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 AmericanSwaption command creates a new American-style swaption with the specified payoff and maturity. The swaption can be exercised at any time between earliestexercise and latestexercise dates. This is the opposite of a European-style swaption, which can only be exercised on the date of expiration.

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

The parameter earliestexercise specifies the earliest time or date when the option can be exercised. It can be given either as a non-negative constant or as a date in any of the formats recognized by the ParseDate command. If earlyexercise is given as a date, then the period between referencedate and earliestexercise will be converted to a fraction of the year according to the day count convention specified by daycounter. Typically the value of this option is $0$, which means that the option can be exercised at any time until the maturity. Note that the time of the earliest exercise must precede the maturity time.

The parameter latestexercise specifies the maturity time of the option. It can be given either as a non-negative constant or as a date in any of the formats recognized by the ParseDate command. If earlyexercise is given as a date, then the period between referencedate and latestexercise will be converted to a fraction of the year according to the day count convention specified by daycounter.

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

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

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

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

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

$\mathrm{nominal}≔1000.0$

$\mathrm{nominal}≔1000.0$

$\mathrm{fixing\_days}≔2$

$\mathrm{fixing\_days}≔2$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\mathrm{atm\_rate}≔0.04995609574$

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

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

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

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

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

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

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

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

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

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

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

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

$\mathrm{itm\_swaption}≔\mathrm{AmericanSwaption}\left(\mathrm{itm\_swap}\,\mathrm{AdvanceDate}\left(\mathrm{start}\,1\,\mathrm{Days}\,\mathrm{EURIBOR}\right)\,\mathrm{AdvanceDate}\left(\mathrm{dates}\left[-2\right]\,1\,\mathrm{Days}\,\mathrm{EURIBOR}\right)\right)$

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

$\mathrm{atm\_swaption}≔\mathrm{AmericanSwaption}\left(\mathrm{atm\_swap}\,\mathrm{AdvanceDate}\left(\mathrm{start}\,1\,\mathrm{Days}\,\mathrm{EURIBOR}\right)\,\mathrm{AdvanceDate}\left(\mathrm{dates}\left[-2\right]\,1\,\mathrm{Days}\,\mathrm{EURIBOR}\right)\right)$

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

$\mathrm{otm\_swaption}≔\mathrm{AmericanSwaption}\left(\mathrm{otm\_swap}\,\mathrm{AdvanceDate}\left(\mathrm{start}\,1\,\mathrm{Days}\,\mathrm{EURIBOR}\right)\,\mathrm{AdvanceDate}\left(\mathrm{dates}\left[-2\right]\,1\,\mathrm{Days}\,\mathrm{EURIBOR}\right)\right)$

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

$a≔0.048696$

$a≔0.048696$

$\mathrm{\sigma }≔0.0058904$

$\mathrm{\sigma }≔0.0058904$

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

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

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

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

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

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

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

$54.64340244$

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

$29.47595615$

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

$15.16252551$

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

$\mathrm{ou\_process}≔\mathrm{\_X0}$

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

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

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

$53.87370143$

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

$21.69892190$

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

$11.69368613$

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[AmericanSwaption] command was introduced in Maple 15.

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