a spot option. It's easier to go through an example (I'll use 5y x 5y vol since I have the sketch below handy...) One decomposition of ...
Helin's user avatar
  • 11.6k
5 votes
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why $f(t,u) \neq E_t^Q [r(u)]$ when $r$ is random?

Your equations are flawed. Also there is no expectation if the process $\{r_s\}$ is deterministic. The correct reasoning is, assuming $\{r_s\}$ is stochastic: $$ f(t,u)=-\frac{d}{du}\ln P(t,u)=-\...
Antoine Conze's user avatar
5 votes
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How to prove martingality of forward rate under T-forward measure

For the instantaneous forward, please see the last page of this note: T-Forward Measure by Fabrice Douglas Rouah (http://www.frouah.com/finance%20notes/The%20T-Forward%20Measure.pdf). For the simple ...
Magic is in the chain's user avatar
5 votes

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

By definition, $$Fo(t,T)=E^T[S_T|F_t]$$ Note that expectation is taken under $T$-forward measure. Now, evaluating at $s<T$: $$E^T[Fo(t,T)|F_s] = E^T[E^T[S_T|F_t]|F_s] = E^T[S_T|F_s] = Fo(s,T)$$ (...
Prabhnoor Duggal's user avatar
5 votes

Instantaneous forward rate within the HJM framework

This is known as the classical Leibniz rule. The link sends to Wikipedia, where you can find a proof. It allows to differentiate under the integral sign. A general statement of the formula is: $$\text{...
Daneel Olivaw's user avatar
4 votes
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continuously compound forward rate formula

In the simple case, you have as per first equation on your last slide: $\frac{P(t,T_0)}{P(t,T)}=1+\delta F(t,T_0, T)$ The continuous time equivalent, assuming constant piecewise rate, as per your ...
Magic is in the chain's user avatar
4 votes
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Approximation of Forward Rates in discrete time

You may show it as follows: \begin{align*} f_{t,T}&= \left[ \frac{(1+r_T)^T}{(1+r_t)^t} \right]^{\frac{1}{T-t}}-1\\ &=e^{\frac{1}{T-t} \left[\ln (1+r_T)^T - \ln (1+r_t)^t \right]} -1\\ &\...
Gordon's user avatar
  • 21.1k
4 votes

How to calculate one-year forward one-year rate?

Let $\{r_t\}_{t>0}$ be the spotrates and $f_{t,T}$ be the forward rate from time $t$ to $T$ for $t<T$. Then the general formula to compute $f_{t,T}$ is $$ (1+r_T)^T=(1+r_t)^t(1+f_{t,T})^{T-t} $$...
Sanjay's user avatar
  • 1,637
4 votes

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

The answer by @Prabhnoor Duggal is correct. Here, I would like to further expand to make it more streamlined (see also Section 2.5 of the book Interest Rate Models - Theory and Practice). Let $Q$ and $...
Gordon's user avatar
  • 21.1k
4 votes

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

Implied FX-OIS basis should be pretty simple to "compute", it is the classical "Cross-currency" basis observed in FX Swaps & FX Forwards, that can be backed out when plugging ...
Jan Stuller's user avatar
  • 6,108
4 votes

Which measure is used to price a swap?

I want to propose a different answer here. I think mathematical expectation (under any measure) is not used in valuing an interest swap. Years ago I used to explain swaps to beginners by speaking in ...
nbbo2's user avatar
  • 11.2k
4 votes
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Pricing a Forward Rate Agreement using QuantLib Python

For a 3x6 FRA, you probably want to write something like: ...
Luigi Ballabio's user avatar
4 votes
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If any zero coupon bond $P(T)$ can be chosen as a numéraire, then why can the rolling bond for any time discretization be chosen as numéraire

The rolling bond $R(t)$ as defined in your question is a valid numéraire. Indeed, this bond can synthetized with the following iterative trading strategy in basic assets: At any time $T_i\in\{T_0,\...
Daneel Olivaw's user avatar
4 votes

Understanding FX forward points and market usage

There are two relevant sections on the help page, the direct links (e.g. if you have it in an IB) look like this: {LPHP FRD:0:1 2898067 }: ON ("Overnight"), TN ("Tomorrow-Next"), ...
AKdemy's user avatar
  • 9,059
3 votes

Dual discounted forward curve

Which currency are you looking at ? Say that your 1y swap would have yearly fixed payments vs 3M floating payments. Your 1.5y swap would probably have: a fixed payment 6m after effective date and ...
Bozothegrey's user avatar
3 votes
Accepted

Long Forward Rate Agreement, short Eurodollar futures

hypothetically if we assume that $R_{fra}=R_{fut}-\frac{1}{2} \cdot \sigma^2\cdot T^2$ holds (convexity adjustment) and you are able to observe $R_{fra}$, $R_{fut}$ and $T$ then you can extract ...
Nicholas's user avatar
  • 732
3 votes

FX Forward pricing with correlation between FX and Zero-Cupon

We consider the expectation \begin{align*} E^{Q_d^{t_f}} \Big(P_d(t_f, T) X_{t_f} \mid \mathcal{F}_t \Big), \end{align*} where $Q_d^{t_f}$ is the $t_f$-forward measure, and $P_d(t_f, T)$ is the ...
Gordon's user avatar
  • 21.1k
3 votes
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forward space vs zero space in finance jargon

In interest rate land you can look at the yield curve in 3 ways: par space (a chart of the par swap rates of different maturities) , zero space (the zero coupon swap rates) and forward space (usually ...
dm63's user avatar
  • 16.9k
3 votes
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Why can a swap option be regarded as a type of Bond option?

Consider a payer swaption with maturity $T_0$ and strike $K$. Here the strike $K$ is the fixed rate paid on the fixed leg of the underlying fixed-for-floating swap with reset dates $T_0, \ldots, T_{n-...
Gordon's user avatar
  • 21.1k
3 votes
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Change of measure between T-forward and T*-forward contract?

By definition $Q^{T_s}$ is risk neutral for the numeraire $P(t,T_s)$, and $Q^{T_e}$ is risk neutral for the numeraire $P(t,T_e)$, hence $$ \left(\frac{dQ^{T_s}}{dQ^{T_e}}\right)_t = \frac{P(t,T_s)}{P(...
Antoine Conze's user avatar
3 votes
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Dynamics of LIBOR foward rate under T-forward measure

We assume that, under the risk-neutral measure $Q$, \begin{align*} dP(t, T) = P(t, T)(r_t + \sigma(t, T)dW_t), \end{align*} where $\{W_t, \, t \ge 0\}$ is a standard Brownian motion. Then \begin{align*...
Gordon's user avatar
  • 21.1k
3 votes

Mid-curve swaption

You are correct. The midcurve swaption expresses the volatility of the forward swap rate , not the "forward volatility". The latter refers to the price of an option whose strike price will be ...
dm63's user avatar
  • 16.9k
3 votes
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Dual Curve Bootstrapping - When to OIS discount?

Modern curve building methodologies, certainly implemented in top tier fixed income trading houses, use a simultaneous non-linear solver to construct all curves at once. Essentially the procedure is: ...
Attack68's user avatar
  • 9,939
3 votes
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Difference between settlement of Eurodollars and FRA

Futures trading are settled on a daily basis meaning in the end of day, your account will be adjusted by your PnL. So of course your payment on T1 is not discounted. However forward is settled only ...
numerairX's user avatar
  • 609
3 votes

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

Recall that the simple forward rate as at time t for lending/borrowing between time T and $T+\tau$ can be written in terms of the discount factors as follows: $F(t,T, T+\tau)= \frac{1}{\tau}\left( \...
Magic is in the chain's user avatar
3 votes

Pricing of compounded swaps

Let us start from your last equation, and focus specifically on the expectation. Assuming that the end date of each period is the start period of the next, the idea is to simplify it using conditional ...
byouness's user avatar
  • 2,210

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