14
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 fixings.
1.1. Observed spot rates
For simplicity, assume Zero Coupon Bonds (ZCBs) are traded with time left to maturity of 10Y, 15Y and 20Y. Hence, by observing ...
11
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 forward rate and zero coupon bond is:
$F(t,T,T+\delta) = \frac{p(t,T) - p(t,T+\delta)}{\delta p(t,T+\delta)}$
We have,
\begin{equation}
f(t,T) = \lim_{\...
9
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 described by a smooth function $r(t,\alpha, \beta,\gamma)$ (mostly of three parameters)
Given a set of market data $Y(t,T_1)\dots Y(t, T_n)$ one looks for ...
8
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 assumptions, leaving just the bones of the fixing (upon which everything else still fixes) and the mantle of risk-free rate proxy was passed on to a family of ...
6
You are not giving the constructor a discountCurve. The constructor is:
ql.ForwardRateAgreement(valueDate, maturityDate, position, strikeForward, notional, iborIndex, discountCurve=ql.YieldTermStructureHandle())
So you should add a the spotCurveHandle as the last parameter:
fra = ql.ForwardRateAgreement(ql.Date(7, 5, 2018), ql.Date(15,12,2020), ql.Position....
5
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 by the Fed during the contract month.
5
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:
$$
f(t,u)=-\frac{d}{du}\ln P(t,u)=-\frac{\frac{d}{du}P(t,u)}{P(t,u)}\\
=-\frac{\frac{d}{du}E^Q_t[e^{-\int_t^u r_s ds}]}{P(t,u)}
=\frac{E^Q_t[e^{-\int_t^u r_s ds} r_u]}{P(t,u)}
=E^Q_t\left[\frac{e^{-\...
5
By definition,
$$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)$$
(using the tower property of expectations). Hence Forwards rate is a martingale under the T-forward measure.
5
This is known as the classical Leibniz rule. The link sends to Wikipedia, where you can find a proof. It allows to differentiate under the integral sign. A general statement of the formula is:
$$\text{d}\left(\int_{g(x)}^{h(x)}f(x,s)\text{d}s\right)=h'(x)f(x,h(x))\text{d}x-g'(x)f(x,g(x))\text{d}x+\int_{g(x)}^{h(x)}\text{d}f(x,s)\text{d}s$$
4
For a swap, we have a sequence of re-setting and payment dates. The # of forward rates corresponding to the # of payment dates. For example, let us assume that we have $n$ payment dates $t_1, \ldots, t_n$, where $0< t_1 < \cdots < t_n$. Then there are $n$ forward rates.
During the simulation, for time steps prior to $t_1$, there exist $n$ "...
4
The price of the zero-coupon bond is the discount factor for this maturity.
In the world of exponential compounding formulas are of the form $\exp(\sum \cdots)$.
With a replication argument if we want to invest money for $n$ years what can we do.
We invest for one year $r_0 = F(0,1)$ then after this year we invest for another year, the rate for this today is ...
4
The flaw is $L(T,S)$ is a future spot rate that is determined at time $T>t$ and unknown at present.
It is correct that
$$F(t,T,S)=\frac{1}{S-T}\left[\frac{P(t,T)}{P(t,S)}-1\right]
\iff
P(t,S)(S-T)F(t,T,S) = P(t,T) - P(t,S),
$$
as this is just the definition of the forward rate.
However, you are saying that
$$\frac1{P(t,T)}\frac1{P(T,S)}=\frac1{P(t,...
4
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 the 5y5y spot vol is as follows:
1y forward 4y x 5y vol: this is the implied vol of an option starting in 1 year, expiring 4 years thereafter, and eventually ...
4
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\\
&\approx e^{\frac{1}{T-t} \left[(1+r_T)^T-1 - \big((1+r_t)^t -1\big)\right]} -1\\
&=e^{\frac{1}{T-t} \left[(1+r_T)^T - (1+r_t)^t\right]} -1\\
&\approx 1+ \...
4
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
$$
(1+r_T)^T=(1+r_t)^t(1+f_{t,T})^{T-t}
$$
Now you can solve for $f_{t,T}$ to obtain:
$f_{t,T}= \left( \frac{(1+r_T)^T}{(1+r_t)^t} \right) ^{1/(T-t)}-1$
In your example: Spot rates are given by the ...
4
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 forward, you know the relationship between the price of the zero coupon and the simple forward:
$ \frac{P \left(t,T_{n}\right)}{P \left(t,T_{n+1}\right) }=1+\...
4
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 $Q^T$ be the risk-neutral and the $T$-forward respective probability measures. Then, for $0\le t \le T$,
\begin{align*}
\frac{dQ}{dQ^T}\big|_{[t, T]} = \frac{...
4
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 (that your trade isn't large enough to impact the market);
there are no other fees or costs.
For example:
Bid-Ask Spreads: Measuring Trade Execution Costs in ...
4
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 terms of expectation (perhaps because that is how I learned it myself, although I am not sure). "We see in this example that the market expects future Libor ...
4
For a 3x6 FRA, you probably want to write something like:
today = ql.Date(30, 6, 2020)
ql.Settings.instance().evaluationDate = today
startDate = calendar.advance(today, ql.Period('3M'))
maturityDate = calendar.advance(startDate, ql.Period('3M'))
That is, the start and maturity dates you pass to the FRA constructor should be the underlying period of the ...
3
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 price at $t_f$ of a domestic zero-coupon bond with maturity $T$. Note that $P_d(t_f, T) X_{t_f}$ is the value at time $t_f$ of the process
\begin{align*}
P_d(t,...
3
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 the 3 month forward rates for various maturities). These are equivalent ways to display the prevailing market rates. Perhaps that is what is being referred to
3
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 $T_0, \ldots, T_{n-1}$ and payment dates
$T_1, \ldots, T_n$, where $0<T_0 < \cdots < T_n$. We assume that the swap exchanges the payments $L(T_{i-1}; T_{i-1}, T_i)\Delta ...
3
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 another fixed payment 18m
after effective date
regular quarterly floating payments
Your curve was built with 1y and 2y swaps, nothing in the middle ? Then yes, ...
3
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 lend itself to Black.
3
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 $Ke^{-rT}$ riskless bonds growing at rate $r.$
So its value today is
$$
S_0 - Ke^{-rT}.
$$
This has zero value if and only if $K= S_0 e^{rT}.$
This value is then called ...
3
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-rates were mostly seen as the fundamental market rates (or the "risk-neutral" rates). After the crisis it turned out that these rates were not completely free of ...
3
You can start with
$$P(t,T)=exp({-\int_t^T f_t(u).du})$$
then take derivative wrt to T
$$R_F(0,T)=f_0(T)=-\frac{\partial} {\partial T}{ ln(P(0,T))} $$
3
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
$$
\left(\frac{dQ^{T_s}}{dQ^{T_e}}\right)_t = \frac{P(t,T_s)}{P(t,T_e)} \frac{P(t_0,T_e)}{P(t_0,T_s)}
$$
In the specific case that you are looking at you are computing the forward in-arrears fixing Libor (in arrears because ...
3
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*}
dL(t) &= \frac{1}{T-S}\bigg(\frac{dP(t, S)}{P(t, T)} -\frac{dP(t, S)}{P^2(t, T)}dP(t, T) \\
&\qquad + \frac{dP(t, S)}{P^3(t, T)} \langle dP(t, T), \,...
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