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$$\frac{1}{(1+r_{02})^2} = E\left(\frac{1}{1+r_{12}}\right)\frac{1}{1+r_{01}}$$ 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.

5

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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I see several problems that might explain those differences: The frequency of the fixed leg on a EONIA swap is Annual and not semi The deposit facility rate is not part of the EONIA curve. Use the Eonia rate. You are calculating rates with simple compounding and not annual compounding Here is an alternative implementation: tenors = [ '1D', '1W', '2W', '...

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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 (...

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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 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 ... 4 1. Theory The Student t distribution does not exhibit a moment generating function M_X(t)=\mathbb{E}\left(e^{tX} \right) $$Hence, there exist no closed form solution for M_X(t=1)=\mathbb{E}\left(e^X\right), i.e. the expected future spot price. Thus, at least theoretically, we are not able to pinpoint the expectation of the future asset value, thereby ... 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://... 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 ... 2 Hopefully clear from the table below. On the left, the NPV of the bond today is 111.2199, exactly as you say. On the right is the same for 12 months time, after the payment of the first coupon. 1 As others have mentioned, the conventional answer is 'no'. However, here is one interesting angle of the new cryptocurrency trading market to me - it operates 24/7/365 (yes even Christmas day!). Now I'm told that the biggest markets are the US and Korea, so perhaps, given sufficient liquidity, we could back out a USDKRW rate over the weekend given the USD ... 1 Well it depends on what pairs you want data for. But in general, no. Even if you can see intraday data in those 'dead' times, the quotes can be very stale (hours old). 1 The confusion arises because “spot” is variousLy used in FX, equities and commodities to refer to the immediate/very-short-term price (before any forward adjustments for interest rates, dividends, contract rolls etc). The same is sort-of the same with interest rates. If the spot 2 year rate is say 10% for a zero-coupon loan, then you would receive 1.21x ... 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 Since SD in this case is usually the 1-day difference of log prices (i.e. 1-day returns) and corr is a dimensionless number, you shouldn't have to keep the units the same. After all that's how you're able to hedge a position using a different commodity that you have access to, for example jet fuel. 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}\...

1

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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