# Tag Info

8

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 ...

8

$\mathbb{P}$ 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 ...

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.

7

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 ...

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 ...

6

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 $\Delta (... 5 You can use the DiscountCurve class, that takes a list of dates and a list of corresponding discount factors. The one exported by default in the Python module uses a log-linear interpolation between the given discounts; using a different interpolation would require adding a line in the corresponding SWIG interface and recompiling the module. 5 Collateral posted in currency XYZ is remunerated at$\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 ... 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 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 ... 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 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 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 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$be some risk ... 3 I believe that Eonia can still be used for discounting derivatives after 2020. Article: https://www.risk.net/derivatives/5848051/esma-eonia-can-be-used-in-csas-after-2020 If it becomes illiquid, counterparties can repaper csa's to Ester. That will cause an economic effect if there is a non zero Eonia/Ester basis. 3 A plethora of instruments, a menagerie of curves Different instruments are traded in different ways, and relate to a collection of curves. Floating rate instruments depend on some index in order to calculate the cashflows, and so trading instruments which depend on different indices is implicitly trading the expectations of those indices in the future. Fed ... 3 This holds due to a change of measure. There is the real-world$\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}$such ... 3 For both cleared and OTC swaps you need to post margin. If you are delivering cash then you will receive OIS in generally in either case. As OTC trades are bespoke you might have a different agreement with your particular counterparty - but that would be unusual. The main advantage of a central clearer is the recycling of margin. If I receive 5y from ... 3 There are two types of discounting approaches of a future payment in your question. Zero rates and forward rates. Let's just briefly consider each in turn. i) ZERO RATES. The zero rate discount factor to time$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$is a ... 2 From$(2)of Piterbarg, \begin{align*} V(t) = \Delta (t) S(t) + \gamma(t), \end{align*} where\Delta (t)= \frac{\partial V(t)}{\partial S}$, and$\gamma(t)is the cash account that satisfies \begin{align*} d\gamma(t) &= \big[r_C(t) C(t) + r_F(t)(V(t)-C(t))-(r_R(t)-r_D(t))\Delta(t)S(t) \big]dt\\ &=\big[r_F(t)V(t) + (r_C(t)-r_F(t)) C(t)-(r_R(t)-r_D(... 2 self financed portfolio will give you : $$dV_t = r_F(t) \underbrace{(V(t)-C(t) - \Delta S_t )}_{\text{cash position}} dt + r_C(t) \underbrace{C(t)}_{\text{posted collateral}} dt + \underbrace{\Delta dS_t}_{\text{market move}}$$ then you retrieve his equation using that under risk-neutral measure : $$\mathbb{E}[dS_t|\mathcal{F}_t]=r_F(t)S_tdt$$ 2 You have\beta_1=\frac{1}{(1+r)^n}\frac{1}{(1+r)^\theta}$and$\beta_2=\frac{1}{(1+r)^n}\frac{1}{(1+\theta r)}$. Both are equal when$\theta=1$. If you consider simple interest then go for$\beta_2$. If you would like compound interest within fraction of year then pick$\beta_1$. However, because$\theta$is between 0 and 1 then values$\beta$'s won't ... 2 1.$V_0$is what the bank would write for it's book value This is only the case for items held on the balance sheet at Fair Value. Most banks will hold many assets/loans at amortized cost (principal outstanding). The book value will therefore be irrespective of any changes in interest rates and credit spreads. The ABA has a good summary of why this is the ... 2 Then I suppose the following will be how it should be done? 2 By no arbitrage, market participants need to agree on the values of the discount factor, even if they are using different conventions (day count, compounding period) to convert the discount factor into a rate. For example, consider two discount factors computed using continuous compounding, where one is computed using the 30/360 day count (year fraction$t_{...

2

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 ...

2

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, ... 2 Recall that the price of your contract is \begin{align*} V_t = e^{-r(T-t)} \mathbb{E}^\mathbb{Q} [H1_{\{S_T>K\}}|\mathcal{F}_t] \end{align*} because your option always paysH$if$S_T>K. Next, \begin{align*} V_t &=He^{-r(T-t)} \mathbb{E}^\mathbb{Q} [1_{\{S_T>K\}}|\mathcal{F}_t] \\ &= He^{-r(T-t)} \mathbb{Q} [{\{S_T>K\}}|\mathcal{F}_t] \\... 2 The digital option paysH$at time$T$if$S_T \geq K$, so its option time at time$t$is given by $$V_t=E_t\left[e^{-r(T-t)}H 1_{\{S_T \geq K\}}\right]=e^{-r(T-t)}H* P_t(S_T \geq K)$$ The model used is Black-model, that $$dS_t=rS_tdt+\sigma dW_t$$ or $$S_T=S_te^{\left(r-\frac12 \sigma^2\right)(T-t)+\sigma (W_T-W_t)}{}$$ Calculate$ P_t(S_T \geq K)\$ ...

2

try this : 4000= 2000/(1+r)^2 + 3000/(1+r)^4 solving this equation for r you'll find equals 7.30274083178438%.

Only top voted, non community-wiki answers of a minimum length are eligible