Discount Curves
Yield curve computed from discount curve
Input:
- : the discount curve
- : the initial time
Output:
The continuously compounded yield curve
Algorithm:
Recall that
To handle the singularity at , we take sufficiently small (e.g. ) and denote
Then the yield is computed as
where the function is a part of STL and improves the standard log function when .
The Nelson-Siegel discount curve
Input:
- : the coefficients of the Nelson-Siegel yield curve
- : the mean-reversion rate
- : the initial time
Output:
The discount curve
where the yield curve has the Nelson-Siegel form:
Algorithm:
Write the yield curve as
where, for and
The functions and have a removable singularity at .
Discount curve for the Vasicek model of interest rates
Input:
- : the constant drift term ( is the mean-reversion level)
- : the mean-reversion rate
- : the volatility
- : the initial short-term interest rate
- : the initial time given as year fraction
Output:
- : the discount curve in the Vasicek model of interest rates
In the Vasicek model, under the risk-neutral probability, the short-term interest rate evolves as
where is a Brownian motion. The discount curve is given by
We can show that the yield curve has the form
where
Algorithm:
The yield curve can be written as
where, for and ,
The functions and have a removable singularity at .
The Svensson discount curve
Input:
- : the constant coefficients of the Svensson model of interest rates
- : the strictly positive mean-reversion rates,
- : the initial time given as year fraction
Output:
the discount curve
where the yield curve has the Svensson form:
Forwards
Forward price curve for a coupon bond
Input:
- : the coupon rate
- : the time interval between coupon payments
- : the maturity
- : the discount curve
- : the initial time
is_clean: the boolean parameter specifying the type of the prices: “clean” or “dirty”. The dirty price is the actual amount paid in a transaction. The clean price is the difference between the dirty price and the accrued interest. If is the settlement time and is the previous coupon or issue time, then the accrued interest is computed as
Output:
- : the curve of forward prices for the bond computed at for maturities .
The bond pays coupons s at times such that
The bond also pays notional at maturity . The buyer of the forward contract pays “dirty” forward price at delivery time and receives the bond, that is, gets the notional at maturity and coupons s at times .
Algorithm:
Use the following notations:
- : the dirty forward bond price computed at for delivery at
- : the dirty spot bond price at (= the value of coupons s at times , and notional at )
- : the discount factor computed at for maturity
At delivery time , the long position in the forward contract has value
At issue time , the long position has value 0. Using replication, we deduce that
It follows that
For the clean price, we have that
where is the time elapsed from the previous coupon (or issue) time:
Forward price curve for a stock that pays dividends
Input:
- : the spot price
- : the dividend times,
- : the cash dividend payments
- : the discount curve
- : the initial time (the issue time) for the forward
Output:
- : the forward prices for the stock.
The buyer pays forward price at delivery time and then receives the stock at . It costs nothing to enter a forward contract at its issue time
Forward price curve for an annuity
Input:
- : the coupon rate
- : the time interval between coupon payments
- : the maturity
- : the discount curve
- : the initial time
is_clean: the boolean parameter specifying the type of the prices: “clean” or “dirty”. The dirty price is the actual amount paid in a transaction. The clean price is the difference between the dirty price and the accrued interest. If is the previous coupon time (or the initial time if no coupons have been paid so far) and is the settlement time, then the accrued interest is given by
Output:
- : the forward prices for the annuity. Here is the maturity of the contract and is the issue time.
The annuity pays coupons at times such that
The buyer pays forward price at delivery time and then receives coupons at payments times .
Forward swap rates
Input:
- : the time interval between payments
- : the number of payments
- : the discount curve; is the initial time
Output:
- : the forward swap rates computed at in the contract with period and number of payments .
It costs nothing to enter the forward swap contract with maturity . At time , the swap is issued with a notional amount , the number payments , the period between payments , and the fixed rate .
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