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Finance Stock Option Valuation using Binomial Method in Pure Elixir
Raw
european.exs
defmodule Options.European do
@moduledoc """
Functions for evaluating European call options.
"""
######################
### default values ###
######################
# volatility
def vol(x \\ 0.3), do: x
# time to maturity (years)
def t(x \\ 365 / 365), do: x
# granularity of binomial tree (levels)
def levels(x \\ 50), do: x
# strike price
def k(x \\ 40.0), do: x
# stock price present value
def s(x \\ 45.0), do: x
# annual risk free rate continuous compounding
def r(x \\ 0.05), do: x
#######################################
### start() used during development ###
#######################################
def start(s, k, v, t, r, n) do
IO.puts("")
IO.puts("For: s=#{s}, k=#{k}, vol=#{v}, t=#{t}, r=#{r}, levels=#{n}")
answer = simplecall(s, k, v, t, n, r)
IO.puts("Value of call option is: #{answer}")
IO.puts("")
end
def test_convergence() do
25..84
|> Enum.map(&callvalue(&1))
|> Enum.chunk_every(10)
|> Enum.map(&avg(&1))
|> Enum.each(&IO.puts(&1))
end
def callvalue(n) do
dt = dt(t(), n)
gu = gu(vol(), dt)
sl = leaves(s(), gu, n)
cl = cleaves(sl)
r = r()
callnode(cl, sl, gu, r, dt)
end
###############
### library ###
###############
@doc """
simplecall/6
stock price, strike price, annual volatility, time yrs, levels, risk-free rate
iex> Options.European.simplecall(100.0, 125.0, 0.5, 1.0, 5, 0.06)
12.627410465807767
"""
def simplecall(s \\ s(), k \\ k(), v \\ vol(), t \\ t(), n \\ levels(), r \\ r()) do
with dt = dt(t, n),
gu = voltorate(v, dt),
sf = leaves(s, gu, n) do
sf
|> cleaves(k)
|> callnode(sf, gu, r, dt)
end
end
@doc """
leaves/3
creates the end of the tree
iex> Options.European.leaves(75.0, 0.8, 6)
[286.102294921875, 183.10546875000003, 117.18750000000006, 75.00000000000004, 48.00000000000004, 30.720000000000027, 19.660800000000023]
"""
def leaves(s \\ s(), gu \\ gu(), levels \\ levels()) do
with bottom = s * :math.pow(gd(gu), levels) do
0..levels
|> Enum.map(&(bottom * :math.pow(gu * gu, &1)))
end
end
@doc """
cleaves/2
creates the end of the tree
iex> [167.7, 128.2, 98.0, 75.0, 57.3, 43.8, 33.5] |> Options.European.cleaves(65.0)
[102.69999999999999, 63.19999999999999, 33.0, 10.0, 0.0, 0.0, 0.0]
"""
def cleaves(sleaves, k \\ k()) do
sleaves
|> Enum.map(&max(&1 - k, 0.0))
end
@doc """
bondsf/2
future value of bonds for hedge
iex> [167.7, 128.2, 98.0, 75.0, 57.3, 43.8, 33.5] |> Options.European.bondsf([102.7, 63.2, 33.0, 10.0, 0.0, 0.0, 0.0])
[65.0, 65.00000000000003, 65.0, 57.3, 0.0, 0.0]
"""
def bondsf(sfl, cfl) do
sfl
|> pairs()
|> Enum.zip(pairs(cfl))
|> Enum.map(&bfhelper(&1))
end
def bfhelper({[_, _], [x, x]}), do: 0.0
def bfhelper({[sfu, sfd], [cfu, cfd]}), do: (cfu * sfd - sfu * cfd) / (cfu - cfd)
@doc """
bondsp/3
discounted bonds
iex> [167.7, 128.2, 98.0, 75.0, 57.3, 43.8, 33.5] |> Options.European.bondsp(0.06, 0.50)
[162.74371597608481, 124.41111740091874, 95.10366228775379, 72.78341501613811, 55.606529072329515, 42.50551436942465, 32.50992537387502]
"""
def bondsp(bf, r \\ r(), dt \\ dt()) do
bf |> Enum.map(&(&1 * :math.exp(-r * dt)))
end
@doc """
htops/2
future value(s) of hedged portfolio
iex> [167.7, 128.2, 98.0, 75.0, 57.3, 43.8, 33.5] |> Options.European.htops([102.7, 63.2, 33.0, 10.0, 0.0, 0.0, 0.0])
[25.499999999999986, 34.8, 42.0, 47.3, 43.8, 33.5]
"""
def htops(sfl, bfl) do
sfl
|> List.delete_at(0)
|> Enum.zip(bfl)
|> Enum.map(&htpshelper(&1))
end
defp htpshelper({x, y}), do: x - y
@doc """
sfratios/2
ratios for adjusting hedged portfolios to call future values
iex> [25.5, 34.8, 41.9, 47.3, 43.8, 33.5] |> Options.European.sfratios([102.7, 63.2, 33.0, 10.0, 0.0, 0.0, 0.0])
[0.40348101265822783, 1.0545454545454545, 4.1899999999999995, 0.0, 0.0, 0.0]
"""
def sfratios(htops, cfl) do
cfl
|> List.delete_at(0)
|> Enum.zip(htops)
|> Enum.map(&sfrhelper(&1))
end
defp sfrhelper({0.0, _}), do: 0.0
defp sfrhelper({x, y}), do: y / x
@doc """
callp/3
call values back one layer
iex> [0.6, 0.55, 0.65] |> Options.European.callp([50.0, 100.0, 200.0], [50.0, 100.0, 100.0])
[0.0, 0.0, 153.84615384615384]
"""
def callp(hr, sp, bp) do
Enum.zip([sp, bp, hr])
|> Enum.map(&cphelper(&1))
end
defp cphelper({_sp, _bp, 0.0}), do: 0.0
defp cphelper({sp, bp, hr}), do: (sp - bp) / hr
@doc """
callagain/5
valuing call prices back one layer
iex> [167.7, 128.2, 98.0, 75.0, 57.3, 43.8, 33.5] |> Options.European.callagain([102.7, 63.2, 33.0, 10.0, 0.0, 0.0, 0.0], 1.1, 0.04, 0.23)
[52.140712161874546, 24.686166707329086, 3.7770757982381724, 0.0, 0.0, 0.0]
"""
def callagain(sf, cf, gu \\ gu(), r \\ r(), dt \\ dt()) do
with bf = bondsf(sf, cf),
sp = sf |> sreduce(gd(gu)),
bp = bf |> bondsp(r, dt) do
sf
|> htops(bf)
|> sfratios(cf)
|> callp(sp, bp)
end
end
@doc """
callnode/5
present value of call option
iex> [0.0, 0.0, 0.0, 14.3, 49.5, 95.5] |> Options.European.callnode([51.1, 66.8, 87.4, 114.3, 149.5, 195.5], 1.14, 0.07, 0.25)
17.61075364545998
"""
def callnode(cf, sf, gu \\ gu(), r \\ r(), dt \\ dt())
def callnode(cf, [sfd, sfu], gu, r, dt) do
with bf = bondsf([sfd, sfu], cf),
bp = bondsp(bf, r, dt),
sp = sreduce([sfd, sfu], gd(gu)) do
[sfd, sfu]
|> htops(bf)
|> sfratios(cf)
|> callp(sp, bp)
|> hd()
end
end
def callnode(cf, sf, gu, r, dt) do
with sp = sreduce(sf, gd(gu)) do
callagain(sf, cf, gu, r, dt)
|> callnode(sp, gu, r, dt)
end
end
@doc """
dt/2
duration of time step given number of levels and time to maturity
iex> Options.European.dt(1.0, 5)
0.2
"""
def dt(t \\ t(), l \\ levels()) do
t / l
end
@doc """
gu/2
up growth rate given volatility and duration of time step
iex> Options.European.gu(0.3, 0.2)
1.1435804413868396
"""
def gu(v \\ vol(), dt \\ dt()) do
voltorate(v, dt)
end
@doc """
gd/1
down growth rate given up growth rate for symetric tree
iex> Options.European.gd(2.0)
0.5
"""
def gd(gu \\ gu()) do
1.0 / gu
end
# annual volatility to growth rate per delta-t in years
def voltorate(volatility, dt) do
(volatility * :math.sqrt(dt))
|> :math.exp()
end
@doc """
iex> [1,2,3,4,5] |> Options.European.pairs()
[[1,2], [2,3], [3,4], [4,5]]
"""
# splits a future price distribution into ordered pairs
def pairs([d, u]), do: [[d, u]]
def pairs(dist), do: [[hd(dist), hd(tl(dist))] | pairs(tl(dist))]
@doc """
reduce/2
steps stock price list down binomial tree by single layer
iex> [0.25, 1.0, 4.0] |> Options.European.sreduce(0.5)
[0.5, 2.0]
"""
def sreduce(slist, gd \\ gd()) do
slist
|> List.delete_at(0)
|> Enum.map(&(&1 * gd))
end
@doc """
clist/2
provides call option values at end of binomial tree
iex> [0.25, 1.0, 4.0] |> Options.European.clist(0.5)
[0.0, 0.5, 3.5]
"""
def clist(slist, k \\ k()) do
slist
|> Enum.map(&max(0.0, &1 - k))
end
@doc """
iex> [1,2,3] |> Options.European.avg()
2.0
"""
def avg(x) do
Enum.reduce(x, fn x, acc -> x + acc end) / length(x)
end
end
# Copyright [2019] [Jonathan Kaufman]
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
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