swapbyzero - Price swap instrument from set of zero curves

Syntax

[Price, SwapRate AI, RecCF, RecCFDates, PayCF, PayCFDates] = swapbyzero(RateSpec, LegRate, Settle, Maturity) [Price, SwapRate AI, RecCF, RecCFDates, PayCF, PayCFDates] = swapbyzero(RateSpec, LegRate, Settle, Maturity, LegReset, Basis, Principal, LegType, EndMonthRule) [Price, SwapRate, AI, RecCF, RecCFDates, PayCF, PayCFDates] = swapbyzero(RateSpec, LegRate, Settle, Maturity, Name, Value)

Description

[Price, SwapRate AI, RecCF, RecCFDates, PayCF, PayCFDates] =swapbyzero(RateSpec, LegRate, Settle, Maturity) prices a swap instrument from a set of zero coupon bond rates. All inputs are either scalars or NINST-by-1 vectors unless otherwise specified. Any date can be a serial date number or date string. An optional argument can be passed as an empty matrix [].

[Price, SwapRate AI, RecCF, RecCFDates, PayCF, PayCFDates] =swapbyzero(RateSpec, LegRate, Settle, Maturity, LegReset, Basis, Principal, LegType, EndMonthRule) prices a swap instrument from a set of zero coupon bond rates with optional input arguments. All inputs are either scalars or NINST-by-1 vectors unless otherwise specified. Any date can be a serial date number or date string. An optional argument can be passed as an empty matrix [].

[Price, SwapRate, AI, RecCF, RecCFDates, PayCF, PayCFDates] = swapbyzero(RateSpec, LegRate, Settle, Maturity, Name, Value) prices a swap instrument from a set of zero coupon bond rates with additional options specified by one or more Name, Value pair arguments.

Input Arguments

`RateSpec`	Structure containing the properties of an interest-rate structure. See `intenvset` for information on creating `RateSpec`. `RateSpec` can be a `NINST`-by-`2` input variable of `RateSpec`s, with the second input being the discount curve for the paying leg if different than the receiving leg. If only one curve is specified, than it is used to discount both legs.
`LegRate`	Number of instruments (`NINST`)-by-`2` matrix, with each row defined as: `[CouponRate Spread]` or `[Spread CouponRate]` `CouponRate` is the decimal annual rate. `Spread` is the number of basis points over the reference rate. The first column represents the receiving leg, while the second column represents the paying leg.
`Settle`	Settlement date. `NINST`-by-`1` vector of serial date numbers or date strings representing the settlement date for each swap. `Settle` must be earlier than `Maturity`.
`Maturity`	Maturity date. `NINST`-by-`1` vector of dates representing the maturity date for each swap.

Ordered Input or Name-Value Pair Arguments

Enter the following optional inputs using an ordered syntax or as name-value pair arguments. You cannot mix ordered syntax with name-value pair arguments.

`LegReset`	`NINST`-by-`2` matrix representing the reset frequency per year for each swap. `NINST`-by-`1` vector representing the frequency of payments per year. Default: `[1 1]`
`Basis`	Day-count basis of the instrument. A vector of integers. 0 = actual/actual 1 = 30/360 (SIA) 2 = actual/360 3 = actual/365 4 = 30/360 (PSA) 5 = 30/360 (ISDA) 6 = 30/360 (European) 7 = actual/365 (Japanese) 8 = actual/actual (ISMA) 9 = actual/360 (ISMA) 10 = actual/365 (ISMA) 11 = 30/360E (ISMA) 12 = actual/365 (ISDA) 13 = BUS/252 For more information, see basis. Default: `0` (actual/actual)
`Principal`	`NINST`-by-`1` vector or `NINST`-by-`1` cell array of the notional principal amounts or principal value schedules. For the latter case, each element of the cell array is a `NumDates`-by-`2` call array where the first column is dates and the second column is its associated notional principal value. The date indicates the last day that the principal value is valid. Default: `100`
`LegType`	`NINST`-by-`2` matrix. Each row represents an instrument. Each column indicates if the corresponding leg is fixed (`1`) or floating (`0`). This matrix defines the interpretation of the values entered in `LegRate`. Default: `[1 0]` for each instrument
`Options`	Derivatives pricing options structure created with `derivset`.
`EndMonthRule`	End-of-month rule. `NINST`-by-`1` vector. This rule applies only when `Maturity` is an end-of-month date for a month having 30 or fewer days. `0` = Ignore rule, meaning that a bond coupon payment date is always the same numerical day of the month. `1` = Set rule on, meaning that a bond coupon payment date is always the last actual day of the month. Default: `1`

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments, where Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

`AdjustCashFlowsBasis`	Adjust the cash flows based on the actual period day count. `NINST`-by-`1` of logicals. Default: `false`
`BusinessDayConvention`	Require payment dates to be business dates. `NINST`-by-`1` cell array with possible choices of business day convention: `actual` `follow` `modifiedfollow` `previous` `modifiedprevious` Default: `actual`
`ForwardRateSpec`	Forward rate spec to use in generating cash flows. Default: If not specified, then the `RateSpec` is used both for discounting cash flows and generating floating cash flows.
`Holidays`	Holidays used for business day convention. `NHOLIDAYS`-by-`1` of MATLAB date numbers. Default: If none specified, `holidays.m` is used.
`LatestFloatingRate`	Rate for the next floating payment, set at the last reset date. `NINST`-by-`1` of scalars. Default: If not specified, then the `RateSpec` must contain this information.

Output Arguments

Price

Number of instruments (NINST) by number of curves (NUMCURVES) matrix of swap prices. Each column arises from one of the zero curves.

SwapRate

NINST-by-NUMCURVES matrix of rates applicable to the fixed leg such that the swap's values are zero at time 0. This rate is used in calculating the swaps' prices when the rate specified for the fixed leg in LegRate is NaN. The SwapRate output is padded with NaN for those instruments in which CouponRate is not set to NaN.

Output cash flows, cash flow dates, and accrued interest.

AI

NINST-by-NUMCURVES matrix of accrued interest.

RecCF

NINST-by-NUMCURVES matrix of cash flows for the receiving leg.

Note If there is more than one curve specified in the RateSpec input, then the first NCURVES row corresponds to the first swap, the second NCURVES row correspond to the second swap, and so on.

RecCFDates

NINST-by-NUMCURVES matrix of payment dates for the receiving leg.

PayCF

NINST-by-NUMCURVES matrix of cash flows for the paying leg.

PayCFDates

NINST-by-NUMCURVES matrix of payment dates for the paying leg.

Definitions

Amortizing Swap

In an amortizing swap, the notional principal decreases periodically because it is tied to an underlying financial instrument with a declining (amortizing) principal balance, such as a mortgage.

Examples

Price an Interest-Rate Swap

Price an interest-rate swap with a fixed receiving leg and a floating paying leg. Payments are made once a year, and the notional principal amount is $100. The values for the remaining arguments are:

Coupon rate for fixed leg: 0.06 (6%)
Spread for floating leg: 20 basis points
Swap settlement date: Jan. 01, 2000
Swap maturity date: Jan. 01, 2003

Based on the information above, set the required arguments and build the LegRate, LegType, and LegReset matrices:

Settle = '01-Jan-2000';
Maturity = '01-Jan-2003';
Basis = 0; 
Principal = 100;
LegRate = [0.06 20]; % [CouponRate Spread] 
LegType = [1 0]; % [Fixed Float] 
LegReset = [1 1]; % Payments once per year

Load the file deriv.mat, which provides ZeroRateSpec, the interest-rate term structure needed to price the bond.

load deriv.mat;

Use swapbyzero to compute the price of the swap.

Price = swapbyzero(ZeroRateSpec, LegRate, Settle, Maturity,... 
LegReset, Basis, Principal, LegType)

Price = 
   3.6923

Using the previous data, calculate the swap rate, which is the coupon rate for the fixed leg, such that the swap price at time = 0 is zero.

LegRate = [NaN 20]; 

[Price, SwapRate] = swapbyzero(ZeroRateSpec, LegRate, Settle,...
Maturity, LegReset, Basis, Principal, LegType)

Price = 

 -1.4211e-014

SwapRate = 
   0.0466

Use swapbyzero with name-value pair arguments for LegRate, LegType, LatestFloatingRate, AdjustCashFlowsBasis, and BusinessDayConvention to calculate output for Price, SwapRate, AI, RecCF, RecCFDates, PayCF, and PayCFDates:

Settle = datenum('08-Jun-2010');
RateSpec = intenvset('Rates', [.005 .0075 .01 .014 .02 .025 .03]',...
'StartDates',Settle, 'EndDates',{'08-Dec-2010','08-Jun-2011',...
'08-Jun-2012','08-Jun-2013','08-Jun-2015','08-Jun-2017','08-Jun-2020'}');
Maturity = datenum('15-Sep-2020');
LegRate = [.025 50];
LegType = [1 0]; % fixed/floating
LatestFloatingRate = .005;
 
[Price, SwapRate, AI, RecCF, RecCFDates, PayCF,PayCFDates] = ...
swapbyzero(RateSpec, LegRate, Settle, Maturity,'LegType',LegType,...
'LatestFloatingRate',LatestFloatingRate,'AdjustCashFlowsBasis',true,...
'BusinessDayConvention','modifiedfollow')

Price =

   -3.3937

SwapRate =

   NaN

AI =

    1.4575


RecCF =

Columns 1 through 10

-1.8219    1.2603    1.2603    1.2740    1.2671   1.2466    1.2534    1.2603   1.2603  1.2740

Columns 11 through 12

1.2671  101.2534


RecCFDates =

Columns 1 through 8

734297      734396      734761     735129     735493      735857    736222   736588

Columns 9 through 12

736953      737320      737684      738049


PayCF =

Columns 1 through 10

-0.3644    0.2521    0.7082    1.0116    1.4423    1.6380   1.9161    2.1038  2.2768  2.2766

Columns 11 through 12

2.4370  102.3432


PayCFDates =

Columns 1 through 8

734297      734396      734761      735129      735493      735857    736222     736588

Columns 9 through 12

736953      737320      737684      738049

Price Swaps By Specifying Multiple Term Structures Using RateSpec

Price three swaps using two interest-rate curves.

Define data for the interest-rate term structure.

Settle = '01-Jan-2000';
Maturity = '01-Jan-2003';
Basis = 0; 
Principal = [100;50;100]; %three notional amounts
LegRate = [0.06 20]; % [CouponRate Spread] 
LegType = [1 0]; % [Fixed Float] 
LegReset = [1 1]; % Payments once per year

Load the data in deriv.mat.

load deriv.mat

Create the RateSpec.

ZeroRateSpecNew = intenvset(ZeroRateSpec, 'Rates', [ZeroRateSpec.Rates,ZeroRateSpec.Rates]);

ZeroRateSpecNew = 

           FinObj: 'RateSpec'
      Compounding: 1
             Disc: [4x2 double]
            Rates: [4x2 double]
         EndTimes: [4x1 double]
       StartTimes: [4x1 double]
         EndDates: [4x1 double]
       StartDates: 730486
    ValuationDate: 730486
            Basis: 0
     EndMonthRule: 1

Price three swaps using one curve.

Price = swapbyzero(ZeroRateSpec, LegRate, Settle, Maturity,... 
LegReset, Basis, Principal, LegType)

Price =

   3.692309149501682
   1.846154574750841
   3.692309149501682

Price three swaps using two curves.

Price = swapbyzero(ZeroRateSpecNew, LegRate, Settle, Maturity,...
LegReset, Basis, Principal, LegType)

Price =

   3.692309149501682   3.692309149501682
   1.846154574750841   1.846154574750841
   3.692309149501682   3.692309149501682

Price an Amortizing Swap

Price an amortizing swap using the Principal input argument to define the amortization schedule.