17
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 ...
- 1,206
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
6
votes
Accepted
FX Forward rate agreement valuation in quantlib
You are not giving the constructor a discountCurve. The constructor is:
...
- 5,375
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
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
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:
$$
f(t,u)=-\frac{d}{du}\ln P(t,u)=-\...
- 5,552
5
votes
How to prove martingality of forward rate under T-forward measure
By definition,
$$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)$$
(...
- 137
5
votes
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Half of the bid-ask spread as transaction cost
The idea of assuming that the transaction cost is one half of the bid-offer spread comes from several assumptions:
the positions are marked-to-market at mid;
you can actually execute at bid or ask (...
- 9,963
5
votes
Instantaneous forward rate within the HJM framework
This is known as the classical Leibniz rule. The link sends to Wikipedia, where you can find a proof. It allows to differentiate under the integral sign. A general statement of the formula is:
$$\text{...
- 7,301
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,\...
- 7,301
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, ...
- 20.5k
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 ...
- 13.3k
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 ...
- 6,074
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\\
&\...
- 20.5k
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 ...
- 6,074
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 $...
- 20.5k
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 ...
- 9,647
4
votes
Accepted
Pricing a Forward Rate Agreement using QuantLib Python
For a 3x6 FRA, you probably want to write something like:
...
- 5,743
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 ...
- 20.5k
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 ...
- 14.3k
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-...
- 20.5k
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 ...
- 14.3k
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 $Ke^{-rT}$ ...
- 6,763
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(...
- 5,552
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*...
- 20.5k
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 ...
- 14.3k
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:
...
- 8,099
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
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