hjmvolspec (Financial Derivatives Toolbox)

Specify an HJM forward rate volatility process

Syntax

```
Volspec = hjmvolspec(varargin)
```

Arguments

The arguments to hjmvolspec vary according to the type and number of volatility factors specified when calling the function. Factors are specified by pairs of names and parameter sets. Factor names can be 'Constant', 'Stationary', 'Exponential', 'Vasicek', or 'Proportional'. The parameter set is specific for each of these factor types:

Constant volatility (Ho-Lee):
VolSpec = hjmvolspec('Constant', Sigma_0)

Stationary volatility:
VolSpec = hjmvolspec('Stationary', CurveVol, CurveTerm)

Exponential volatility:
VolSpec = hjmvolspec('Exponential', Sigma_0, Lambda)

Vasicek, Hull-White: VolSpec = hjmvolspec('Vasicek', Sigma_0, CurveDecay, CurveTerm)

Nearly proportional stationary:
VolSpec = hjmvolspec('Proportional', CurveProp, CurveTerm, MaxSpot)

You can specify more than one factor by concatenating names and parameter sets.

The table below defines the various arguments to hjmvolspec.

Argument
Description

Sigma_0
Scalar base volatility over a unit time.

Lambda
Scalar decay factor.

CurveVol
Number of curves (NCURVES) -by-1 vector of Vol values at sample points.

CurveDecay
NCURVES-by-1 vector of Decay values at sample points.

CurveProp
NCURVES-by-1 vector of Prop values at sample points.

CurveTerm
NCURVES-by-1 vector of Term sample points.

Note: See the volatility specifications formulas below for a description of Vol, Decay, Prop, and Term.

Argument	Description
`Sigma_0`	Scalar base volatility over a unit time.
`Lambda`	Scalar decay factor.
`CurveVol`	Number of curves (`NCURVES`) -by-`1` vector of `Vol` values at sample points.
`CurveDecay`	`NCURVES`-by-`1` vector of `Decay` values at sample points.
`CurveProp`	`NCURVES`-by-`1` vector of `Prop` values at sample points.
`CurveTerm`	`NCURVES`-by-`1` vector of `Term` sample points.
Note: See the volatility specifications formulas below for a description of `Vol`, `Decay`, `Prop`, and `Term`.

Description

Volspec = hjmvolspec(varargin) computes VolSpec, a structure that specifies the volatility model for hjmtree.

hjmvolspec specifies a HJM forward rate volatility process. Each factor is specified with one of the functional forms:

Volatility Specification
Formula

Constant
= Sigma_0

Stationary
= Vol(T-t) = Vol(Term)

Exponential
= Sigma_0*exp(-Lambda*(T-t))

Vasicek, Hull-White
= Sigma_0*exp(-Decay(T-t))

Proportional
= Prop(T-t)*max(SpotRate(t), MaxSpot)

Volatility Specification	Formula
Constant	`= Sigma_0`
Stationary	`= Vol(T-t) = Vol(Term)`
Exponential	`= Sigma_0exp(-Lambda(T-t))`
Vasicek, Hull-White	`= Sigma_0*exp(-Decay(T-t))`
Proportional	`= Prop(T-t)*max(SpotRate(t), MaxSpot)`

The volatility process is , where is the observation time and is the starting time of a forward rate. In a stationary process the volatility term is . Multiple factors can be specified sequentially.

The time values , , and Term are in coupon interval units specified by the Compounding input of hjmtimespec. For instance if Compounding = 2, Term = 1 is a semiannual period (six months).

Examples

Example 1.

Volatility is single-factor proportional.

CurveProp = [0.11765; 0.08825; 0.06865];
CurveTerm = [1; 2; 3];
VolSpec = hjmvolspec('Proportional', CurveProp, CurveTerm, 1e6)

 VolSpec = 
          FinObj: 'HJMVolSpec'
    FactorModels: {'Proportional'}
      FactorArgs: {{1x3 cell}}
      SigmaShift: 0
      NumFactors: 1
       NumBranch: 2
         PBranch: [0.5000 0.5000]
     Fact2Branch: [-1 1]

Example 2.

Volatility is two-factor exponential and constant.

VolSpec = hjmvolspec('Exponential', 0.1, 1, 'Constant', 0.2)

VolSpec = 
          FinObj: 'HJMVolSpec'
    FactorModels: {'Exponential'  'Constant'}
      FactorArgs: {{1x2 cell}  {1x1 cell}}
      SigmaShift: 0
      NumFactors: 2
       NumBranch: 3
         PBranch: [0.2500 0.2500 0.5000]
     Fact2Branch: [2x3 double]

See Also

hjmtimespec, hjmtree

hjmtree instadd