yrnd-functions

library(yrnd)

yrnd

The package ‘“yrnd”’ estimates a parametric form of the Risk Neutral Density (RND) of the price of fixed-income futures such as Short Term Interest Rates futures and government bond futures, using options written on those futures. The RND is estimated at options’ maturity for any type of option’s style. The futures price is modeled either as a mixture of two or three lognormal laws. It provides with the options prices predicted with the fitted parameters, some statistics of the distribution as well as a density and a cumulative density plot. Leveraging on this, the package provides with several functions. First, it provides with the distribution of the STIR rate and the government bond yield themselves at options’ maturity, using the RND of respectively the futures price of the STIR futures and the bond futures. Then, using the RND of the bond futures price, it provides with the probability attached to each bond in the delivery basket of a government bond futures contract to be the cheapest at maturity of the futures and at maturity of the options, using the forward net basis. Then, the package extracts from the RND of the bond futures price two RNDs simultaneously: for each bond in the delivery basket of the futures contract, it provides with the RND of its forward yield and the RND of its forward repo rate. At last, it also provides with the non parametric RND of the bond yield spread between two issuers or two maturities, using RNDs on two bond futures with the same maturity date and a Gaussian copula.

stir_future_price provides with the risk neutral density at options’ maturity of the STIR futures price, using options on the STIR futures

stir_future_price( c(1.44500, 1.32000, 1.19750, 1.07500, 0.95750,
0.84250, 0.78750, 0.73250, 0.68000, 0.62750, 0.57750, 0.53000, 0.48500,
0.44000, 0.39750, 0.35750, 0.32000, 0.28500, 0.25250, 0.22250, 0.19500,
0.17000, 0.14750, 0.12750, 0.10750, 0.09250, 0.07750, 0.06500, 0.05500,
0.04500, 0.03750, 0.03000, 0.02500, 0.02000, 0.01500, 0.01250, 0.01000,
0.00750, 0.00500, 0.00500, 0.00250, 0.00250, 0.00250, 0.00250,
rep(0.00024, 47)),
c(seq(93.25, 93.875, 0.125),  seq(93.9375, 98.8125, 0.0625),
seq(98.875, 99.5, 0.125)),
c(0.0025, 0.0050, 0.0075, 0.0125, 0.0175, 0.0300, 0.0350, 0.0425, 0.0525,
0.0625, 0.0750, 0.0900, 0.1050, 0.1225, 0.1425, 0.1650, 0.1900, 0.2175,
0.2450, 0.2775, 0.3125, 0.3500, 0.3875, 0.4300, 0.4725, 0.5175, 0.5675,
0.6150, 0.6675, 0.7200, 0.7750, 0.8300, 0.8850, 0.9425, 1.0025, 1.0625,
1.1225, 1.1825, 1.2425, 1.3050, 1.3675, 1.4300, 1.4925, 1.5550, 1.6175,
1.6800, 1.7425, 1.8050, 1.8675, 1.9300, 1.9925, 2.0550, 2.1175, 2.1800,
2.2425, 2.3050, 2.3675, 2.4300, 2.4925, 2.5550, 2.6175, 2.6800, 2.7425,
2.8050, 2.8675, 2.9300, 2.9925, 3.0550, 3.1175, 3.1800, 3.2425, 3.3050,
3.3675, 3.4300, 3.4925, 3.5550, 3.6175, 3.6800, 3.7425, 3.8050, 3.8675,
3.9300, 3.9925, 4.0550, 4.1175, 4.1800, 4.3050, 4.4300, 4.5550, 4.6800,
4.8050),
c(seq(93.25, 93.875, 0.125),  seq(93.9375, 98.8125, 0.0625),
seq(98.875, 99.5, 0.125)),
2,
0.0537,
1,
3,
94.7,
as.Date("2024-02-29"),
as.Date("2024-02-25"),
as.Date("2023-12-18"),
"fed_fund_rate",
"USD")
#> $params
#>            [,1]
#> [1,] 4.55465456
#> [2,] 4.54735937
#> [3,] 0.00530515
#> [4,] 0.00555419
#> [5,] 0.45000375
#> attr(,"names")
#> [1] "m1"  "m2"  "s1"  "s2"  "pi1"
#> 
#> $discretized_rnd
#> # A tibble: 10,015 × 2
#>    domain      rnd
#>     <dbl>    <dbl>
#>  1   90.5 1.23e-13
#>  2   90.5 1.25e-13
#>  3   90.5 1.27e-13
#>  4   90.5 1.28e-13
#>  5   90.5 1.30e-13
#>  6   90.5 1.32e-13
#>  7   90.5 1.34e-13
#>  8   90.5 1.36e-13
#>  9   90.5 1.39e-13
#> 10   90.5 1.41e-13
#> # ℹ 10,005 more rows
#> 
#> $discretized_cdf
#> # A tibble: 10,015 × 2
#>    domain      cdf
#>     <dbl>    <dbl>
#>  1   90.5 7.98e-15
#>  2   90.5 8.10e-15
#>  3   90.5 8.23e-15
#>  4   90.5 8.36e-15
#>  5   90.5 8.49e-15
#>  6   90.5 8.62e-15
#>  7   90.5 8.75e-15
#>  8   90.5 8.89e-15
#>  9   90.5 9.03e-15
#> 10   90.5 9.16e-15
#> # ℹ 10,005 more rows
#> 
#> $CV
#> [1] 0
#> 
#> $moments
#>          mean        stddev      skewness      kurtosis 
#> 94.6952305688  0.6195098073 -0.0002829645  2.8051083501 
#> 
#> $mode
#> [1] 94.69638
#> 
#> $quantiles
#>       q0.1     q0.5       q1       q5      q10      q25      q50      q75
#> 1 91.73288 93.14838 93.28988 93.67838 93.89238 94.26488 94.69538 95.12588
#>        q90      q95      q99    q99.5    q99.9
#> 1 95.49738 95.71138 96.10088 96.24238 97.84338
#> 
#> $model_prices
#> $model_prices$model_call_price
#>  [1] 1.446733e+00 1.323138e+00 1.200595e+00 1.079686e+00 9.611771e-01
#>  [6] 8.460129e-01 7.900207e-01 7.352796e-01 6.819366e-01 6.301377e-01
#> [11] 5.800255e-01 5.317360e-01 4.853962e-01 4.411212e-01 3.990123e-01
#> [16] 3.591552e-01 3.216187e-01 2.864533e-01 2.536909e-01 2.233438e-01
#> [21] 1.954051e-01 1.698483e-01 1.466280e-01 1.256804e-01 1.069239e-01
#> [26] 9.026107e-02 7.557992e-02 6.275603e-02 5.165504e-02 4.213530e-02
#> [31] 3.405082e-02 2.725420e-02 2.159955e-02 1.694524e-02 1.315634e-02
#> [36] 1.010658e-02 7.679958e-03 5.771769e-03 4.289135e-03 3.151101e-03
#> [41] 2.288303e-03 1.642310e-03 1.164726e-03 8.161350e-04 5.649573e-04
#> [46] 3.863095e-04 2.609012e-04 1.740187e-04 1.146192e-04 7.454612e-05
#> [51] 4.787047e-05 3.034985e-05 1.899613e-05 1.173735e-05 7.158954e-06
#> [56] 4.310071e-06 2.561282e-06 1.502286e-06 8.696734e-07 4.968850e-07
#> [61] 2.801823e-07 1.559197e-07 8.563053e-08 4.641048e-08 2.482317e-08
#> [66] 1.310229e-08 6.824664e-09 3.507974e-09 1.779384e-09 8.906758e-10
#> [71] 4.399515e-10 2.144493e-10 1.031524e-10 4.896317e-11 2.293486e-11
#> [76] 1.060135e-11 4.835775e-12 2.176777e-12 9.669574e-13 4.238867e-13
#> [81] 1.833770e-13 7.828819e-14 3.298440e-14 1.371470e-14 5.627740e-15
#> [86] 2.279067e-15 3.592945e-16 5.374173e-17 7.627617e-18 1.027386e-18
#> [91] 1.313407e-19
#> 
#> $model_prices$model_put_price
#>  [1] 0.001502534 0.002907377 0.005364906 0.009455727 0.015946519 0.025782346
#>  [7] 0.032290126 0.040049074 0.049206014 0.059907105 0.072294901 0.086505468
#> [13] 0.102665656 0.120890628 0.141281722 0.163924661 0.188888142 0.216222766
#> [19] 0.245960296 0.278113206 0.312674489 0.349617735 0.388897472 0.430449792
#> [25] 0.474193316 0.520030503 0.567849351 0.617525462 0.668924468 0.721904735
#> [31] 0.776320256 0.832023631 0.888868980 0.946714676 1.005425769 1.064876009
#> [37] 1.124949390 1.185541200 1.246558566 1.307920533 1.369557734 1.431411741
#> [43] 1.493434158 1.555585566 1.617834389 1.680155741 1.742530332 1.804943450
#> [49] 1.867384050 1.929843977 1.992317302 2.054799781 2.117288427 2.179781169
#> [55] 2.242276590 2.304773741 2.367271993 2.429770934 2.492270301 2.554769928
#> [61] 2.617269711 2.679769587 2.742269517 2.804769478 2.867269456 2.929769444
#> [67] 2.992269438 3.054769435 3.117269433 3.179769432 3.242269432 3.304769431
#> [73] 3.367269431 3.429769431 3.492269431 3.554769431 3.617269431 3.679769431
#> [79] 3.742269431 3.804769431 3.867269431 3.929769431 3.992269431 4.054769431
#> [85] 4.117269431 4.179769431 4.304769431 4.429769431 4.554769431 4.679769431
#> [91] 4.804769431
#> 
#> 
#> $rnd_plot

#> 
#> $cdf_plot

stir_rate provides with the risk neutral density at options’ maturity of the STIR rate implicit in the STIR futures price, using options on the STIR futures

stir_rate( c(1.44500, 1.32000, 1.19750, 1.07500, 0.95750,
0.84250, 0.78750, 0.73250, 0.68000, 0.62750, 0.57750, 0.53000, 0.48500,
0.44000, 0.39750, 0.35750, 0.32000, 0.28500, 0.25250, 0.22250, 0.19500,
0.17000, 0.14750, 0.12750, 0.10750, 0.09250, 0.07750, 0.06500, 0.05500,
0.04500, 0.03750, 0.03000, 0.02500, 0.02000, 0.01500, 0.01250, 0.01000,
0.00750, 0.00500, 0.00500, 0.00250, 0.00250, 0.00250, 0.00250,
rep(0.00024, 47)),
c(seq(93.25, 93.875, 0.125),  seq(93.9375, 98.8125, 0.0625),
seq(98.875, 99.5, 0.125)),
c(0.0025, 0.0050, 0.0075, 0.0125, 0.0175, 0.0300, 0.0350, 0.0425,
0.0525, 0.0625, 0.0750, 0.0900, 0.1050, 0.1225, 0.1425, 0.1650, 0.1900,
0.2175, 0.2450, 0.2775, 0.3125, 0.3500, 0.3875, 0.4300, 0.4725, 0.5175,
0.5675, 0.6150, 0.6675, 0.7200, 0.7750, 0.8300, 0.8850, 0.9425, 1.0025,
1.0625, 1.1225, 1.1825, 1.2425, 1.3050, 1.3675, 1.4300, 1.4925, 1.5550,
1.6175, 1.6800, 1.7425, 1.8050, 1.8675, 1.9300, 1.9925, 2.0550, 2.1175,
2.1800, 2.2425, 2.3050, 2.3675, 2.4300, 2.4925, 2.5550, 2.6175, 2.6800,
2.7425, 2.8050, 2.8675, 2.9300, 2.9925, 3.0550, 3.1175, 3.1800, 3.2425,
3.3050, 3.3675, 3.4300, 3.4925, 3.5550, 3.6175, 3.6800, 3.7425, 3.8050,
3.8675, 3.9300, 3.9925, 4.0550, 4.1175, 4.1800, 4.3050, 4.4300, 4.5550,
4.6800, 4.8050),
c(seq(93.25, 93.875, 0.125),  seq(93.9375, 98.8125, 0.0625),
seq(98.875, 99.5, 0.125)),
2,
0.0537,
1,
3,
94.7,
as.Date("2024-02-29"),
as.Date("2024-02-25"),
as.Date("2023-12-18"),
"fed_fund_rate",
"USD")
#> $discretized_rnd
#> # A tibble: 10,015 × 2
#>    domain      rnd
#>     <dbl>    <dbl>
#>  1 -0.494 6.41e-25
#>  2 -0.493 6.54e-25
#>  3 -0.492 6.67e-25
#>  4 -0.491 6.80e-25
#>  5 -0.490 6.94e-25
#>  6 -0.489 7.07e-25
#>  7 -0.488 7.21e-25
#>  8 -0.487 7.36e-25
#>  9 -0.486 7.50e-25
#> 10 -0.485 7.65e-25
#> # ℹ 10,005 more rows
#> 
#> $discretized_cdf
#> # A tibble: 10,014 × 2
#>    domain      cdf
#>     <dbl>    <dbl>
#>  1 -0.493 6.48e-28
#>  2 -0.492 1.31e-27
#>  3 -0.491 1.98e-27
#>  4 -0.490 2.67e-27
#>  5 -0.489 3.37e-27
#>  6 -0.488 4.08e-27
#>  7 -0.487 4.81e-27
#>  8 -0.486 5.56e-27
#>  9 -0.485 6.31e-27
#> 10 -0.484 7.09e-27
#> # ℹ 10,004 more rows
#> 
#> $CV
#> [1] 0
#> 
#> $moments
#>         mean       stddev     skewness     kurtosis 
#> 5.3047694312 0.6195098073 0.0002829645 2.8051083501 
#> 
#> $mode
#> [1] 5.303618
#> 
#> $quantiles
#>       q0.1     q0.5       q1       q5      q10      q25      q50      q75
#> 1 1.549118 3.756618 3.898118 4.287618 4.501618 4.873118 5.303618 5.734118
#>        q90      q95      q99    q99.5    q99.9
#> 1 6.106618 6.320618 6.709118 6.850618 8.266118
#> 
#> $rnd_r_plot

#> 
#> $cdf_r_plot

bond_future_price provides with the risk neutral density at options’ maturity of the government bond futures price, using options on the government bond futures

bond_future_price(c(10.39,9.92,9.46,9.00,8.55,8.10,7.66,7.23,
6.81,6.39,5.98,5.58,5.20,4.82,4.46,4.10,3.76,3.44,3.13,2.83,2.56,
2.29,2.05,1.82,1.61,1.42,1.25,1.09,0.95,0.82,0.71,0.61,0.53,0.45,
0.38,0.33,0.28,0.23,0.20,0.17,0.14,0.12,0.10,0.08),
seq(106, 127.5, 0.5),
c(0.22,0.25,0.29,0.33,0.38,0.43,0.49,0.56,0.64,0.72,0.81,0.91,
1.03,1.15,1.29, 1.43,1.59,1.77,1.96,2.16,2.39,2.62,2.88,3.15,
3.44,3.75,4.08, 4.42,4.78,5.15,5.54,5.94,6.36,6.78,7.21,7.66,
8.11,8.56,9.03, 9.50,9.97,10.45,10.93,11.41),
seq(106, 127.5, 0.5),
2,
0.0344,
1,
3,
as.Date("2033-11-01"),
116.17,
as.Date("2024-12-10"),
as.Date("2024-11-22"),
as.Date("2024-06-14"),
"Italian",
"EUR")
#> $params
#>            [,1]
#> [1,] 4.73733566
#> [2,] 4.77141589
#> [3,] 0.06384427
#> [4,] 0.03548329
#> [5,] 0.52406783
#> attr(,"names")
#> [1] "m1"  "m2"  "s1"  "s2"  "pi1"
#> 
#> $discretized_rnd
#> # A tibble: 62,806 × 2
#>    domain        rnd
#>     <dbl>      <dbl>
#>  1   86.7 0.00000356
#>  2   86.7 0.00000356
#>  3   86.7 0.00000356
#>  4   86.7 0.00000357
#>  5   86.7 0.00000357
#>  6   86.7 0.00000357
#>  7   86.7 0.00000357
#>  8   86.7 0.00000358
#>  9   86.7 0.00000358
#> 10   86.7 0.00000358
#> # ℹ 62,796 more rows
#> 
#> $discretized_cdf
#> # A tibble: 62,806 × 2
#>    domain        cdf
#>     <dbl>      <dbl>
#>  1   86.7 0.00000436
#>  2   86.7 0.00000436
#>  3   86.7 0.00000437
#>  4   86.7 0.00000437
#>  5   86.7 0.00000437
#>  6   86.7 0.00000438
#>  7   86.7 0.00000438
#>  8   86.7 0.00000438
#>  9   86.7 0.00000439
#> 10   86.7 0.00000439
#> # ℹ 62,796 more rows
#> 
#> $CV
#> [1] 0
#> 
#> $moments
#>        mean      stddev    skewness    kurtosis 
#> 116.1689003   6.3208524  -0.2302168   3.4592909 
#> 
#> $mode
#> [1] 117.2891
#> 
#> $quantiles
#>       q0.1     q0.5       q1       q5      q10      q25      q50      q75
#> 1 91.47664 98.22664 99.97014 104.9681 107.8101 112.3656 116.5646 120.3046
#>        q90      q95      q99    q99.5    q99.9
#> 1 123.6886 125.8661 130.6676 132.7656 142.4341
#> 
#> $model_prices
#> $model_prices$model_call_price
#>  [1] 10.38193573  9.91668738  9.45589567  8.99996208  8.54931355  8.10440636
#>  [7]  7.66573067  7.23381544  6.80923319  6.39260421  5.98459920  5.58593993
#> [13]  5.19739686  4.81978325  4.45394511  4.10074698  3.76105342  3.43570694
#> [19]  3.12550315  2.83116433  2.55331301  2.29244707  2.04891802  1.82291398
#> [25]  1.61444843  1.42335563  1.24929298  1.09175014  0.95006438  0.82344105
#> [31]  0.71097786  0.61169171  0.52454616  0.44847859  0.38242536  0.32534432
#> [37]  0.27623371  0.23414721  0.19820499  0.16760090  0.14160605  0.11956947
#> [43]  0.10091615  0.08514326
#> 
#> $model_prices$model_put_price
#>  [1]  0.2117292  0.2464808  0.2856891  0.3297555  0.3791070  0.4341998
#>  [7]  0.4955241  0.5636089  0.6390266  0.7223976  0.8143926  0.9157333
#> [13]  1.0271903  1.1495767  1.2837385  1.4305404  1.5908468  1.7655004
#> [19]  1.9552966  2.1609577  2.3831064  2.6222405  2.8787114  3.1527074
#> [25]  3.4442419  3.7531491  4.0790864  4.4215436  4.7798578  5.1532345
#> [31]  5.5407713  5.9414851  6.3543396  6.7782720  7.2122188  7.6551377
#> [37]  8.1060271  8.5639406  9.0279984  9.4973943  9.9713995 10.4493629
#> [43] 10.9307096 11.4149367
#> 
#> 
#> $rnd_plot

#> 
#> $cdf_plot

ctd_bond_yield provides with the risk neutral density at options’ maturity of the yield to maturity of the Cheapest-to-Deliver Bond in a futures contract, using options on the government bond futures. The current CtD is assumed to be the CtD at futures’ maturity. This is achieved by considering a “term cash and carry relationship” between the Cheapest-to-Deliver Bond and the futures contract, holding constant the CtD repo rate at options’ maturity.

ctd_bond_yield(c(10.39,9.92,9.46,9.00,8.55,8.10,7.66,7.23,
6.81,6.39,5.98,5.58,5.20,4.82,4.46,4.10,3.76,3.44,3.13,2.83,2.56,
2.29,2.05,1.82,1.61,1.42,1.25,1.09,0.95,0.82,0.71,0.61,0.53,0.45,
0.38,0.33,0.28,0.23,0.20,0.17,0.14,0.12,0.10,0.08),
seq(106, 127.5, 0.5),
c(0.22,0.25,0.29,0.33,0.38,0.43,0.49,0.56,0.64,0.72,0.81,0.91,
1.03,1.15,1.29, 1.43,1.59,1.77,1.96,2.16,2.39,2.62,2.88,3.15,
3.44,3.75,4.08, 4.42,4.78,5.15,5.54,5.94,6.36,6.78,7.21,7.66,
8.11,8.56,9.03, 9.50,9.97,10.45,10.93,11.41),
seq(106, 127.5, 0.5),
2,
0.0344,
0.035,
1,
3,
0.893,
0.0435, 
as.Date("2033-11-01"),
2,
100,
2,
116.17,
as.Date("2024-12-10"),
as.Date("2024-11-22"),
as.Date("2024-06-14"),
"Italian",
"EUR")
#> $discretized_rnd
#> # A tibble: 62,805 × 2
#>     domain     rnd
#>      <dbl>   <dbl>
#>  1 0.00518 0.00327
#>  2 0.00518 0.00328
#>  3 0.00518 0.00328
#>  4 0.00519 0.00328
#>  5 0.00519 0.00328
#>  6 0.00519 0.00328
#>  7 0.00519 0.00328
#>  8 0.00519 0.00328
#>  9 0.00519 0.00329
#> 10 0.00519 0.00329
#> # ℹ 62,795 more rows
#> 
#> $discretized_cdf
#> # A tibble: 62,804 × 2
#>     domain           cdf
#>      <dbl>         <dbl>
#>  1 0.00518 0.00000000286
#>  2 0.00518 0.00000000572
#>  3 0.00519 0.00000000857
#>  4 0.00519 0.0000000114 
#>  5 0.00519 0.0000000143 
#>  6 0.00519 0.0000000172 
#>  7 0.00519 0.0000000200 
#>  8 0.00519 0.0000000229 
#>  9 0.00519 0.0000000258 
#> 10 0.00519 0.0000000286 
#> # ℹ 62,794 more rows
#> 
#> $CV
#> [1] 0
#> 
#> $moments
#>        mean      stddev    skewness    kurtosis 
#> 0.039163805 0.007595362 0.477981866 3.657946486 
#> 
#> $mode
#> [1] 0.03728708
#> 
#> $quantiles
#>         q0.1       q0.5         q1         q5       q10        q25        q50
#> 1 0.01170344 0.02082377 0.02294875 0.02797913 0.0303401 0.03411341 0.03843945
#>          q75        q90        q95        q99      q99.5      q99.9
#> 1 0.04350652 0.04927819 0.05303589 0.05996477 0.06248657 0.07305418
#> 
#> $rnd_y_plot

#> 
#> $cdf_y_plot

proba_ctd provides, for each bond in the delivery basket of a government bond futures contact, its probability to be the cheapest to deliver bond at futures’ maturity, based on a ranking of the net basis of each bond in the delivery basket at futures’ maturity. Here it is assumed that the RND of the bond futures price at futures maturity is the same as its RND at options’ maturity.

proba_ctd( c(24.10, 23.10, 22.12, 21.12, 20.12, 19.14, 18.14, 17.16, 16.18, 15.20,
14.22, 13.24, 12.28, 11.32, 10.36, 9.44, 8.50, 7.60, 6.72, 5.86, 5.04, 4.28,
3.56, 2.88, 2.30, 1.78,  1.36, 1.02, 0.76, 0.56, 0.42, 0.30, 0.22, 0.18, 0.14,
0.10,  0.08, 0.06, 0.06, 0.04, 0.04, 0.02, 0.02, 0.02, 0.02, 0.02, 0.02),
seq(85, 131),
c(0.02,  0.02, 0.02, 0.02, 0.02, 0.04, 0.04, 0.06, 0.08, 0.10, 0.12, 0.14, 0.18, 0.22,
0.26, 0.34,  0.40, 0.50, 0.62, 0.76, 0.94, 1.18, 1.46, 1.78, 2.20, 2.68, 3.26, 3.92,
4.66,  5.46, 6.32, 7.20, 8.12, 9.08, 10.04, 11.00, 11.98, 12.96, 13.96, 14.94, 15.94,
16.92, 17.92, 18.92, 19.92, 20.92, 21.92),
seq(85, 131),
2,
0.0187,
1,
3,
as.Date("2054-08-15"),
109.1,
as.Date("2026-09-08"),
as.Date("2026-08-21"),
as.Date("2026-05-28"),
c("DE0001102572", "DE0001102614", "DE0001030757", "DE000BU2D004", "DE000BU2D012"),
c(0.000, 0.018, 0.018, 0.025, 0.029),
rep(1, 5),
as.Date(c("2052-08-15", "2053-08-15", "2053-08-15", "2054-08-15", "2056-08-15")),
100,
c(0.361698, 0.641260, 0.641260, 0.750372, 0.809987),
c(0.03500, 0.03507, 0.03492, 0.03510, 0.03514),
2)
#>           ISIN proba_ctd_matu
#> 1 DE000BU2D004          0.673
#> 2 DE000BU2D012          0.261
#> 3 DE0001102572          0.066
#> 4 DE0001102614          0.000
#> 5 DE0001030757          0.000

proba_ctd_opt provides, for each bond in the delivery basket of a government bond futures contact, its probability to be the cheapest to deliver bond at options’ maturity, based on a ranking of the net basis of each bond in the delivery basket at options’ maturity

proba_ctd_opt( c(24.10, 23.10, 22.12, 21.12, 20.12, 19.14, 18.14, 17.16, 16.18, 15.20,
14.22, 13.24, 12.28, 11.32, 10.36, 9.44, 8.50, 7.60, 6.72, 5.86, 5.04, 4.28,
3.56, 2.88, 2.30, 1.78,  1.36, 1.02, 0.76, 0.56, 0.42, 0.30, 0.22, 0.18, 0.14,
0.10,  0.08, 0.06, 0.06, 0.04, 0.04, 0.02, 0.02, 0.02, 0.02, 0.02, 0.02),
seq(85, 131),
c(0.02,  0.02, 0.02, 0.02, 0.02, 0.04, 0.04, 0.06, 0.08, 0.10, 0.12, 0.14, 0.18, 0.22,
0.26, 0.34,  0.40, 0.50, 0.62, 0.76, 0.94, 1.18, 1.46, 1.78, 2.20, 2.68, 3.26, 3.92,
4.66,  5.46, 6.32, 7.20, 8.12, 9.08, 10.04, 11.00, 11.98, 12.96, 13.96, 14.94, 15.94,
16.92, 17.92, 18.92, 19.92, 20.92, 21.92),
seq(85, 131),
2,
0.0187,
0.019,
1,
3,
as.Date("2054-08-15"),
109.1,
as.Date("2026-09-08"),
as.Date("2026-08-21"),
as.Date("2026-05-28"),
c("DE0001102572", "DE0001102614", "DE0001030757", "DE000BU2D004", "DE000BU2D012"),
c(0.000, 0.018, 0.018, 0.025, 0.029),
rep(1, 5),
as.Date(c("2052-08-15", "2053-08-15", "2053-08-15", "2054-08-15", "2056-08-15")),
100,
c(0.361698, 0.641260, 0.641260, 0.750372, 0.809987),
c(0.03500, 0.03507, 0.03492, 0.03510, 0.03514),
2)
#>           ISIN proba_ctd_matu
#> 1 DE000BU2D004          0.586
#> 2 DE000BU2D012          0.354
#> 3 DE0001102572          0.060
#> 4 DE0001102614          0.000
#> 5 DE0001030757          0.000

bond_yield_spread provides, at a given maturity, 10,000 realizations of the bond yield spread between two different issuers’ yield or two yields of the same issuer at different maturities, using options on two bond futures with the same maturity and using the correlation coefficient between returns of the corresponding bond futures’ prices. The realizations are generated with the joint distribution of the two bond futures prices which are converted into bond yields, using the bonds in the delivery basket of each futures contract. The joint distribution is modeled with a gaussian copula and the marginal RNDs of bond futures price at option’s maturity. The bond futures prices are modeled as a mixture of two lognormal laws. The function also returns a density plot of the spread, in bps

bond_yield_spread(
c(12.64, 12.14, 11.65, 11.15, 10.65, 10.15,  9.65,  9.16, 8.66,
8.16, 7.67,  7.17, 6.68, 6.19, 5.70, 5.21, 4.73, 4.25, 3.78, 3.32,
2.87,  2.45, 2.05, 1.67, 1.33, 1.03, 0.77, 0.57, 0.41, 0.29,
0.21,  0.15, 0.10, 0.07, 0.06, 0.04, 0.04, 0.03, 0.02, 0.02, 0.02,
0.02,  0.01, 0.01, 0.01, 0.01, 0.01),
c(seq(114, 136.5, 0.5), 137.5),
c(0.01, 0.01, 0.02, 0.02, 0.02, 0.02, 0.02, 0.03, 0.03, 0.03, 0.04, 0.04,
0.05, 0.06, 0.07, 0.08, 0.10, 0.12, 0.15, 0.19, 0.24, 0.32, 0.42, 0.54,
0.70, 0.90, 1.14, 1.44, 1.78, 2.16, 2.58, 3.02, 3.47, 3.94, 4.43, 4.91,
5.41, 5.90, 6.39, 6.89, 7.39, 7.89, 8.38, 8.88, 9.38, 9.88, 10.88),
c(seq(114, 136.5, 0.5), 137.5),
c(13.46, 12.96, 12.46, 11.96, 11.47, 10.97, 10.47, 9.98, 9.48, 8.99, 8.49,
8.00,  7.51, 7.02, 6.53, 6.04, 5.56, 5.08, 4.61, 4.15, 3.69, 3.25, 2.82,
2.41,  2.02, 1.67, 1.34, 1.05, 0.80, 0.59, 0.43, 0.30, 0.21, 0.14, 0.09,
0.06, 0.04,  0.02, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01),
seq(105.5, 129, 0.5),
c(0.02, 0.02, 0.02, 0.02, 0.03, 0.03, 0.03, 0.04, 0.04, 0.05, 0.05, 0.06,
0.07, 0.08, 0.09, 0.10, 0.12, 0.14, 0.17, 0.21, 0.25, 0.31, 0.38, 0.47,
0.58, 0.73, 0.90, 1.11, 1.36, 1.65, 1.99, 2.36, 2.77, 3.20, 3.65, 4.12,
4.60, 5.08, 5.57, 6.07, 6.57, 7.06, 7.56, 8.06, 8.56, 9.06, 9.56, 10.06),
seq(105.5, 129, 0.5),
0.0187,
0.019,
1,
1,
3,
3,
as.Date("2035-08-15"),
as.Date("2035-08-01"),
126.66,
119.04,
as.Date("2026-09-08"),
as.Date("2026-08-21"),
as.Date("2026-06-17"),
c("DE000BU2Z056", "DE000BU2Z064"),
c(0.026, 0.029),
rep(1, 2),
as.Date(c("2035-08-15", "2036-02-15")),
c(0.770088, 0.781249),
c(0.02898, 0.02926),
2,
c("IT0005631590", "IT0005648149", "IT0005676504", "IT0005402117",
"IT0005706285", "IT0005433195"),
c(0.0365, 0.0360, 0.0345, 0.0145, 0.0380, 0.0095),
rep(2, 6),
as.Date(c("2035-08-01", "2035-10-01", "2036-02-01", "2036-03-01",
"2036-07-01", "2037-03-01")),
c(0.845212, 0.839556, 0.824415, 0.679587, 0.844354, 0.616457),
c(0.03567, 0.03587, 0.03620, 0.03614, 0.03665, 0.03712),
2,
100,
0.4,
"Germany",
"Italy")
#> $spread_distrib
#>              spread
#> 1       62.88911013
#> 2       96.63384660
#> 3       24.09730603
#> 4       44.61879568
#> 5       79.53967694
#> 6       35.41309745
#> 7      169.05205936
#> 8       72.75206782
#> 9       86.24933724
#> 10      -4.82851400
#> 11     128.29636501
#> 12     140.73278893
#> 13      78.47717407
#> 14      86.61799283
#> 15      63.60856050
#> 16      54.45754025
#> 17      10.80887521
#> 18      74.67444080
#> 19      47.04197610
#> 20      94.26930772
#> 21      54.71453383
#> 22      96.16244390
#> 23     123.15127348
#> 24     158.70357154
#> 25      69.63557706
#> 26      86.50438652
#> 27      80.66714827
#> 28      66.12498258
#> 29      93.18129641
#> 30      23.73589330
#> 31      52.53032021
#> 32      75.61523224
#> 33      59.94067088
#> 34      25.48006563
#> 35      94.66381591
#> 36      72.10310787
#> 37      54.80294732
#> 38      57.92013311
#> 39      63.22643139
#> 40      65.05411162
#> 41      15.56845387
#> 42      54.78666901
#> 43     -11.17814812
#> 44      62.69028266
#> 45      60.45772590
#> 46      72.92761237
#> 47      79.70814770
#> 48      65.14064549
#> 49      64.20098598
#> 50      50.95518630
#> 51      13.81583523
#> 52      97.38219291
#> 53      47.08497268
#> 54      38.67610432
#> 55      58.03488419
#> 56      40.88550984
#> 57      51.88590849
#> 58      37.24956441
#> 59      81.38947949
#> 60      69.88363616
#> 61      45.15144779
#> 62      90.02819285
#> 63      56.20028635
#> 64      51.54803062
#> 65      43.13373907
#> 66      56.17454241
#> 67      65.43139055
#> 68     103.23277493
#> 69      45.02082224
#> 70     113.73822542
#> 71      80.38370658
#> 72      36.16775362
#> 73     112.52458438
#> 74      63.67089538
#> 75     137.85008671
#> 76      70.13140366
#> 77     -15.60169750
#> 78      73.12339082
#> 79      67.63638702
#> 80     120.53809112
#> 81      72.30177446
#> 82      84.01929342
#> 83     105.67274852
#> 84      78.73062271
#> 85     102.29682369
#> 86     101.13518014
#> 87      25.24946872
#> 88      96.35643700
#> 89      48.07296019
#> 90      70.67228159
#> 91      78.77217359
#> 92      39.86689539
#> 93      85.59367273
#> 94      63.50941503
#> 95     -23.06746272
#> 96      62.89422956
#> 97      88.16078716
#> 98     -40.45416361
#> 99      58.38063609
#> 100     36.51270782
#> 101     72.59636952
#> 102     67.61373230
#> 103     84.33105993
#> 104     88.74158992
#> 105    144.78814697
#> 106     66.35889944
#> 107     84.88834027
#> 108    125.84406335
#> 109     40.93847203
#> 110     73.80891830
#> 111    116.93702131
#> 112    154.05058928
#> 113     49.09980140
#> 114     60.32537239
#> 115     12.76675469
#> 116     77.42379382
#> 117     98.69901181
#> 118     60.71157792
#> 119     80.80710744
#> 120     76.13295664
#> 121     77.90015349
#> 122     23.22473383
#> 123     48.19085424
#> 124    271.46473565
#> 125     56.35472885
#> 126     58.25393565
#> 127     82.91560777
#> 128     64.83185744
#> 129     89.36152061
#> 130     54.51451457
#> 131     64.73244279
#> 132     34.14959828
#> 133     75.36548913
#> 134    121.81840628
#> 135    106.67627705
#> 136     63.99825764
#> 137     78.28075582
#> 138    126.03817124
#> 139     64.03446378
#> 140     19.96836062
#> 141    128.02636920
#> 142    188.36634782
#> 143     67.90012987
#> 144     77.05773858
#> 145     44.86794134
#> 146     72.17844405
#> 147    131.60549122
#> 148     81.55537952
#> 149     56.87695702
#> 150     40.62718443
#> 151     71.99345267
#> 152    103.52559061
#> 153     41.87797290
#> 154    103.59675272
#> 155     15.14918288
#> 156     42.23490944
#> 157     84.96723711
#> 158    101.77657660
#> 159     84.86472085
#> 160     37.06805370
#> 161    128.97480765
#> 162    142.00162650
#> 163     57.57683917
#> 164     25.77824818
#> 165     31.14384379
#> 166    124.50837765
#> 167     36.08367542
#> 168     98.89791785
#> 169     77.36478116
#> 170     71.63855657
#> 171    133.15398905
#> 172     89.91579117
#> 173     60.86093318
#> 174     98.92103355
#> 175     37.55479376
#> 176     57.60491964
#> 177    102.17532047
#> 178     52.14969070
#> 179    118.66984410
#> 180     67.80724135
#> 181    105.89456054
#> 182     61.30745722
#> 183     83.00959637
#> 184    123.19598666
#> 185     74.53556703
#> 186     90.96531226
#> 187     79.95635757
#> 188     85.01629288
#> 189     64.53238295
#> 190    136.97632402
#> 191    111.08803328
#> 192     33.79749958
#> 193     87.14233817
#> 194     61.81236704
#> 195     55.24073719
#> 196     86.86121762
#> 197     41.74832719
#> 198     97.79313766
#> 199     49.62180862
#> 200     27.47389568
#> 201     94.56564490
#> 202     70.60784316
#> 203    -17.60309337
#> 204     26.76528975
#> 205     44.48448268
#> 206     54.25379677
#> 207     74.74763253
#> 208     70.71797912
#> 209     55.96514866
#> 210     44.24427709
#> 211     72.63255820
#> 212     68.65768883
#> 213     98.47363412
#> 214     94.32814406
#> 215    105.25995746
#> 216    116.24645387
#> 217     58.14733524
#> 218     17.26606564
#> 219     59.44889392
#> 220     67.55032372
#> 221    174.21797570
#> 222     32.48481255
#> 223     46.15360577
#> 224     45.70589562
#> 225     18.47819466
#> 226     68.72279471
#> 227     64.48811147
#> 228    130.25863186
#> 229     69.03076326
#> 230     67.93559694
#> 231     29.83431036
#> 232    135.00252786
#> 233     70.82656541
#> 234     77.51431263
#> 235     13.11991033
#> 236     61.56043692
#> 237    111.65927620
#> 238     37.78402499
#> 239     74.29271711
#> 240    112.28742619
#> 241    112.51584882
#> 242     87.43360479
#> 243     89.88906090
#> 244     55.92857452
#> 245     40.33518568
#> 246     59.41140415
#> 247     55.65565224
#> 248    123.16748450
#> 249    112.07560753
#> 250     58.01052660
#> 251     53.58838692
#> 252     73.43318000
#> 253    106.30955134
#> 254    123.90435502
#> 255     72.74086507
#> 256     23.42996288
#> 257    130.69235479
#> 258     18.47452680
#> 259     53.54659140
#> 260     45.26800882
#> 261     52.27125643
#> 262     81.40000300
#> 263     30.66094028
#> 264     45.52812590
#> 265     63.39597673
#> 266     74.56864556
#> 267    106.14789335
#> 268     30.00065992
#> 269     10.28705746
#> 270    126.34947166
#> 271     76.49806002
#> 272     61.89457001
#> 273     60.33836775
#> 274     70.70047692
#> 275     21.63163156
#> 276     94.86149422
#> 277     75.21903999
#> 278     96.97989103
#> 279     95.98123204
#> 280     59.50412139
#> 281    108.45189109
#> 282     96.69979562
#> 283     99.78295272
#> 284     88.65390056
#> 285     74.53533215
#> 286     71.97382840
#> 287     72.99410055
#> 288    113.97518781
#> 289     71.49377897
#> 290     77.77984678
#> 291     55.98445601
#> 292     79.25408402
#> 293     18.20250048
#> 294     42.43796531
#> 295     49.62894121
#> 296     71.91226406
#> 297     26.13226790
#> 298    131.62811631
#> 299     75.07409423
#> 300    -21.60233067
#> 301     49.05447097
#> 302     16.44980772
#> 303    111.38139578
#> 304     59.20660702
#> 305     80.18380875
#> 306    107.68679581
#> 307     74.80826043
#> 308     50.51173593
#> 309     74.74867641
#> 310     46.44749785
#> 311     66.33325415
#> 312     87.21570532
#> 313    132.95326083
#> 314     74.89616478
#> 315     77.84316237
#> 316     56.63937831
#> 317     66.22184147
#> 318     88.73523239
#> 319     42.65188398
#> 320     80.15502508
#> 321     58.52019728
#> 322     11.76525547
#> 323     30.27995418
#> 324     53.37176919
#> 325     69.19230403
#> 326     28.39779485
#> 327     97.51771178
#> 328    105.55559430
#> 329     63.68934335
#> 330     50.50278802
#> 331     84.66872536
#> 332     49.56876662
#> 333      6.64380399
#> 334     92.93464066
#> 335     92.85879491
#> 336     65.97276091
#> 337     67.20913600
#> 338     23.69785530
#> 339     33.67595343
#> 340     59.29438242
#> 341     57.27862850
#> 342     86.71832589
#> 343     96.47927176
#> 344     55.55218905
#> 345     74.64026881
#> 346     27.10354505
#> 347     54.62512171
#> 348     96.47108029
#> 349     55.45941250
#> 350     98.25253492
#> 351     32.17081390
#> 352      4.20537552
#> 353      8.04080443
#> 354     43.76793659
#> 355     88.46694673
#> 356    114.73593456
#> 357     93.41484960
#> 358     82.78283803
#> 359     53.58195417
#> 360     68.95182499
#> 361    132.32085494
#> 362     46.16980571
#> 363     98.96006438
#> 364    143.22328072
#> 365     59.24232675
#> 366    104.93057814
#> 367     64.73519609
#> 368     24.23962739
#> 369     23.21934377
#> 370     28.41887975
#> 371    100.22172947
#> 372     54.84761550
#> 373    253.11381932
#> 374    162.14642577
#> 375    111.32996332
#> 376    -11.31275660
#> 377     14.37116782
#> 378    152.20839621
#> 379     18.86921004
#> 380     70.68169436
#> 381     91.51521418
#> 382     68.31516232
#> 383     39.18758306
#> 384     62.12770154
#> 385    155.62463396
#> 386     77.45208855
#> 387    106.19809014
#> 388     75.80337497
#> 389    184.31825208
#> 390     40.84613648
#> 391     58.22133647
#> 392     77.36568462
#> 393     47.58302829
#> 394     82.67713984
#> 395     72.97183522
#> 396     53.15465925
#> 397     81.07971606
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#> 1723    64.82833853
#> 1724    80.65628449
#> 1725   111.72910194
#> 1726    74.77451140
#> 1727    53.12171578
#> 1728   117.86411625
#> 1729    28.24809315
#> 1730    93.58116047
#> 1731    33.62992049
#> 1732    75.49819062
#> 1733    65.26080842
#> 1734    48.43402468
#> 1735   226.58065916
#> 1736   128.95982921
#> 1737    79.12892718
#> 1738    46.57652048
#> 1739    79.12386995
#> 1740   151.91172975
#> 1741    54.88008683
#> 1742    94.45571542
#> 1743    87.38286724
#> 1744    55.20993311
#> 1745    86.42499810
#> 1746    47.54642324
#> 1747    44.14426626
#> 1748    74.77620525
#> 1749   122.20796507
#> 1750   123.76183318
#> 1751    51.05900046
#> 1752    64.12290747
#> 1753    54.20939734
#> 1754    73.39400962
#> 1755   -26.06159726
#> 1756   114.54618083
#> 1757    71.00116392
#> 1758    51.26295569
#> 1759    95.83002738
#> 1760    58.56629406
#> 1761    61.08598012
#> 1762    64.67347579
#> 1763    78.16440190
#> 1764   104.85267339
#> 1765    94.12735964
#> 1766    95.22998209
#> 1767    38.77522577
#> 1768    21.67288031
#> 1769    59.31992190
#> 1770    50.15200061
#> 1771    97.50164548
#> 1772    48.00841743
#> 1773   126.28261209
#> 1774    86.17864491
#> 1775   122.08872516
#> 1776   109.04008812
#> 1777   100.99670696
#> 1778   162.23089093
#> 1779    56.13593073
#> 1780    93.23846252
#> 1781    44.75361162
#> 1782    74.60025864
#> 1783    92.70273289
#> 1784    64.53399832
#> 1785    31.44264948
#> 1786    12.02170593
#> 1787    69.18056297
#> 1788     9.43157042
#> 1789    70.41771128
#> 1790    83.82753369
#> 1791    52.56953355
#> 1792   102.40881620
#> 1793    31.25998654
#> 1794   140.35737403
#> 1795    84.60567242
#> 1796    88.65792478
#> 1797    57.67117980
#> 1798    48.84827507
#> 1799   247.67397592
#> 1800    55.84420394
#> 1801    77.03588369
#> 1802    64.69529171
#> 1803    40.11102860
#> 1804    59.32643945
#> 1805    39.38293247
#> 1806    74.90333490
#> 1807    58.94848996
#> 1808    46.33028454
#> 1809    24.53715625
#> 1810    77.14876023
#> 1811   101.24036786
#> 1812    96.73076025
#> 1813    10.08341906
#> 1814    50.87822564
#> 1815    92.65471704
#> 1816    32.16970385
#> 1817   118.46223465
#> 1818    56.98749640
#> 1819   132.33511972
#> 1820    53.42349998
#> 1821    -0.39850309
#> 1822    87.45870835
#> 1823    67.49935499
#> 1824    40.04958859
#> 1825    79.24704698
#> 1826    47.38868116
#> 1827    58.65238046
#> 1828    63.40560286
#> 1829    58.48036050
#> 1830    95.46515481
#> 1831    39.88583191
#> 1832    46.93230614
#> 1833    41.06729623
#> 1834   106.07372325
#> 1835    76.03918384
#> 1836    35.04207446
#> 1837   155.94638326
#> 1838    96.20357900
#> 1839    75.13588915
#> 1840    65.27174978
#> 1841    65.25654991
#> 1842    58.47161494
#> 1843    34.32185939
#> 1844    87.18449274
#> 1845    49.38252288
#> 1846    -5.05381915
#> 1847   140.08910392
#> 1848    37.11116501
#> 1849   127.44767499
#> 1850    68.27931902
#> 1851    51.77824812
#> 1852    74.69777449
#> 1853   -37.17053941
#> 1854    45.39105612
#> 1855    74.58286156
#> 1856    54.17678087
#> 1857   160.76937495
#> 1858    56.69154686
#> 1859    30.98393249
#> 1860    84.68115867
#> 1861    69.90114862
#> 1862    50.83273981
#> 1863    74.03202047
#> 1864    93.32069912
#> 1865    37.22546057
#> 1866   113.09131024
#> 1867    41.92970905
#> 1868    91.17884240
#> 1869    38.89117348
#> 1870    51.99284115
#> 1871    80.02565991
#> 1872    64.60266093
#> 1873    43.15205886
#> 1874   114.52999908
#> 1875    30.23886416
#> 1876    66.19058634
#> 1877   162.71849173
#> 1878    91.84023985
#> 1879    87.31839771
#> 1880    90.78969514
#> 1881    90.14885532
#> 1882    71.20340069
#> 1883    73.36861998
#> 1884    82.02070657
#> 1885   153.91748914
#> 1886    55.11516522
#> 1887    41.86304422
#> 1888   143.04132447
#> 1889    42.66091807
#> 1890   113.32609525
#> 1891    22.26974263
#> 1892   100.12431702
#> 1893    69.35984794
#> 1894    54.10516293
#> 1895    94.61181056
#> 1896   106.38154294
#> 1897    93.81411389
#> 1898    91.39432394
#> 1899    65.70397816
#> 1900    96.77965383
#> 1901    83.11084436
#> 1902    86.06605659
#> 1903    94.68332453
#> 1904    25.26560340
#> 1905    77.64258181
#> 1906    66.71220745
#> 1907    61.06917394
#> 1908    91.76665282
#> 1909    73.01163296
#> 1910   112.36148610
#> 1911    49.13030108
#> 1912    73.81813320
#> 1913    84.79480839
#> 1914    94.14268755
#> 1915    38.89799024
#> 1916   125.01961911
#> 1917   101.90887537
#> 1918    55.76830834
#> 1919    44.62662478
#> 1920    34.10974519
#> 1921    91.99875191
#> 1922    49.48292154
#> 1923    54.82208461
#> 1924   -23.07113565
#> 1925   112.02314006
#> 1926   117.03787838
#> 1927   117.47123655
#> 1928    77.84076029
#> 1929    72.07071388
#> 1930    46.31466632
#> 1931    82.46019117
#> 1932    72.95251885
#> 1933    45.61029635
#> 1934    59.75317757
#> 1935    75.12946775
#> 1936    72.44563556
#> 1937   146.65185824
#> 1938    48.99579374
#> 1939    40.68356904
#> 1940    53.37488791
#> 1941    90.52598201
#> 1942    62.08347353
#> 1943    99.85750305
#> 1944    58.56631756
#> 1945    91.32982446
#> 1946    48.37277092
#> 1947    48.63403844
#> 1948   134.03066602
#> 1949    62.70630585
#> 1950    47.94974597
#> 1951    57.72125701
#> 1952    74.68492970
#> 1953    76.76293585
#> 1954    59.10666414
#> 1955    99.57078949
#> 1956    37.30286856
#> 1957    71.22151214
#> 1958    61.49928844
#> 1959    96.93869820
#> 1960    55.12717261
#> 1961    94.17386598
#> 1962   100.51469334
#> 1963    92.08844511
#> 1964   107.04454150
#> 1965   108.95312913
#> 1966    59.45294793
#> 1967    49.43865057
#> 1968    29.89443976
#> 1969    58.96965500
#> 1970    90.39957059
#> 1971    60.82338356
#> 1972    94.96962263
#> 1973    63.97859477
#> 1974     5.53619021
#> 1975    98.59369232
#> 1976   -69.37801262
#> 1977    57.84602233
#> 1978    45.53567440
#> 1979   142.32627908
#> 1980   226.12974274
#> 1981   101.35668141
#> 1982    92.81314129
#> 1983   210.74979469
#> 1984   105.36243979
#> 1985    74.69066973
#> 1986    79.66910285
#> 1987   -57.97085212
#> 1988    32.44935548
#> 1989    77.47800234
#> 1990   100.03371831
#> 1991    67.89809568
#> 1992    52.03690362
#> 1993    87.87764647
#> 1994    76.72853739
#> 1995   127.83836477
#> 1996    53.34518223
#> 1997    26.52262351
#> 1998   -18.75523787
#> 1999    86.66047471
#> 2000    31.73004200
#> 2001    18.31203117
#> 2002    70.56239962
#> 2003   114.32502426
#> 2004   101.66594813
#> 2005    91.98367320
#> 2006    78.12218519
#> 2007    83.18473646
#> 2008    79.51994189
#> 2009    92.50976021
#> 2010    72.26310129
#> 2011    46.63590109
#> 2012    62.97465012
#> 2013    37.33070701
#> 2014    70.14298123
#> 2015   170.06119537
#> 2016    84.77355933
#> 2017    26.62960344
#> 2018    85.81204190
#> 2019    77.65765863
#> 2020    46.82426735
#> 2021   109.99775572
#> 2022    54.39924350
#> 2023    31.63674231
#> 2024    55.96791098
#> 2025    72.68657280
#> 2026    78.75439871
#> 2027    86.94165885
#> 2028    35.00092300
#> 2029    40.55484307
#> 2030    47.85628264
#> 2031   117.20550823
#> 2032    80.03909571
#> 2033   101.50373409
#> 2034    72.46924240
#> 2035    37.25720318
#> 2036    39.30224523
#> 2037    40.24240551
#> 2038    69.03385296
#> 2039    40.61513659
#> 2040    44.64028644
#> 2041   -36.09448890
#> 2042    82.27469541
#> 2043    53.05950816
#> 2044    81.68849210
#> 2045    70.41976117
#> 2046    51.98252645
#> 2047    59.35225018
#> 2048    93.13282925
#> 2049    76.47183053
#> 2050    92.22189733
#> 2051    27.10182109
#> 2052    63.15939624
#> 2053    60.16235107
#> 2054   135.35971826
#> 2055    53.50337747
#> 2056    76.70973616
#> 2057    49.30877942
#> 2058    24.45998782
#> 2059    90.24734905
#> 2060    42.79188247
#> 2061    39.15152955
#> 2062    89.42178590
#> 2063   187.84150891
#> 2064   214.80237282
#> 2065    54.79251707
#> 2066    63.81599413
#> 2067    85.10367742
#> 2068   102.34047538
#> 2069    88.92721299
#> 2070    48.16691790
#> 2071   100.57508862
#> 2072     3.71462473
#> 2073    16.65135526
#> 2074     7.92985712
#> 2075    49.17971570
#> 2076    43.02013389
#> 2077    74.98506471
#> 2078    82.65871657
#> 2079    31.36639734
#> 2080    47.33274902
#> 2081   106.42962324
#> 2082    29.31413879
#> 2083    67.73496519
#> 2084    97.11695289
#> 2085    53.04569687
#> 2086    46.81432623
#> 2087    83.79404350
#> 2088    90.07839481
#> 2089    97.56322566
#> 2090    63.23998373
#> 2091    62.86431360
#> 2092    20.75279676
#> 2093    40.21529441
#> 2094    95.03292135
#> 2095   117.21218233
#> 2096    67.10035844
#> 2097    81.47509547
#> 2098    64.76752552
#> 2099    53.51595887
#> 2100    22.65746311
#> 2101    56.10118049
#> 2102    56.50823120
#> 2103    56.90326163
#> 2104    79.05542363
#> 2105    90.41174702
#> 2106    19.10999542
#> 2107    67.52660213
#> 2108    40.44805276
#> 2109    41.49729306
#> 2110    80.95318184
#> 2111    50.35914534
#> 2112    70.65581363
#> 2113   235.47976865
#> 2114    60.12443940
#> 2115    83.56186101
#> 2116    27.82523303
#> 2117   101.83936366
#> 2118    79.72619759
#> 2119    71.75943669
#> 2120    87.87935838
#> 2121    40.82660460
#> 2122   107.53311177
#> 2123    36.61881287
#> 2124    43.21249889
#> 2125    85.37019050
#> 2126   102.51907150
#> 2127    78.09900858
#> 2128    72.56311394
#> 2129    40.95905603
#> 2130    37.98168648
#> 2131    88.50327029
#> 2132   111.60902258
#> 2133    97.08406085
#> 2134    21.13345643
#> 2135   109.31561650
#> 2136   102.76418024
#> 2137    58.74954781
#> 2138   113.55073451
#> 2139    82.68902592
#> 2140    80.24596250
#> 2141    29.31735301
#> 2142    96.55642242
#> 2143    71.21868578
#> 2144    53.58197866
#> 2145    89.97460339
#> 2146    75.96000349
#> 2147    55.69221757
#> 2148    98.68890208
#> 2149    38.02336387
#> 2150    -2.87371279
#> 2151    87.40539937
#> 2152    38.99684139
#> 2153    46.75080357
#> 2154    51.43491324
#> 2155    69.33177712
#> 2156    86.17838747
#> 2157    67.56113937
#> 2158    46.93747847
#> 2159   107.53595262
#> 2160    67.44907114
#> 2161    99.84870670
#> 2162    62.19283093
#> 2163    86.87388834
#> 2164    58.83350776
#> 2165    60.32775216
#> 2166    83.44281589
#> 2167    64.90971288
#> 2168    45.61701421
#> 2169    47.45316244
#> 2170    52.90084869
#> 2171    57.87995753
#> 2172    80.98416932
#> 2173     6.44842400
#> 2174    73.05688678
#> 2175    80.79730976
#> 2176    99.32370350
#> 2177    81.18634209
#> 2178    91.90083398
#> 2179    94.90282686
#> 2180   105.94305956
#> 2181    80.21092232
#> 2182    69.60043595
#> 2183   107.94905196
#> 2184    59.77597931
#> 2185    29.82679003
#> 2186    89.31823064
#> 2187    96.99228226
#> 2188    36.64196628
#> 2189    46.02566188
#> 2190   101.91044126
#> 2191    58.27244629
#> 2192    34.96883140
#> 2193    74.81600351
#> 2194    69.33787392
#> 2195    47.45305438
#> 2196    78.67266650
#> 2197    97.06819431
#> 2198    92.65697829
#> 2199    53.71324403
#> 2200    45.99888496
#> 2201    92.48314449
#> 2202    94.89149627
#> 2203    90.00726783
#> 2204    42.02500350
#> 2205    94.26298792
#> 2206    89.86381337
#> 2207   174.66677388
#> 2208    43.50139328
#> 2209   126.35988302
#> 2210   101.73966534
#> 2211    26.01648779
#> 2212   142.44342672
#> 2213    44.64890554
#> 2214    94.15892702
#> 2215    69.10080147
#> 2216    49.33078153
#> 2217    77.35241955
#> 2218   -12.40703022
#> 2219    65.68081280
#> 2220    98.35145299
#> 2221   239.17661183
#> 2222   116.97441649
#> 2223   109.61781257
#> 2224    68.93396655
#> 2225    73.55008028
#> 2226    61.18720232
#> 2227    47.66570768
#> 2228    72.29291159
#> 2229    31.20493875
#> 2230   155.23811277
#> 2231    86.43614752
#> 2232    45.74666628
#> 2233    14.73413628
#> 2234   109.92548772
#> 2235    33.84338802
#> 2236    46.83219328
#> 2237    83.81433040
#> 2238    90.76252807
#> 2239    60.38649046
#> 2240    17.17263013
#> 2241    92.08386520
#> 2242   151.45493538
#> 2243   125.54671989
#> 2244    84.95753646
#> 2245    63.44041545
#> 2246    80.84565184
#> 2247   186.80477806
#> 2248    56.86969058
#> 2249    74.17029100
#> 2250    40.68840833
#> 2251    64.35386272
#> 2252    39.03018679
#> 2253    78.08551716
#> 2254    63.93809560
#> 2255    37.36852619
#> 2256    70.73795107
#> 2257    53.68874317
#> 2258    64.09797950
#> 2259    57.94669983
#> 2260    90.59628478
#> 2261   103.18779670
#> 2262    66.60882746
#> 2263    48.47941555
#> 2264    77.93714216
#> 2265    46.53917652
#> 2266   131.92420660
#> 2267    89.91842273
#> 2268    59.62995356
#> 2269    91.38268706
#> 2270    62.82238905
#> 2271    41.79586119
#> 2272    43.72374408
#> 2273    92.67764146
#> 2274   104.90694528
#> 2275    76.87145791
#> 2276   105.80354214
#> 2277    50.18502082
#> 2278    51.03030871
#> 2279    47.78927467
#> 2280    26.47489683
#> 2281   132.02188196
#> 2282    73.99328502
#> 2283    83.77166315
#> 2284   168.02305535
#> 2285    32.10459152
#> 2286    69.78100183
#> 2287   107.85910170
#> 2288   112.10306361
#> 2289   119.64959296
#> 2290    89.00864875
#> 2291    96.50461528
#> 2292    66.85726386
#> 2293    44.59665987
#> 2294    68.60576044
#> 2295    94.14815123
#> 2296    59.09968753
#> 2297    51.31819252
#> 2298    47.45703725
#> 2299    75.90637990
#> 2300    88.36707048
#> 2301    30.64005365
#> 2302    20.00014901
#> 2303    21.84969139
#> 2304   124.65371424
#> 2305   116.09199231
#> 2306    77.26027118
#> 2307    17.93511696
#> 2308    43.68627478
#> 2309    93.45396833
#> 2310    40.66322854
#> 2311    74.04244135
#> 2312    85.66166031
#> 2313   198.74062762
#> 2314    82.02379804
#> 2315   147.90318237
#> 2316    44.34824034
#> 2317   214.03845855
#> 2318   136.86379956
#> 2319    69.12213374
#> 2320    98.28973252
#> 2321    28.31325100
#> 2322    45.02431431
#> 2323    97.72593313
#> 2324    93.19221482
#> 2325    81.07216302
#> 2326    71.61613038
#> 2327    57.40432462
#> 2328   120.87792414
#> 2329    61.48190455
#> 2330    68.39287208
#> 2331    77.24457905
#> 2332    74.93360904
#> 2333    21.39264233
#> 2334   125.50853524
#> 2335    78.47501937
#> 2336    82.15379043
#> 2337   137.12976597
#> 2338   113.31047076
#> 2339     4.09542770
#> 2340    60.60505729
#> 2341    32.41407089
#> 2342    40.82541842
#> 2343    53.53663378
#> 2344    25.45425041
#> 2345   104.04920714
#> 2346    99.13249150
#> 2347   107.20281535
#> 2348    33.35076380
#> 2349    65.55713469
#> 2350    72.09885350
#> 2351    53.27868245
#> 2352    64.65592696
#> 2353    32.04181431
#> 2354   -11.27504578
#> 2355    11.17367679
#> 2356    89.35073024
#> 2357    22.05910788
#> 2358    88.06205681
#> 2359    58.22724085
#> 2360   124.45726324
#> 2361    51.48388631
#> 2362    75.04718139
#> 2363    82.12035674
#> 2364    58.95643509
#> 2365    29.07879842
#> 2366    89.00225713
#> 2367    78.07772089
#> 2368   133.06621483
#> 2369    65.87816533
#> 2370    90.14095570
#> 2371    97.13849945
#> 2372    72.99973791
#> 2373   201.01131053
#> 2374    96.74438898
#> 2375   -93.22073822
#> 2376   108.74765149
#> 2377    54.06565201
#> 2378    95.69719694
#> 2379    40.99577323
#> 2380    77.03728853
#> 2381   110.92385477
#> 2382    27.25224435
#> 2383     9.81267484
#> 2384    78.89726915
#> 2385    72.75422301
#> 2386    38.67235151
#> 2387    18.34643597
#> 2388    83.82619708
#> 2389    48.12585039
#> 2390    43.29244098
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#> 6892   110.50566179
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#> 6908   105.62030666
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#> 6953   115.05962175
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#> 6967    68.52024238
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#> 9584   123.71380156
#> 9585    91.34845485
#> 9586   -14.79236327
#> 9587    60.32308022
#> 9588    58.66496431
#> 9589    31.55437939
#> 9590    49.40947020
#> 9591    57.14532681
#> 9592    59.03587349
#> 9593   131.29968305
#> 9594   123.25771527
#> 9595    94.80527277
#> 9596   101.59105972
#> 9597    88.91466177
#> 9598   245.86521468
#> 9599    74.00467745
#> 9600   112.60506980
#> 9601   129.83356086
#> 9602    88.44902370
#> 9603    35.88178926
#> 9604   120.88409757
#> 9605    74.62799173
#> 9606    -2.12210202
#> 9607    76.26723070
#> 9608    -2.68349913
#> 9609    34.59636137
#> 9610   107.42218658
#> 9611    73.83038032
#> 9612    93.58435888
#> 9613   102.51489358
#> 9614    53.90688725
#> 9615    66.99274282
#> 9616    61.07253713
#> 9617    94.65377827
#> 9618   116.25437532
#> 9619    63.65837960
#> 9620    74.52185300
#> 9621    68.41070224
#> 9622   115.75397384
#> 9623    64.48038730
#> 9624    44.58643478
#> 9625   166.91554373
#> 9626    62.78116364
#> 9627    68.95367326
#> 9628   100.31073487
#> 9629    87.84942265
#> 9630    19.09413950
#> 9631   116.97615460
#> 9632    71.20494532
#> 9633   126.53333964
#> 9634    74.10575620
#> 9635    18.38271045
#> 9636    59.21985531
#> 9637    79.53895448
#> 9638    29.24921338
#> 9639    59.51403594
#> 9640    67.88874146
#> 9641    82.12184995
#> 9642    41.27918315
#> 9643    70.39491043
#> 9644    62.04850919
#> 9645    80.46180368
#> 9646   107.77647220
#> 9647   142.31332078
#> 9648    31.01320365
#> 9649    25.14799128
#> 9650    69.31178852
#> 9651   -21.50262324
#> 9652    51.50520025
#> 9653    54.90855012
#> 9654    11.83668579
#> 9655    86.32109654
#> 9656   102.61794781
#> 9657   120.20303776
#> 9658    49.77098312
#> 9659    51.52596187
#> 9660   103.77704615
#> 9661    50.68895207
#> 9662   114.86910468
#> 9663    52.26874940
#> 9664    70.86673949
#> 9665    73.43267174
#> 9666    69.53305422
#> 9667    77.20714644
#> 9668    76.31287301
#> 9669    66.71829274
#> 9670    81.68588613
#> 9671    52.82213078
#> 9672    93.56220600
#> 9673    79.38318170
#> 9674   130.24826845
#> 9675    45.95491263
#> 9676    55.49598996
#> 9677    62.03170037
#> 9678    60.55867441
#> 9679    22.81933017
#> 9680    57.26204144
#> 9681    95.12196158
#> 9682    36.91859899
#> 9683    63.29867837
#> 9684    58.80472661
#> 9685    43.77689588
#> 9686    27.76768613
#> 9687    42.35625981
#> 9688    40.64296747
#> 9689    88.43731850
#> 9690    95.61763742
#> 9691   101.92753988
#> 9692   -13.18353340
#> 9693    70.13397762
#> 9694    54.05618892
#> 9695    16.84228594
#> 9696    98.41613344
#> 9697   136.03272876
#> 9698    40.51421337
#> 9699    89.70357744
#> 9700    21.46544022
#> 9701    64.24400746
#> 9702    83.34353240
#> 9703    79.25888343
#> 9704   137.63030955
#> 9705    32.30586470
#> 9706   208.85553309
#> 9707    63.60231878
#> 9708    46.78362558
#> 9709    64.40768635
#> 9710    80.87576363
#> 9711    47.85147059
#> 9712    68.87238154
#> 9713    71.15394942
#> 9714   118.74709534
#> 9715    98.93166621
#> 9716    49.25107352
#> 9717    71.95009200
#> 9718    54.61902153
#> 9719    56.18642469
#> 9720    80.94858328
#> 9721    39.72162667
#> 9722    46.48085755
#> 9723    63.74196994
#> 9724    94.65751430
#> 9725   111.47563001
#> 9726    74.71496133
#> 9727    63.78548277
#> 9728    37.18684901
#> 9729   104.23720761
#> 9730   121.39983415
#> 9731    47.06367563
#> 9732    48.15482852
#> 9733    25.50676621
#> 9734    77.03577357
#> 9735    56.18182821
#> 9736    77.17092633
#> 9737    66.79383715
#> 9738    93.98267381
#> 9739    64.93986709
#> 9740    37.14489056
#> 9741    76.42360747
#> 9742    50.99501010
#> 9743    44.23970558
#> 9744   144.82906724
#> 9745    14.25939362
#> 9746    96.82284287
#> 9747    70.18929915
#> 9748    75.92908404
#> 9749    55.62457293
#> 9750    80.42746161
#> 9751     8.41700653
#> 9752    64.53306373
#> 9753   115.32930188
#> 9754    77.23560500
#> 9755    71.70978822
#> 9756    31.83333452
#> 9757    40.75808829
#> 9758    52.51546240
#> 9759   107.79454710
#> 9760    34.58124651
#> 9761    63.23084455
#> 9762   139.42620396
#> 9763    52.20796222
#> 9764    70.81793755
#> 9765    77.33865312
#> 9766    47.54792325
#> 9767    53.91351530
#> 9768    75.06541383
#> 9769    57.69443527
#> 9770   -16.03020800
#> 9771    68.86753777
#> 9772    64.27188797
#> 9773    45.10260654
#> 9774    69.29183893
#> 9775    91.56835583
#> 9776    47.65224626
#> 9777    75.10399537
#> 9778   108.30412225
#> 9779    73.59749555
#> 9780   -49.61272352
#> 9781    57.49922896
#> 9782    71.39456411
#> 9783    99.84065355
#> 9784    79.63335935
#> 9785    60.86681807
#> 9786    84.04088889
#> 9787     0.13225226
#> 9788    65.55672380
#> 9789    54.79561829
#> 9790   102.10312840
#> 9791   112.06884453
#> 9792    26.09773177
#> 9793    66.48611012
#> 9794   141.60388105
#> 9795    60.88151969
#> 9796    60.73458629
#> 9797    75.20813393
#> 9798   115.41094848
#> 9799    66.44286337
#> 9800    60.30160545
#> 9801    61.75281406
#> 9802   107.29113884
#> 9803    74.83000808
#> 9804   106.03741448
#> 9805    78.34553858
#> 9806    41.74383652
#> 9807    90.77611255
#> 9808     5.35181126
#> 9809    -6.50838912
#> 9810   108.69622103
#> 9811    24.37730524
#> 9812    58.46020250
#> 9813    70.56173180
#> 9814   214.74296482
#> 9815    31.74855748
#> 9816    54.33660463
#> 9817    67.63315308
#> 9818   109.10992492
#> 9819    89.16969824
#> 9820    61.85271386
#> 9821    26.96761186
#> 9822   130.15500246
#> 9823    95.50441413
#> 9824    52.74032369
#> 9825    56.19669897
#> 9826   -45.13219105
#> 9827    38.02085682
#> 9828    44.58360481
#> 9829    55.70172612
#> 9830   102.62506995
#> 9831    67.63480556
#> 9832    40.12719158
#> 9833    73.03080874
#> 9834    79.19570919
#> 9835    86.34318955
#> 9836    59.38014600
#> 9837    48.17905874
#> 9838    42.89997419
#> 9839    66.93860115
#> 9840    67.96233880
#> 9841    76.04158536
#> 9842    80.05642835
#> 9843   115.65077448
#> 9844    19.85077484
#> 9845    55.15932909
#> 9846    93.43646561
#> 9847    87.07050095
#> 9848     2.32825756
#> 9849   118.15553401
#> 9850    75.89679120
#> 9851   118.19961288
#> 9852    80.30794641
#> 9853    50.23923922
#> 9854   115.03823273
#> 9855    49.75632756
#> 9856    76.47008252
#> 9857    76.19230469
#> 9858    46.31099880
#> 9859    77.77786203
#> 9860   158.82101868
#> 9861   122.78924928
#> 9862    76.20594339
#> 9863   104.21458363
#> 9864    37.04828813
#> 9865   135.92309110
#> 9866    96.85450283
#> 9867    90.64628532
#> 9868   105.60232499
#> 9869    97.97037991
#> 9870    63.58049147
#> 9871    24.53921010
#> 9872   106.78996942
#> 9873    77.15350784
#> 9874    68.92438859
#> 9875    71.40381556
#> 9876    83.00731037
#> 9877    85.18519998
#> 9878    68.39341655
#> 9879    53.18746480
#> 9880    59.96012451
#> 9881   104.31039312
#> 9882    51.63987897
#> 9883   112.97133601
#> 9884    72.77302619
#> 9885   132.68825361
#> 9886    84.78400647
#> 9887    26.36962024
#> 9888    44.61860161
#> 9889   -19.80254992
#> 9890    79.03147977
#> 9891   102.48415567
#> 9892    59.98131792
#> 9893    55.76426168
#> 9894    52.26458397
#> 9895   151.31262087
#> 9896    77.96921390
#> 9897    65.85667372
#> 9898   107.95069460
#> 9899   104.19808980
#> 9900    70.31767164
#> 9901   105.10111258
#> 9902   109.70121962
#> 9903    61.92642532
#> 9904    93.16770001
#> 9905    74.51133390
#> 9906    28.30177693
#> 9907    47.98970605
#> 9908    41.38312890
#> 9909    99.50049697
#> 9910   111.20111301
#> 9911    70.99856247
#> 9912    49.32444072
#> 9913   101.20148735
#> 9914    93.60941359
#> 9915   114.94005914
#> 9916    63.56935950
#> 9917    47.51653183
#> 9918    56.99537693
#> 9919    83.53241602
#> 9920    39.96957213
#> 9921   210.52726802
#> 9922   108.66510440
#> 9923    33.02934734
#> 9924    49.42143583
#> 9925    88.04907633
#> 9926    84.78564985
#> 9927    53.65797423
#> 9928    69.19769475
#> 9929    33.00964584
#> 9930    36.66939813
#> 9931    79.39061287
#> 9932    37.01079860
#> 9933   125.78462794
#> 9934   106.50929227
#> 9935    57.05928721
#> 9936    84.09511291
#> 9937    91.45298078
#> 9938    68.50571779
#> 9939    47.30346055
#> 9940    39.41265259
#> 9941    53.59460899
#> 9942   121.81709882
#> 9943    75.54296442
#> 9944   219.29061616
#> 9945    30.62404021
#> 9946    58.47741083
#> 9947    53.65966123
#> 9948    10.59596412
#> 9949    55.09719878
#> 9950    60.17551553
#> 9951    40.74939411
#> 9952    28.92165438
#> 9953    81.57593901
#> 9954     9.67010997
#> 9955    84.07867187
#> 9956    66.59759313
#> 9957    93.38135363
#> 9958    29.06729258
#> 9959    14.61319479
#> 9960    78.15643874
#> 9961    70.55726035
#> 9962    64.89455195
#> 9963    55.23092874
#> 9964    51.01872142
#> 9965    93.94885855
#> 9966   101.28094137
#> 9967    62.64458519
#> 9968   110.00511615
#> 9969    49.69292985
#> 9970    19.58549647
#> 9971    58.66597720
#> 9972    41.37511055
#> 9973    58.16869550
#> 9974    59.10588855
#> 9975    63.85328512
#> 9976   208.03740577
#> 9977    56.86330611
#> 9978    44.81559864
#> 9979   -16.16278437
#> 9980    33.37393733
#> 9981   112.43630398
#> 9982   125.60503918
#> 9983    57.90183362
#> 9984    23.21837933
#> 9985    75.10044152
#> 9986   121.57135666
#> 9987    61.80067148
#> 9988   134.91012388
#> 9989    92.83810744
#> 9990    47.66921341
#> 9991    77.79611073
#> 9992    85.71889272
#> 9993   140.89155645
#> 9994    65.61291618
#> 9995    12.54856363
#> 9996    56.12536795
#> 9997    31.10397828
#> 9998    90.47923721
#> 9999    66.03326469
#> 10000  124.09915057
#> 
#> $spread_plot

bond_fut_irr_ytm fits to options on a bond futures the RND of the bond futures price with a mixture of two lognormal laws. Then it extracts from this RND, for the current Cheapest-to-Deliver bond in the delivery basket of the futures contract, the RND of its forward yield and the RND of its forward repo rate which are again decribed by a mixture of two lognormal laws each. The current CtD is assumed to be the CtD at futures’ maturity. This is achieved again by considering a “term cash and carry relationship” between the current Cheapest-to-Deliver bond and the futures contract, allowing here the repo rate of the bond at options’ maturity to have a distribution. The RND of the STIR futures contract of the same currency and approximate same maturity is used to fix the bounds of the volatility of the forward repo rate density.

bond_fut_irr_ytm(c(12.64, 12.14, 11.65, 11.15, 10.65, 10.15, 9.65,
9.16, 8.66, 8.16, 7.67,  7.17, 6.68, 6.19, 5.70, 5.21, 4.73, 4.25,
3.78, 3.32, 2.87, 2.45, 2.05, 1.67, 1.33, 1.03, 0.77, 0.57, 0.41,
0.29, 0.21, 0.15, 0.10, 0.07, 0.06, 0.04, 0.04, 0.03, 0.02, 0.02,
0.02, 0.02, 0.01, 0.01, 0.01, 0.01, 0.01),
c(seq(114, 136.5, 0.5), 137.5),
c(0.01, 0.01, 0.02, 0.02, 0.02, 0.02, 0.02, 0.03, 0.03, 0.03, 0.04,
0.04, 0.05, 0.06, 0.07, 0.08, 0.10, 0.12, 0.15, 0.19, 0.24, 0.32,
0.42, 0.54, 0.70, 0.90, 1.14, 1.44, 1.78, 2.16, 2.58, 3.02, 3.47,
3.94, 4.43, 4.91, 5.41, 5.90, 6.39, 6.89, 7.39, 7.89, 8.38, 8.88,
9.38, 9.88, 10.88),
c(seq(114, 136.5, 0.5), 137.5),
c(1.704999924, 1.579999924, 1.454999924, 1.329999924, 1.204999924,
1.079999924, 0.954999983, 0.892499983, 0.829999983, 0.767499983,
0.704999983, 0.642499983, 0.582499981, 0.519999981, 0.457499981,
0.397499979, 0.334999979, 0.275000006, 0.217500001, 0.162499994,
0.112499997, 0.072499998, 0.039999999, 0.02, 0.0075, 0.0025, 0.0025,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),
c(seq(95.75, 96.5, 0.125), seq(96.5625, 98.75, 0.0625),
seq(98.875, 99.5, 0.125)),
c(0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0.0025,  0.0025,
0.0025,  0.005, 0.005,  0.0075,  0.012499999,  0.02, 0.029999999,
0.055, 0.085000001, 0.127499998, 0.177499995, 0.234999999,
0.297499985, 0.357499987, 0.419999987, 0.482499987, 0.544999957,
0.607499957, 0.669999957, 0.732499957, 0.794999957, 0.857499957,
0.919999957, 0.982499957, 1.044999957, 1.107499957, 1.169999957,
1.232499957, 1.294999957, 1.419999957, 1.544999957, 1.669999957,
1.794999957, 1.919999957, 2.044999838),
c(seq(95.75, 96.5, 0.125), seq(96.5625, 98.75, 0.0625),
seq(98.875, 99.5, 0.125)),
0.0224,
0.02232,
1,
3,
126.66,
0.026,
1,
0.770088,
2,
bond_N = 100,
as.Date("2035-08-15"),
as.Date("2026-09-08"),
as.Date("2026-08-21"),
3,
97.45,
as.Date("2026-09-14"),
as.Date("2026-08-14"),
as.Date("2026-06-17"),
ref_rate = "3-mth Euribor",
country = "Germany",
currency = "EUR")
#> $params_bond.unlist.marginal_bond.
#> [1] 4.50280960 4.58045023 0.01384493 0.01786785 0.01869879
#> 
#> $params_repo.unlist.marginal_repo.
#> [1] 8.691728e-04 8.691726e-04 9.713988e-05 9.713914e-05 4.983387e-01
#> 
#> $moments_repo
#>         mean       stddev     skewness     kurtosis 
#> 1.762489e-02 1.969773e-03 4.345881e-06 3.000000e+00 
#> 
#> $mode_repo
#> [1] 0.01761888
#> 
#> $discretized_rnd_repo
#> # A tibble: 37,658 × 2
#>     domain      rnd
#>      <dbl>    <dbl>
#>  1 -0.0230 5.84e-91
#>  2 -0.0230 6.25e-91
#>  3 -0.0230 6.70e-91
#>  4 -0.0230 7.18e-91
#>  5 -0.0230 7.69e-91
#>  6 -0.0230 8.24e-91
#>  7 -0.0230 8.83e-91
#>  8 -0.0230 9.46e-91
#>  9 -0.0230 1.01e-90
#> 10 -0.0230 1.09e-90
#> # ℹ 37,648 more rows
#> 
#> $rnd_plot_repo

#> 
#> $cdf_plot_repo

#> 
#> $moments_ytm
#>         mean       stddev     skewness     kurtosis 
#>  0.029300556  0.001164204  8.209143596 69.038114057 
#> 
#> $mode_ytm
#> [1] 0.0291617
#> 
#> $discretized_rnd_ytm
#> # A tibble: 9,230 × 2
#>    domain      rnd
#>     <dbl>    <dbl>
#>  1 0.0269 1.44e-89
#>  2 0.0269 1.83e-89
#>  3 0.0269 2.32e-89
#>  4 0.0269 2.95e-89
#>  5 0.0269 3.75e-89
#>  6 0.0269 4.76e-89
#>  7 0.0269 6.05e-89
#>  8 0.0269 7.68e-89
#>  9 0.0269 9.75e-89
#> 10 0.0269 1.24e-88
#> # ℹ 9,220 more rows
#> 
#> $rnd_plot_ytm

#> 
#> $cdf_plot_ytm

#> 
#> $CV
#> [1] 0