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

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 ...


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

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

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 Overnight fixings, called Sonia, Eonia and -ahem-...


4

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.


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 Let $X_i$ 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.


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

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

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 ...


1

You can synthesise this with the single currency IBOR-OIS basis swap (SBS) in each currency. For example paying the EUR/USD 10Y XCS @ -40bps, represents paying 3M Euribor -40 versus receiving 3M USD Libor flat. If you then buy a EUR 10Y OIS/IBOR SBS @ 8bps, this represents receiving 3M Euribor -8bps and paying EONIA flat. If you then sell a USD 10Y OIS/IBOR ...


1

For floating-rate bonds, it is difficult to compute z-spread if you don't know their cash flows. As a result, you would use discount margin where you have an assumption for the future cash flows of the floater.


1

The discount rate for 2) should be a risky rate $r + \lambda$, although we must talk about how to determine $\lambda$. If you have sold an uncollateralized put option, the put option is an unsecured obligation of yours. Hence it should carry a similar discount rate to other unsecured obligations that you have issued. Thus, $\lambda$ is your credit spread. ...


1

OIS is based on overnight Fed Funds, which as you say is an unsecured overnight rate between banks in the Federal funds market. This is not technically risk-free, although pretty close (what are the chances of Citibank defaulting by tomorrow?). The OIS swap market thus provides an almost-risk-free rate for any desired term. For example, the 5yr OIS swap ...


1

For a fixed interest rate or dividend payment/yield, integrating the function becomes: $\large{P_T = } \Large{e^{\int_t^T r_s ds} = \Large{e^{r_s (T - t)}}}$ $P_T$ is usually understood to be the future value of a zero-coupon bond with a face value of \$1. The present value, $P_t$, is simply then: $\large{P_t = } \Large{e^{\int_t^T -r_s ds} = \Large{e^{...


1

So, you have this: $$ \sum_{k=1}^{k=15} 0.8 \cdot 1.06^{2k} = 5.3456\ldots $$ And you want to know if there's a formula, or closed form. Yes there is. $$ \sum_{k=1}^{k=n} x^{k} = x \frac{x^{n+1}-1}{x-1} $$ Where, we're going to set $x=1.06^{-2}$, and sum from 1 to 15 (since you're not including the first period in the valuations). $$ 0.8 \sum_{k=1}^{k=...


1

Then I suppose the following will be how it should be done?


1

TL;DR: Looks like the book is crap. Switching from annual to semi-annual compounding should give a rather small change in PV. A payment of 1 at time T, with an annually compounded rate r, is worth $1/(1+r)^T$. With a semi-annual compounded rate s, it is $1/(1+s/2)^{2T}$, which incidentally is $1/(1+s+s^2/4)^T$, so the same as annually compounded to first ...


1

This is a "cheapest-to-deliver" option - in the absence of any restrictions, the rational investor would post whichever collateral class offers the best rate of return at each moment in time (of course, this could vary over the life of the trade). Thus, to price in the presence of this option, you need to model the optimal posting strategy. You mentioned ...


1

You have to be very careful with terminology here. In particular "yield" is being thrown around carelessly by both of you. The textbook is correct if the (meaningless) phrase "at a 6% yield (rate)" is crossed out and replaced by "at a 6% discount rate". And this is how Tbill's are handled when they are issued (the press release by the US Treasury speaks of ...


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