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 ...
6
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_{\...
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
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 Overnight fixings, called Sonia, Eonia and -ahem-...
5
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 ...
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:
$$
f(t,u)=-\frac{d}{du}\ln P(t,u)=-\frac{\frac{d}{du}P(t,u)}{P(t,u)}\\
=-\frac{\frac{d}{du}E^Q_t[e^{-\int_t^u r_s ds}]}{P(t,u)}
=\frac{E^Q_t[e^{-\int_t^u r_s ds} r_u]}{P(t,u)}
=E^Q_t\left[\frac{e^{-\...
4
The price of the zero-coupon bond is the discount factor for this maturity.
In the world of exponential compounding formulas are of the form $\exp(\sum \cdots)$.
With a replication argument if we want to invest money for $n$ years what can we do.
We invest for one year $r_0 = F(0,1)$ then after this year we invest for another year, the rate for this today is ...
4
The flaw is $L(T,S)$ is a future spot rate that is determined at time $T>t$ and unknown at present.
It is correct that
$$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 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 ...
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
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
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
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
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 $n$ "...
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), \,...
3
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 ...
3
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 once at expiration, hence you discount the whole duration.
3
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 question, is:
$\frac{P(t,T_0)}{P(t,T)}=e^{y (T_0,T) \delta}$
Taking log of both sides, and rearranging:
$\frac{1}{\delta} \ln {\frac{P(t,T_0)}{P(t,T)}}=y (T_0,...
3
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+\...
2
The FRA
A FRA is an agreement to exchange cash flows; the FRA in question is:
Start 15/9/14
End 15/5/15
which is 242 days. USD Money Market quoting is Actual/360, so the accrual factor here is 242/360 = 0.6722.
The FRA cashflows, therefore, are: on 15/9/14, Fix pays $\$1m * (0.6722 * 0.05) = \$33,611.11$, and Float pays $\$1m * (0.6722 * L)$, where L ...
2
The forward price $F$ for a forward contract, determined at the contract inception time today, is the price that the holder will pay at maturity $T$ to buy the underlying equity. Then the payoff, at maturity $T$, of the forward contract is given by
\begin{align*}
S_T-F.
\end{align*}
The present value of the contract is then
\begin{align*}
e^{-rT} \mathbb{E}\...
2
When you buy a forward you don't have to invest any money, so that's to your advantage in a world of positive interest rates. To charge you the same as the spot rate would be unfair, you would be "getting something for nothing", that is why the appropriate price for a forward is higher. It takes the interest rate into account, balancing things out. In ...
2
If you want to discount the CF3 from 3 years in the future to today you should use (1 + 3yr spot rate)^3. There's no reason to use forward rates for that purpose. The forward rates should only be used for period-by-period discounting - for example, if you wanted to find the value after 3 years of a CF4 which occurs after 4 years, you would use (1+ 1yr ...
2
The IRR cannot be used to move the Present Value to a Future Value, because it doesn't represent the rate for that interval of time. It is a complicated average of rates for 1,2,3 years, not the rate for 3 years which you require.
2
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
2
The question is asking if there is a way to create arbitrage by borrowing in one currency, exchanging at the current spot rate, lending in another currency and converting the future payments back to the original currency at the forward exchange rate.
Specifically, given the assumption above, if there were no arbitrage the inequality above could not hold. ...
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