oasbycir

Determine option adjusted spread using Cox-Ingersoll-Ross model

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Syntax

[OAS,OAD,OAC] = oasbycir(CIRTree,Price,CouponRate,Settle,Maturity,OptSpec,Strike,ExerciseDates)
[OAS,OAD,OAC] = oasbycir(___,Name,Value)

Description

[OAS,OAD,OAC] = oasbycir(CIRTree,Price,CouponRate,Settle,Maturity,OptSpec,Strike,ExerciseDates) calculates the option adjusted spread from a Cox-Ingersoll-Ross (CIR) interest-rate tree using a CIR++ model with the Nawalka-Beliaeva (NB) approach.

oasbycir computes prices of vanilla bonds with embedded options, stepped coupon bonds with embedded options, amortizing bonds with embedded options, and sinking fund bonds with embedded option. For more information, see More About.

example

[OAS,OAD,OAC] = oasbycir(___,Name,Value) adds optional name-value pair arguments.

example

Examples

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Open Live Script

Create a RateSpec using the intenvset function. 

ValuationDate = datetime(2018,10,25);
Rates = [0.0355; 0.0382; 0.0427; 0.0489];
StartDates = ValuationDate;
EndDates = datemnth(ValuationDate, 12:12:48)';
Compounding = 1;

RateSpec = intenvset('ValuationDate', ValuationDate, 'StartDates', ValuationDate, 'EndDates',EndDates,'Rates', Rates, 'Compounding', Compounding);

Create a CIR tree.

NumPeriods = length(EndDates); 
Alpha = 0.03; 
Theta = 0.02;  
Sigma = 0.1;    
Maturity = datetime(2023,10,25); 
CIRTimeSpec = cirtimespec(ValuationDate, Maturity, NumPeriods); 
CIRVolSpec = cirvolspec(Sigma, Alpha, Theta); 

CIRT = cirtree(CIRVolSpec, RateSpec, CIRTimeSpec)
CIRT = struct with fields:
      FinObj: 'CIRFwdTree'
     VolSpec: [1×1 struct]
    TimeSpec: [1×1 struct]
    RateSpec: [1×1 struct]
        tObs: [0 1.2500 2.5000 3.7500]
        dObs: [737358 737814 738271 738727]
     FwdTree: {[1.0454]  [1.0952 1.0574 1.0312]  [1.1706 1.1188 1.0802 1.0534 1.0376]  [1.2285 1.1624 1.1110 1.0726 1.0460 1.0304 1.0252]}
     Connect: {[3×1 double]  [3×3 double]  [3×5 double]}
       Probs: {[3×1 double]  [3×3 double]  [3×5 double]}

Define the OAS instrument.

CouponRate = 0.045;
Settle = ValuationDate;
Maturity = '25-October-2019';
OptSpec = 'call';
Strike = 100;
ExerciseDates = {'25-October-2018','25-October-2019'};
Period = 1;
AmericanOpt = 0;
Price = 97;

Compute the OAS.

[OAS,OAD] = oasbycir(CIRT,Price,CouponRate,Settle,Maturity,OptSpec,Strike,ExerciseDates,'Period',Period,'AmericanOpt',AmericanOpt)
OAS = 
411.4425

OAD = 
0.9282
Open Live Script

his example shows how to compute the OAS for an amortizing callable bond using a CIR lattice model.

Create a RateSpec using the intenvset function. 

Rates = [0.025; 0.032; 0.037; 0.042]; 
Dates = [datetime(2017,1,1) ; datetime(2018,1,1) ; datetime(2019,1,1) ; datetime(2020,1,1) ; datetime(2021,1,1)]; 
ValuationDate = datetime(2016,1,1); 
EndDates = Dates(2:end)'; 
Compounding = 1; 
RateSpec = intenvset('ValuationDate', ValuationDate, 'StartDates', ValuationDate, 'EndDates',EndDates,'Rates', Rates, 'Compounding', Compounding);

Create a CIR tree.

NumPeriods = length(EndDates); 
Alpha = 0.03; 
Theta = 0.02;  
Sigma = 0.1;
Maturity = datetime(2019,1,1); 
CIRTimeSpec = cirtimespec(ValuationDate, Maturity, NumPeriods); 
CIRVolSpec = cirvolspec(Sigma, Alpha, Theta); 
CIRT = cirtree(CIRVolSpec, RateSpec, CIRTimeSpec)
CIRT = struct with fields:
      FinObj: 'CIRFwdTree'
     VolSpec: [1×1 struct]
    TimeSpec: [1×1 struct]
    RateSpec: [1×1 struct]
        tObs: [0 0.7500 1.5000 2.2500]
        dObs: [736330 736604 736878 737152]
     FwdTree: {[1.0187]  [1.0338 1.0188 1.0083]  [1.0577 1.0380 1.0230 1.0124 1.0061]  [1.0964 1.0716 1.0517 1.0364 1.0257 1.0193 1.0172]}
     Connect: {[3×1 double]  [3×3 double]  [3×5 double]}
       Probs: {[3×1 double]  [3×3 double]  [3×5 double]}

Define the callable bond.

BondSettlement = datetime(2016,1,1);
BondMaturity   = datetime(2020,1,1); 
CouponRate = 0.035;
Period = 1;
OptSpec = 'call'; 
Strike = 100;  

 Face = { 
                 {datetime(2018,1,1) 100; 
                  datetime(2019,1,1) 70; 
                  datetime(2020,1,1) 50};
                 };

ExerciseDates = [datetime(2018,1,1) ; datetime(2019,1,1)];

Compute OAS for a callable amortizing bond using the CIR tree.

Price = 99;
BondType = 'amortizing';
OAS = oasbycir(CIRT, Price, CouponRate, BondSettlement, Maturity,...
OptSpec, Strike, ExerciseDates, 'Period', Period, 'Face', Face,'BondType', BondType)
OAS = 2×1

   80.4801
   84.3684

Input Arguments

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Interest-rate tree structure, specified by using cirtree.

Data Types: struct

Market prices of bonds with embedded options, specified as an NINST-by-1 vector.

Data Types: double

Bond coupon rate, specified as an NINST-by-1 decimal annual rate.

Data Types: double

Settlement date for the bond option, specified as a NINST-by-1 vector using a datetime array, string array, or date character vectors.

Note

The Settle date for every bond with an embedded option is set to the ValuationDate of the CIR tree. The bond argument Settle is ignored.

To support existing code, oasbycir also accepts serial date numbers as inputs, but they are not recommended.

Maturity date, specified as an NINST-by-1 vector using a datetime array, string array, or date character vectors.

To support existing code, oasbycir also accepts serial date numbers as inputs, but they are not recommended.

Definition of option, specified as a NINST-by-1 cell array of character vectors or string arrays with values 'call' or 'put'.

Data Types: char | cell | string

Option strike price value, specified as a NINST-by-1 or NINST-by-NSTRIKES depending on the type of option:

  • European option — NINST-by-1 vector of strike price values.

  • Bermuda option — NINST by number of strikes (NSTRIKES) matrix of strike price values. Each row is the schedule for one option. If an option has fewer than NSTRIKES exercise opportunities, the end of the row is padded with NaNs.

  • American option — NINST-by-1 vector of strike price values for each option.

Data Types: double

Option exercise dates, specified as a NINST-by-1, NINST-by-2, or NINST-by-NSTRIKES vector using a datetime array, string array, or date character vectors, depending on the type of option:

  • For a European option, use a NINST-by-1 vector of dates. For a European option, there is only one ExerciseDates on the option expiry date.

  • For a Bermuda option, use a NINST-by-NSTRIKES vector of dates.

  • For an American option, use a NINST-by-2 vector of exercise date boundaries. The option can be exercised on any date between or including the pair of dates on that row. If only one non-NaN date is listed, or if ExerciseDates is a NINST-by-1 vector, the option can be exercised between ValuationDate of the stock tree and the single listed ExerciseDates.

.

To support existing code, oasbycir also accepts serial date numbers as inputs, but they are not recommended.

Name-Value Arguments

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Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: OAS = oasbycir(CIRTree,Price,CouponRate,Settle,Maturity,OptSpec,Strike,ExerciseDates,'Period',4)

Option type, specified as the comma-separated pair consisting of 'AmericanOpt' and a NINST-by-1 positive integer flags with values:

  • 0 — European/Bermuda

  • 1 — American

Data Types: double

Coupons per year, specified as the comma-separated pair consisting of 'Period' and a NINST-by-1 vector.

Data Types: double

Day-count basis, specified as the comma-separated pair consisting of 'Basis' and a NINST-by-1 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 (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

End-of-month rule flag, specified as the comma-separated pair consisting of 'EndMonthRule' and a nonnegative integer using a 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.

Data Types: double

Bond issue date, specified as the comma-separated pair consisting of 'IssueDate' and a NINST-by-1 vector using a datetime array, string array, or date character vectors.

To support existing code, oasbycir also accepts serial date numbers as inputs, but they are not recommended.

Irregular first coupon date, specified as the comma-separated pair consisting of 'FirstCouponDate' and a NINST-by-1 vector using a datetime array, string array, or date character vectors.

To support existing code, oasbycir also accepts serial date numbers as inputs, but they are not recommended.

When FirstCouponDate and LastCouponDate are both specified, FirstCouponDate takes precedence in determining the coupon payment structure. If you do not specify a FirstCouponDate, the cash flow payment dates are determined from other inputs.

Irregular last coupon date, specified as the comma-separated pair consisting of 'LastCouponDate' and a NINST-by-1 vector using a datetime array, string array, or date character vectors.

To support existing code, oasbycir also accepts serial date numbers as inputs, but they are not recommended.

In the absence of a specified FirstCouponDate, a specified LastCouponDate determines the coupon structure of the bond. The coupon structure of a bond is truncated at the LastCouponDate, regardless of where it falls, and is followed only by the bond's maturity cash flow date. If you do not specify a LastCouponDate, the cash flow payment dates are determined from other inputs.

Forward starting date of payments (the date from which a bond cash flow is considered), specified as the comma-separated pair consisting of 'StartDate' and a NINST-by-1 vector using a datetime array, string array, or date character vectors.

To support existing code, oasbycir also accepts serial date numbers as inputs, but they are not recommended.

If you do not specify StartDate, the effective start date is the Settle date.

Face or par value, specified as the comma-separated pair consisting of 'Face' and a NINST-by-1 vector or a NINST-by-1 cell array where each element is a NumDates-by-2 cell array where the first column is dates using a datetime, string, or date character vector, and the second column is associated face value. The date indicates the last day that the face value is valid.

Data Types: double | char | string | datetime

Type of underlying bond, specified as the comma-separated pair consisting of 'BondType' and a NINST-by-1 cell array of character vectors or string array specifying if the underlying is a vanilla bond, an amortizing bond, or a callable sinking fund bond. The supported types are:

  • 'vanilla' is a standard callable or puttable bond with a scalar Face value and a single coupon or stepped coupons.

  • 'callablesinking' is a bond with a schedule of Face values and a sinking fund call provision with a single or stepped coupons.

  • 'amortizing' is an amortizing callable or puttable bond with a schedule of Face values with single or stepped coupons.

Data Types: char | string

Output Arguments

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Option adjusted spread in basis points, returned as a NINST-by-1 vector.

Option adjusted duration, returned as a NINST-by-1 vector.

Option adjusted convexity, returned as a NINST-by-1 vector.

More About

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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.

Version History

Introduced in R2018a

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See Also

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