16
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 ...
12
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 ...
- 136
9
votes
Accepted
Calculating instantaneous forward rate from zero-coupon yield curve
Your overall approach is correct. However to my knowledge it is formally more appealing to work with a parameterized and smoothed yield curve.
Basically one assumes that the yield curve can be ...
- 3,337
8
votes
Accepted
Why are multiple custom curves (swap) built for one desk?
Chapter 1: Goldilocks is ousted by the bears
Once upon a time, the banks used a fixing called LIBOR as a measure of the risk-free interest rate. Then the big hairy crisis came along and ate all our ...
- 3,589
6
votes
Accepted
FX Forward rate agreement valuation in quantlib
You are not giving the constructor a discountCurve. The constructor is:
...
- 5,285
5
votes
Accepted
U.S. Rate Hike Prediction
The CME' Fed Fund Futures are what you are looking for.
http://www.cmegroup.com/trading/interest-rates/stir/30-day-federal-fund.html
On settlement day they settle at the average overnight rate set ...
- 739
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:
$$
f(t,u)=-\frac{d}{du}\ln P(t,u)=-\...
- 5,532
5
votes
How to prove martingality of forward rate under T-forward measure
By definition,
$$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)$$
(...
- 137
5
votes
Instantaneous forward rate within the HJM framework
This is known as the classical Leibniz rule. The link sends to Wikipedia, where you can find a proof. It allows to differentiate under the integral sign. A general statement of the formula is:
$$\text{...
- 7,091
5
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,\...
- 7,091
4
votes
Accepted
Practical implementation of Libor Market Model
For a swap, we have a sequence of re-setting and payment dates. The # of forward rates corresponding to the # of payment dates. For example, let us assume that we have $n$ payment dates $t_1, \ldots, ...
- 20.4k
4
votes
Accepted
Formula for the forward rates?
The price of the zero-coupon bond is the discount factor for this maturity.
In the world of exponential compounding formulas are of the form $\exp(\sum \cdots)$.
With a replication argument if we want ...
- 13.2k
4
votes
forward vs spot simply-compounded spot interest rate
The flaw is $L(T,S)$ is a future spot rate that is determined at time $T>t$ and unknown at present.
It is correct that
$$F(t,T,S)=\frac{1}{S-T}\left[\frac{P(t,T)}{P(t,S)}-1\right]
\iff
P(t,S)(S-...
- 3,250
4
votes
Mid-curve swaption
You can only infer forward vol by pairing a mid-curve option with a spot option. It's easier to go through an example (I'll use 5y x 5y vol since I have the sketch below handy...) One decomposition of ...
- 10.9k
4
votes
Accepted
continuously compound forward rate formula
In the simple case, you have as per first equation on your last slide:
$\frac{P(t,T_0)}{P(t,T)}=1+\delta F(t,T_0, T)$
The continuous time equivalent, assuming constant piecewise rate, as per your ...
- 6,044
4
votes
Accepted
Approximation of Forward Rates in discrete time
You may show it as follows:
\begin{align*}
f_{t,T}&= \left[ \frac{(1+r_T)^T}{(1+r_t)^t} \right]^{\frac{1}{T-t}}-1\\
&=e^{\frac{1}{T-t} \left[\ln (1+r_T)^T - \ln (1+r_t)^t \right]} -1\\
&\...
- 20.4k
4
votes
How to calculate one-year forward one-year rate?
Let $\{r_t\}_{t>0}$ be the spotrates and $f_{t,T}$ be the forward rate from time $t$ to $T$ for $t<T$. Then the general formula to compute $f_{t,T}$ is
$$
(1+r_T)^T=(1+r_t)^t(1+f_{t,T})^{T-t}
$$...
- 1,585
4
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 ...
- 6,044
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 $...
- 20.4k
4
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 (...
- 9,229
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 ...
- 9,347
4
votes
Accepted
Pricing a Forward Rate Agreement using QuantLib Python
For a 3x6 FRA, you probably want to write something like:
...
- 5,543
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 ...
- 20.4k
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 ...
- 13.7k
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-...
- 20.4k
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
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 ...
- 13.7k
3
votes
Why is the spot price not used as the forward price when a forward contract is created?
it's easiest to see in terms of replication. The pay-off of a forward contract
is
$$
S_T - K.
$$
We can replicate this precisely and statically by buying one unit of stock, $S_0,$ and $Ke^{-rT}$ ...
- 6,743
3
votes
Accepted
meaning of discount term in FRA value
A very good and up-to-date question.
Whether you use the LIBOR-rate or any other rate for discounting depends on what you decide to be the fundamental rates in the market.
Before the crisis LIBOR-...
- 3,337
3
votes
Derive instantaneous forward rate
You can start with
$$P(t,T)=exp({-\int_t^T f_t(u).du})$$
then take derivative wrt to T
$$R_F(0,T)=f_0(T)=-\frac{\partial} {\partial T}{ ln(P(0,T))} $$
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