Let denote the discount factor for the discount rate and the discount factor for the forecast rate. The currency of the discount rate must be the same as the settlement currency (Currency). Note that the currency of the forecast rate may in future be different to the settelment currency but is currently not implemented.
The Distrubution Type on the volatility price factor can only be set to Lognormal currently. This assumes that the price factors are log-normally distributed (hence have implied Black volatilities). Note that this can be extended to Normal resulting in the price factor having implied Bachelier volatilities.
Cashflows
Fixed Interest Cashflows
A cashflow with
- principal
- fixed interest rate
- accrual start date
- accrual end date
- accrual day count convention with the accrual year fraction from to
- payment date
- Fixed Amount
has a standard payoff:
with value at time of .
Floating Interest Cashflows
In addtion to the Fixed Interest Cashflow, a Floating Interest Cashflow also has
- reset date
- reset start
- reset end
- margin rate
The cashflow dates must satisfy and and the payoff is with is the simply-compounded forward rate at time given by
where is the accrual year fraction from to using the rate day count convention.
The value of a standard floating interest rate cashflow at time is,
If and the discount and forecast rate are the same, then the value is
For , the valuation needs a convexity correction but this is yet to be implemented. The standard payoff is:
where
- is a simply-compounded forward rate
- is the swaplet multiplier
- is a caplet multiplier, with the caplet strike
- is a floorlet multiplier, with the floorlet strike
Caplets/Floorlets
A caplet/floorlet is a call/put option on a simply compounded rate. The option payoff at time is:
where is the strike and is either for caplets and for floorlets. If then the option value at time is
where is the Black function and is the volatility of the forecast rate at time for expiry , tenor and strike . Note that if then the above formula is still used as no covexity correction has been applied.
Averaging
A cashflow with averaging depends on a sequence of simply compounded rates with the same nominal tenor . Each rate jas a positive weight . Let be the reset date of . The average rate at time is
Cashflow Lists
Consider a fixed or floating cashflow list with payment dates and notional principal amounts . If denotes the value of the cashflow at time , then the value of the cashflow list is
There may also be an optional Settlement Date and Settlement Amount . If not specified, then . Cashflow payment dates must be after the settlement date ().
The time value of the deal is
,
where either for a Buy, else for a Sell. If , the deal is treated as a forward contract on the underling cashflow list. If is the accrued interest up to , then if Settlement Amount Is Clean else . The value at time is
For cash settled deals, the valuation profile terminates at with a corresponding cashflow . If physically settled, then the cashflow is at and the profile continues until .
Fixed Compounding Cashflow Lists
For cashflow lists with the interest frequency greater than its payment frequency with payment dates ,let be the index of the last cashflow with payment date (with ). For groups of cashflows with the same payment date, interest is compounded as follows: the cashflow at time pays with
where
- is the principal amount$
- is the fixed rate
- is the accrual year fraction
- is the Fixed Amount of the cashflow.
The final value of the cashflows with payment date is
Floating Compounding Cashflow Lists
Similar to the Fixed Compounding Cashflow Lists, where being the reset date. Let be the value at time of an amount paid at the accrual end date (as opposed to the actual payment date ). The estimated interest is when otherwise .
The compounding method can be:
Include Margin where the cashflow pays at time with a value at of
Flat where the cashflow pays at time with . Its value at is
None in which case the cashflow pays at time .
CFFixedInterestListDeal
A series of fixed interest cashflows as described here
CFFloatingInterestListDeal
A series of floating interest cashflows as described here
FixedCashflowDeal
The time value of a fixed cashflow amount paid at time is .
MtMCrossCurrencySwapDeal
This currency swap adjusts the notional of one leg to capture any changes in the FX Spot rate since the last reset. At each reset, the principal of the adjusted leg is set to the principal of the unadjusted leg multiplied by the spot FX rate. MtM cross currency swaps are path dependent.
The unadjusted leg is either a fixed or floating interest rate list and is valued as such, however, the floating adjusted leg is valued as
where
- for and
- is the expected principal ,
- is the unadjusted leg principal for the period.
- is the forward FX rate for settlement at time .
SwaptionDeal
Let and be the Option Expiry Date, Swap Effective Date and Swap Maturity Date respectively of the swaption deal (). If the deal is cash settled, then let be the Settlement Date.
The value of the underlying swap is
where is the value of floating interest rate cashflows, the value of fixed interest cashflows and is either for payer swaptions and for receiver swaptions.
If the fixed leg has payments at times , then the Present value of a Basis Point is
where is the principal amount and is the accrual year fraction for the fixed interest cashflow. The forward swap rate is
Define the effective strike rate as
Note that presently only zero-margin floating cashflow lists are supported (but this can be extended). The value of the underlying swap is given by . If both fixed and floating cashflows have the same payment and accrual dates, then where is the constant fixed rate on the fixed interest cashflow list.
Physically Settled Swaptions
If the Settlement Style is Physical and , then the option holder receives the underlying swap and the value of the deal for is . Note that physical settlement has significant path dependency.
Cash Settled Swaptions
If the Settlement Style is Cash, then the option holder receives on settlement date . The value of the deal at is . Note that this assumes a lognormal distribution of the forecast rate and uses the Black Model as usual.
Swap Rate Volatility
Forward starting (where the effective date of the underlying swap is several months or years after the option expiry) and amortizing swaptions are not currently supported. This can be extended as needed. Otherwise, is the volatility of the underlying rate at time for expiry , tenor and strike