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Finance[BermudanSwaption] - create a new Bermudan swaption

Calling Sequence

BermudanSwaption(irswap, exercisetimes, opts)

BermudanSwaption(irswap, exercisedates, opts)

Parameters

irswap

-

interest rate swap data structures; interest rate swap

exercisedates

-

list of dates given in a format recognized by ParseDate or a date data structure; dates when the option can be exercised

exercisetimes

-

list of non-negative constants; times when the option can be exercised

opts

-

(optional) equation(s) of the form option = value where option is one of referencedate or daycounter; specify options for the BermudanSwaption command

Description

• 

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.

Examples

withFinance:

SetEvaluationDateNovember 17, 2006:

EvaluationDate

November 17, 2006

(1)

nominal:=1000.0

nominal:=1000.0

(2)

fixing_days:=2

fixing_days:=2

(3)

start:=AdvanceDate1,Years,EURIBOR

start:=November 17, 2007

(4)

maturity:=AdvanceDatestart,5,Years,EURIBOR

maturity:=November 17, 2012

(5)

discount_curve:=ForwardCurve0.04875825,'daycounter'=Actual365Fixed

discount_curve:=moduleend module

(6)

fixed_schedule:=Schedulestart,maturity,Annual,'convention'=Unadjusted,'calendar'=EURIBOR

fixed_schedule:=moduleend module

(7)

floating_schedule:=Schedulestart,maturity,Semiannual,'convention'=ModifiedFollowing,'calendar'=EURIBOR

floating_schedule:=moduleend module

(8)

benchmark:=BenchmarkRate6,Months,EURIBOR,0.04875825

benchmark:=moduleend module

(9)

Construct an interest rate swap receiving the fixed-rate payments in exchange for the floating-rate payment.

swap:=InterestRateSwapnominal,0.0,fixed_schedule,benchmark,floating_schedule,0.0

swap:=moduleend module

(10)

Compute the at-the-money rate for this interest rate swap.

atm_rate:=FairRateswap,discount_curve

atm_rate:=0.04995609574

(11)

Construct three swaps.

itm_swap:=InterestRateSwapnominal,0.8atm_rate,fixed_schedule,benchmark,floating_schedule,0.0

itm_swap:=moduleend module

(12)

atm_swap:=InterestRateSwapnominal,1.0atm_rate,fixed_schedule,benchmark,floating_schedule,0.0

atm_swap:=moduleend module

(13)

otm_swap:=InterestRateSwapnominal,1.2atm_rate,fixed_schedule,benchmark,floating_schedule,0.0

otm_swap:=moduleend module

(14)

Here are cash flows for the paying leg of your interest rate swap.

cash_flows:=CashFlowsitm_swap,paying

cash_flows:=39.97833882 on 'November 17, 2008',39.95141436 on 'November 17, 2009',39.96487659 on 'November 17, 2010',39.96487659 on 'November 17, 2011',39.97833882 on 'November 19, 2012'

(15)

Here are cash flows for the receiving leg of your interest rate swap.

CashFlowsitm_swap,receiving

24.55793340 on 'May 19, 2008',24.54222773 on 'November 17, 2008',24.59383300 on 'May 18, 2009',24.74716833 on 'November 17, 2009',24.47342475 on 'May 17, 2010',24.88406756 on 'November 17, 2010',24.47342475 on 'May 17, 2011',24.88406756 on 'November 17, 2011',24.55868130 on 'May 17, 2012',25.08832826 on 'November 19, 2012'

(16)

These are days when coupon payments are scheduled to occur.

dates:=mapt→tdate,cash_flows

dates:=date,date,date,date,date

(17)

Set up exercise dates.

exercise_dates:=mapt→AdvanceDatet,1,Days,EURIBOR,start,op1..2,dates

exercise_dates:=November 19, 2007,November 18, 2008,November 18, 2009,November 18, 2010,November 18, 2011

(18)

Construct three swaptions that can be exercised on any of the previous dates.

itm_swaption:=BermudanSwaptionitm_swap,exercise_dates

itm_swaption:=moduleend module

(19)

atm_swaption:=BermudanSwaptionatm_swap,exercise_dates

atm_swaption:=moduleend module

(20)

otm_swaption:=BermudanSwaptionotm_swap,exercise_dates

otm_swaption:=moduleend module

(21)

Price these swaptions using the Hull-White trinomial tree.

a:=0.048696

a:=0.048696

(22)

σ:=0.0058904

σ:=0.0058904

(23)

model:=HullWhiteModeldiscount_curve,a,σ

model:=moduleend module

(24)

time_grid:=TimeGridYearFractionmaturity+0.5,100

time_grid:=moduleend module

(25)

short_rate_tree:=ShortRateTreemodel,time_grid

short_rate_tree:=moduleend module

(26)

Price our swaptions using the tree constructed above.

LatticePriceitm_swaption,short_rate_tree,discount_curve

42.37568409

(27)

LatticePriceatm_swaption,short_rate_tree,discount_curve

13.41345287

(28)

LatticePriceotm_swaption,short_rate_tree,discount_curve

2.795526880

(29)

You can also price these swaptions using an explicitly constructed trinomial tree.

ou_process:=OrnsteinUhlenbeckProcess0.04875,0.04875,1.0,0.3

ou_process:=_X

(30)

tree:=ShortRateTreeou_process,time_grid

tree:=moduleend module

(31)

Price the swaptions using the second tree.

LatticePriceitm_swaption,tree,discount_curve

41.39592398

(32)

LatticePriceatm_swaption,tree,discount_curve

1.528598823

(33)

LatticePriceotm_swaption,tree,discount_curve

0.

(34)

See Also

Finance[BermudanSwaption], Finance[BinomialTree], Finance[BlackScholesBinomialTree], Finance[BlackScholesTrinomialTree], Finance[EuropeanSwaption], Finance[GetDescendants], Finance[GetProbabilities], Finance[GetUnderlying], Finance[ImpliedBinomialTree], Finance[ImpliedTrinomialTree], Finance[LatticeMethods], Finance[LatticePrice], Finance[SetProbabilities], Finance[SetUnderlying], Finance[StochasticProcesses], Finance[TreePlot], Finance[TrinomialTree]

References

  

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.


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