oasbycir

Determine option adjusted spread using Cox-Ingersoll-Ross model

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.

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

Examples

Compute OAS Using a CIR Interest-Rate Tree

Create a RateSpec using the intenvset function.

ValuationDate = 'October-25-2018';
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 = '01-Jan-2023'; 
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 1.0462 2.0924 3.1386]
        dObs: [737358 737740 738122 738504]
     FwdTree: {[1.0373]  [1.0750 1.0443 1.0229]  [1x5 double]  [1x7 double]}
     Connect: {[3x1 double]  [3x3 double]  [3x5 double]}
       Probs: {[3x1 double]  [3x3 double]  [3x5 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 = 416.9457

OAD = 0.9282

Compute OAS for an Amortizing Callable Bond Using a CIR Interest-Rate Tree

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 = {'Jan-1-2017'; 'Jan-1-2018'; 'Jan-1-2019'; 'Jan-1-2020'; 'Jan-1-2021'}; 
ValuationDate = 'Jan-1-2016'; 
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 = '01-Jan-2019'; 
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 0.7500 1.5000 2.2500]
        dObs: [736330 736604 736878 737152]
     FwdTree: {[1.0187]  [1.0338 1.0188 1.0083]  [1x5 double]  [1x7 double]}
     Connect: {[3x1 double]  [3x3 double]  [3x5 double]}
       Probs: {[3x1 double]  [3x3 double]  [3x5 double]}

Define the callable bond.

BondSettlement = 'Jan-1-2016';
BondMaturity   = 'Jan-1-2020'; 
CouponRate = 0.035;
Period = 1;
OptSpec = 'call'; 
Strike = 100;  

 Face = { 
                 {'Jan-1-2018'  100; 
                  'Jan-1-2019'   70; 
                  'Jan-1-2020'   50};
                 };

ExerciseDates = {'Jan-1-2018' '01-Jan-2019'}; 

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 = 80.4801

Input Arguments

CIRTree — Interest-rate tree structure

structure

Interest-rate tree structure, specified by using cirtree.

Data Types: struct

Price — Market prices of bonds with embedded options

numeric

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

Data Types: double

CouponRate — Bond coupon rate

positive decimal value

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

Data Types: double

Settle — Settlement date

serial date number | date character vector | string array | datetime

Settlement date for the bond option, specified as a NINST-by-1 vector of serial date numbers, date character vectors, string arrays, or datetime arrays.

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.

Data Types: double | char | string | datetime

Maturity — Maturity date

serial date number | date character vector | string array | datetime

Maturity date, specified as an NINST-by-1 vector of serial date numbers, date character vectors, string arrays, or datetime arrays.

Data Types: double | char | string | datetime

OptSpec — Definition of option

character vector with value 'call' or 'put' | cell array of character vectors with values 'call' or 'put' | string array with values "call" or "put"

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

Strike — Option strike price values

nonnegative integer

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

ExerciseDates — Option exercise dates

serial date number | date character vector | string array | datetime

Option exercise dates, specified as a NINST-by-1, NINST-by-2, or NINST-by-NSTRIKES using serial date numbers, date character vectors, string arrays, or datetime arrays 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.

.

Data Types: double | char | 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: OAS = oasbycir(CIRTree,Price,CouponRate,Settle,Maturity,OptSpec,Strike,ExerciseDates,'Period',4)

'AmericanOpt' — Option type

0 European/Bermuda (default) | integer with values 0 or 1

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

'Period' — Coupons per year

2 per year (default) | vector

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

Data Types: double

'Basis' — Day-count basis

0 (actual/actual) (default) | integer from 0 to 13

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

'EndMonthRule' — End-of-month rule flag

1 (in effect) (default) | nonnegative integer with values 0 or 1

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

'IssueDate' — Bond issue date

serial date number | date character vector | string array | datetime

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

Data Types: double | char | string | datetime

'FirstCouponDate' — Irregular first coupon date

serial date number | date character vector | string array | datetime

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

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.

Data Types: double | char | string | datetime

'LastCouponDate' — Irregular last coupon date

serial date number | date character vector | string array | datetime

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

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.

Data Types: char | double | string | datetime

'StartDate' — Forward starting date of payments

serial date number | date character vector | string array | datetime

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 serial date numbers, date character vectors, string arrays, or datetime array.

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

Data Types: char | double | string | datetime

'Face' — Face value

100 (default) | NINST-by-1 vector | NINST-by-1 cell array

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 and the second column is associated face value. The date indicates the last day that the face value is valid.

Data Types: double

'BondType' — Type of underlying bond

'vanilla' for scalar Face values, 'callablesinking' for scheduled Face values (default) | cell array of character vectors with values 'vanilla','amortizing', or 'callablesinking' | string array with values "vanilla", "amortizing", or "callablesinking"

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

OAS — Option adjusted spread

vector

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

OAD — Option adjusted duration

vector

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

OAC — Option adjusted convexity

vector

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

More About

Vanilla Bond with Embedded Option

A vanilla coupon bond is a security representing an obligation to repay a borrowed amount at a designated time and to make periodic interest payments until that time.

The issuer of a bond makes the periodic interest payments until the bond matures. At maturity, the issuer pays to the holder of the bond the principal amount owed (face value) and the last interest payment. A vanilla bond with an embedded option is where an option contract has an underlying asset of a vanilla bond.

Stepped Coupon Bond with Callable and Puttable Features

A step-up and step-down bond is a debt security with a predetermined coupon structure over time.

With these instruments, coupons increase (step up) or decrease (step down) at specific times during the life of the bond. Stepped coupon bonds can have options features (call and puts).

Sinking Fund Bond with Call Embedded Option

A sinking fund bond is a coupon bond with a sinking fund provision.

This provision obligates the issuer to amortize portions of the principal prior to maturity, affecting bond prices since the time of the principal repayment changes. This means that investors receive the coupon and a portion of the principal paid back over time. These types of bonds reduce credit risk, since it lowers the probability of investors not receiving their principal payment at maturity.

The bond may have a sinking fund call option provision allowing the issuer to retire the sinking fund obligation either by purchasing the bonds to be redeemed from the market or by calling the bond via a sinking fund call, whichever is cheaper. If interest rates are high, then the issuer buys back the requirement amount of bonds from the market since bonds are cheap, but if interest rates are low (bond prices are high), then most likely the issuer is buying the bonds at the call price. Unlike a call feature, however, if a bond has a sinking fund call option provision, it is an obligation, not an option, for the issuer to buy back the increments of the issue as stated. Because of this, a sinking fund bond trades at a lower price than a non-sinking fund bond.

Amortizing Callable or Puttable Bond

Amortizing callable or puttable bonds work under a scheduled Face.

An amortizing callable bond gives the issuer the right to call back the bond, but instead of paying the Face amount at maturity, it repays part of the principal along with the coupon payments. An amortizing puttable bond, repays part of the principal along with the coupon payments and gives the bondholder the right to sell the bond back to the issuer.

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.