Discount Curve
Discount curve obtained by fitting of yields
Input:
- : the maturities,
- : the discount factors
- : the initial time,
- : a fitting method for yields, a class
cfl::Fit
Output:
- : the fitted discount curve
- : the error. function of fit for the discount curve
- : the parameters of the fit for the yield curve containing the fitted constants , the covariance matrix , and the total fitting error
The discount curve has the form
where the yield curve is the result of the -fit of the market yields
Algorithm:
Step 1: Compute the market yields
Step 2: Fit of the yields
Compute the least-squares fit for the yield curve:
Also compute the error function of this fit and the fitting parameters .
Step 3: Fit of the discount curve
Compute
The formula for holds by the first-order expansion w.r.t. :
Discount curve with constant yield obtained by the least-squares fitting of market yields
Input:
- : the maturities of the market discount factors,
- : the market discount factors
- : the initial time given as year fraction,
Output:
- : the fitted discount curve with constant yield
- : the error function of the fit for the discount curve
- : the fitting parameters for the yield curve containing the fitted constant yield, its variance, and the total fitting error.
The fitted discount curve has the form:
where the constant yield is the best least-squares fit to the market yields:
Algorithm:
Compute the discount curve and its error function by fitting the market yields with linear least-squares method and the constant basis function
As part of the result, we obtain the fitting parameters for the yield curve containing the fitted constant yield, its variance, and the total fitting error.
Discount curve obtained by the Nelson-Siegel fitting of market yields
Input:
- : the maturities,
- : the discount factors
- : the mean-reversion rate
- : the initial time,
Output:
- : the fitted discount curve
- : the error function of the fit for the discount curve
- : the fitting parameters for the yield containing the fitted constants, their covariance matrix, and the total fitting error
The discount curve is given by
where the yield curve has the Nelson-Siegel form:
and the constants are the result of the least-squares fit of the market yields:
Algorithm:
Step 1: Basis functions
Compute the basis functions on :
where
Step 2: Fit of yield curve
Compute the discount curve and its error function by fitting the market yields with linear least-squares method and basis functions . As part of the result, we obtain the fitting parameters for the yield.
Discount curve for the Vasicek model obtained by the least-squares fitting of market yields
Input:
- : the maturities of market discount factors,
- : the market discount factors
- : the mean reversion rate
- : the short-term volatility
- : the initial time,
Output:
- : the fitted discount curve
- : the error function of the fit for the discount curve
- : the fitted constants, their covariance matrix, and the total fitting error
The discount curve is given by
In the Vasicek model, the yield curve has the form:
where
The initial short-term rate and the drift are the result of the least-squares fit of the market yields:
Algorithm:
Step 1: Basis functions
Compute the basis functions on :
and the free function
so that the yield curve is given by
We have that
where, for and
Step 2: Fit of yield curve
Compute the discount curve and its error function by fitting the market yields with linear least-squares method based on the basis functions and free function . As part of the result, we obtain the fitting parameters for the yield.
(TODO) Discount curve by an approximate fit of discount factors
Input:
- : the maturities of discount factors,
- : the discount factors
- : the initial time given as year fraction
- : a fitting method for the yields, a class
cfl::Fit
Output:
- : the fitted discount curve
- : the error function of the fit for the discount curve
- : the parameters of the fit for the yield curve. They consist of the fitted constants , the covariance matrix , and the total fitting error
The fitted discount curve has the form:
where the yield curve is the result of minimization:
with the fitting method and the weights
Here are the market yields:
Equation is the first order approximation to the (non-linear) least-squares fitting of the discount curve to the market discount factors . The underlying statistical model has the form:
where are independent standard gaussian random variables with mean 0 and variance 1, is the family of fitting functions from method , is the true value of the parameter, and is the unknown normalizing volatility.
(TODO) Discount curve by the Svensson fit of yields
Input:
- : the maturties,
- : the discount factors
- : the first mean-reversion rate
- : the second mean-reversion rate,
- : the initial time,
Output:
- : the fitted discount curve
- : the error function of the fit for the discount curve
- : the fitted constants, their covariance matrix, and the total fitting error
The discount curve is given by
where the yield curve has the Svensson form:
and constants are the result of the least-squares fit of the market yields:
Forward Curve
(TODO) Forward curve by fitting of cost-of-carry rates
Input:
- : the market spot price
- : the maturities of the forward contracts,
- : the market forward prices
- : the initial time,
- : a fitting method for cost of carry rates, a class
cfl::Fit
Output:
- : the fitted forward curve
- : the error function of the fit for the forward curve
- : the parameters of the fit for the cost-of-carry rate curve. They contain the fitted constants , the covariance matrix , and the total fitting error
The forward curve has the form:
where the cost-of-carry function is the result of the -fit of the market cost-of-carry rates:
(TODO) Forward curve obtained by fitting of cost-of-carry rates in the BS model
Input:
- : the market spot price
- : the maturities of forward contracts,
- : the market forward prices
- : the mean-reversion rate
- : the short-term volatility
- : the initial time,
Output:
- : the fitted forward curve
- : the error function of the fit for the forward curve
- : the parameters of the fit for the cost-of-carry rate curve. They contain the fitted constant , its variance , and the total fitting error
Returns the forward curve in the form:
where the cost-of-carry rate fits the market cost-of-carry rates in the Black-Scholes model. Under the BS model, the log of spot price has the form:
where is an OU processs:
Here is a Brownian motion and the drift is the fitting parameter. Computations show that
(TODO) Forward curve by fitting of cost-of-carry rates in the 2-factor BS model
Input:
- : the market spot price
- : the maturities of forward contracts,
- : the market forward prices
- : the second mean-reversion rate
- : the first short-term volatility
- : the second short-term volatility;
- : the initial time,
Output:
- : the fitted forward curve
- : the error function of the fit for the forward curve
- : the fitted constants, their covariance matrix, and the total fitting error
Returns the forward curve in the form:
where the cost-of-carry rate curve fits the market cost-of-carry rates in the 2-factor BS model. Under this model, the log of spot price has the form:
where and are independent OU processes:
Here and are independet Brownian motions and drifts and are the fitting parameters. Computations show that
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