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 ...
- 1,242
13
votes
Accepted
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 ...
- 146
6
votes
Accepted
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 (...
- 11.9k
6
votes
Accepted
FX Forward rate agreement valuation in quantlib
You are not giving the constructor a discountCurve. The constructor is:
...
- 5,685
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 ...
- 11.4k
5
votes
Accepted
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:
$$
f(t,u)=-\frac{d}{du}\ln P(t,u)=-\...
- 5,622
5
votes
Accepted
How to prove martingality of forward rate under T-forward measure
For the instantaneous forward, please see the last page of this note: T-Forward Measure by Fabrice Douglas Rouah (http://www.frouah.com/finance%20notes/The%20T-Forward%20Measure.pdf).
For the simple ...
- 6,204
5
votes
How to prove martingality of forward rate under T-forward measure
By definition,
$$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)$$
(...
- 146
5
votes
Instantaneous forward rate within the HJM framework
This is known as the classical Leibniz rule. The link sends to Wikipedia, where you can find a proof. It allows to differentiate under the integral sign. A general statement of the formula is:
$$\text{...
- 7,979
4
votes
Accepted
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 ...
- 6,204
4
votes
Accepted
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\\
&\...
- 21k
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
$$
(1+r_T)^T=(1+r_t)^t(1+f_{t,T})^{T-t}
$$...
- 1,627
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 $...
- 21k
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 ...
- 5,998
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 ...
- 10.9k
4
votes
Accepted
Pricing a Forward Rate Agreement using QuantLib Python
For a 3x6 FRA, you probably want to write something like:
...
- 6,948
4
votes
Accepted
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,\...
- 7,979
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"), ...
- 8,143
3
votes
Accepted
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(...
- 5,622
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 ...
- 21k
3
votes
Accepted
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 ...
- 16.5k
3
votes
Accepted
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-...
- 21k
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 ...
- 121
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 ...
- 722
3
votes
Why is the forward rate used for the underlying in Black's model?
As a trader I used Black model (amongst others) to value swaptions, where the forward swap rate is the key observable underlying rate. Any market where the forward is the traded instrument would ...
- 16.5k
3
votes
Accepted
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*...
- 21k
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 ...
- 16.5k
3
votes
Accepted
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:
...
- 9,195
3
votes
Accepted
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 ...
- 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( \...
- 6,204
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