swapbycir

Price swap instrument from Cox-Ingersoll-Ross interest-rate tree

Description

example

[Price,PriceTree,SwapRate] = swapbycir(CIRTree,LegRate,Settle,Maturity) prices a swap instrument from a Cox-Ingersoll-Ross (CIR) interest-rate tree. swapbycir computes prices of vanilla swaps, amortizing swaps, and forward swaps using a CIR++ model with the Nawalka-Beliaeva (NB) approach.

example

[Price,PriceTree,SwapRate] = swapbycir(___,Name,Value) adds additional name-value pair arguments.

Examples

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Define 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.

Basis = 0; 
Principal = 100;
LegRate = [0.06 20]; % [CouponRate Spread] 
LegType = [1 0]; % [Fixed Float] 
LegReset = [1 1]; % Payments once per year

Create a RateSpec using the intenvset function.

Rates = [0.035; 0.042147; 0.047345; 0.052707]; 
Dates = {'Jan-1-2017'; 'Jan-1-2018'; 'Jan-1-2019'; 'Jan-1-2020'; 'Jan-1-2021'}; 
ValuationDate = 'Jan-1-2017'; 
EndDates = Dates(2:end)'; 
Compounding = 1; 
RateSpec = intenvset('ValuationDate', ValuationDate, 'StartDates', ValuationDate, 'EndDates',EndDates,'Rates', Rates, 'Compounding', Compounding); 

Create a CIR tree.

NumPeriods = 5;  
Alpha = 0.03; 
Theta = 0.02;  
Sigma = 0.1;   
Settle = '01-Jan-2017'; 
Maturity = '01-Jan-2022'; 
CIRTimeSpec = cirtimespec(ValuationDate, Maturity, NumPeriods); 
CIRVolSpec = cirvolspec(Sigma, Alpha, Theta); 

CIRT = cirtree(CIRVolSpec, RateSpec, CIRTimeSpec)
CIRT = struct with fields:
      FinObj: 'CIRFwdTree'
     VolSpec: [1x1 struct]
    TimeSpec: [1x1 struct]
    RateSpec: [1x1 struct]
        tObs: [0 1 2 3 4]
        dObs: [736696 737061 737426 737791 738156]
     FwdTree: {1x5 cell}
     Connect: {[3x1 double]  [3x3 double]  [3x5 double]  [3x7 double]}
       Probs: {[3x1 double]  [3x3 double]  [3x5 double]  [3x7 double]}

Price the interest-rate swap.

[Price,PriceTree] = swapbycir(CIRT,LegRate,Settle,Maturity,'LegReset',LegReset,'Basis',3,'Principal',100,'LegType',LegType) 
Price = 2.5522
PriceTree = struct with fields:
     FinObj: 'CIRPriceTree'
       tObs: [0 1 2 3 4 5]
      PTree: {1x6 cell}
    Connect: {[3x1 double]  [3x3 double]  [3x5 double]  [3x7 double]}

Input Arguments

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Interest-rate tree structure, created by cirtree

Data Types: struct

Leg rate, specified as a NINST-by-2 matrix, with each row defined as one of the following:

  • [CouponRate Spread] (fixed-float)

  • [Spread CouponRate] (float-fixed)

  • [CouponRate CouponRate] (fixed-fixed)

  • [Spread Spread] (float-float)

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.

Data Types: double

Settlement date, specified either as a scalar or NINST-by-1 vector of serial date numbers, date character vectors, string arrays, or datetime arrays.

The Settle date for every swap is set to the ValuationDate of the CIR tree. The swap argument Settle is ignored.

Data Types: char | double | string | datetime

Maturity date, specified as a NINST-by-1 vector of serial date numbers, date character vectors, string arrays, or datetime arrays representing the maturity date for each swap.

Data Types: char | double | string | datetime

Name-Value Pair Arguments

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

Example: [Price,PriceTree,SwapRate] = swapbycir(CIRTree,LegRate,Settle,Maturity,LegReset,Basis,Principal,LegType)

Reset frequency per year for each swap, specified as the comma-separated pair consisting of 'LegReset' and a NINST-by-2 vector.

Data Types: double

Day-count basis representing the basis for each leg, specified as the comma-separated pair consisting of 'Basis' and a NINST-by-1 array (or NINST-by-2 if Basis is different for each leg).

  • 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 (ICMA)

  • 9 = actual/360 (ICMA)

  • 10 = actual/365 (ICMA)

  • 11 = 30/360E (ICMA)

  • 12 = actual/365 (ISDA)

  • 13 = BUS/252

For more information, see Basis.

Data Types: double

Notional principal amounts or principal value schedules, specified as the comma-separated pair consisting of 'Principal' and a vector or cell array.

Principal accepts a NINST-by-1 vector or NINST-by-1 cell array (or NINST-by-2 if Principal is different for each leg) of the notional principal amounts or principal value schedules. For schedules, each element of the cell array is a NumDates-by-2 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.

Data Types: cell | double

Leg type, specified as the comma-separated pair consisting of 'LegType' and a NINST-by-2 matrix with values:

  • [1 1] (fixed-fixed) swap

  • [1 0] (fixed-float) swap

  • [0 1] (float-fixed) swap

  • [0 0] (float-float) swap

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.

Data Types: double

End-of-month rule flag for generating dates when Maturity is an end-of-month date for a month having 30 or fewer days, specified as the comma-separated pair consisting of 'EndMonthRule' and a nonnegative integer [0, 1] using a NINST-by-1 (or NINST-by-2 if EndMonthRule is different for each leg).

  • 0 = Ignore rule, meaning that a payment date is always the same numerical day of the month.

  • 1 = Set rule on, meaning that a payment date is always the last actual day of the month.

Data Types: logical

Flag to adjust cash flows based on actual period day count, specified as the comma-separated pair consisting of 'AdjustCashFlowsBasis' and a NINST-by-1 (or NINST-by-2 if AdjustCashFlowsBasis is different for each leg) of logicals with values of 0 (false) or 1 (true).

Data Types: logical

Business day conventions, specified as the comma-separated pair consisting of 'BusinessDayConvention' and a character vector or a N-by-1 (or NINST-by-2 if BusinessDayConvention is different for each leg) cell array of character vectors of business day conventions. The selection for business day convention determines how nonbusiness days are treated. Nonbusiness days are defined as weekends plus any other date that businesses are not open (e.g. statutory holidays). Values are:

  • actual — Nonbusiness days are effectively ignored. Cash flows that fall on nonbusiness days are assumed to be distributed on the actual date.

  • follow — Cash flows that fall on a non-business day are assumed to be distributed on the following business day.

  • modifiedfollow — Cash flows that fall on a non-business day are assumed to be distributed on the following business day. However if the following business day is in a different month, the previous business day is adopted instead.

  • previous — Cash flows that fall on a non-business day are assumed to be distributed on the previous business day.

  • modifiedprevious — Cash flows that fall on a non-business day are assumed to be distributed on the previous business day. However if the previous business day is in a different month, the following business day is adopted instead.

Data Types: char | cell

Holidays used in computing business days, specified as the comma-separated pair consisting of 'Holidays' and MATLAB date numbers using a NHolidays-by-1 vector.

Data Types: double

Date swap actually starts, specified as the comma-separated pair consisting of 'StartDate' and a NINST-by-1 vector of dates using a serial date number, character vector, string array, or string array.

Use this argument to price forward swaps, that is, swaps that start in a future date

Data Types: char | double | string | datetime

Output Arguments

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Expected swap prices at time 0, returned as a NINST-by-1 vector.

Tree structure of instrument prices, returned as a MATLAB structure of trees containing vectors of swaption instrument prices and a vector of observation times for each node. Within PriceTree:

  • PriceTree.tObs contains the observation times.

  • PriceTree.PTree contains the clean prices.

Rates applicable to the fixed leg, returned as a NINST-by-1 vector of rates applicable to the fixed leg such that the swaps’ 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.

More About

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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.

Forward Swap

Agreement to enter into an interest-rate swap arrangement on a fixed date in future.

References

[1] Cox, J., Ingersoll, J., and S. Ross. "A Theory of the Term Structure of Interest Rates." Econometrica. Vol. 53, 1985.

[2] Brigo, D. and F. Mercurio. Interest Rate Models - Theory and Practice. Springer Finance, 2006.

[3] Hirsa, A. Computational Methods in Finance. CRC Press, 2012.

[4] Nawalka, S., Soto, G., and N. Beliaeva. Dynamic Term Structure Modeling. Wiley, 2007.

[5] Nelson, D. and K. Ramaswamy. "Simple Binomial Processes as Diffusion Approximations in Financial Models." The Review of Financial Studies. Vol 3. 1990, pp. 393–430.

Introduced in R2018a