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7

$$\frac{1}{(1+r_{02})^2} = E\left(\frac{1}{1+r_{12}}\right)\frac{1}{1+r_{01}}$$ Indeed, in the pricing measure, the distribution of $r_{12}$ has to be such that this relation holds. If you look at drift derivations for the LIBOR market model, a lot of work goes into making this sort of equation hold.

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Completely depends on the asset class. For currencies (including GBP/USD) the spot market is an order of magnitude more liquid than forwards, futures or options. However, some currencies with trading restrictions have a non-deliverable forward contract which can be much more liquid than the spot market for offshore investors (e.g. INR, KRW, TWD). ...

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As you noted, this is a Riccati type ODE and it can thus be simplified using the standard transformations for this class - see e.g. Wikipedia. We start by defining \begin{equation} C(t, T) = \frac{1}{2} \alpha B(t, T) \qquad \Rightarrow \qquad C_t(t, T) = \frac{1}{2} \alpha B_t(t, t) \end{equation} and get \begin{eqnarray} C_t(t, T) & = & C^2(t, T)...

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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 There is no conflict here. In the identity, \begin{align*} \frac{1}{(1+r_{02})^2} = E\left(\frac{1}{1+r_{12}}\right)\frac{1}{1+r_{01}}, \end{align*} the expectation is under the year-1 forward measure. However, in the identity \begin{align*} (1+r_{01})E(1+r_{12})=(1+r_{02})^2, \end{align*} the expectation is under the year-2 forward measure. For ... 4 The paper is generally correct, but it is not a general statement, as in a general truth of options hedging in a theoretical context, rather a statement regarding how the structured derivs market is typically set up: retail and institutional investors buy a large number of products that at their core entail the dealer buying (from the investor) long-dated (... 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 ... 3 To add to the above on a more practical note: In general, SP desks make money on the individual product when the underlying declines. Dividends make the underlying decline, hence they are naturally long dividends. Take an auto-callable product which is exercised if the spot is above a pre-determined strike each year and say the SP desk sells this ... 3 First question can't be answered without knowing what you are discounting. Second question you are asking whether the rate of a 16 day interbank loan can be obtained from interpolating an overnight rate and a 1 month bank loan. I would say it is a reasonable guess. 3 It is very difficult to outperform the "random walk without drift" benchmark. The forward rate is not a particularly good predictor as it is often biased. Nevertheless some economists claim it is possible. Here is a literature review (Barbara Rossi: Exchange Rate Predictability, Journal of Economic Literature vol. 51, no. 4, December 2013): https://... 3 You can create the data using the procedure described in the reference manual on pages 31 and 32. The necessary code is copied below: # The following code may be used to generate an empty data set, # which can then be filled with bond data: ISIN <- vector() MATURITYDATE <- vector() STARTDATE <- vector() COUPONRATE <- vector() PRICE <- vector()... 2 You are going to need to interpolate in some way shape or form.... Linear is the easiest and most basic, however it may not capture the curvature, you can use splines to better capture the curve. A nice guide to doing so is here: It's a guide to bootstrapping and it has all the components. http://www.business.mcmaster.ca/finance/deavesr/yieldcur.pdf 2 You could compare 3 month fx forward points versus realised 3 month fx differentials to see if interest rate differentials are a good predictor. I looked at a 1 year horizon and concluded that you cannot ignore the interest rate differential, but on the other hand the best prediction might lie between zero drift and the fx forward predicted rate. 2 The reason for the bid and ask twisting is that you can think of a long AUD forward as three transactions: Borrow USD Sell USD, buy AUD spot Lend AUD As a result, there are three sources of bid/offer cost for a forward. In contrast, for an interest rate, it's just one transaction (borrow or lend). This is why they twist those equations. They are trying to ... 1 It seems you are using the same curve for forward and discounting. The EUR Vanilla Swaps vs 6M actually have yearly payments, so to obtain the discount factors, and after having the DF for year 1, you can sequentially solve for them just using the par swap Rates.DF_n = \frac{1-par_n \times \sum^{n-1}_{i=1} DF_i}{1+par_n}So the DF for year 2 would be:... 1 Yes you can access spot rates on the Treasury website here: https://www.treasury.gov/resource-center/economic-policy/corp-bond-yield/Pages/TNC-YC.aspx https://www.treasury.gov/resource-center/data-chart-center/interest-rates/pages/textview.aspx?data=yield 1 The question really is what is the discount factor for a payment in one year assuming semiannual compounding? Because then your present value is simply 1.75 times this discount factor. If you have k periods of compounding, a payment of \1 in n years worth today \begin{align*} \frac{1}{\left(1+\frac{r}{k}\right)^{n\cdot k}}, \end{align*} where r is ... 1 If you want to calculate the forward rate given semi-annual compounding then the answer should be: \begin{equation} 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 \end{equation} This is derived by the fact that : \begin{equation} \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 t P(t,t+1) is the price of zero-coupon bond with maturity t+... 1 When you want to consider arbitrary (i.e. non parallel) movements of the yield curve, the duration ( a scalar) is replaced by a vector of 'key rate durations' one for each maturity you wish to consider. investopedia.com/terms/k/keyrateduration.asp 1 Intuitively, this is the "coupon effect" at work – when the yield curve is upward sloping, lower coupon bonds have higher yield and their yields move up more when the overall curve shifts up (all else equal). The opposite is true when the yield curve is downward sloping. We'll focus on when the curve is upward sloping below. I think it's probably best to ... 1 Let's go for a detailed and rigorous proof. Let us define our local currency Y as the numéraire, i.e. the asset in terms of whose price the relative prices of all other tradeables are expressed. X is therefore the foreign currency, whose price in terms of Y is X(t) at any time t. Let r_X(t,T) be the risk-free interest rate in currency X and ... 1 Just to add a remark on top of Ivan’s excellent answer, note that the core reason IB package those KI puts in the autocallables and other SP for investors is not just in order to make the coupons more attractive to the investors but fundamentally to buy back the volatility skew that the vanilla desk is structurally seller of. 1 The Milstein scheme for the following CIR modeldr_t=(\eta-\gamma r_t)dt+\sqrt{\alpha r_t} dW_t$$should be$$r_{t+1}=r_t+(\eta-\gamma r_t)\delta t+\sqrt{\alpha r_t}\cdot\sqrt{\delta t}\phi +\frac{1}{2}\sqrt{\alpha r_t}\cdot\left(\frac{1}{2}\sqrt{\frac{\alpha}{r_t}}\right)[\delta t(\phi^2-1)]r_{t+1}=r_t+(\eta-\gamma r_t)\delta t+\sqrt{\alpha r_t}\...

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Yes. The map $R(\cdot;S,T):\mathbb{R}^{2}\to\mathbb{R}$ completely describes the forward rate/spot rate term interest rate structure for each $t\geq0$. (You can think of it as the market interest rate surface for the rate $R$ at time $t$). The notation $R(t;S,T)$ is meant to remind you that $R$ is a stochastic process for $t>0$, the periods of time ...

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I am note $100\%$ sure that I understand the question. But yes. More formally one could write $R(t,S,T)$ for the rate from $S$ to $T$ observed at $t$ and $R(t,t,T)$ for the spot.

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If you're bootstrapping and if there are bonds maturing on the same date, you should use only one. A good rule is to discard the older issue and keep the more recently issued securities. If you're building a spline, then it really doesn't matter since you're building a best fit curve that best approximates the prices of all bonds. Assuming the quotes you ...

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I'm not expert on this field so may not able to answer your question precisely, but I can try the best to offer you some hints. According to the pure expectations hypothesis(PEH), forward rates provide unbiased predictions about future spot rates. Even if the PEH can be rejected, various scholars including Fama has provided evidence for the weaker form of ...

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Assume we have $r(t)$ continuously compounded spot rate for maturity $t$. The price of the 2-year bond with semi-annual coupon $C$ is known to be $P$. We already have $r(0.5)$ and $r(1)$. We need $r(2)$ and $r(1.5) = f(r(1), r(2))$. Then $$P = C [e^{-0.5 \times r(0.5)} + e^{-r(1)}+e^{-1.5 \times r(1.5)}] + (1+C)e^{-2 \times r(2)}$$ Using linear ...

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It turned out to be more simple than I thought. First, be sure to replace "STARTDATE" with "ISSUEDATE" when building the list. Once the list is build simply reclassify it using the following command: class(mybonds)="couponbonds" That's it!

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