compounding = Simple, Continuous, Annual, Semiannual, EveryFourthMonth, Quarterly, Bimonthly, Monthly, SimpleThenAnnual, SimpleThenSemiannual, SimpleThenEveryFourthMonth, SimpleThenQuarterly, SimpleThenBimonthly, or SimpleThenMonthly -- This option specifies the compounding type for the given discount factor(s).

daycounter = Actual360, Actual365Fixed, AFB, Bond, Euro, Historical, ISDA, ISMA, OneDay, Simple, Thirty360BondBasis, Thirty360EuroBondBasis, Thirty360European, Thirty360Italian, Thirty360USA, or a day counter data structure -- This option specifies the convention used to convert the amount of time between two dates to year fractions.

interpolation = BackwardFlat, Cubic, ForwardFlat, Linear, or LogLinear -- This option specifies the type of interpolation used to build a discount curve from a discrete set of discount factors. The LogLinear interpolation is used by default.

referencedate = date in any of the formats recognized by the ParseDate command -- This option specifies the reference date (when the discount factor is equal to 1).

The DiscountCurve command creates a new yield curve based on the specified discount factors; the resulting curve is represented as a module. This module can be passed to other commands of the Finance package that expect a yield term structure as one of the parameters; it can also be used as if it were a procedure. Assume for example that the module returned by DiscountCurve was assigned to the name R. Then for any positive constant t, $R\left(t\right)$ will return a discount factor for the maturity t based on the term structure R. If d is a date given in any of the formats recognized by the ParseDate command, then the $R\left(d\right)$ command will return the discount factor for the corresponding maturity.

The DiscountCurve(rate, opts) command creates a zero curve based on the specified interest rate. The parameter rate can be either a real constant, a Maple procedure or an algebraic expression. If rate is a real constant then the DiscountCurve command contracts a flat term structure based on the specified interest rate. If rate is a procedure it should accept one parameter (the time) and return the corresponding rate as a floating-point number. Finally, if rate is an algebraic expression, it should depend on a single variable. This variable will be taken as time.

The DiscountCurve(times, rates, opts) and DiscountCurve(dates, rates, opts) commands create a term structure based on piecewise interpolation of specified discount factors. The parameters rates and times can be either a list or a Vector containing numeric values and must have the same number of elements. The parameter dates is a list of dates in one of the formats recognized by ParseDate.

Objects created using the DiscountCurve command will be of Maple type YieldTermStructure.

with(Finance):

SetEvaluationDate("November 25, 2006"):

discountfactors := [1., 0.995, 0.989, 0.981, 0.975, 0.964, 0.944, 0.930, 0.914, 0.886, 0.860, 0.842, 0.832, 0.831, 0.827, 0.823, 0.815, 0.795, 0.778, 0.770]:

times := [.25, .50, .75, 1.0, 1.2, 1.5, 1.8, 2.0, 2.2, 2.5, 2.8, 3.0, 3.2, 3.5, 3.8, 4.0, 4.2, 4.5, 4.8, 5.0]:

discountcurve := DiscountCurve(times, discountfactors):

ZeroRate(discountcurve, .5);

$0.01002508365$

ForwardRate(discountcurve, .5);

$0.02419362214$

DiscountFactor(discountcurve, .5);

$0.9950000000$

date2 := AdvanceDate(6, Months);

$\mathrm{date2}≔''May\; 25,\; 2007''$

time2 := YearFraction(date2);

$\mathrm{time2}≔0.4958904110$

discountcurve(date2);

$0.9950819893$

discountcurve(time2);

$0.9950819893$

discountcurve2 := DiscountCurve(times, discountfactors, interpolation = BackwardFlat):

discountcurve3 := DiscountCurve(times, discountfactors, interpolation = ForwardFlat):

discountcurve4 := DiscountCurve(times, discountfactors, interpolation = Cubic):

plot([discountcurve, discountcurve2, discountcurve3, discountcurve4], 0..5, color = [red, blue, green, cyan], thickness = 2, axes = BOXED, gridlines = true);

f := CurveFitting:-Spline(times, discountfactors, t, degree = 4);

$f≔\left\{\begin{array}{cc}-0.01189941604t^4+0.01189941604t^3-0.004462281015t^2-0.01905099728t+1.004902195& t<0.3750000000\\ -0.03172492545t^4+0.04163768016t^3-0.02119005458t^2-0.01486905389t+1.004510138& t<0.6250000000\\ 0.1039594708t^4-0.2975733104t^3+0.2968202491t^2-0.1473733471t+1.025213935& t<0.8750000000\\ -0.08853890118t^4+0.3761709898t^3-0.5874691433t^2+0.3684621307t+0.9123749242& t<1.100000000\\ -0.1197353883t^4+0.5134355333t^3-0.813955640t^2+0.5345522282t+0.8667001473& t<1.350000000\\ 0.2499934793t^4-1.483100351t^3+3.229029530t^2-3.104134427t+2.094756886& t<1.650000000\\ -0.2653533436t^4+1.918188680t^3-5.189160821t^2+6.155874957t-1.724996984& t<1.900000000\\ 0.0787759053t^4-0.6971936111t^3+2.264678708t^2-3.285655115t+2.759729805& t<2.100000000\\ 0.1910343021t^4-1.640164145t^3+5.235035885t^2-7.444155159t+4.942942325& t<2.350000000\\ -0.2857346683t^4+2.841464179t^3-10.56270395t^2+17.30563722t-9.59756069& t<2.650000000\\ 0.5174704068t^4-5.672509617t^3+23.28034189t^2-42.48374377t+30.01290422& t<2.900000000\\ -0.4144089660t^4+5.137291107t^3-23.74229127t^2+48.42668042t-35.89715344& t<3.100000000\\ -0.2814239993t^4+3.488277521t^3-16.07437810t^2+32.57965988t-23.61571253& t<3.350000000\\ 0.3604175437t^4-5.112399153t^3+27.14402216t^2-63.94143401t+57.22070354& t<3.650000000\\ -0.2866325131t^4+4.334531678t^3-24.57792413t^2+61.91530196t-57.62356812& t<3.900000000\\ 0.1353767710t^4-2.248813156t^3+13.93464317t^2-38.21737304t+40.00579001& t<4.100000000\\ 0.1815123228t^4-3.005436205t^3+18.58787492t^2-50.93620650t+53.04259433& t<4.350000000\\ -0.1245093110t^4+2.319340223t^3-16.15629127t^2+49.82187548t-56.53181988& t<4.650000000\\ -0.09220898271t^4+1.718554117t^3-11.96580818t^2+36.83137790t-41.43036638& t<4.900000000\\ 0.2218548599t^4-4.437097198t^3+33.27822899t^2-110.9658102t+139.6211887& \mathrm{otherwise}\end{array}\right.$

discountcurve5 := DiscountCurve(f):

plot([discountcurve, discountcurve5], 0..2, color = [red, blue], thickness = 2, axes = BOXED, gridlines = true);

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[DiscountCurve] command was introduced in Maple 15.

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