daycounter =  a string containing a date specification in a format recognized by ParseDate or a Date data structure -- This option specifies a day counter or day counting convention.

referencedate =  a string containing a date specification in a format recognized by ParseDate or a Date data structure -- This option specifies the reference date, that is, the date when the discount factor is 1. By default this is set to the global evaluation date.

The LatticePrice command computes the net present value of the specified financial instrument using the specified lattice approximation for the underlying stochastic process.

The parameter instrument is a financial instrument to be valued. At the present the following instruments are supported:

a one-asset option of the American, Bermudan, or European type (see AmericanOption, BermudanOption, or EuropeanOption)

a swaption of the American, Bermudan, or European type (see AmericanSwaption, BermudanSwaption, or EuropeanSwaption)

interest rate cap, floor, or collar (see Cap, Floor, or Collar)

The parameter model is a binomial or trinomial tree.

The parameter discountrate is the discount rate. It can be either a non-negative constant or a yield term structure. In the former case the reference date and the day count convention for the underlying term structure can be provided using the options daycounter and referencedate.

Note that all internal computations are performed at the hardware precision.

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

$\mathrm{SetEvaluationDate}\left(''January\; 3,\; 2006''\right)\:$

$\mathrm{Settings}\left(\mathrm{daycounter}=\mathrm{Thirty360European}\right)\:$

$T≔\mathrm{BlackScholesBinomialTree}\left(100\,0.1\,0.\,0.4\,0.6\,1000\right)\:$

$P≔S\mapsto \max \left(100-S\,0\right)$

$P≔S\mapsto \max \left(100-S\,0\right)$

$\mathrm{A1}≔\mathrm{AmericanOption}\left(P\,0\,0.5\right)\:$

$\mathrm{Maturity}≔\mathrm{AdvanceDate}\left(\mathrm{EvaluationDate}\left(\right)\,6\,\mathrm{Months}\,\mathrm{output}=\mathrm{formatted}\right)$

$\mathrm{Maturity}≔''July\; 3,\; 2006''$

$\mathrm{YearFraction}\left(\mathrm{Maturity}\right)$

$0.5000000000$

$\mathrm{A2}≔\mathrm{AmericanOption}\left(P\,\mathrm{EvaluationDate}\left(\right)\,\mathrm{Maturity}\right)\:$

$\mathrm{LatticePrice}\left(\mathrm{A1}\,T\,0.1\right)$

$9.220859736$

$\mathrm{LatticePrice}\left(\mathrm{A2}\,T\,0.1\right)$

$9.220859736$

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

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

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

$\mathrm{nominalamt}≔1000.0$

$\mathrm{nominalamt}≔1000.0$

$\mathrm{fixing\_days}≔2$

$\mathrm{fixing\_days}≔2$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\mathrm{atm\_rate}≔0.04996048632$

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

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

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

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

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

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

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

$\mathrm{cash\_flows}≔\left[\mathrm{39.96838906\; on\; \text{'}November\; 17,\; 2008\text{'}}\,\mathrm{39.96838906\; on\; \text{'}November\; 17,\; 2009\text{'}}\,\mathrm{39.96838906\; on\; \text{'}November\; 17,\; 2010\text{'}}\,\mathrm{39.96838906\; on\; \text{'}November\; 17,\; 2011\text{'}}\,\mathrm{39.96838906\; on\; \text{'}November\; 19,\; 2012\text{'}}\right]$

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

$\left[\mathrm{24.67872558\; on\; \text{'}May\; 19,\; 2008\text{'}}\,\mathrm{24.40119905\; on\; \text{'}November\; 17,\; 2008\text{'}}\,\mathrm{24.81751704\; on\; \text{'}May\; 18,\; 2009\text{'}}\,\mathrm{24.53995292\; on\; \text{'}November\; 17,\; 2009\text{'}}\,\mathrm{24.67872558\; on\; \text{'}May\; 17,\; 2010\text{'}}\,\mathrm{24.67872558\; on\; \text{'}November\; 17,\; 2010\text{'}}\,\mathrm{24.67872558\; on\; \text{'}May\; 17,\; 2011\text{'}}\,\mathrm{24.67872558\; on\; \text{'}November\; 17,\; 2011\text{'}}\,\mathrm{24.67872558\; on\; \text{'}May\; 17,\; 2012\text{'}}\,\mathrm{24.95632730\; on\; \text{'}November\; 19,\; 2012\text{'}}\right]$

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

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

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

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

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

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

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

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

$a≔0.048696$

$a≔0.048696$

$\mathrm{\sigma }≔0.0058904$

$\mathrm{\sigma }≔0.0058904$

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

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

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

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

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

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

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

$54.62494411$

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

$29.46563575$

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

$15.16495357$

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

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