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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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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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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: $\frac{P \left(t,T_{n}\right)}{P \left(t,T_{n+1}\right) }=1+\... 3 Assume the (annualised, continuously compounded) forward rate between two nodes, say$t_{10}$and$t_{12}$, is constant, say$ f_{10,12}$, then the discount factors of the two consecutive knots will be linked as follows:$D_{12}=D_{10}e^{-f_{10,12} \left(t_{12}-t_{10}\right)}=D_{10}e^{-2f_{10,12}}$From which is then easy to infer the formula for$t_{11}$, ... 3 I will try a simplified approach: Let$P(t,T)$represent the price at time t of a zero coupon that pays 1 at time T. If you divide the period between t and T into n sub-intervals, assume$F \left( t; t_{i-1}, t_{i}\right)$represent the simple forward rate at time t for the interval between$i-1$and$i$, where we assume the length of each interval is equal ... 3 Let us start from your last equation, and focus specifically on the expectation. Assuming that the end date of each period is the start period of the next, the idea is to simplify it using conditional expectations. Since$t < t_{n-2}, we can write using the tower property of conditional expectations: \begin{aligned} \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{... 3 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 ... 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)... 2 The first equation is the result of the effort to show that the product of the forward and the relevant zero coupon, F \left(t,T_n \right)P \left(t,T_{n+1}\right), can be treated as a traded asset. And once you have established that it is the price of a traded asset, then you can write its price using the valuation formula, which in way holds for all ... 2 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 ... 1 The forward rate is estimated at the Libor valuation date = reset date + the reset gap (namely 1, 2 or 3 days) (not the accrual start date). In the case of standard vanilla swaps, the accrual start date is equal to the reset date + the reset gap: this I believe is the case that Davide Duarte talks about. However, for the general case, the accrual date could ... 1 Actually it is not just the long end of the swap curve it is any part of the curve that needs some form of basis swaps to be calibrated. A set of curves in any currency usually encompasses the following: { OIS curve, 1M IBOR curve, 3M Ibor curve, 6M Ibor curve } at a minimum. It is not practical for interbank markets to trade completely bespoke products so ... 1 The measure \mathbb{Q} is associated to the money market account t \mapsto \beta_t = \exp \int_{0}^{t} r_s d s . The measure \mathbb{Q}^T is associated to the zero coupon bond t \mapsto P_{tT} where P_{tT}:=\mathbb{E}^{\mathbb{Q}}_t \left[ \frac{\beta_t}{\beta_T} \right]. We know that the Radon-Nikodym derivative between \mathbb{Q}^{T_1} and \... 1 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 isccb/4 =\frac{F}{S}(1+y_f/4)-(1+y_{usd}/4)$$Now the calculation. If the conventional quotation (i.e. in USDJPY terms) S is 103.00 and the forward ... 1 A forward rate is not the same as a forward price. A forward price is the price you need to pay at time t to receive (purchase) an asset at a future date T. This forward price can be derived from no-arbitrage arguments and is, in its simplest form, given by$$F_t=S_te^{r(T-t)}.$$You can of course incorporate coupons (or dividends), accrued interests ... 1 A definition We note that f(t;T) is defined as$$ 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}\... 1 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. 1 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. ... 1 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 ... 1 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 ... 1 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. 1 If you want to calculate the forward rate given semi-annual compounding then the answer should be: $$F(0,t_a,t_b)=\Bigg(\sqrt[2*(t_b-t_a)]{\frac{(1 + \frac{r_b}{2})^{2*t_b}}{(1 + \frac{r_a}{2})^{2*t_a}}}-1\Bigg)*2$$ This is derived by the fact that : \Bigg(1+\frac{r_b}{2}\Bigg)^{2*t_b} = \Bigg(1+\frac{r_a}{2}\... 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$tP(t,t+1)$is the price of zero-coupon bond with maturity$t+...

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