Pricing floating legs of interest rate swaps

(This article was first published on R on The CuRious Financial Risk ManageR, and kindly contributed to R-bloggers)

In this post we will close the trilogy on (old style) swap pricing. In particular, we will look at how downloading the data for the variable rate needed to calculate the variable leg accrual.

  • Part 1
    gave the general idea behind tidy pricing interest rate swaps using a 7 lines
    pipe

  • Part 2
    went much more into detail and priced some real world contract comparing the results
    obtained vs Bloomberg and showing significantly good results.

The only part missing was calculating the accrual for the floating leg. To do this
we need the information about the historical level of the interest rate to which the leg
is linked.

For standard EUR contracts, this rate is the 6 months’ EURIBOR. For those of you
who are interested in understanding more what this rate is can go this link

Sourcing the data is not a problem, but doing it the R way (ie. for free), can be.

Luckily the awesome Quandl data provider actually
provides the information we need. Quandl is a premier source for financial, economic, and alternative datasets, serving investment professionals. Quandl’s platform is used by over 400,000 people, including analysts from the world’s top hedge funds, asset managers and investment banks.

Pulling data from Quandl is very easy using the Quandl package available on CRAN
Its logic is quite simple:

Quandl::Quandl(code, start_date, end_date)

where code is a unique identifier that can be sourced from the Quandl website.

In particular, we shoudl be grateful to the Bank of France which provides all
the interest rate information we need. Going on their
page we can in fact find that
the code for the 6 months EURIBOR rate is “BOF/QS_D_IEUTIO6M”.

Let’s then see what the output of this very simple function is with our code:

rates <- Quandl::Quandl("BOF/QS_D_IEUTIO6M", 
               start_date = lubridate::dmy("01-01-2018"), 
               end_date = lubridate::dmy("31-12-2018"))

rates %>% 
  knitr::kable(caption = "Output from Quandl", "html") %>% 
  kableExtra::kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% 
   kableExtra::scroll_box(height = "500px")
Table 1: Output from Quandl
Date Value
2018-12-31 -0.237
2018-12-27 -0.237
2018-12-24 -0.237
2018-12-21 -0.238
2018-12-20 -0.238
2018-12-19 -0.238
2018-12-18 -0.238
2018-12-17 -0.237
2018-12-14 -0.238
2018-12-13 -0.239
2018-12-12 -0.241
2018-12-11 -0.244
2018-12-10 -0.245
2018-12-07 -0.246
2018-12-06 -0.246
2018-12-05 -0.246
2018-12-04 -0.247
2018-12-03 -0.248
2018-11-30 -0.251
2018-11-29 -0.253
2018-11-28 -0.256
2018-11-27 -0.256
2018-11-26 -0.256
2018-11-23 -0.257
2018-11-22 -0.257
2018-11-21 -0.257
2018-11-20 -0.257
2018-11-19 -0.257
2018-11-16 -0.257
2018-11-15 -0.257
2018-11-14 -0.257
2018-11-13 -0.257
2018-11-12 -0.257
2018-11-09 -0.257
2018-11-08 -0.257
2018-11-07 -0.258
2018-11-06 -0.258
2018-11-05 -0.257
2018-11-02 -0.258
2018-11-01 -0.258
2018-10-31 -0.259
2018-10-30 -0.259
2018-10-29 -0.259
2018-10-26 -0.259
2018-10-25 -0.259
2018-10-24 -0.259
2018-10-23 -0.259
2018-10-22 -0.261
2018-10-19 -0.262
2018-10-18 -0.265
2018-10-17 -0.266
2018-10-16 -0.265
2018-10-15 -0.266
2018-10-12 -0.267
2018-10-11 -0.267
2018-10-10 -0.268
2018-10-09 -0.268
2018-10-08 -0.267
2018-10-05 -0.267
2018-10-04 -0.268
2018-10-03 -0.267
2018-10-02 -0.268
2018-10-01 -0.268
2018-09-28 -0.268
2018-09-27 -0.268
2018-09-26 -0.267
2018-09-25 -0.267
2018-09-24 -0.267
2018-09-21 -0.268
2018-09-20 -0.268
2018-09-19 -0.267
2018-09-18 -0.268
2018-09-17 -0.268
2018-09-14 -0.269
2018-09-13 -0.269
2018-09-12 -0.269
2018-09-11 -0.269
2018-09-10 -0.269
2018-09-07 -0.269
2018-09-06 -0.269
2018-09-05 -0.269
2018-09-04 -0.269
2018-09-03 -0.268
2018-08-31 -0.268
2018-08-30 -0.268
2018-08-29 -0.266
2018-08-28 -0.266
2018-08-27 -0.265
2018-08-24 -0.266
2018-08-23 -0.266
2018-08-22 -0.266
2018-08-21 -0.266
2018-08-20 -0.266
2018-08-17 -0.266
2018-08-16 -0.266
2018-08-15 -0.266
2018-08-14 -0.266
2018-08-13 -0.266
2018-08-10 -0.267
2018-08-09 -0.266
2018-08-08 -0.268
2018-08-07 -0.268
2018-08-06 -0.268
2018-08-03 -0.268
2018-08-02 -0.269
2018-08-01 -0.269
2018-07-31 -0.268
2018-07-30 -0.268
2018-07-27 -0.269
2018-07-26 -0.269
2018-07-25 -0.269
2018-07-24 -0.269
2018-07-23 -0.269
2018-07-20 -0.269
2018-07-19 -0.269
2018-07-18 -0.269
2018-07-17 -0.269
2018-07-16 -0.269
2018-07-13 -0.268
2018-07-12 -0.271
2018-07-11 -0.271
2018-07-10 -0.269
2018-07-09 -0.270
2018-07-06 -0.270
2018-07-05 -0.269
2018-07-04 -0.269
2018-07-03 -0.270
2018-07-02 -0.269
2018-06-29 -0.270
2018-06-28 -0.270
2018-06-27 -0.270
2018-06-26 -0.270
2018-06-25 -0.269
2018-06-22 -0.268
2018-06-21 -0.268
2018-06-20 -0.268
2018-06-19 -0.268
2018-06-18 -0.268
2018-06-15 -0.268
2018-06-14 -0.268
2018-06-13 -0.268
2018-06-12 -0.267
2018-06-11 -0.267
2018-06-08 -0.267
2018-06-07 -0.269
2018-06-06 -0.269
2018-06-05 -0.269
2018-06-04 -0.269
2018-06-01 -0.269
2018-05-31 -0.269
2018-05-30 -0.269
2018-05-29 -0.269
2018-05-28 -0.269
2018-05-25 -0.271
2018-05-24 -0.271
2018-05-23 -0.270
2018-05-22 -0.271
2018-05-18 -0.271
2018-05-17 -0.270
2018-05-16 -0.272
2018-05-15 -0.271
2018-05-14 -0.271
2018-05-11 -0.271
2018-05-10 -0.271
2018-05-09 -0.269
2018-05-08 -0.269
2018-05-07 -0.269
2018-05-04 -0.269
2018-05-03 -0.269
2018-05-02 -0.269
2018-04-30 -0.269
2018-04-27 -0.269
2018-04-26 -0.269
2018-04-25 -0.270
2018-04-24 -0.270
2018-04-23 -0.270
2018-04-20 -0.271
2018-04-19 -0.270
2018-04-18 -0.271
2018-04-17 -0.270
2018-04-16 -0.270
2018-04-13 -0.271
2018-04-12 -0.271
2018-04-11 -0.270
2018-04-10 -0.271
2018-04-09 -0.270
2018-04-06 -0.270
2018-04-05 -0.271
2018-04-04 -0.271
2018-04-03 -0.270
2018-03-29 -0.271
2018-03-28 -0.271
2018-03-27 -0.271
2018-03-26 -0.271
2018-03-23 -0.270
2018-03-22 -0.271
2018-03-21 -0.272
2018-03-20 -0.273
2018-03-19 -0.272
2018-03-16 -0.272
2018-03-15 -0.271
2018-03-14 -0.271
2018-03-13 -0.271
2018-03-12 -0.271
2018-03-09 -0.271
2018-03-08 -0.272
2018-03-07 -0.272
2018-03-06 -0.271
2018-03-05 -0.272
2018-03-02 -0.271
2018-03-01 -0.271
2018-02-28 -0.270
2018-02-27 -0.271
2018-02-26 -0.271
2018-02-23 -0.271
2018-02-22 -0.270
2018-02-21 -0.271
2018-02-20 -0.273
2018-02-19 -0.274
2018-02-16 -0.274
2018-02-15 -0.276
2018-02-14 -0.276
2018-02-13 -0.276
2018-02-12 -0.278
2018-02-09 -0.278
2018-02-08 -0.278
2018-02-07 -0.278
2018-02-06 -0.279
2018-02-05 -0.278
2018-02-02 -0.278
2018-02-01 -0.278
2018-01-31 -0.279
2018-01-30 -0.278
2018-01-29 -0.278
2018-01-26 -0.278
2018-01-25 -0.278
2018-01-24 -0.278
2018-01-23 -0.276
2018-01-22 -0.277
2018-01-19 -0.276
2018-01-18 -0.275
2018-01-17 -0.274
2018-01-16 -0.272
2018-01-15 -0.274
2018-01-12 -0.271
2018-01-11 -0.271
2018-01-10 -0.271
2018-01-09 -0.271
2018-01-08 -0.271
2018-01-05 -0.271
2018-01-04 -0.271
2018-01-03 -0.271
2018-01-02 -0.271

We now have to incorporate this code into the overall coding strategy described
in the previous post

Before we dive in how I coded this, let’s see how I changed the data structure of
how a swap is defined:

swap.25y <- list(notional = 10000000,
                  start.date = lubridate::ymd(20070119),
                  maturity.date = lubridate::ymd(20320119), 
                  strike = 0.00059820,
                  type = "receiver", 
                  time.unit = list(pay = 6, receive = 12), 
                  dcc = list(pay = 0, receive = 6), 
                  calendar = "TARGET")

This new data structure now allows to define – as it should be – different
characteristics (like coupon frequency or day count convention) for the different
legs (payer and receiver) of the swap. In particular, we can specify in the
type variable whether the swap is a receiver or a payer one.

Let’s look atthe code now. The function that gets modified the most is the
SwapCashflowYFCalculation which I have re-named as CashFlowPricing one
which now looks as follows:

CashFlowPricing  <- function(today, start.date, maturity.date, type,
                                     time.unit, dcc, calendar) {
  # Part 1: Calculate the whole cashflow dates
  cashflows <- seq(from = 0,
                   to = (lubridate::year(maturity.date) -
                           lubridate::year(start.date)) * 12,
                   by = time.unit) %>%
    purrr::map_dbl(~RQuantLib::advance(calendar = calendar,
                                       dates = start.date,
                                       n = .x,
                                       timeUnit = 2,
                                       bdc = 1,
                                       emr = TRUE)) %>%
    lubridate::as_date() %>%
    {if (start.date < today) append(today, .) else .}

  # Part 2: calculate accrual and rate fixing days
  accrual.date <- cashflows[today - cashflows > 0]

  if (!identical(as.double(accrual.date), double(0))) {
    accrual.date  %<>%  max()
    if (stringr::str_detect(type, "floating")) {
      fixing.date <- accrual.date %>%
        {RQuantLib::advance(calendar = calendar,
                            dates = .,
                            n = -2,
                            timeUnit = 0,
                            bdc = 1,
                            emr = TRUE)}
    } else {
      fixing.date <- NULL
    }
    accrual.yf <- accrual.date %>%
      {RQuantLib::yearFraction(today, ., dcc)} %>%
      `*`(-1)
  } else {
    fixing.date <- NULL
    accrual.yf <- 0
  }

  # Part 3: Tidy and return the list of relevant dates
  cashflows %<>%
    purrr::map_dbl(~RQuantLib::yearFraction(today, .x, dcc)) %>%
    tibble::tibble(yf = .) %>%
    dplyr::filter(yf >= 0)

  return(list(cashflows = cashflows, accrual.yf = accrual.yf,
              fixing.date = fixing.date))
}

Let’s analyise the code:

  • Part 1 is actually the core of the code previously described

  • Part 2 is the new code. accrual.date is the date from which the accrual starts
    to be calculated. This gets converted into a year fraction and saved into accrual.yf.
    The if statement calculates the date at which the floating EURIBOR rate has to
    be snapped from Quandl only for the floating rate. This date is stored in the
    fixing.date variable and it considers a 2 days lag which is standard for the
    European market.

  • Part 3 finally converts and returns all the future cashflows in terms of
    year fraction

You can note that we now calculate the cashflows for the floating leg even if
it will note be used by the OLDParSwapRate function. This will be needed for
future developments when we will introduce the OIS discounting…(stay tuned!!)

I can now calculate the accrual and for this purpose I developed a brand new function
called CalculateAccrual

CalculateAccrual <- function(swap.dates, leg.type, swap, direction) {
  # Part 1: calculate the accrual rate
  if (!is.null(swap.dates$fixing.date)) {
    rate <- Quandl::Quandl("BOF/QS_D_IEUTIO6M",
                           start_date = swap.dates$fixing.date,
                           end_date = swap.dates$fixing.date) %>%
      tibble::as_tibble(.) %>%
      dplyr::select(Value) %>%
      as.double %>%
      `/`(100)
  } else {
    rate <- swap$strike
  }
  # Part 2: Calculate the value of the accrual
  swap.dates %>%
    purrr::pluck("accrual.yf") %>%
    `*`(swap$notional * rate * switch(leg.type, "pay" = -1, "receive" = 1))
}

This is smaller and easier function:

  • Part 1: for floating legs we use Quandl to download the needed data and extract
    the rate information. For the fixed one, we just use the strike of the swap.

  • Part 2: we calculate the actual accrual amount using the classical function

Let’s see the final result on the 25 years’ swap we use as test:

## # A tibble: 1 x 7
##   swap.id  clean.mv dirty.mv accrual.pay accrual.receive     par    pv01
##                                      
## 1 Swap 25y -881815. -874994.       5441.           1379. 0.00771 -12394.

Let’s compare the result with the Bloomberg screenshot


You can see that we now perfectly price also the receiving accrual and, of course,
also the total one.

We finish today’s post by looking at the pricing of the basket. You can see that
now the algorithm correctly identifies what are the floating and the fixed legs
depending on the type of the swap (ie receiver or payer).

Table 2: Pricing results
swap.id clean.mv dirty.mv accrual.pay accrual.receive par pv01
Swap 25y -881,814.61 -874,994.32 5,441.11 1,379.18 0.77% -12,393.65
Swap 30y 233,691.75 123,999.52 -97,222.22 -12,470.00 1.11% 20,867.00
Swap 10y 222,083.28 236,146.61 6,702.22 7,361.11 -0.14% -5,724.42
Swap 2y16y 360,095.21 360,095.21 -0.00 0.00 1.18% -11,163.37

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