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Price European swaption instrument using Black model

`Price = swaptionbyblk(RateSpec,OptSpec,Strike,Settle,ExerciseDates,Maturity,Volatility)`

`Price = swaptionbyblk(___,Name,Value)`

prices swaptions using the Black option pricing model. `Price`

= swaptionbyblk(`RateSpec`

,`OptSpec`

,`Strike`

,`Settle`

,`ExerciseDates`

,`Maturity`

,`Volatility`

)

adds optional name-value pair arguments. `Price`

= swaptionbyblk(___,`Name,Value`

)

Price a European swaption that gives the holder the right to enter in five years into a three-year paying swap where a fixed-rate of 6.2% is paid and floating is received. Assume that the yield curve is flat at 6% per annum with continuous compounding, the volatility of the swap rate is 20%, the principal is $100, and payments are exchanged semiannually.

Create the `RateSpec`

.

Rate = 0.06; Compounding = -1; ValuationDate = 'Jan-1-2010'; EndDates = 'Jan-1-2020'; Basis = 1; RateSpec = intenvset('ValuationDate', ValuationDate,'StartDates', ValuationDate, ... 'EndDates', EndDates, 'Rates', Rate, 'Compounding', Compounding, 'Basis', Basis);

Price the swaption using the Black model.

Settle = 'Jan-1-2011'; ExerciseDates = 'Jan-1-2016'; Maturity = 'Jan-1-2019'; Reset = 2; Principal = 100; Strike = 0.062; Volatility = 0.2; OptSpec = 'call'; Price= swaptionbyblk(RateSpec, OptSpec, Strike, Settle, ExerciseDates, Maturity, ... Volatility, 'Reset', Reset, 'Principal', Principal, 'Basis', Basis)

Price = 2.0710

This example shows Price a European swaption with receiving and paying legs that gives the holder the right to enter in five years into a three-year paying swap where a fixed-rate of 6.2% is paid and floating is received. Assume that the yield curve is flat at 6% per annum with continuous compounding, the volatility of the swap rate is 20%, the principal is $100, and payments are exchanged semiannually.

Rate = 0.06; Compounding = -1; ValuationDate = 'Jan-1-2010'; EndDates = 'Jan-1-2020'; Basis = 1;

**Define the RateSpec.**

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

**Define the swaption arguments.**

Settle = 'Jan-1-2011'; ExerciseDates = 'Jan-1-2016'; Maturity = 'Jan-1-2019'; Reset = [2 4]; % 1st column represents receiving leg, 2nd column represents paying leg Principal = 100; Strike = 0.062; Volatility = 0.2; OptSpec = 'call'; Basis = [1 3]; % 1st column represents receiving leg, 2nd column represents paying leg

**Price the swaption.**

Price= swaptionbyblk(RateSpec,OptSpec,Strike,Settle,ExerciseDates,Maturity,Volatility, ... 'Reset',Reset,'Principal',Principal,'Basis',Basis)

Price = 1.6494

Price a European swaption that gives the holder the right to enter into a 5-year receiving swap in a year, where a fixed rate of 3% is received and floating is paid. Assume that the 1-year, 2-year, 3-year, 4-year and 5- year zero rates are 3%, 3.4%, 3.7%, 3.9% and 4% with continuous compounding. The swap rate volatility is 21%, the principal is $1000, and payments are exchanged semiannually.

Create the `RateSpec`

.

ValuationDate = 'Jan-1-2010'; EndDates = {'Jan-1-2011';'Jan-1-2012';'Jan-1-2013';'Jan-1-2014';'Jan-1-2015'}; Rates = [0.03; 0.034 ; 0.037; 0.039; 0.04;]; Compounding = -1; Basis = 1; RateSpec = intenvset('ValuationDate', ValuationDate, 'StartDates', ValuationDate, ... 'EndDates', EndDates, 'Rates', Rates, 'Compounding', Compounding,'Basis', Basis)

`RateSpec = `*struct with fields:*
FinObj: 'RateSpec'
Compounding: -1
Disc: [5x1 double]
Rates: [5x1 double]
EndTimes: [5x1 double]
StartTimes: [5x1 double]
EndDates: [5x1 double]
StartDates: 734139
ValuationDate: 734139
Basis: 1
EndMonthRule: 1

Price the swaption using the Black model.

Settle = 'Jan-1-2011'; ExerciseDates = 'Jan-1-2012'; Maturity = 'Jan-1-2017'; Strike = 0.03; Volatility = 0.21; Principal =1000; Reset = 2; OptSpec = 'put'; Price = swaptionbyblk(RateSpec, OptSpec, Strike, Settle, ExerciseDates, ... Maturity, Volatility,'Basis', Basis, 'Reset', Reset,'Principal', Principal)

Price = 0.5771

Define the OIS and Libor curves.

```
Settle = datenum('15-Mar-2013');
CurveDates = daysadd(Settle,360*[1/12 2/12 3/12 6/12 1 2 3 4 5 7 10],1);
OISRates = [.0018 .0019 .0021 .0023 .0031 .006 .011 .017 .021 .026 .03]';
LiborRates = [.0045 .0047 .005 .0055 .0075 .0109 .0162 .0216 .0262 .0309 .0348]';
```

Create an associated `RateSpec`

for the OIS and Libor curves.

OISCurve = intenvset('Rates',OISRates,'StartDate',Settle,'EndDates',CurveDates,'Compounding',2,'Basis',1); LiborCurve = intenvset('Rates',LiborRates,'StartDate',Settle,'EndDates',CurveDates,'Compounding',2,'Basis',1);

Define the swaption instruments.

ExerciseDate = '15-Mar-2018'; Maturity = {'15-Mar-2020';'15-Mar-2023'}; OptSpec = 'call'; Strike = 0.04; BlackVol = 0.2;

Price the swaption instruments using the term structure `OISCurve`

both for discounting the cash flows and generating the future forward rates.

`Price = swaptionbyblk(OISCurve, OptSpec, Strike, Settle, ExerciseDate, Maturity, BlackVol,'Reset',1)`

`Price = `*2×1*
1.0956
2.6944

Price the swaption instruments using the term structure `LiborCurve`

to generate the future forward rates. The term structure `OISCurve`

is used for discounting the cash flows.

PriceLC = swaptionbyblk(OISCurve, OptSpec, Strike, Settle, ExerciseDate, Maturity, BlackVol,'ProjectionCurve',LiborCurve,'Reset',1)

`PriceLC = `*2×1*
1.5346
3.8142

Create the `RateSpec`

.

ValuationDate = 'Jan-1-2016'; EndDates = {'Jan-1-2017';'Jan-1-2018';'Jan-1-2019';'Jan-1-2020';'Jan-1-2021'}; Rates = [-0.02; 0.024 ; 0.047; 0.090; 0.12;]/100; Compounding = 1; Basis = 1; RateSpec = intenvset('ValuationDate',ValuationDate,'StartDates',ValuationDate, ... 'EndDates',EndDates,'Rates',Rates,'Compounding',Compounding,'Basis',Basis)

`RateSpec = `*struct with fields:*
FinObj: 'RateSpec'
Compounding: 1
Disc: [5x1 double]
Rates: [5x1 double]
EndTimes: [5x1 double]
StartTimes: [5x1 double]
EndDates: [5x1 double]
StartDates: 736330
ValuationDate: 736330
Basis: 1
EndMonthRule: 1

Price the swaption with a negative strike using the Shifted Black model.

Settle = 'Jan-1-2016'; ExerciseDates = 'Jan-1-2017'; Maturity = 'Jan-1-2020'; Strike = -0.003; % Set -0.3 percent strike. ShiftedBlackVolatility = 0.31; Principal = 1000; Reset = 1; OptSpec = 'call'; Shift = 0.008; % Set 0.8 percent shift. Price = swaptionbyblk(RateSpec,OptSpec,Strike,Settle,ExerciseDates, ... Maturity,ShiftedBlackVolatility,'Basis',Basis,'Reset',Reset,... 'Principal',Principal,'Shift',Shift)

Price = 12.8301

Create the `RateSpec`

.

ValuationDate = 'Jan-1-2016'; EndDates = {'Jan-1-2017';'Jan-1-2018';'Jan-1-2019';'Jan-1-2020';'Jan-1-2021'}; Rates = [-0.02; 0.024 ; 0.047; 0.090; 0.12;]/100; Compounding = 1; Basis = 1; RateSpec = intenvset('ValuationDate',ValuationDate,'StartDates',ValuationDate, ... 'EndDates',EndDates,'Rates',Rates,'Compounding',Compounding,'Basis',Basis)

`RateSpec = `*struct with fields:*
FinObj: 'RateSpec'
Compounding: 1
Disc: [5x1 double]
Rates: [5x1 double]
EndTimes: [5x1 double]
StartTimes: [5x1 double]
EndDates: [5x1 double]
StartDates: 736330
ValuationDate: 736330
Basis: 1
EndMonthRule: 1

Price the swaptions with using the Shifted Black model.

Settle = 'Jan-1-2016'; ExerciseDates = 'Jan-1-2017'; Maturities = {'Jan-1-2018';'Jan-1-2019';'Jan-1-2020'}; Strikes = [-0.0034;-0.0032;-0.003]; ShiftedBlackVolatilities = [0.33;0.32;0.31]; % A vector of volatilities. Principal = 1000; Reset = 1; OptSpec = 'call'; Shifts = [0.0085;0.0082;0.008]; % A vector of shifts. Prices = swaptionbyblk(RateSpec,OptSpec,Strikes,Settle,ExerciseDates, ... Maturities,ShiftedBlackVolatilities,'Basis',Basis,'Reset',Reset, ... 'Principal',Principal,'Shift',Shifts)

`Prices = `*3×1*
4.1117
8.0577
12.8301

`RateSpec`

— Interest-rate term structurestructure

Interest-rate term structure (annualized and continuously compounded),
specified by the `RateSpec`

obtained from `intenvset`

. For information on the interest-rate
specification, see `intenvset`

.

If the paying leg is different than the receiving leg, the `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 only one curve is specified, then it
is used to discount both legs.

**Data Types: **`struct`

`OptSpec`

— Definition of option character vector with values

`'call'`

or `'put'`

| cell array of character vector with values `'call'`

or `'put'`

Definition of the option as `'call'`

or `'put'`

,
specified as a `NINST`

-by-`1`

cell
array of character vectors.

A `'call'`

swaption, or *Payer
swaption*, allows the option buyer to enter into an interest-rate
swap in which the buyer of the option pays the fixed rate and receives
the floating rate.

A `'put'`

swaption, or *Receiver
swaption*, allows the option buyer to enter into an interest-rate
swap in which the buyer of the option receives the fixed rate and
pays the floating rate.

**Data Types: **`char`

| `cell`

`Strike`

— Strike swap rate valuesdecimal

Strike swap rate values, specified as a `NINST`

-by-`1`

vector
of decimal values.

**Data Types: **`double`

`Settle`

— Settlement dateserial date number | date character vector | cell array of date character vectors

Settlement date (representing the settle date for each swaption),
specified as a `NINST`

-by-`1`

vector
of serial date numbers or date character vectors. `Settle`

must
not be later than `ExerciseDates`

.

The `Settle`

date input for `swaptionbyblk`

is
the valuation date on which the swaption (an option to enter into
a swap) is priced. The swaption buyer pays this price on this date
to hold the swaption.

**Data Types: **`double`

| `char`

`ExerciseDates`

— Dates on which swaption expires and underlying swap startsserial date number | date character vector | cell array of date character vectors

Dates, specified as serial date numbers or date character vectors, on which the swaption expires and the underlying swap starts. The swaption holder can choose to enter into the swap on this date if the situation is favorable.

For a European option, `ExerciseDates`

are
a `NINST`

-by-`1`

vector of exercise
dates. Each row is the schedule for one option. When using a European
option, there is only one `ExerciseDate`

on the option
expiry date.

**Data Types: **`double`

| `char`

| `cell`

`Maturity`

— Maturity date for each forward swapserial date number | date character vector | cell array of date character vectors

Maturity date for each forward swap, specified as a `NINST`

-by-`1`

vector
of dates using serial date numbers or date character vectors.

**Data Types: **`double`

| `char`

| `cell`

`Volatility`

— Annual volatilities valuesnumeric

Annual volatilities values, specified as a
`NINST`

-by-`1`

vector of numeric values.

**Data Types: **`double`

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`

.

`Price = swaptionbyblk(OISCurve,OptSpec,Strike,Settle,ExerciseDate,Maturity,BlackVol,'Reset',1,'Shift',.5)`

`'Basis'`

— Day-count basis of instrument`0`

(actual/actual) (default) | integer from `0`

to `13`

Day-count basis of the instrument, specified as the comma-separated pair
consisting of `'Basis'`

and a
`NINST`

-by-`1`

vector or
`NINST`

-by-`2`

matrix representing the basis for
each leg. If `Basis`

is
`NINST`

-by-`2`

, the first column represents the
receiving leg, while the second column represents the paying leg. Default is
`0`

(actual/actual).

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`

`'Principal'`

— Notional principal amount`100`

(default) | numericNotional principal amount, specified as the comma-separated pair consisting of
`'Principal'`

and a
`NINST`

-by-`1`

vector.

**Data Types: **`double`

`'Reset'`

— Reset frequency per year for underlying forward swap`1`

(default) | numericReset frequency per year for the underlying forward swap, specified as the comma-separated
pair consisting of `'Reset'`

and a
`NINST`

-by-`1`

vector or
`NINST`

-by-`2`

matrix representing the reset
frequency per year for each leg. If `Reset`

is
`NINST`

-by-`2`

, the first column represents the
receiving leg, while the second column represents the paying leg.

**Data Types: **`double`

`'ProjectionCurve'`

— Rate curve used in generating future forward ratesif

`ProjectionCurve`

is not
specified, then `RateSpec`

is used both for discounting
cash flows and projecting future forward rates (default) | structureThe rate curve to be used in generating the future forward rates, specified as the
comma-separated pair consisting of `'ProjectionCurve'`

and a
structure created using `intenvset`

. Use this optional input
if the forward curve is different from the discount curve.

**Data Types: **`struct`

`'Shift'`

— Shift in decimals for shifted Black model`0`

(no shift) (default) | positive decimalShift in decimals for the shifted Black model, specified as the comma-separated pair
consisting of `'Shift'`

and a scalar or
`NINST`

-by-`1`

vector of rate shifts in positive
decimals. Set this parameter to a positive rate shift in decimals to add a positive
shift to the forward swap rate and strike, which effectively sets a negative lower
bound for the forward swap rate and strike. For example, a `Shift`

of
`0.01`

is equal to a 1% shift.

**Data Types: **`double`

`Price`

— Prices for swaptions at time 0vector

Prices for the swaptions at time 0, returned as a `NINST`

-by-`1`

vector
of prices.

A *forward swap* is a
swap that starts at a future date.

The *Shifted Black* model
is essentially the same as the Black’s model, except that it
models the movements of (*F* + *Shift*)
as the underlying asset, instead of *F* (which is
the forward swap rate in the case of swaptions).

This model allows negative rates, with a fixed negative lower bound defined by the amount of shift; that is, the zero lower bound of Black’s model has been shifted.

$$\begin{array}{l}dF={\sigma}_{Black}Fdw\\ call={e}^{-\gamma T}\left[FN({d}_{1})-KN({d}_{2})\right]\\ put={e}^{-\gamma T}\left[KN(-{d}_{2})-FN(-{d}_{1})\right]\\ {d}_{1}=\frac{\mathrm{ln}\left(\frac{F}{K}\right)+\left(\frac{{\sigma}_{B}{}^{2}}{2}\right)T}{{\sigma}_{B}\sqrt{T}},\text{}{d}_{2}={d}_{1}-{\sigma}_{B}\sqrt{T}\\ {\sigma}_{B}={\sigma}_{Black}\end{array}$$

Where *F* is the forward value and *K* is
the strike.

$$\begin{array}{l}dF={\sigma}_{Shifted\_Black}\left(F+Shift\right)dw\\ call={e}^{-\gamma T}\left[\left(F+Shift\right)N({d}_{s1})-\left(K+Shift\right)N({d}_{s2})\right]\\ put={e}^{-\gamma T}\left[\left(K+Shift\right)N(-{d}_{s2})-\left(F+Shift\right)N(-{d}_{s1})\right]\\ {d}_{s1}=\frac{\mathrm{ln}\left(\frac{F+Shift}{K+Shift}\right)+\left(\frac{{\sigma}_{sB}{}^{2}}{2}\right)T}{{\sigma}_{sB}\sqrt{T}},\text{}{d}_{s2}={d}_{s1}-{\sigma}_{sB}\sqrt{T}\\ {\sigma}_{sB}={\sigma}_{Shifted\_Black}\end{array}$$

Where *F*+*Shift* is the forward
value and *K*+*Shift* is the strike
for the shifted version.

`blackvolbysabr`

| `bondbyzero`

| `capbyblk`

| `cfbyzero`

| `fixedbyzero`

| `floatbyzero`

| `floorbyblk`

| `intenvset`

| `swaptionbynormal`

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