Nelson-Siegel model using nloptr R package
In this post, the non-linear least squares problem is solved by using nloptr R package. As a proper example, Nelson-Siegel model is used as it is a representative non-linear problem in financial application.
Nelson-Siegel model as a non-linear optimization problem
Nelson-Siegel model is a non-linear least square problem with 4 parameters with some inequality constraints.
\[\begin{align} y(\tau) &= \beta_1 + \beta_2 \left( \frac{1-e^{- \tau \lambda }}{\tau \lambda }\right) \\ &+ \beta_3 \left(\frac{1-e^{- \tau \lambda }}{\tau \lambda }-e^{- \tau \lambda }\right) \end{align}\] \[\begin{align} \beta_1 > 0, \beta_1 + \beta_2 >0, \lambda > 0 \end{align}\]
Here, \(\tau\) is a maturity and \(y(\tau)\) is a continuously compounded spot rate with \(\tau\) maturity. \(\beta_1, \beta_2, \beta_3\) are coefficient parameters and \(\lambda\) is the time decay parameter.
nloptr() function
After installing nloptr R package, nloptr() function with the following specification is available to use.
1 2 3 4 5 6 7 8 9 10 | nloptr( x0 = initial guess, eval_f = objective function, lb = lower bounds, ub = upper bounds,
eval_grad_f = gradient function,
eval_g_ineq = inequality constraints,
eval_g_eq = equality constraints, opts = optimization options ) | cs |
As the names of arguments indicate, we can understand the meanings of each arguments. Here, initial guesses, objective function, lower and upper bounds, options must be provided but the rest are optional.
Options
nloptr() function should takes opts as an argument since at least algorithm is needed to be provided. nloptr R package provides a list of algorithms, which are categorized into two classes : with and without gradient.
As there are many algorithms included in this package, for the detailed information of algorithms or other settings, refer to https://nlopt.readthedocs.io/en/latest/.
In particular, if the number of evaluation is too small, the non-linear optimization result tends to be poor so that "maxeval" is set be larger and also "xtol_rel" is set smaller to avoid local minima. In this perspective, I set maxeval=500000 and xtol_rel=1.0e-16.
1 2 3 4 5 6 | opts <- list( "algorithm" = "NLOPT_GN_ISRES", "xtol_rel" = 1.0e-16, "maxeval" = 500000, "print_level" = 1 ) | cs |
I used minimal options. If you try to perform the full-fledged optimization work, you need to apply additional settings to options.
R code
The R code below estimates parameters of Nelson-Siegel model with two yield curves. It is worth noting that when an objective function has additional arguments such as data or maturities, these are included in nloptr() function like optim() function.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 | #========================================================# # Quantitative ALM, Financial Econometrics & Derivatives # ML/DL using R, Python, Tensorflow by Sang-Heon Lee # # https://shleeai.blogspot.com #--------------------------------------------------------# # Nelson-Siegel using nloptr function #========================================================# graphics.off(); rm(list = ls()) library(nloptr) #----------------------------------------------- # objective function #----------------------------------------------- objfunc <- function(para, y, m) { beta <- para[1:3]; la <- para[4] C <- cbind( rep(1,length(m)), (1-exp(-la*m))/(la*m), (1-exp(-la*m))/(la*m)-exp(-la*m)) return(sqrt(mean((y - C%*%beta)^2))) } #----------------------------------------------- # constraint function ( <= : less than equal ) #----------------------------------------------- # beta0 + beta1 > 0 constfunc <- function(para, y, m) { beta <- para[1:3] lhs <- -beta[1]-beta[2] return(lhs) } #======================================================= # 1. Read data #======================================================= # b1, b2, b3, lambda, rmse for comparisons ns_reg_para_rmse1 <- c( 4.26219396, -4.08609206, -4.90893865, 0.02722607, 0.04883786) ns_reg_para_rmse2 <- c( 4.97628654, -4.75365297, -6.40263059, 0.05046789, 0.04157326) str.zero <- " mat rate1 rate2 3 0.0781 0.0591 6 0.1192 0.0931 9 0.1579 0.1270 12 0.1893 0.1654 24 0.2669 0.3919 36 0.3831 0.8192 48 0.5489 1.3242 60 0.7371 1.7623 72 0.9523 2.1495 84 1.1936 2.4994 96 1.4275 2.7740 108 1.6424 2.9798 120 1.8326 3.1662 144 2.1715 3.4829 180 2.5489 3.7827 240 2.8093 3.9696" df <- read.table(text = str.zero, header=TRUE) m <- df$mat y1 <- df$rate1; y2 <- df$rate2 #======================================================= # 2. Nonlinear least squares with constraints #======================================================= # Set optimization options. opts <- list("algorithm" = "NLOPT_GN_ISRES", "xtol_rel" = 1.0e-16, "maxeval" = 500000, "print_level" = 1 ) #--------------------------------------------- # NS estimation with 1st data #--------------------------------------------- y <- y1 x_init <- c(y[16], y[1]-y[16], 2*y[6]-y[1]-y[16], 0.0609) m_nloptr <- nloptr( x0 = x_init, y = y, m = m, eval_f = objfunc, eval_g_ineq = constfunc, # beta0 > 0, 0.01 < lambda < 0.1 lb = c( 0,-30,-30, 0.001), ub = c(30, 30, 30, 0.1), opts = opts ) ns_nloptr_out1 <- c(m_nloptr$solution, m_nloptr$objective) #--------------------------------------------- # NS estimation with 2nd data #--------------------------------------------- y <- y2 x_init <- c(y[16], y[1]-y[16], 2*y[6]-y[1]-y[16], 0.0609) m_nloptr <- nloptr( x0 = x_init, y = y, m = m, eval_f = objfunc, eval_g_ineq = constfunc, # beta0 > 0, 0.01 < lambda < 0.1 lb = c( 0,-30,-30, 0.001), ub = c(30, 30, 30, 0.1), opts = opts ) ns_nloptr_out2 <- c(m_nloptr$solution, m_nloptr$objective) #======================================================= # 3. Results and Comparisons #======================================================= ns_reg_para_rmse1 ns_nloptr_out1 ns_reg_para_rmse2 ns_nloptr_out2 | cs |
The following estimation results show that non-linear optimization results from nloptr are close to the answers.
Using these estimated parameters at two dates, fitted yield curves are generated as follows.
Concluding Remarks
This post shows how to use nloptr R package to perform non-linear optimization problem with or without equality or inequality constraints. In fact, as non-linear optimization is vulnerable to the type or magnitude of constraints, careful consideration is needed to handle this kind of problem. \(\blacksquare\)
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