Swaptions (OTC) 
The market price calculator for swaptions calculates current market values, time values, and future market values (the future point in time is the horizon).
A swaption is an option on an interest swap with a purely fixed and a purely floating side, and is either a call option (buy) or put option (sell). The buyer can switch to a swap at a particular time. Swaptions are priced only if the maturity date is after the horizon.
To price a swaption, it must first be divided into two parts – a fixed-rate side, and a floating-rate side. Based on the settings in the evaluation type and the valuation rule, the system prices swaptions as follows:
As options on bonds when the Black-Scholes model is used
As options on bonds when the Hull-White model is used
As interest-rate options when the Black-Scholes model is used
As interest-rate options when the model for normally distributed interest rates is used
Integration
The following describes what happens when the Black-Scholes model is used to price swaptions as options on bonds.
To price a swaption, the transaction data - or, alternatively, a par coupon or zero coupon yield curve - must exist in the transaction currency for the evaluation date.
You also need an interest volatility curve for the term of the option. The reference interest rate you enter is the one corresponding to the reference interest rate in the yield curve whose term most closely parallels that of the fixed side of the swap. For example, if the term of the fixed side of the swap is 4.25 years, the term for the reference interest rate in a yield curve with annual grid points has to be set to four years.
Depending on the calculation procedure, the following input parameters are calculated for later determining the risk-free interest rate and the spot rate (both of which go into the price formula):
Zero coupon rates and zero bond discounting factors which are calculated using the zero and par coupon calculation methods.
Spot:
The spot is the NPV of the price (standardized to the nominal volume) of the fixed side of the swap on the expiration date of the option. Using the Treasury and Risk Management component, a cash flow is generated when the fixed side of the swap is created. The cash flow consists of interest and principal payments, which “flow” at particular points in time. Both the times and the amounts of the individual cash flows are known. The NPV of the individual cash flows is calculated on the horizon according to either the par or zero coupon calculation methods. This is done using the yield curves, dependent on the transaction currency of the fixed side of the swap. It is the equivalent of the NPV of the individual cash flows first on the expiration date of the option, and subsequently on the horizon. The value of the spot (in the transaction currency) is the absolute sum of the NPVs of the cash flows standardized to the nominal value of the swap.
The following definitions apply:
ti: Maturity date of the cash flows
Ci: Cash flow at time ti
NPV(Ci): Net present value on the horizon of the cash flow due on ti
NV: Nominal volume of the swap
Volatility:
The interest volatility is determined for the term, according to the difference between the expiration date and horizon of the option (if necessary linearly interpolated from the values of neighboring option terms). The price volatility calculator then converts it to a price volatility using the modified duration (based on the interest and principal payment flows from the fixed side of the swap), and the forward zero yield (calculated from the given zero bond discounting factors during the term of the swap), based on the following formula:
Price volatility = Forward rate * interest volatility * modified duration
Call/Put Indicator:
If the swap in the underlying of the swaption is a fixed interest recipient swap, the swaption can be viewed as a call on a fixed-interest bond corresponding to the fixed side of the swap. The "Call/Put" indicator is set to "Call". If the swap in the underlying of the swaption is a fixed interest payer swap, the swaption can be viewed as a put on a fixed-interest bond corresponding to the fixed side of the swap. The "Call/Put" indicator is set to "Put".
If the transaction currency differs from the display currency of the swap as an underlying, the transaction currency is translated into the display currency using the currency rate from the horizon. If the horizon is later than the evaluation date, the corresponding forward currency rate (bid or ask price) is calculated for the evaluation date using the yield curves from the transaction and display currencies.
The input parameters are calculated using function modules, such as the following:
Features
The option price calculator uses the following parameters (some of which are taken from the input parameters) when pricing European options (Black-Scholes formula):
Call/Put: See input parameters
Term: Term of the option in days (expiration date of the option to the horizon)
Spot: See input parameters
Strike: 1 (the value of the swap standardized to the nominal volume from the floating side)
Interest rate 1: 0 (no forex option)
Interest rate 2: Risk-free interest rate = zero interest rate of the interest yield curve with a term equivalent to the option term.
Volatility: See input parameters
The price difference (DZ) between the standardized price of the fixed side of the swap and 1 (standardized price of the floating side of the swap) on the expiration date of the option is standardized to the nominal volume of the swap, with a lower limit of zero. The NPV of the simulated DZ is then calculated.
The net present value of the swaption is the nominal volume (NV) multiplied by the interest difference DZ.
In addition to the formulas given already, the following are used:
NPV=K*NV*DZ
where
K = Call/put indicator
If the display currency differs from the transaction currency, the NPV is calculated using the forward currency rate.