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日期:2019-12-14 10:53

FRE6831

COMPUTATIONAL FINANCE LABORATORY (PYTHON)

Dept. of Finance and Risk Engineering

Office Hours by appointment

PROJECT: IMPLEMENTING A LIBOR YIELD CURVE OBJECT

The class project is to write a Python program that infers the short end of the USD LIBOR yield curve

from market observations, via the methodology in use prior to 2008 (More modern methodologies are too

complex for an introductory course.). It is to support two calculations:

1. The “discount factor,” or present value of a US dollar for a given future date, up to 3 years after a

given “spot” date (See below for a definition of the spot date.).

2. The “forward rate,” which is the simple interest rate, expressed as a percentage, over a given

period between two dates, computed from the above discount factor curve as explained below.

Because this calculation uses the above discount factor calculation, both of the input dates must

also be less than approximately 3 years after the spot date.

The yield curve is to be implemented as a Python class that I will import into my environment and run via

the commands

import USDYieldCurve

usdCurve = USDYieldCurve(“depoRates.txt”, “futuresPrices.txt”,

“tradeDate.txt”, “holidayCalendar.txt”)

print(usdCurve.getDfToDate(d1)) #d1 is a string in YYYY-MM-DD format

print usdCurve.getFwdRate(d2, d3) #inputs are in YYYY-MM-DD firnat

Inputs

Typical inputs to such a calculation – and the inputs to assume here – are to be read from the four text

files named above.

1. depoRates.txt contains a series of annualized cash deposit rates. These cash deposits are

effectively jumbo versions of the certificates of deposit that are on offer at any bank. These

deposits should be read from a text file, each line of which has the following format:

<5 character instrument code> white space <rate as percentage, rounded to nearest basis point>

The instrument code will always have the prefix “USD” and the suffix “D”, “W”, or “M”, which

will indicate whether the deposit is for a period of days, weeks or months. In between this prefix

and suffix will be a number, indicating the number of days, weeks, or months to maturity. For

example, “USD1W” would be the code for the cash deposit maturing in one week; “USD2M”

would be the cash deposit maturing in 2 months, etc. The start date of these cash deposits is

always assumed to be the “spot date” (More about the “spot date” later.).

The first few lines of such a file might read

USD1D 2.73

USD1W 2.74

USD1M 2.75

USD3M 2.80

USD6M 2.85

You can assume that the number in the above codes is only a single digit (Although 12 month

LIBOR deposits do exist, they are seldom used to construct yield curves.). Some of the above

tenors may be omitted; in fact, the only one that is really necessary is USD6M (Why?).

2. futuresPrices.txt contains a series of prices of Eurodollar futures, which are futures on 3

month cash LIBOR deposits. You can assume that the futures on successive lines of the file

mature on successively later dates. Each line of the file has the following format:

<4 character instrument code> <price, given as a four digit integer>

The instrument code will always have the prefix “ED”, followed by one of the letters “H”, “M”,

“U”, or “Z”, followed by a single numeric character. The letters “H”, “M”, “U”, and “Z”

correspond to the maturity date of the future: the respective 3rd Wednesdays of March, June,

September, and December of the year indicated by the numeric character. Thus, “EDZ2” would

be the code for the future expiring in December, 2022. Necessarily, these maturity dates are dates

in the future, so there is no ambiguity, given that all futures must mature prior to ten years from

the current date. To see this, suppose

Thus, the first few lines of such a file might read

EDH9 97.25

EDM9 97.20

EDU9 97.19

3. tradeDate.txt contains the date at which the rates/prices are being observed, given as

character strings in the form YYYY-MM-DD.

4. holidayCalendar.txt contains the dates of Federal Reserve holidays, given as character

strings in the form YYYY-MM-DD, for the next five years (available from, among other places,

the Federal Reserve website).

As we will see in more detail in the calculation below, it may be impossible to compute the yield curve if

the input files are incomplete. In this case, an error condition should be raised and the error message

“Cannot build curve from given inputs” should be generated.

Calculation

“Modified Following” date adjustment

USD payments can only be made on days in which the US Federal Reserve (the “Fed”) is open. If the

date of a scheduled payment falls on a weekend or a Federal Reserve holiday, it must be adjusted

according to the so-called modified following business convention. Under this convention, if a payment

would otherwise be scheduled on a non-business day, the payment date “rolls forward” to the next

business day unless the next business day falls into the subsequent month. If so, the payment date “rolls

backward” to the next previous month. Thus, a payment that would be scheduled for December 30, 2018

(a Sunday) would be rolled forward to December 31, 2018 (a Monday and a business day), but a payment

scheduled for January 1, 2019 (New Year’s Day and a bank holiday) would be rolled backward to

December 31.

There are two additions to this convention. If the spot date is the end of a month, the maturity date of

deposits for one month would be the last date of the next month (So, when spot is February 28, the one

month deposit matures on March 31.). Same with 2, 3, … month deposits.

Also, if the spot date is January 29 or January 30, the one month deposit matures on February 28 (or

February 29 on leap years.).

The spot date

The beginning of the period over which USD interest is computed is usually the so-called “spot date,”

which is always two business days after the trade date (The intervening time period is required for

clearing and settlement.). Given this interest computation convention, discount factors are also only

defined from the spot date. The spot date will be denoted as s below.

Date arithmetic

Here and subsequently, if dates are to appear in a formula at all, one date will be subtracted from a later

date. Use the Python built-in library datetime to do the date arithmetic. If d1 and d2 are two such

dates, d1 – d2 is interpreted as the number of calendar days between them (More precisely, this difference

calculation results in a timedelta object, which can then be converted to a Python int.). As we

will see, this is just what is needed for the calculations below.

Computing discount factors from cash deposit rates

LIBOR interest calculations for periods less than a year always assume simple interest applied to the

relevant fraction of a year, computed as the actual number of days that a loan is outstanding, including

weekends and holidays, divided by 360 (Remember, all of this was standardized before there were

computers to facilitate the calculations!). Thus, if the given rate is R and the maturity date of the deposit

is d and the spot date is s, the amount to be repaid for every dollar lent is 1 + R(d - s)/36000. It follows

that the discount factor, or present value of a dollar n days from the spot date is (1 + Rn/36000)-1

. If a

discount factors df1 and df2 have been computed for dates d1 and d2 after the spot date, the discount factor,

df, for a date d between d1 and d2 is given by logarithmic interpolation

ln(df) = ln(df1) + [(d – d1)/(d2 – d1)] [ln(df2) – ln(df1)].

Computing discount factors from futures prices

Interest rate futures are futures on forward rates, i.e. rates over some pair of fixed future dates (In

actuality, 3 month LIBOR really means 3 calendar months, but assume, for simplicity, that it is the period

between successive futures maturity dates.). Thus, these futures are the market’s best guess as to what

these rates will actually be when the beginning of the forward period coincides with the spot date, and 3

month “cash” LIBOR becomes known. In the meantime, the rate implied by a futures price P is

computed as a percentage according to the formula

R = 100.0 – P/100.0

As before, simple interest is assumed, with the number of days in the three calendar months starting with

the futures expiration date.

However, the discount factor thus computed is a forward discount factor; that is, it is the discount factor

from the expiration of the future to the expiration of the underlying 3 month LIBOR deposit. To get the

discount factor from spot to the expiration of the LIBOR deposit, this forward discount factor must be

multiplied by the discount factor from spot to the start date of the LIBOR deposit, which is also the

maturity date of the future. Thus, given the discount factor, dfs, from spot to the earliest futures expiration

date and the forward discount factor, df1, computed as above, the discount factor to the expiration date of

the first future is dfs x df1. In general, the maturity dates of the any of the futures will not coincide with

any of the expiration dates of the cash deposits, so the interpolation formula given above will be required.

This computation cannot be done if the rates for cash LIBOR deposits are unavailable for the required

maturities. If this is the case, an error condition should be raised, and the message “Insufficient LIBOR

cash rate data” should be printed.

Again, we simplify things by ignoring the awkward fact that the futures maturity dates are almost never

exactly three months apart; instead, we assume that the maturity date of the 3 month LIBOR deposit

underlying one futures contract is also the expiration date of the next futures contract. Thus, the discount

factor to the maturity date of the underlying of the second future is dfs x df1x df2, where df2 is the forward

discount factor computed from the price of second future. This process can be continued to calculate the

discount factor from spot to the n

th futures underlying expiry date as dfs x df1x df2 x … x dfn.

The df method

This method should return the value of the df curve for the date, d, that has been passed as an argument.

In general, d will not coincide with any of the dates for which the above calculation will have produced

discount factors. In that case, determine the required df curve value by finding dates d1 and d2 from the

spot date, the discount factor, df, for a number, d, of days between d1 and d2 is given by logarithmic

interpolation

ln(df) = ln(df1) + [( d – d1)/(d2 – d1)] [ln(df2) – ln(df1)]

If d is before the spot date or d is after the last date for which the discount factor curve is defined, an error

condition should be raised, and an appropriate error message should be printed to STDOUT.

The fwdRate method

Given two dates, d1 < d2, this method should return the forward rate, rf , between these two dates, inferred

from the discount curve computed above, according to the formula

???? =360??2 ? ??1[????(??1)????(??2)? 1]

Use Python’s built-in datetime library to do the date arithmetic. This formula will compute the rate as

a decimal fraction, rather than a percentage (e.g. something like 0.03, rather than 3%.).

If d1 is before the spot date, d1 ≥ d2, or d2 is after the last date for which the discount factor curve is

defined, an error condition should be raised, and an appropriate error message should be printed to

STDOUT.

Testing

In the “Project” folder on the Course website, I have posted the spreadsheet “YCtestcase.xlsx”,

which implements the calculations for the spot date April 24, 2015.

Notes:

1. More modern methodologies take into consideration the changes in the LIBOR markets that arose

from the debt crisis of 2008, which involves the use of a truly risk free curve for discounting (As

the financial community learned to its chagrin, interbank lending, the “I” in LIBOR, is not risk

free!). Thus, one curve is needed for discounting and another is needed for predicting forward

rates.

2. There is considerable disagreement about exactly which mix of the available cash deposits,

futures, and swaps – or even whether instruments besides these – should be used in the above

calculation. What is generally agreed upon is that the most liquid instruments should be used, as

these are the best indicators of the market.

3. Although you are to use the interpolation methods given above, they are often replaced in practice

by more sophisticated methods, such as splines.

4. Because owners of futures contracts are required to post margin daily as their futures prices vary,

the rates inferred from futures are actually biased estimates. In practice, a so-called “convexity

adjustment” is often applied to these rates. For simplicity, we will ignore this adjustment.

5. The above conventions and instrument details are only applicable to USD. The construction of

the LIBOR discount curve for other currencies differs in detail, but not in general, from this

outline. For example, GBP has a spot date one day after the trade date, and 365 replaces 360 in

the above calculations.


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