Appendix B — Interest Rates

References

HULL, John. Options, futures, and other derivatives. Ninth edition. Harlow: Pearson, 2018. ISBN 978-1-292-21289-0.
- Chapter 4 - Interest Rates

Learning Outcomes:

Understand the different types of interest rates.
Define the risk-free rate and its significance in financial derivatives.
Explain the concept of continuous compounding and its importance in pricing financial derivatives.

B.1 Types of Rates

Unsecured interbank overnight rate of interest.
Allows banks to adjust the cash (i.e., reserves) on deposit with the Federal Reserve at the end of each day.
The effective fed funds rate is the average rate on brokered transactions.
The central bank may intervene with its own transactions to raise or lower the rate.
Similar arrangements in other countries.

Repurchase agreement is an agreement where a financial institution that owns securities agrees to sell them for X and buy them bank in the future (usually the next day) for a slightly higher price, Y.
The financial institution obtains a loan.
The rate of interest is calculated from the difference between X and Y and is known as the repo rate.

Detailed information about LIBOR: https://www.theice.com/iba/libor
LIBOR is the rate of interest at which a AA bank can borrow money on an unsecured basis from another bank.
Based on submissions from a panel of contributor banks (16 for each of USD and GBP).
It is calculated daily for 5 currencies and 7 maturities.
There have been some suggestions that banks manipulated LIBOR during certain periods.
Why would they do this?

Country/Currency/CODE	IBOR Rate	New Reference Rate
USA/Dollars/USD	USD ICE LIBOR	SOFR
UK/Pounds Sterling/GBP	GBP ICE LIBOR	SONIA
Switzerland/Swiss Francs/CHF	CHF ICE LIBOR	SARON
Japan/Yen/JPY	JPY ICE LIBOR, Tibor	TONAR
EU/Euro/EUR	Euribor	ESTER

CME Group Education
Administered by Federal Reserve Bank of New York (link)
Transaction-based, calculated from overnight US Treasury repurchase (repo) activity.
SOFR is a broad measure of the cost of borrowing USD cash overnight, collateralized by U.S. Treasury securities.
SOFR is a good representation of general funding conditions in the overnight Treasury repo market.
As such, it will reflect an economic cost of lending and borrowing relevant to the wide array of market participants active in the market.

CME Group Education
Administered by Bank of England (link)
Unsecured transaction-based index, wholesale based (beyond Interbank)
It has been endorsed by the Sterling Risk-Free Reference Rate Working Group (Working Group) as the preferred risk-free reference rate for Sterling Overnight Indexed Swaps (OIS).
In January 2018, the Working Group added banks, dealers, investment managers, non-financial corporates, infrastructure providers, trade associations and professional services firms.
In April 2018, the BOE introduced a series of reforms of the SONIA benchmark.

Administered by European Central Bank (link)
It is based on the unsecured market segment.
The ECB developed an unsecured rate, because it is intended to complement the EONIA.
Furthermore, a secured rate would be affected by the type of the collaterals.
The money market statistical reporting covers the 50 largest banks in the euro area in terms of balance sheet size.
While the EONIA (link) reflects the interbank market, the €STR extends the scope to money market funds, insurance companies and other financial corporations because banks developed significant money market activity with those entities.

An overnight indexed swap is swap where a fixed rate for a period (e.g. 3 months) is exchanged for the geometric average of overnight rates (or overnight rate compounded over the term of the swap).
The underlying floating rate is typically the rate for overnight lending between banks, either non-secured or secured (SOFR, SONIA, €STR).
For maturities up to one year there is a single exchange (swap term is not overnight).
For maturities beyond one year there are periodic exchanges, e.g. every quarter.
The OIS rate is a continually refreshed overnight rate.
The fixed rate of OIS is typically an interest rate considered less risky than the corresponding interbank rate (LIBOR) because there is limited counterparty risk.

The Treasury rate is considered to be artificially low because:
- Banks are not required to keep capital for Treasury instruments
- Treasury instruments are given favorable tax treatment in the US
OIS rates are now used as a proxy for risk-free rates in derivatives valuation.

When we compound $m$ times per year at rate $r$ an amount $P$ grows to $P(1+r/m)^m$ in one year.
The compounding frequency used for an interest rate is the unit of measurement.
The difference between quarterly and annual compounding is analogous to the difference between miles and kilometers.
Effect of the compounding frequency on the value of $100 at the end of 1 year when the interest rate is 10% per annum.

Rates used in option pricing are nearly always expressed with continuous compounding.
In the limit as we compound more and more frequently we obtain continuously compounded interest rates.
Notation:
- $r$: continuously compounded annual interest rate
- $T$: time to maturity in years
- $e$: Euler’s number (mathematical constant)

\[ \text{Future value} = P \times e^{rT} \]

\[ \text{Present value} = P \times e^{-rT} \]

USD 100 grows to $100 \times e^{rT}$ when invested at a continuously compounded rate $r$ for time $T$.
USD 100 received at time $T$ discounts to $100 \times e^{-rT}$ at time zero when the continuously compounded discount rate is $r$.

\[ r_c = m \ln (1+ \frac{r_m}{m}) \]

\[ r_m = m(e^{r_c/m} - 1) \]

Examples:

10% with semiannual compounding is equivalent to $2 \ln (1.05) = 9.758\%$ with continuous compounding.
8% with continuous compounding is equivalent to $4(e^{0.08/4} - 1) = 8.08\%$ with quarterly compounding.