Caps (OTC) 
The market price calculator for caps calculates current market values as well as market/time values for a future date (horizon).
A cap contains a series of hedges to secure a particular reference interest rate Rk against an increase in the interest rate above a given value Rx (strike or cap rate). If the start of the term of the cap is before the horizon, the cap contains an existing fixed-rate transaction on the horizon. The value of the reference interest rate Rk is set at regular intervals of length t (for example, every six months). If the value of the reference interest rate Rk at time point kt is above the highest interest rate Rx, the buyer of the cap receives the difference at time point (k + 1) t.
The seller of the cap makes the following payment at time point (k+1)t: t*NV*max(Rk-Rx,0)
Notation:
Rx = Cap rate
NV = Nominal volume
Interest payments at time points: t, 2t, 3t,...,nt
If Fk is the forward rate for the period between kt and (k+1)t, and the interest rates Rx, Rk, and Fk are based on a compounding frequency t , then Fk can be taken as an approximation of the discount rate for the period between kt and (k + 1)t. Therefore, the payment made at time point (k+t)t is the discounted difference from (k + 1)t after kt
The advantage of this point of view is that it allows you to see every caplet as a European call on a t-periodic interest rate. The pay out is on the maturity of the option, though, and not a period later. The nominal value of the underlying for every option is:
Integration
To valuate a cap or a caplet, the transaction data - or, alternatively, a par coupon or zero coupon yield curve - must exist in the transaction currency for the evaluation date.
In addition, you need to specify a yield curve to calculate the forward rates Fk. If the start of the term of the cap is based on the evaluation date in the past, you have to specify the interest rate Rf of the fixing date for the current caplet. If this interest rate is not available, the value of the interest rate Rf is set at zero.
You also need an interest volatility curve for the term of the option. The specified reference interest rate is the reference interest rate Rk used in the analysis.
The first step restricts the cap to those caplets that mature after the horizon.
Zero bond discounting factors are calculated from the yield curve of the transaction currency. These, along with zero interest rates, are used for determining the discounting factors at a later date. You can use the zero coupon calculation method for this.
In the first step, the spot is removed for each individual caplet. The spot is used later on in the option price formula. To calculate the spot, the forward calculator calculates the forward rate Fk of the agreed reference interest rate Rk. The run-up period is the time up to the start date of the individual caplet (or the term of the option).
If the transaction currency differs from the display currency of the cap, the transaction currency is translated into the display currency using the currency rate at the horizon. If the horizon is later than the evaluation date, the corresponding forward currency rate is calculated for the evaluation date using the yield curves of the transaction and display currencies.
The input parameters are calculated using function modules, such as the following:
Features
The option price calculator for European options (applying either the Black Scholes formula or the normal distribution model) is called up for each caplet whose term begins after the horizon. It is called with the following parameters:
Call/Put | Call |
Term | Term up to the start of the caplet (same as the term of the respective option), given in days (start of the caplet - horizon). |
Spot | Forward rate Fk |
Strike | Cap rate Rx |
Interest rate 1 | 0 (no forex option) |
Interest rate 2 | 0 |
Volatility | Interest rate volatility of the reference interest rate Rk from the volatility curve with the term corresponding to that of the caplet. |
The result is interest rate DI that is the difference between the reference interest rate Rk and the strike rate Rx. If this difference is less than 0, then 0 is used in all subsequent calculations. The rate is calculated on the due date of settlement payment (that is, at the end of the caplet).
The net present value of a caplet (in the transaction currency) is the nominal volume multiplied by the interest rate difference DI, discounted from the due date of the settlement payment to the horizon:
NPV (caplet (i)) = K*t*NV*NPV(DI)
where
NPV (DI): Discounted interest rate difference
NV = Nominal volume
t = Compounding frequency
K = Call/put indicator
The NPV of the current caplet is calculated on the horizon (in the transaction currency). If the term of this caplet begins before the evaluation date, the interest difference (with zero as the floor), calculated from the given fixed rate of interest and the cap rate, is discounted to the horizon using the yield curve (according to the transaction currency of the current caplet). This value is then multiplied by the nominal volume NV and the compounding frequency t. If the beginning of the term of this caplet is after the evaluation date, the forward rate Fk is calculated from the agreed reference interest rate Rk using the forward calculator. The interest difference (with zero as the floor) from the calculated forward interest rate Fk and the cap rate is discounted to the horizon using the yield curve from the horizon (according to the transaction currency of the current caplet). This is then multiplied by the nominal volume NV and the compounding frequency t.
In addition to the formulas given already, the following are used:
If the start of the term of the caplet < the evaluation date:
NPV (caplet (current)) = K*t*NV*NPV(max(Rf—Rx),0)
If the start of the term of the caplet > the evaluation date:
NPV (caplet (current)) = K*t*NV*NPV(max(Fk—Rx),0)
The NPV of the cap is the sum of the NPVs of the individual caplets:
If the display currency differs from the transaction currency, the NPV is calculated using the forward currency rate.