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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_{\...


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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 ...


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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^{-\...


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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: $$\text{d}\left(\int_{g(x)}^{h(x)}f(x,s)\text{d}s\right)=h'(x)f(x,h(x))\text{d}x-g'(x)f(x,g(x))\text{d}x+\int_{g(x)}^{h(x)}\text{d}f(x,s)\text{d}s$$


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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 ...


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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 ...


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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+\...


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


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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)...


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It is just an application of the Leibniz integral rule, written in differential form. Please see here: https://en.m.wikipedia.org/wiki/Leibniz_integral_rule Capital T is constant, t is changing, so the second term on the right hand side is the exchange of integral and differential, the first term on the right hand side is the function value at the lower ...


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Since we are dealing with quarterly returns we have to use the returns over one quarter (one period) not the annualized returns that are commonly quoted and that you used in your formula. So the formula is $$ccb/4 =\frac{F}{S}(1+y_f/4)-(1+y_{usd}/4)$$ Now the calculation. If the conventional quotation (i.e. in USDJPY terms) S is 103.00 and the forward ...


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A forward rate is not the same as a forward price. A forward price is the price you need to pay at time $t$ to receive (purchase) an asset at a future date $T$. This forward price can be derived from no-arbitrage arguments and is, in its simplest form, given by $$F_t=S_te^{r(T-t)}.$$ You can of course incorporate coupons (or dividends), accrued interests ...


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A definition We note that $f(t;T)$ is defined as $$ f(t;T) = \lim_{\delta \to 0} f(t;T,t+\delta) \equiv -\frac{1}{Z(t;T)} \frac{\partial}{\partial T}Z(t;T). $$ We know the solution for $Z$ We know that the solution for the ZCB is given by the stochastic/Dolean exponential $$ Z(t;T) = Z(t_0;T)\exp\left(\int_{t_0}^t \left(r(s) - \frac{\sigma^2(s;T)}{2}\...


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Agree with oronimbus. If that's the case just set it for 3mfwd3m, 6mfwd3m, 9mfwd3m etc. Or for 3m libor you can use eurdollar futures but they only go out for a few years. Don't do the classic forward math on the swap curve like you might do for treasury zero rates to get the forward. It's now way more complicated because they're discounted on ois.


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Bloomberg helpdesk should be able to help with this. In any case, you can get the forward rates for both swaps and treasuries. I‘d recommend to look at ICVS > Curve Analysis > Forward Analysis for swaps (this is what their pricers use) and FWCV/FWCM for treasuries. You can get both a full forward curve and a matrix of different term/tenor combinations. ...


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Let's say $ I $ is the Libor index for our underlying swap and $ D $ is our discount curve. If at time $ t $ we have a forecast of all the relevant future Libor fixings $ L_{I}(t, T_{i}, T_{i}+\tau) $ for our swap, where $ \tau $ is the accrual factor for our Libor index and $ T_{i} $ is the time of the $ i $th fixing for our swap, we just need to solve ...


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it requires a model to do it correctly but often i might just do a simple forward math calculation especially if it's not very far forward. So for 1yr fwd 2yr i'd do ((1+yield(3yr))^3 /(1+yield(1yr)^1)^(1/2)-1. It's better to do this with zero coupon bonds but often those yields aren't that different these days anyway.


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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+...


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You won't get a arbitrage-free estimate of 6M curve volatility based on 3m curve. That point is actually a relief -- meaning that if you make reasonable assumptions, it should be fine. The basis is all that matters here along with the correlation between the basis and the 3m curve. It becomes an empirical problem. Having said those, if you have to have ...


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