Smith-Wilson-Yield_Curves.jl

### A Pluto.jl notebook ###
# v0.16.1

using Markdown
using InteractiveUtils

# ╔═╡ 3cf8bdda-9376-4165-857d-563205bb5d12
using LinearAlgebra

# ╔═╡ 941cdea6-8f8b-4d96-a386-c48b03d44fa2
using Plots

# ╔═╡ df1f6071-90d6-426c-adad-94826f660401
md"
This follows the example at [http://gli.lu/2017/12/smith-wilson-yield-curves/](http://gli.lu/2017/12/smith-wilson-yield-curves/)
"

# ╔═╡ f9d756be-2175-11ec-0858-7f781618d90e
U = repeat(1:5,1,5)

# ╔═╡ d6991c71-5514-4cae-b83b-3cea26194a5b
t = repeat(1:5,1,5)' 

# ╔═╡ 135418aa-a52a-4728-955e-841553fdc6fb
C = [
		1.010 0.020 0.026 0.034
		0.000 1.020 0.026 0.034
		0.000 0.000 1.026 0.034
		0.000 0.000 0.000 0.034
		0.000 0.000 0.000 1.034
	  
	]

# ╔═╡ 1baccf3b-fb92-4e70-97da-3d540d7ce806
function W(t,u;ultimate_forward=log(1+0.042),α=0.1) 
	l = min.(t,u)
	h = max.(t,u)
	
	return (exp.(-ultimate_forward.*(t+u)) 
	.* (α .* l - 0.5.*exp.(-α.*h).*(exp.(α.*l)-exp.(-α.*l))))
end

# ╔═╡ d7860389-479e-420c-b8a5-eaf1b7c32e56
function ζ(C,W,m,μ) 
	inv(C'*W*C)*(m .- C' * μ)
end

# ╔═╡ 716440c3-9665-44a8-8e50-17fc25568404
W(t,U)

# ╔═╡ 0fb6f17a-3b9b-4371-bb27-f1c5e4dbf8d3
C

# ╔═╡ 06f1830b-9009-4821-bfce-5af501607900
price = [1,1,1,1]

# ╔═╡ 5c1fa349-07b8-46a8-a333-e062f0de5bf7
z = let ufr = log(1+0.042)
	ζ(C,W.(t,U),price,exp.(-ufr .* [1:5...]))
end

# ╔═╡ 1e7243b3-99bd-440d-9dee-17e439f6eb4b
μ = exp.(-log(1+0.042) .* [1:5...])

# ╔═╡ 453bd819-62bb-4ec3-97d4-7880539dfa6a
price .- C' * μ

# ╔═╡ 7985a141-dca9-4c83-b7bb-301e765522a4
md"## Visualizing components"

# ╔═╡ 6ccdde74-57ef-4f92-bfd1-8a4535c4ec02
J = 5

# ╔═╡ 93421ad3-c0c0-4227-8409-c65b1de75fe7
K = 50

# ╔═╡ 4b09ec48-4d41-4b36-b1c6-61363767327d
T_big = repeat(1:J,1,K)'

# ╔═╡ d8b302ac-bbda-4660-9624-563cda467f57
U_big = repeat(1:K,1,J)

# ╔═╡ 4f4e20e1-c0ba-4d86-8e18-09a773f4d787
plot(W(T_big,U_big))

# ╔═╡ 3c5d8ab3-3a8a-47e3-8653-ff7b227d30f8
"""
the kernel function
"""
function kernel(c,w)
	return [sum(c[:,bond_i] .* w[t,:]) for t in 1:size(w,1),bond_i in 1:size(c,2)]
end


# ╔═╡ 9b69ac80-3779-4c1b-960e-cb27f53023fc
plot(kernel(C,W(T_big,U_big)))

# ╔═╡ de977178-08ff-4cd2-96a9-05b4406a201c
function P(c,w,ζ,ufr=log(1+0.042))
	A = exp.(-ufr.* [1:size(w,1)...])
	B = [sum(ζ .* kernel(c,w)[t,:] ) for t in 1:size(w,1)]
	return A .+ B
end


# ╔═╡ b9f13dc7-ac0d-40be-9dbb-81993f87b1b5
zero_price = P(C,W(T_big,U_big),z)

# ╔═╡ 0bfe990c-06df-418c-8dcb-411fac7a8789
md"
The zero interest curve:
"

# ╔═╡ cabff7dc-7ef3-465d-945c-ee0a91c105a2
plot([1-zero_price[t] ^ (1/t) for t in 1:length(zero_price)])

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git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "cc4bf3fdde8b7e3e9fa0351bdeedba1cf3b7f6e6"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "c45f4e40e7aafe9d086379e5578947ec8b95a8fb"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+0"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

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