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

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

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 ...


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

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+\...


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

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

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 ...


2

Because you are keeping the 6m rate constant. Therefore, if the spot 3m rate goes down, the forward must go up.


2

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 determined at a future date.


2

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: $F(t,T, T+\tau)= \frac{1}{\tau}\left( \frac{B(t,T)}{B(t,T+\tau)}-1\right)$ Think of $\tau$ as 6 months or 3 months, and simple forward rate as LIBOR. You can also write it as follows: $F(t,T, T+\tau)...


1

Forward Rate = $\frac {(1+(0.5) 2\%)^{2 * 2}} {(1+(0.5) 1\%)^{2 *1}} -1$ The above works fine when the day count convention is 30/360. General formula - $F(t,t+1,t+2)= \frac {P(t,t+1) - P(t,t+2)} {\tau P(t,t+2)}$ where $F(t,t+1,t+2)$ is the forward rate between $t+1$ and $t+2$, as seen at $t$ $P(t,t+1)$ is the price of zero-coupon bond with maturity $t+...


1

Your expression for the RN derivative is correct indeed $$ \left. \frac{d\Bbb{Q}}{d\Bbb{Q}^{T_1}} \right\vert_{\mathcal{F}_t} = \frac{P(0,T_1)}{P(t,T_1)} \frac{B(t)}{B(0)} $$ Your problem comes the application of the (abstract) Bayes rule. More specifically you should have $$ \Bbb{E}_t^{T_1}[ X_T ] = \frac{ \Bbb{E}_t \left[ X_T \left. \frac{d\Bbb{Q}^T_1}{ d\...


1

You know the zero coupon bond price can be written in terms of spot rate, say y, or forward rates: $B(0,t)=e^{-y(0,t) \, t}=e^{-\int_0^t{f(0,u) \, du}}$ Which means: $y(0,t) \, t=\int_0^t{f(0,u) \, du}$ Differentiating with respect to t, you get your answer: $y(0,t)+ y\prime (0,t) t=f(0,t)$


1

I think one way to approach the answer is thinking what are these two rates used for. Starting with zero coupon rates, it's aiming for getting the par value back at maturity (similar to a bank's loan, where in the end payments are all up). For forward rates however, is calculated under the risk neutral measure and is mostly used for option pricing in fixed ...


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