Transcript Pricing of Forward Rate Agreements

Pricing of Forward Rate
Agreements
ANALYTICAL FINANCE II
MICHAEL AUSSIEKER, OLIVER GRACE
Step One

Calculate the discount factors:
𝑑𝑂𝑁
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𝑑 𝑇𝑁
𝑝𝑎𝑟
𝐷𝑇𝑁 = 𝐷𝑂𝑁 ÷ 1 + 𝑟𝑇𝑁 ×
360
𝑑𝑖
𝑝𝑎𝑟
𝐷𝑖 = 𝐷𝑇𝑁 ÷ 1 + 𝑟𝑖
×
; 𝑖 = { 1𝑊, 1𝑀, 2𝑀, 3𝑀}
360
𝑝𝑎𝑟
𝐷𝑂𝑁 = 1 ÷ 1 + 𝑟𝑂𝑁 ×
Step Two

Calculate the Zero Coupon Rates:
𝑑𝑂𝑁
365
𝑑 𝑇𝑁
÷
365
𝑍𝑂𝑁 = −100 × 𝐿𝑛 𝐷𝑂𝑁 ÷
𝑍𝑇𝑁 = −100 × 𝐿𝑛 𝐷𝑇𝑁
𝑑𝑖
𝑍𝑖 = −100 × 𝐿𝑁 𝐷𝑖 ÷
; 𝑖 = { 1𝑊, 1𝑀, 2𝑀, 3𝑀}
365
The Nelson–Siegel–Svensson model
𝛽1 1 − 𝑒
𝑟 𝑡 = 𝛽0 +
𝑡/𝜏1
𝑡
−
𝜏1
𝛽2 1 − 𝑒
+
𝑡/𝜏1
𝑡
−
𝜏1
− 𝛽2 𝑒
𝑡
−
𝜏1
𝛽3 1 − 𝑒
+
𝑡/𝜏2
𝑡
−
𝜏2
− 𝛽3 𝑒
𝑡
−
𝜏2
where 𝛽0 𝛽1 𝛽2 𝛽3 𝜏1 𝜏2 are parameters calculated via least squares or a solver
such as the one found in Microsoft Excel.
Calculate Parameters
Used R language to calculate parameters below:
Parameters
β0
β1
β2
β3
τ1
τ2
3 month Forward Rate
1.861033
-1.707564
-6.009837
8.069004
3.908024
6.417405
6 month Forward Rate
-0.6770534
0.7609579
1.115105
10.88655
0.5622814
12.64441
OIS Forward Rate
-1.349865
1.470678
-1.418929
12.62976
1.119861
6.417405
OIS Zero Rate
1.914176
-1.813437
-4.888361
5.597794
3.350408
6.417405
Step Three

Calculate Zero Rate Curve for EUR 3M, EUR 6M, EUR OIS 3M, EUR OIS 6M
with NSS

Find 3x6 etc Forward Rates from EUR 3M and 3x9 etc Rates from EUR 6M

Do same for EUR OIS 3M and EUR OIS 6M curves
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2027-11-01
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2013-11-01
3 Month FRA
4,0000000
3,5000000
3,0000000
2,5000000
2,0000000
1,5000000
Zero Rate Curve via NSS
3MOT Forward
1,0000000
0,5000000
0,0000000
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2038-11-01
2037-12-01
2037-01-01
2036-02-01
2035-03-01
2034-04-01
2033-05-01
2032-06-01
2031-07-01
2030-08-01
2029-09-01
2028-10-01
2027-11-01
2026-12-01
2026-01-01
2025-02-01
2024-03-01
2023-04-01
2022-05-01
2021-06-01
2020-07-01
2019-08-01
2018-09-01
2017-10-01
2016-11-01
2015-12-01
2015-01-01
2014-02-01
6 Month FRA
4,000000
3,500000
3,000000
2,500000
2,000000
1,500000
NSS 3MOT Zero Rate
6mo FRA
1,000000
0,500000
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2019-09-01
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3
2,5
2
1,5
NSS-3moZeor
1
NSS-3moFor
0,5
0
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2023-04-01
2024-02-01
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2025-10-01
2026-08-01
2027-06-01
2028-04-01
2029-02-01
2029-12-01
2030-10-01
2031-08-01
2032-06-01
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2034-02-01
2034-12-01
2035-10-01
2036-08-01
2037-06-01
2038-04-01
2039-02-01
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3,000000
2,500000
2,000000
1,500000
NSS-6moZeor
1,000000
NSS-6moFor
0,500000
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