referencedate = a string containing a date specification in a format recognized by Finance[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 EuropeanSwaption command creates a new European-style swaption with the specified payoff and maturity. The swaption can be exercised only at the time or date specified by the exercise parameter. This is the opposite of an American-style swaption, which can be exercised at any time before the expiration.

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

The parameter exercise specifies the 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 Finance[ParseDate] command.

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

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

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

$\mathrm{nominal}≔1000.0$

$\mathrm{nominal}≔1000.0$

$\mathrm{fixing\_days}≔2$

$\mathrm{fixing\_days}≔2$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\mathrm{atm\_rate}≔0.04995609574$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$a≔0.048696$

$a≔0.048696$

$\mathrm{\sigma }≔0.0058904$

$\mathrm{\sigma }≔0.0058904$

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

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

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

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

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

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

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

$9.794855431$

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

$4.523445349$

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

$1.502550032$

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

$\mathrm{ou\_process}≔\mathrm{\_X0}$

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

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

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

$8.995305075$

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

$1.528598823$

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

$0.$

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

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

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