referencedate = a string containing a date specification in a format recognized by ParseDate or a date data structure -- This option provides the evaluation date. It is set to the global evaluation date by default.

daycounter = a name representing a supported day counter (e.g. ISDA, Simple) or a day counter data structure created using the DayCounter constructor -- This option provides a day counter that will be used to convert the period between two dates to a fraction of the year. This option is used only if one of earliestexercise or latestexercise is specified as a date.

The BermudanSwaption command creates a new Bermudan-style swaption on the specified interest rate swap. This swaption can be exercised at any time or date given in the exercisetimes or exercisedates list.

The parameter irswap is the underlying interest rate swap (see InterestRateSwap for more details).

The parameter exercisedates specifies the dates when the swaption can be exercised. It has to be given as a list of dates in any of the formats recognized by the ParseDate command. The exercise times will be computed by converting the period between referencedate and the corresponding exercise date to a fraction of the year according to the day count convention specified by daycounter. Alternatively, one can specify exercise times directly using the exercisetimes parameter. In this case the referencedate and daycounter options are ignored.

The LatticePrice command can be used to price a Bermudan-style swaption using any given binomial or trinomial tree.

$\mathrm{with}\left(\mathrm{Finance}\right)\:$

$\mathrm{SetEvaluationDate}\left(''November\; 17,\; 2006''\right)\:$

$\mathrm{EvaluationDate}\left(\right)$

$''November\; 17,\; 2006''$

$\mathrm{nominal}≔1000.0$

$\mathrm{nominal}≔1000.0$

$\mathrm{fixing\_days}≔2$

$\mathrm{fixing\_days}≔2$

$\mathrm{start}≔\mathrm{AdvanceDate}\left(1\,\mathrm{Years}\,\mathrm{EURIBOR}\right)$

$\mathrm{start}≔''November\; 17,\; 2007''$

$\mathrm{maturity}≔\mathrm{AdvanceDate}\left(\mathrm{start}\,5\,\mathrm{Years}\,\mathrm{EURIBOR}\right)$

$\mathrm{maturity}≔''November\; 17,\; 2012''$

$\mathrm{discount\_curve}≔\mathrm{ForwardCurve}\left(0.04875825\,'\mathrm{daycounter}'=\mathrm{Actual365Fixed}\right)$

$\mathrm{discount\_curve}≔\mathbf{module}\left(\right)\mathbf{end\; module}$

$\mathrm{fixed\_schedule}≔\mathrm{Schedule}\left(\mathrm{start}\,\mathrm{maturity}\,\mathrm{Annual}\,'\mathrm{convention}'=\mathrm{Unadjusted}\,'\mathrm{calendar}'=\mathrm{EURIBOR}\right)$

$\mathrm{fixed\_schedule}≔\mathbf{module}\left(\right)\mathbf{end\; module}$

$\mathrm{floating\_schedule}≔\mathrm{Schedule}\left(\mathrm{start}\,\mathrm{maturity}\,\mathrm{Semiannual}\,'\mathrm{convention}'=\mathrm{ModifiedFollowing}\,'\mathrm{calendar}'=\mathrm{EURIBOR}\right)$

$\mathrm{floating\_schedule}≔\mathbf{module}\left(\right)\mathbf{end\; module}$

$\mathrm{benchmark}≔\mathrm{BenchmarkRate}\left(6\,\mathrm{Months}\,\mathrm{EURIBOR}\,0.04875825\right)$

$\mathrm{benchmark}≔\mathbf{module}\left(\right)\mathbf{end\; module}$

$\mathrm{swap}≔\mathrm{InterestRateSwap}\left(\mathrm{nominal}\,0.\,\mathrm{fixed\_schedule}\,\mathrm{benchmark}\,\mathrm{floating\_schedule}\,0.\right)$

$\mathrm{swap}≔\mathbf{module}\left(\right)\mathbf{end\; module}$

$\mathrm{atm\_rate}≔\mathrm{FairRate}\left(\mathrm{swap}\,\mathrm{discount\_curve}\right)$

$\mathrm{atm\_rate}≔0.04995609574$

$\mathrm{itm\_swap}≔\mathrm{InterestRateSwap}\left(\mathrm{nominal}\,0.8\mathrm{atm\_rate}\,\mathrm{fixed\_schedule}\,\mathrm{benchmark}\,\mathrm{floating\_schedule}\,0.\right)$

$\mathrm{itm\_swap}≔\mathbf{module}\left(\right)\mathbf{end\; module}$

$\mathrm{atm\_swap}≔\mathrm{InterestRateSwap}\left(\mathrm{nominal}\,1.0\mathrm{atm\_rate}\,\mathrm{fixed\_schedule}\,\mathrm{benchmark}\,\mathrm{floating\_schedule}\,0.\right)$

$\mathrm{atm\_swap}≔\mathbf{module}\left(\right)\mathbf{end\; module}$

$\mathrm{otm\_swap}≔\mathrm{InterestRateSwap}\left(\mathrm{nominal}\,1.2\mathrm{atm\_rate}\,\mathrm{fixed\_schedule}\,\mathrm{benchmark}\,\mathrm{floating\_schedule}\,0.\right)$

$\mathrm{otm\_swap}≔\mathbf{module}\left(\right)\mathbf{end\; module}$

$\mathrm{cash\_flows}≔\mathrm{CashFlows}\left(\mathrm{itm\_swap}\,\mathrm{paying}\right)$

$\mathrm{cash\_flows}≔\left[\mathrm{39.97833882\; on\; \text{'}November\; 17,\; 2008\text{'}}\,\mathrm{39.95141436\; on\; \text{'}November\; 17,\; 2009\text{'}}\,\mathrm{39.96487659\; on\; \text{'}November\; 17,\; 2010\text{'}}\,\mathrm{39.96487659\; on\; \text{'}November\; 17,\; 2011\text{'}}\,\mathrm{39.97833882\; on\; \text{'}November\; 19,\; 2012\text{'}}\right]$

$\mathrm{CashFlows}\left(\mathrm{itm\_swap}\,\mathrm{receiving}\right)$

$\left[\mathrm{24.55793340\; on\; \text{'}May\; 19,\; 2008\text{'}}\,\mathrm{24.54222773\; on\; \text{'}November\; 17,\; 2008\text{'}}\,\mathrm{24.59383300\; on\; \text{'}May\; 18,\; 2009\text{'}}\,\mathrm{24.74716833\; on\; \text{'}November\; 17,\; 2009\text{'}}\,\mathrm{24.47342475\; on\; \text{'}May\; 17,\; 2010\text{'}}\,\mathrm{24.88406756\; on\; \text{'}November\; 17,\; 2010\text{'}}\,\mathrm{24.47342475\; on\; \text{'}May\; 17,\; 2011\text{'}}\,\mathrm{24.88406756\; on\; \text{'}November\; 17,\; 2011\text{'}}\,\mathrm{24.55868130\; on\; \text{'}May\; 17,\; 2012\text{'}}\,\mathrm{25.08832826\; on\; \text{'}November\; 19,\; 2012\text{'}}\right]$

$\mathrm{dates}≔\mathrm{map}\left(t\mapsto t\left[\mathrm{date}\right]\,\mathrm{cash\_flows}\right)$

$\mathrm{dates}≔\left[\mathrm{date}\,\mathrm{date}\,\mathrm{date}\,\mathrm{date}\,\mathrm{date}\right]$

$\mathrm{exercise\_dates}≔\mathrm{map}\left(t\mapsto \mathrm{AdvanceDate}\left(t\,1\,\mathrm{Days}\,\mathrm{EURIBOR}\right)\,\left[\mathrm{start}\,\mathrm{op}\left(1..-2\,\mathrm{dates}\right)\right]\right)$

$\mathrm{exercise\_dates}≔\left[''November\; 19,\; 2007''\,''November\; 18,\; 2008''\,''November\; 18,\; 2009''\,''November\; 18,\; 2010''\,''November\; 18,\; 2011''\right]$

$\mathrm{itm\_swaption}≔\mathrm{BermudanSwaption}\left(\mathrm{itm\_swap}\,\mathrm{exercise\_dates}\right)$

$\mathrm{itm\_swaption}≔\mathbf{module}\left(\right)\mathbf{end\; module}$

$\mathrm{atm\_swaption}≔\mathrm{BermudanSwaption}\left(\mathrm{atm\_swap}\,\mathrm{exercise\_dates}\right)$

$\mathrm{atm\_swaption}≔\mathbf{module}\left(\right)\mathbf{end\; module}$

$\mathrm{otm\_swaption}≔\mathrm{BermudanSwaption}\left(\mathrm{otm\_swap}\,\mathrm{exercise\_dates}\right)$

$\mathrm{otm\_swaption}≔\mathbf{module}\left(\right)\mathbf{end\; module}$

$a≔0.048696$

$a≔0.048696$

$\mathrm{\sigma }≔0.0058904$

$\mathrm{\sigma }≔0.0058904$

$\mathrm{model}≔\mathrm{HullWhiteModel}\left(\mathrm{discount\_curve}\,a\,\mathrm{\sigma }\right)$

$\mathrm{model}≔\mathbf{module}\left(\right)\mathbf{end\; module}$

$\mathrm{time\_grid}≔\mathrm{TimeGrid}\left(\mathrm{YearFraction}\left(\mathrm{maturity}\right)+0.5\,100\right)$

$\mathrm{time\_grid}≔\mathbf{module}\left(\right)\mathbf{end\; module}$

$\mathrm{short\_rate\_tree}≔\mathrm{ShortRateTree}\left(\mathrm{model}\,\mathrm{time\_grid}\right)$

$\mathrm{short\_rate\_tree}≔\mathbf{module}\left(\right)\mathbf{end\; module}$

$\mathrm{LatticePrice}\left(\mathrm{itm\_swaption}\,\mathrm{short\_rate\_tree}\,\mathrm{discount\_curve}\right)$

$42.37568409$

$\mathrm{LatticePrice}\left(\mathrm{atm\_swaption}\,\mathrm{short\_rate\_tree}\,\mathrm{discount\_curve}\right)$

$13.41345287$

$\mathrm{LatticePrice}\left(\mathrm{otm\_swaption}\,\mathrm{short\_rate\_tree}\,\mathrm{discount\_curve}\right)$

$2.795526880$

$\mathrm{ou\_process}≔\mathrm{OrnsteinUhlenbeckProcess}\left(0.04875\,0.04875\,1.0\,0.3\right)$

$\mathrm{ou\_process}≔\mathrm{\_X0}$

$\mathrm{tree}≔\mathrm{ShortRateTree}\left(\mathrm{ou\_process}\,\mathrm{time\_grid}\right)$

$\mathrm{tree}≔\mathbf{module}\left(\right)\mathbf{end\; module}$

$\mathrm{LatticePrice}\left(\mathrm{itm\_swaption}\,\mathrm{tree}\,\mathrm{discount\_curve}\right)$

$41.39592398$

$\mathrm{LatticePrice}\left(\mathrm{atm\_swaption}\,\mathrm{tree}\,\mathrm{discount\_curve}\right)$

$1.528598823$

$\mathrm{LatticePrice}\left(\mathrm{otm\_swaption}\,\mathrm{tree}\,\mathrm{discount\_curve}\right)$

$0.$

Brigo, D., Mercurio, F., Interest Rate Models: Theory and Practice. New York: Springer-Verlag, 2001.

Hull, J., Options, Futures, and Other Derivatives, 5th. edition. Upper Saddle River, New Jersey: Prentice Hall, 2003.

The Finance[BermudanSwaption] command was introduced in Maple 15.

For more information on Maple 15 changes, see Updates in Maple 15.