# Tag Info

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$\mathbb{P}$ is the true probability measure. Measure $\mathbb{Q}$ is a measure of convenience that allows risk neutral pricing. Stochastic discount factor $M$ takes you between the two. If you care about prices you can either: (1) work under $\mathbb{Q}$ or (2) work under $\mathbb{P}$ with a stochastic discount factor $M$. There's an isomorphic ...

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There are two parts to this question: 1) Is OIS a good risk-free proxy? and 2) Why is OIS used to discount cash flows of derivatives. First, overnight indexed swaps, in the US, are indexed to the Fed funds effective rate, which in turn tracks the Fed funds target rate. The Fed Fund target rate is directly set by the Federal Reserve, while the Fed Funds ...

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Collateral posted in currency XYZ is remunerated at $\text{OIS}_{\text{XYZ}}$, which translates, using the XYZUSD basis, into a synthetic USD rate $\text{OIS}_{\text{USD}}^{\text{XYZ}} = \text{OIS}_{\text{USD}} + \text{basis}_{\text{XYZUSD}}$. If you post collateral you want to choose the currency XYZ that has the highest equivalent synthetic USD rate, and ...

8

Chapter 1: Goldilocks is ousted by the bears Once upon a time, the banks used a fixing called LIBOR as a measure of the risk-free interest rate. Then the big hairy crisis came along and ate all our assumptions, leaving just the bones of the fixing (upon which everything else still fixes) and the mantle of risk-free rate proxy was passed on to a family of ...

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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If they were a bank, or insurer, utility etc, then some regulator would likely encourage them do everything that others do, whether they like it or not, or whether it makes any sense. But if no regulator tells them to... and if if they don't have exposure to interest rates beyond 1 year... well, let's look at USD 1 year swap rate, for example, https://fred....

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One derivation is to replace $V_u$ in Equation $(5)$ using the expression given by Equation $(3)$ and then work out to reach $(5)$; see Appendix A in this paper for more details. Here, we provide another derivation. See also this question. We recall that, from $(2)$ of Piterbarg, \begin{align*} V_t = \Delta (t) S(t) + \gamma(t), \end{align*} where \Delta (... 6 Better yet, don't use LIBOR for discounting at all. Since LIBOR involves credit spread over the risk free rate, using LIBOR for discounting would adjust the deal's market value to reflect some amount of credit risk. Hull and White argue it's not generally the best idea, since it would mean double-counting, as one also normally computes the CVA to handle ... 5 You can use the DiscountCurve class, that takes a list of dates and a list of corresponding discount factors. The one exported by default in the Python module uses a log-linear interpolation between the given discounts; using a different interpolation would require adding a line in the corresponding SWIG interface and recompiling the module. 5 The price of most (not all) bonds is quoted as a percentage of face value (par). For most amortizing bonds that have already amortized, the percentage is of the face value now, after amortizations, not the initial face value. (Bonds that are quoted / trading dirty / flat / on proceeds are different and I won't go there.) Suppose, for concreteness, than some ... 5 In a past life, I was an equity strategist at a sell-side bulge bracket firm. In 2008 (obviously) the bank decided to take a long hard look at the funding costs of its derivatives books. So they appointed an MD in IBD/corporate finance who would obviously lack “skin in that game”, and I was his chosen “research guy” (deliberately not then from a fixed income ... 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 That formula is algebraically equivalent to saying different, stochastic assets can have different expected returns. $$\mathbb{E} \left[ R_i \right] = r_f + \gamma_i$$ Some simple algebra LetX_i$be a random variable denoting a risky cash flow,$p_i$be today's price of that risky cash flow,$r_f$be the risk free rate, and$\gamma_i$be some risk ... 4 You are basically just arguing semantics from two models, neither of which are necessarily precisely accurate. If you observe the assumptions regarding yield to maturity, you have; 1) Coupons can be reinvested at the same yield through the life of the bond, 2) The payment dates all have consistent amount of time between each one, i.e. nothing falls on a ... 4 Since pretty much all trades (at least interbank) are collateralized nowadays, you would follow the principle of CSA discounting and use the interest rate on the collateral as a discount rate. Typically it's an overnight rate, for example SONIA in GBP, EONIA in EUR, Fed Funds in USD (broad switch to SOFR hasn't yet taken place I think), so you would use OIS ... 4 This is not a bug, just how computers work. Would have been better to ask in a non finance forum though. It is called (Integer) Overflow. If you are into reading humorous chats, you can have a look here. You can find an explanation in most documentations. Numpy is a funny one as Python does not have this issue. I assume it is fair to say the author's did ... 4 The computer programs are evaluating the following expression:$FV(i,N,PMT,PV)=-PMT[\frac{(1+i)^N-1}{i}]-PV(1+i)^N$the test case you are running is the special case where you choose$PMT=-i\cdot PV$If we evaluate the formula symbolically (as opposed to numerically) we get a fortunate cancellation:$FV(i,N,-i \cdot PV,PV)= PV(1+i)^N-PV(1+i)^N-PV=-PV$the ... 3 Ideally, you would discount a certain cash flow by its appropriate curve. For example: 1) you would discount the cash flow of your fixed income investments by the yield curve of the Treasury market; 1) you would discount the cash flow of an oil company by the interest curve that the it encounters when financing its ventures. If such a ('ideal') curve is ... 3 The curve Bloomberg EUR swaps curve (YCSW0045 Index) is indeed the euro equivalent of the Bloomberg USD swaps curve (YCSW0023 Index). By equivalent I mean that each curves are constructed in the same manner : using sames types of instruments (deposits, FRAs, futures, swaps) with the same bootstrapping/implying method (exact fit vs best fit). For each you ... 3 Reformatting for an answer: OIS (vols) - vols backed out of/for pricing in the presence of multiple curves LIBOR (vols) - vols backed out of/for pricing in the 'old' way where discount=forward and basis is negligible Black (vols) - Black-76 inverted volatilities Normal (vols) - Normal/Bachelier (?) inverted volatilities FINCAD primer on the 'new curves ... 3 Which currency are you looking at ? Say that your 1y swap would have yearly fixed payments vs 3M floating payments. Your 1.5y swap would probably have: a fixed payment 6m after effective date and another fixed payment 18m after effective date regular quarterly floating payments Your curve was built with 1y and 2y swaps, nothing in the middle ? Then yes, ... 3 I believe that Eonia can still be used for discounting derivatives after 2020. Article: https://www.risk.net/derivatives/5848051/esma-eonia-can-be-used-in-csas-after-2020 If it becomes illiquid, counterparties can repaper csa's to Ester. That will cause an economic effect if there is a non zero Eonia/Ester basis. 3 The first statement is kind of clear. If all investors are risk-neutral, they simply do not care about risk and do not pay more or less regardless how risky an asset is. As a consequence, the return of all assets is the risk-free rate. Regarding the second statement. What does risk-neutral pricing really do? We change the probabilities from the real world,$...

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A plethora of instruments, a menagerie of curves Different instruments are traded in different ways, and relate to a collection of curves. Floating rate instruments depend on some index in order to calculate the cashflows, and so trading instruments which depend on different indices is implicitly trading the expectations of those indices in the future. Fed ...

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This holds due to a change of measure. There is the real-world $\mathbb{P}$ and the risk-neutral world $\mathbb{Q}$. (I am going to assume constant interest rate $r$) The first fundamental theorem of asset pricing states that if there are no arbitrage strategies in a market, then there exists at least one probability measure $\mathbb{Q}\sim\mathbb{P}$ such ...

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For both cleared and OTC swaps you need to post margin. If you are delivering cash then you will receive OIS in generally in either case. As OTC trades are bespoke you might have a different agreement with your particular counterparty - but that would be unusual. The main advantage of a central clearer is the recycling of margin. If I receive 5y from ...

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There are two types of discounting approaches of a future payment in your question. Zero rates and forward rates. Let's just briefly consider each in turn. i) ZERO RATES. The zero rate discount factor to time $T$ is $df(T) = (1+R(T)/f)^{-Tf}$ where $f$ is the compounding frequency associated with $T$-year zero rate $R(T)$. The choice of $f$ is a ...

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The last edition (10th, 2017) of Hull's book explains it fairly well. Basically, there is indeed a theoretical arbitrage within the dual curve framework: you could borrow at the overnight rate (Fed funds, SONIA, EONIA, etc.), lend at LIBOR and cash-in the spread in all your dynamic derivatives replication trades. However, such arbitrage is only theoretical : ...

3

Libor, besides its name, is not a good proxy for uncollateralised funding. In fact, a large bank I worked for had an uncollateralised funding curve which was a a spread of +100bps to Libor. The fact that it was a spread to Libor was for legacy reasons and easier to adopt to existing systems. However, a spread of say +115bps to CSA was equally appropriate. ...

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I think it's possible. When you say the RFRs are flat, I think we can interpret that as flat for all maturities, including the 1y. So from the 1y Forward rate, we can back out the spot rate via the following relationship between the Spot, Forwards and the RFRs: $$S_{EUR/USD}(1+r_{USD})^n=(1+r_{EUR}+r_{Basis})^nF_{EUR/USD}$$ Above, $r_{Basis}$ stands for the ...

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