# Relation between OIS rate and discounting rate

This is from book Modern Derivatives Pricing and Credit Exposure Analysis page 22

https://www.amazon.com/Modern-Derivatives-Pricing-Exposure-Analysis/dp/1137494832

In an OIS, two parties exchange a fixed coupon (paid annually for longer-dated swaps and as a single payment at maturity otherwise) against the daily fixed and compounded overnight rate. Daily compounding means that the rate paid at the end of period i = 1, . . . ,n is given by $$R_i = \dfrac{1}{\delta_i}\left[\prod\limits_{k=1}^{n_i}(1+F^{ON}(t_{i,k-1},t_{i,k})\delta_{i,k})-1\right]$$

Here $$df^{ON}$$ is dicounting factor.

I cannot understand that

1. Here is the $$F^{ON}(t_{i,k-1},t_{i,k})$$ the overnight index rate (like Fderal funds rate)?

2. What the OIS rate or OIS curve mentioned at last in the formula? I think that it is the swap rate $$c_0.$$

3. What is OIS quoting? Do we first have the OIS quoting or first have the overnight index rate quoting in the market? From the final formula we can see that it is not related to the overnight index rate $$F^{ON}(t_{i,k-1},t_{i,k}).$$ And if we already had a the quoting of OIS rate, then from the relation between OIS and overnight rate, there will be a restriction of series of overnight rate? Or OIS rate is quoted from the forward overnight rate?

Can anyone clarify those of my confusions?

1. No its not the Fed Funds Rate, or the Bank of England Base Rate or the ECB Refi Rate, it is the forecast, published OIS fixing index determined by the relevant authority in the currency. I.e in USD it is FFOIS, in GBP it is SONIA and in EUR it is EONIA. (Actually these names may in fact be transitioning to other in index definitions now, especially in EUR)

2. An OIS contract is a 'swap' but there are other types of 'swaps' that settle to IBOR based indexes and not OIS indexes. If you want to build an OIS curve from market prices you can take OIS-swap rates longer than 1 year and use a bootstrapping (note that more modern multivariate simultaneous solving methods are used not bootstrapping) process to backward filling all the forecast forward rates (subject to some interpolation scheme) that will make sure your curve reprices every swap correctly.

3. Quotes are just stated prices, i.e. bids and offers. Since forecast rates are undetermined they give rise to uncertainty and scope for disagreement between counterparties and market-makers. Quotes provide the mechanism for people to strike deals when they wish to trade at another's price level. For the layperson quotes provide a means to calibrate the mid-market, i.e. expected collection of forward rates making up a contract or swap, and in turn one can then use the above bootstrapping process to generate a curve of expected forward rates.

Edit:

Say you observe some market quoted prices for OIS swaps:

bid .  offer .  mid
1-month 1%  .  1.02%    1.01%
2-month 1.1% . 1.12% .  1.11%
3-month 1.2% . 1.22% .  1.21%

To determine these prices the trader has essentially forecast every OIS fixing in the 3month period, i.e 63 business days. So 63 datapoints have gone into their calculation. So now you want to get the values of the fixings on these 63 days what do you do? Use assumptions:

a) you know what yesterday's fixing was so that gives you some info about the start of your curve.

b) you know that compounded the first 21 or so must compound to the 1M rate.

c) you know the first 42 or so must compound to the 2M rate.

d) you know the first 63 equal the 3M rate of 1.21%.

e) you assume some relation between days, e.g. they are interpolated in some smooth manner or only jump on certain dates (central bank policy meetings).

Then you take all the that infomation and incorporate it into a numerical solver which determines all the rates for the 63 dates.

• 2. So $c_0$ is the OIS swap rate (fixing rate makes OIS swap zero value) right? But what's the OIS curve? Is it the discounting factor $df_i^{ON}$ striped from market quote OIS rate? – user6703592 Oct 15 '18 at 7:21
• 3. Does that mean there is a mathematical relation between OIS rate and overnight index rate, but quote OIS rate is just the expected average of overnight index rate, two quotes are not necessary matching (just like the forward rate and spot rate)? – user6703592 Oct 15 '18 at 7:24
• A curve is a collection of datapoints for each and every date. The OIS forecast curve will contain the OIS overnight forward rate for each forecastable data. Since you can derive overnight forward rates from discount factors this is equivalent to saying the curve lists all discounts factors for each date – Attack68 Oct 15 '18 at 8:09
• So from the quoting of overnight forward rate (forward or spot) from current market, we can not determine the quoting of OIS rate, yes? – user6703592 Oct 17 '18 at 6:43