oasbyhw

Determine option adjusted spread using Hull-White model

Syntax

[OAS,OAD,OAC] = oasbyhw(HWTree,Price,CouponRate,Settle,Maturity,OptSpec,Strike,ExerciseDates)

[OAS,OAD,OAC] = oasbyhw(___,Name,Value)

Description

[OAS,OAD,OAC] = oasbyhw(HWTree,Price,CouponRate,Settle,Maturity,OptSpec,Strike,ExerciseDates) calculates option adjusted spread using a Hull-White model.

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

Note

Alternatively, you can use oas with an OptionEmbeddedFixedBond instrument using an IRTree pricer and a HullWhite model. For more information, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.

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

Examples

Compute OAS and OAD Using the Hull-White (HW) Model

This example shows how to compute OAS and OAD using the Hull-White (HW) model using the following data.

ValuationDate = datetime(2010,10,25);
Rates = [0.0355; 0.0382; 0.0427; 0.0489];
StartDates = ValuationDate;
EndDates = datemnth(ValuationDate, 12:12:48)';
Compounding = 1;

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

% specify VolsSpec and TimeSpec
Sigma = 0.05;
Alpha = 0.01;
VS = hwvolspec(ValuationDate, EndDates, Sigma*ones(size(EndDates)),...
EndDates, Alpha*ones(size(EndDates)));
TS = hwtimespec(ValuationDate, EndDates, Compounding);

% build the HW tree
HWTree = hwtree(VS, RateSpec, TS);

% instrument information
CouponRate = 0.045;
Settle = ValuationDate;
Maturity = datetime(2014,10,25);
OptSpec = 'call';
Strike = 100;
ExerciseDates = [datetime(2010,10,25) ; datetime(2013,10,25)];
Period = 1;
AmericanOpt = 0;
Price = 97;

% compute the OAS
[OAS, OAD] = oasbyhw(HWTree, Price, CouponRate, Settle, Maturity,...
OptSpec, Strike, ExerciseDates, 'Period', Period, 'AmericanOpt', AmericanOpt)

OAS = 2×1

   52.3118
  -12.8538

OAD = 2×1

    3.5507
    3.2910

Compute the OAS to Measure Cost of an Embedded Option Relative to a Risk-Free Curve

This example shows how to compute the price of a callable bond using a Hull-White tree.

Use the following bond data:

Settle = datetime(2014,8,20);

% Bond Properties
Maturity = datetime(2034,4,1);
CouponRate = .0625;
CallDates = datemnth('01-Oct-2014',6*(0:19));
CallStrikes = [102.85 102.7 102.55 102.4 102.25 102.1 101.95 101.8 ...
    101.65 101.5 101.35 101.2 101.05 100.9 100.75 100.6 100.45 100.3 ...
    100.15 100];

Use the following zero-curve data:

CurveDates = datemnth(Settle,12*[1 2 3 5 7 10 20 30]');
ZeroRates = [.11 0.30 0.64 1.44 2.07 2.61 3.29 3.55]'/100;

Define the RateSpec and build the HW tree.

RateSpec = intenvset('StartDate',Settle,'EndDates',CurveDates,'Rates',ZeroRates);

% HW Model Properties
alpha = .1;
sigma = .01;

TimeSpec = hwtimespec(Settle,cfdates(Settle,Maturity,12),2);
VolSpec = hwvolspec(Settle,Maturity,sigma,Maturity,alpha);

HWTree = hwtree(VolSpec,RateSpec,TimeSpec,'method','HW2000')

HWTree = struct with fields:
      FinObj: 'HWFwdTree'
     VolSpec: [1×1 struct]
    TimeSpec: [1×1 struct]
    RateSpec: [1×1 struct]
        tObs: [0 0.0652 0.2295 0.3967 0.5628 0.7283 0.8967 1.0652 1.2295 1.3967 1.5628 1.7283 1.8967 2.0652 2.2295 2.3967 2.5628 2.7283 2.8967 3.0652 3.2295 3.3967 3.5628 3.7283 3.8967 4.0652 4.2295 4.3967 4.5628 4.7283 4.8967 5.0652 … ] (1×236 double)
        dObs: [735831 735843 735873 735904 735934 735965 735996 736024 736055 736085 736116 736146 736177 736208 736238 736269 736299 736330 736361 736390 736421 736451 736482 736512 736543 736574 736604 736635 736665 736696 736727 … ] (1×236 double)
      CFlowT: {1×236 cell}
       Probs: {1×235 cell}
     Connect: {1×235 cell}
     FwdTree: {1×236 cell}

Compute the OAS for the bond.

Price = 103.25;
OAS = oasbyhw(HWTree, Price, CouponRate, Settle, Maturity, 'call', CallStrikes, CallDates)

OAS = 
234.8209

If you want to compute an OAS that only measures the option cost, you can pass in an issuer-specific curve instead of a risk-free curve (this would be done in the RateSpec argument).

Compute OAS for an Amortizing Callable Bond Using an HW Interest-Rate Tree Model

This example shows how to compute the OAS for an amortizing callable bond using an HW lattice model.

Create a RateSpec.

Rates = [0.035; 0.042147; 0.047345; 0.052707];
ValuationDate = 'Jan-1-2012';
StartDates = ValuationDate;
EndDates = {'Jan-1-2013'; 'Jan-1-2014'; 'Jan-1-2015'; 'Jan-1-2016'};
Compounding = 1;
RS = intenvset('ValuationDate', ValuationDate, 'StartDates', StartDates,...
'EndDates', EndDates,'Rates', Rates, 'Compounding', Compounding);

Build a HW tree.

VolDates = ['1-Jan-2013'; '1-Jan-2014'; '1-Jan-2015'; '1-Jan-2016'];
VolCurve = 0.01;
AlphaDates = '01-01-2016';
AlphaCurve = 0.1;

HWVolSpec = hwvolspec(RS.ValuationDate, VolDates, VolCurve,... 
AlphaDates, AlphaCurve);
HWTimeSpec = hwtimespec(RS.ValuationDate, EndDates, Compounding);
HWT = hwtree(HWVolSpec, RS, HWTimeSpec);

Define the callable bond.

CouponRate = 0.05;
Settle = 'Jan-1-2012';
Maturity =  'Jan-1-2016';
Period = 1;
    
    Face = { 
                 {'Jan-1-2014'  100; 
                  'Jan-1-2015'   70;
                  'Jan-1-2016'   50};
                 };
 
OptSpec = 'call'; 
Strike = [97 95 93];
ExerciseDates ={'Jan-1-2014' 'Jan-1-2015' 'Jan-1-2016'};

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

Price = 55;
BondType = 'amortizing';
OAS = oasbyhw(HWT,Price, CouponRate, Settle, Maturity,...
OptSpec, Strike, ExerciseDates, 'Period', Period, 'Face', Face,'BondType', BondType)

OAS = 
2.4023e+03

Compute OAS for a Stepped Coupon Bond Using an HW Interest-Rate Tree Model

This example shows how to compute the OAS for a stepped callable bond using an HW lattice model.

Create a RateSpec for a zero curve using intenvset.

Settle = datenum('15-Jun-2020');
ZeroRates = [0.0052 0.0055 0.0061 0.0073 0.0094 0.0119 0.0168 0.0222 0.0293 0.0307]';
ZeroDates = datemnth(Settle,[6 12*[1 2 3 4 5 7 10 20 30]]');
myRC = intenvset('Rates',ZeroRates,'StartDate',Settle,'EndDates',ZeroDates,...
'Basis',3,'Compounding',1);

Define the stepped callable bond.

CouponRate = {{'15-Jun-2025' .02;'15-Jun-2040' .03}};
Maturity = datetime(2040,6,15);
Price = 90;
ExerciseDates = [datetime(2034,6,15) datetime(2035,6,15) datetime(2036,6,15)];
Strikes = 100;
Period = 1;
OptSpec = 'call';

Build an HW tree using hwtree.

AlphaHW = .1;
VolHW = .01;
myVolSpec = hwvolspec(Settle,Maturity,VolHW,Maturity,AlphaHW);
myTimeSpec = hwtimespec(Settle,cfdates(Settle,Maturity,Period),Period);
myHWTree = hwtree(myVolSpec,myRC,myTimeSpec);

Compute the OAS for a callable stepped bond using the HW tree.

[OAS,OAD,OAC] = oasbyhw(myHWTree,Price,CouponRate,Settle,Maturity,OptSpec,Strikes, ...
ExerciseDates,'Period',Period)

OAS = 
55.4323

OAD = 
14.3161

OAC = 
125.6324

Input Arguments

HWTree — Interest-rate tree structure

structure

Interest-rate tree structure, specified by using hwtree.

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

datetime array | string array | date character vector

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 HW tree. The bond argument Settle is ignored.

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

Maturity — Maturity date

datetime array | string array | date character vector

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

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

OptSpec — Definition of option

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

Definition of option, specified as a NINST-by-1 cell array of character vectors.

Data Types: char | cell

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

datetime array | string array | date character vector

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. Each row is the schedule for one option.

• 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 is exercised between the underlying bond Settle date and the single listed exercise date.

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

Name-Value Arguments

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 = oasbybk(BDTTree,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 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

datetime array | string array | date character vector

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, oasbyhw also accepts serial date numbers as inputs, but they are not recommended.

FirstCouponDate — Irregular first coupon date

datetime array | string array | date character vector

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, oasbyhw 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.

LastCouponDate — Irregular last coupon date

datetime array | string array | date character vector

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, oasbyhw 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.

StartDate — Forward starting date of payments

datetime array | string array | date character vector

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, oasbyhw 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 — 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 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

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

Options — Derivatives pricing options

structure

Derivatives pricing options, specified as the comma-separated pair consisting of 'Options' and a structure that is created with derivset.

Data Types: struct

Output Arguments

OAS — Option adjusted spread in basis points

vector

Option adjusted spread in basis points, 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

Option Adjusted Spread

Option adjusted spread (OAS) adjusts a bond spread for the option's value and is the standard measure for valuing and comparing bonds with different redemption structures.

OAS is a measure of yield spread that accounts for embedded call or put options in the valuation of bonds. The computation of OAS is similar to computing the bond spread, with the difference being that the cash flows are nondeterministic. In other words, the OAS computation considers the possibility of a change in the bond’s cash flows due to early redemptions. To compute an OAS, you must model the future behavior of interest rates.

In general, bonds with similar characteristics and credit risks should have the same OAS. If a bond has an OAS higher than the OAS of its peers (bond with similar characteristics and credit quality), it is considered undervalued. Conversely, a bond with a low OAS relative to its peers is considered overvalued.

Option Adjusted Duration

Option adjusted duration (OAD) accounts for the effect of the call option on the expected life of a bond.

OAD weighs the probability that the bond will be called based on the spread between its coupon rate and its yield, as well as the volatility of interest rates. Generally speaking, option adjusted duration (OAD) is longer than modified duration when a bond is priced to a call date, and shorter than modified duration when a bond is priced to maturity.

Option Adjusted Convexity

Option adjusted convexity (OAC) is a measure of a bond's convexity, which account for the convexity of options embedded within the bond.

OAC captures the curvature of the price and yield relationship observed in bonds. Low values mean the relationship is near to linearity (a change in the price leads to a proportional change in the yield). The OAC can vary from the negative to the positive, depending on the yield’s amount and the time to call or time to put. In contrast with modified convexity, OAC assumes that the cash flows of a bond change when yields change.

References

[1] Fabozzi, F. Handbook of Fixed Income Securities. 7th Edition. McGraw-Hill, 2005.

[2] Windas, T. Introduction to Option-Adjusted Spread Analysis. 3rd Edition. Bloomberg Press, 2007.

Version History

Introduced in R2011a

R2022b: Serial date numbers not recommended

Although oasbyhw supports serial date numbers, datetime values are recommended instead. The datetime data type provides flexible date and time formats, storage out to nanosecond precision, and properties to account for time zones and daylight saving time.

To convert serial date numbers or text to datetime values, use the datetime function. For example:

t = datetime(738427.656845093,"ConvertFrom","datenum");
y = year(t)

y =

        2021

There are no plans to remove support for serial date number inputs.