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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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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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The risk-neutral probability measure is defined in terms of its numeraire. For the usual risk-neutral probability measure the numeraire is the bank account, $\beta(t)$. If we have a tradeable asset $X(t)$ then $\tilde{X}(t)=\frac{X(t)}{\beta(t)}$ is a martingale under $\mathbb{Q}$ meaning that $$\tilde{X}(t)=\mathbb{E}_{t}^{\mathbb{Q}}(\tilde{X}(T))$$ So we ...


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You do not need zero rates to estimate a parametric model of the yield curve, such as Nelson-Siegel. Suppose for instance that you have a cross-section of bond prices. Then: For given parameters for your yield-curve model, compute yield curve; with this yield curve, calculate theoretical bond prices; compute discrepancy between theoretical bond prices and ...


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A forward rate agreement is an agreement to exchange a fixed for a floating rate over one period, with the payment being made at the start of the period. A zero coupon swap (with both legs paid at maturity) is an agreement to exchange a fixed for floating rate over one or more periods, with the payments being made at the end of the final period. So the two ...


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(Assuming that the two coupon bonds have exactly the same schedules, and that you're settling when the accrueds are 0.) Consider a portfolio consisting of \$7 long 3% bond and $3 short 7% bond. This portfolio costs 7 * 89 - 3 * 97 = 332. Every time you receive a 7 * 3% coupon from the 3% bond position, you pay out the same 3 * 7% amount for the 7% bond ...


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I notice you mention GBP. This effect is particularly apparent there since a large number of insurance, pension and asset management companies like to trade ZCS. They do this because the forward risk profile of a ZCS more accurately reflects the increasing notional of their portfolio and avoids them having to deal with interim coupon payments. They almost ...


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The value of the bond would be the first case, because you have to discount each cashflow with the relevant spot rate for that payment date. Although, because rates are normally expressed in annual terms, you would have to adjust for the days: $(1+R)^{n}$ or $(1 + R \times n)$ What you might be confused with, is the yield of the bond, which would be the ...


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In general futures contracts are leverage instruments. They never require the investment of principal. They do however require margin: you need to fund your account at a futures exchange so that they have insurance against any losses you incur, as an example this might be 2 days standard volatility. On 1 ED contract for 5bps a day thats probably 10bps margin ...


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(1) corresponds to simple interest and (2) to compound interest. For instance, Canadian treasury bills are based on simple interest (see Broverman's book Mathematics of investment and credit).


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In case you're still looking: https://data.snb.ch/en/topics/ziredev#!/cube/rendoblid


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By definition of the $T$-forward measure $P_T$, the process $\Big\{\frac{P(t,S)}{P(t,T)} \mid t\geq 0\Big\}$ is a martingale under the measure $P_T$, without assuming any specific models of the short rate $r_t$. That is, this martingale property is model independent. However, as a good exercise, you can also do the following: Given the CIR interest rate ...


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In part (a) use discount rate $e^.07 -1 = .072508181$ to get the right answer. For part (b) I am just giving you hint: Calculate bond price at the end of 1st year and 2nd year in the same way as you did in part (a). Use the above calculated price to buy bond from the dividend at the end of first and second year. You may assume bond can be purchased in ...


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Under the Hull-White interest rate model, the short rate $r_t$ satisfies a risk-neutral SDE of the form \begin{align*} dr_t = (\theta(t)-a r_t)dt+ \sigma dW_t. \end{align*} The price at time $t$ of a zero-coupon bond with maturity $T$ and unit face value is then given by \begin{align*} P(t, T) &= A(t, T) e^{-B(t, T) r_t}, \end{align*} where \begin{...


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Who knows what the 5 year zero coupon rate is in that case, there could be an event 4.5 years out that will have serious interest rate implications that we don't know about. The only thing you can do with these three numbers is extrapolate and say the rate should 9%. You should be aware of what assumptions you're making when you do something like that, but I'...


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Here is a general proof for all parameters in an open domain. $$dr = adt+bdW:=r\big(k(\theta-x)+\frac12\sigma^2\big)dt+\sigma rdW.$$ Let $$u(r(s),s):=e^{-\int_t^sr}B(r(s),s,T)=:\phi(s) B.$$ Then $$u(r(t),t)=\mathbf E\big[u(r(s),s)\big|r(t)\big],\, \forall t<s. \tag{1}$$ So, by Ito's Lemma, \begin{align} du(r(s),s) &= Bd\phi +\phi dB \\ &= \phi \...


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The CMT yields published by the Fed/US Treasury are par yields calculated using a cubic spline model. In other words, these are the yields to maturity as well as coupon rates on bonds whose theoretic prices are 100. With this information in mind, you can linearly interpolate between these yields, or use a cubic spline to fill in rates at other tenors, ...


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Yes. The time p/l can be found by leaving all the inputs the same and allowing a day to pass. I prefer not to call it theta - that term is used to describe the time decay of options.


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Let $P(t,T)$ denote the time $t$ price of a zero-coupon bond (with unit face value) maturing at time $T$. Firstly, recall that for every $s\leq t$, we have \begin{align*} r_t = r_s e^{-\kappa(t-s)}+\theta\left(1-e^{-\kappa(t-s)}\right)+\sigma \int_s^t e^{-\kappa(t-u)}\mathrm{d}W_u. \end{align*} Thus, the short rate $(r_t)$ is normally distributed for every ...


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Maybe you should start with a simple example, because you have so many moving parts that it's hard to figure out where the difference is. Most likely some different convention between your helpers and the instruments you are trying to price. import QuantLib as ql today = ql.Date().todaysDate() calendar = ql.TARGET() spot = calendar.advance(today, 2, ql....


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Just adding the valuation date/time to the bonds identifier will make it clearer. See below:


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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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The website below shows how to price bonds from curves, currently it only supports fixed rate and zero-coupon bonds, but it might give you an idea how to price a floater using similar concept: Goto: https://www.opencminc.com Switch to Yield Curves under the Market Data section Click on any curve point. For example: click on a rate under 10Y https://www....


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The delta is (value of bond) - (value of bond if rates go up 1bp) =5mm/(1.0210)^5 - 5mm/(1.0211)^5 =$2206


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Your answer is correct. You included .5 in the exponent and therefore got an annualized result. 6.118% divided by 2 is your bootstrapped 6 month spot rate.


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You think you make a mistake where you actually don´t make one. The exercise is just like it is. Resulting in $$r_{6m}>r_{12m}$$ The difference in your both answers, based on the same rounding, lays in the different basis for the logarithm. $$r_{6m} = - 2 \log_e \left( \frac{99.8-102.5 e^{-r_{1y}} }{2.5} \right) = \textbf{6.118%}$$ $$r_{6m} = - 2 \log_{...


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If you have a negative yield which cannot be ruled out nowadays the issueprice will be above par.


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There is only one cashflow for the zero-coupon bond. At maturity, it pays the par value. They are always issued below par, as the buyer is paying the NPV for the bond that matures in the future. Here is a brief reference at Investopedia. A zero-coupon bond, also known as an "accrual bond," is a debt security that doesn't pay interest (a coupon) but is ...


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