Module: OptionLab::BlackScholes

Defined in:
lib/option_lab/black_scholes.rb

Class Method Summary collapse

Class Method Details

.get_bs_info(s, x, r, vol, years_to_maturity, y = 0.0) ⇒ Models::BlackScholesInfo

Get all Black-Scholes info

Parameters:

  • s (Float) —

    Spot price

  • x (Float, Numo::DFloat) —

    Strike price(s)

  • r (Float) —

    Risk-free interest rate

  • vol (Float) —

    Volatility

  • years_to_maturity (Float) —

    Time to maturity in years

  • y (Float) (defaults to: 0.0) —

    Dividend yield

Returns:



226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
 
# File 'lib/option_lab/black_scholes.rb', line 226

def get_bs_info(s, x, r, vol, years_to_maturity, y = 0.0)
  d1 = get_d1(s, x, r, vol, years_to_maturity, y)
  d2 = get_d2(s, x, r, vol, years_to_maturity, y)

  call_price = get_option_price('call', s, x, r, years_to_maturity, d1, d2, y)
  put_price = get_option_price('put', s, x, r, years_to_maturity, d1, d2, y)
  call_delta = get_delta('call', d1, years_to_maturity, y)
  put_delta = get_delta('put', d1, years_to_maturity, y)
  call_theta = get_theta('call', s, x, r, vol, years_to_maturity, d1, d2, y)
  put_theta = get_theta('put', s, x, r, vol, years_to_maturity, d1, d2, y)
  gamma = get_gamma(s, vol, years_to_maturity, d1, y)
  vega = get_vega(s, years_to_maturity, d1, y)
  call_rho = get_rho('call', x, r, years_to_maturity, d2)
  put_rho = get_rho('put', x, r, years_to_maturity, d2)
  call_itm_prob = get_itm_probability('call', d2, years_to_maturity, y)
  put_itm_prob = get_itm_probability('put', d2, years_to_maturity, y)

  Models::BlackScholesInfo.new(
    call_price: call_price,
    put_price: put_price,
    call_delta: call_delta,
    put_delta: put_delta,
    call_theta: call_theta,
    put_theta: put_theta,
    gamma: gamma,
    vega: vega,
    call_rho: call_rho,
    put_rho: put_rho,
    call_itm_prob: call_itm_prob,
    put_itm_prob: put_itm_prob,
  )
end

.get_d1(s0, x, r, vol, years_to_maturity, y = 0.0) ⇒ Float, Numo::DFloat

Get d1 parameter for Black-Scholes formula

Parameters:

  • s0 (Float, Numo::DFloat) —

    Spot price(s)

  • x (Float, Numo::DFloat) —

    Strike price(s)

  • r (Float) —

    Risk-free interest rate

  • vol (Float, Numo::DFloat) —

    Volatility

  • years_to_maturity (Float) —

    Time to maturity in years

  • y (Float) (defaults to: 0.0) —

    Dividend yield

Returns:

  • (Float, Numo::DFloat) —

    d1 parameter(s)



20
21
22
23
24
25
26
27
 
# File 'lib/option_lab/black_scholes.rb', line 20

def get_d1(s0, x, r, vol, years_to_maturity, y = 0.0)
  # Handle edge cases
  return 0.0 if years_to_maturity <= 0.0 || vol <= 0.0

  numerator = Math.log(s0 / x) + (r - y + 0.5 * vol * vol) * years_to_maturity
  denominator = vol * Math.sqrt(years_to_maturity)
  numerator / denominator
end

.get_d2(s0, x, r, vol, years_to_maturity, y = 0.0) ⇒ Float, Numo::DFloat

Get d2 parameter for Black-Scholes formula

Parameters:

  • s0 (Float, Numo::DFloat) —

    Spot price(s)

  • x (Float, Numo::DFloat) —

    Strike price(s)

  • r (Float) —

    Risk-free interest rate

  • vol (Float, Numo::DFloat) —

    Volatility

  • years_to_maturity (Float) —

    Time to maturity in years

  • y (Float) (defaults to: 0.0) —

    Dividend yield

Returns:

  • (Float, Numo::DFloat) —

    d2 parameter(s)



37
38
39
40
41
42
43
 
# File 'lib/option_lab/black_scholes.rb', line 37

def get_d2(s0, x, r, vol, years_to_maturity, y = 0.0)
  # Handle edge cases
  return 0.0 if years_to_maturity <= 0.0 || vol <= 0.0

  d1 = get_d1(s0, x, r, vol, years_to_maturity, y)
  d1 - vol * Math.sqrt(years_to_maturity)
end

.get_delta(option_type, d1, years_to_maturity, y = 0.0) ⇒ Float, Numo::DFloat

Get option delta

Parameters:

  • option_type (String) —

    'call' or 'put'

  • d1 (Float, Numo::DFloat) —

    d1 parameter(s)

  • years_to_maturity (Float) —

    Time to maturity in years

  • y (Float) (defaults to: 0.0) —

    Dividend yield

Returns:

  • (Float, Numo::DFloat) —

    Option delta(s)



81
82
83
84
85
86
87
88
89
90
91
92
 
# File 'lib/option_lab/black_scholes.rb', line 81

def get_delta(option_type, d1, years_to_maturity, y = 0.0)
  yfac = Math.exp(-y * years_to_maturity)

  case option_type
  when 'call'
    yfac * Distribution::Normal.cdf(d1)
  when 'put'
    yfac * (Distribution::Normal.cdf(d1) - 1.0)
  else
    raise ArgumentError, "Option type must be either 'call' or 'put'!"
  end
end

.get_gamma(s0, vol, years_to_maturity, d1, y = 0.0) ⇒ Float, Numo::DFloat

Get option gamma

Parameters:

  • s0 (Float) —

    Spot price

  • vol (Float) —

    Volatility

  • years_to_maturity (Float) —

    Time to maturity in years

  • d1 (Float, Numo::DFloat) —

    d1 parameter(s)

  • y (Float) (defaults to: 0.0) —

    Dividend yield

Returns:

  • (Float, Numo::DFloat) —

    Option gamma(s)



101
102
103
104
105
106
107
108
 
# File 'lib/option_lab/black_scholes.rb', line 101

def get_gamma(s0, vol, years_to_maturity, d1, y = 0.0)
  yfac = Math.exp(-y * years_to_maturity)

  # PDF of d1
  cdf_d1_prime = Math.exp(-0.5 * d1 * d1) / Math.sqrt(2.0 * Math::PI)

  yfac * cdf_d1_prime / (s0 * vol * Math.sqrt(years_to_maturity))
end

.get_implied_vol(option_type, oprice, s0, x, r, years_to_maturity, y = 0.0) ⇒ Float

Get implied volatility

Parameters:

  • option_type (String) —

    'call' or 'put'

  • oprice (Float) —

    Option price

  • s0 (Float) —

    Spot price

  • x (Float) —

    Strike price

  • r (Float) —

    Risk-free interest rate

  • years_to_maturity (Float) —

    Time to maturity in years

  • y (Float) (defaults to: 0.0) —

    Dividend yield

Returns:

  • (Float) —

    Implied volatility



200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
 
# File 'lib/option_lab/black_scholes.rb', line 200

def get_implied_vol(option_type, oprice, s0, x, r, years_to_maturity, y = 0.0)
  # Start with volatilities from 0.001 to 1.0 in steps of 0.001
  volatilities = (1..1000).map { |i| i * 0.001 }

  # Calculate option prices for each volatility
  prices = volatilities.map do |vol|
    d1 = get_d1(s0, x, r, vol, years_to_maturity, y)
    d2 = get_d2(s0, x, r, vol, years_to_maturity, y)
    get_option_price(option_type, s0, x, r, years_to_maturity, d1, d2, y)
  end

  # Calculate absolute differences from market price
  diffs = prices.map { |price| (price - oprice).abs }

  # Return volatility with minimal difference
  volatilities[diffs.index(diffs.min)]
end

.get_itm_probability(option_type, d2, years_to_maturity, y = 0.0) ⇒ Float, Numo::DFloat

Get in-the-money probability

Parameters:

  • option_type (String) —

    'call' or 'put'

  • d2 (Float, Numo::DFloat) —

    d2 parameter(s)

  • years_to_maturity (Float) —

    Time to maturity in years

  • y (Float) (defaults to: 0.0) —

    Dividend yield

Returns:

  • (Float, Numo::DFloat) —

    ITM probability(ies)



178
179
180
181
182
183
184
185
186
187
188
189
 
# File 'lib/option_lab/black_scholes.rb', line 178

def get_itm_probability(option_type, d2, years_to_maturity, y = 0.0)
  yfac = Math.exp(-y * years_to_maturity)

  case option_type
  when 'call'
    yfac * Distribution::Normal.cdf(d2)
  when 'put'
    yfac * Distribution::Normal.cdf(-d2)
  else
    raise ArgumentError, "Option type must be either 'call' or 'put'!"
  end
end

.get_option_price(option_type, s0, x, r, years_to_maturity, d1, d2, y = 0.0) ⇒ Float, Numo::DFloat

Get option price using Black-Scholes formula

Parameters:

  • option_type (String) —

    'call' or 'put'

  • s0 (Float, Numo::DFloat) —

    Spot price(s)

  • x (Float, Numo::DFloat) —

    Strike price(s)

  • r (Float) —

    Risk-free interest rate

  • years_to_maturity (Float) —

    Time to maturity in years

  • d1 (Float, Numo::DFloat) —

    d1 parameter(s)

  • d2 (Float, Numo::DFloat) —

    d2 parameter(s)

  • y (Float) (defaults to: 0.0) —

    Dividend yield

Returns:

  • (Float, Numo::DFloat) —

    Option price(s)



55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
 
# File 'lib/option_lab/black_scholes.rb', line 55

def get_option_price(option_type, s0, x, r, years_to_maturity, d1, d2, y = 0.0)
  # First validate option type
  unless ['call', 'put'].include?(option_type)
    raise ArgumentError, "Option type must be either 'call' or 'put'!"
  end

  # Calculate normally
  s = s0 * Math.exp(-y * years_to_maturity)
  discount_factor = Math.exp(-r * years_to_maturity)

  case option_type
  when 'call'
    # Call option price: S * N(d1) - X * e^(-rT) * N(d2)
    (s * Distribution::Normal.cdf(d1)) - (x * discount_factor * Distribution::Normal.cdf(d2))
  when 'put'
    # Put option price: X * e^(-rT) * N(-d2) - S * N(-d1)
    (x * discount_factor * Distribution::Normal.cdf(-d2)) - (s * Distribution::Normal.cdf(-d1))
  end
end

.get_rho(option_type, x, r, years_to_maturity, d2) ⇒ Float, Numo::DFloat

Get option rho

Parameters:

  • option_type (String) —

    'call' or 'put'

  • x (Float, Numo::DFloat) —

    Strike price(s)

  • r (Float) —

    Risk-free interest rate

  • years_to_maturity (Float) —

    Time to maturity in years

  • d2 (Float, Numo::DFloat) —

    d2 parameter(s)

Returns:

  • (Float, Numo::DFloat) —

    Option rho(s)



161
162
163
164
165
166
167
168
169
170
 
# File 'lib/option_lab/black_scholes.rb', line 161

def get_rho(option_type, x, r, years_to_maturity, d2)
  case option_type
  when 'call'
    x * years_to_maturity * Math.exp(-r * years_to_maturity) * Distribution::Normal.cdf(d2) / 100
  when 'put'
    -x * years_to_maturity * Math.exp(-r * years_to_maturity) * Distribution::Normal.cdf(-d2) / 100
  else
    raise ArgumentError, "Option type must be either 'call' or 'put'!"
  end
end

.get_theta(option_type, s0, x, r, vol, years_to_maturity, d1, d2, y = 0.0) ⇒ Float, Numo::DFloat

Get option theta

Parameters:

  • option_type (String) —

    'call' or 'put'

  • s0 (Float) —

    Spot price

  • x (Float, Numo::DFloat) —

    Strike price(s)

  • r (Float) —

    Risk-free interest rate

  • vol (Float) —

    Volatility

  • years_to_maturity (Float) —

    Time to maturity in years

  • d1 (Float, Numo::DFloat) —

    d1 parameter(s)

  • d2 (Float, Numo::DFloat) —

    d2 parameter(s)

  • y (Float) (defaults to: 0.0) —

    Dividend yield

Returns:

  • (Float, Numo::DFloat) —

    Option theta(s)



121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
 
# File 'lib/option_lab/black_scholes.rb', line 121

def get_theta(option_type, s0, x, r, vol, years_to_maturity, d1, d2, y = 0.0)
  s = s0 * Math.exp(-y * years_to_maturity)

  # PDF of d1
  cdf_d1_prime = Math.exp(-0.5 * d1 * d1) / Math.sqrt(2.0 * Math::PI)

  common_term = -(s * vol * cdf_d1_prime / (2.0 * Math.sqrt(years_to_maturity)))

  case option_type
  when 'call'
    common_term - (r * x * Math.exp(-r * years_to_maturity) * Distribution::Normal.cdf(d2)) + (y * s * Distribution::Normal.cdf(d1))
  when 'put'
    common_term + (r * x * Math.exp(-r * years_to_maturity) * Distribution::Normal.cdf(-d2)) - (y * s * Distribution::Normal.cdf(-d1))
  else
    raise ArgumentError, "Option type must be either 'call' or 'put'!"
  end
end

.get_vega(s0, years_to_maturity, d1, y = 0.0) ⇒ Float, Numo::DFloat

Get option vega

Parameters:

  • s0 (Float) —

    Spot price

  • years_to_maturity (Float) —

    Time to maturity in years

  • d1 (Float, Numo::DFloat) —

    d1 parameter(s)

  • y (Float) (defaults to: 0.0) —

    Dividend yield

Returns:

  • (Float, Numo::DFloat) —

    Option vega(s)



145
146
147
148
149
150
151
152
 
# File 'lib/option_lab/black_scholes.rb', line 145

def get_vega(s0, years_to_maturity, d1, y = 0.0)
  s = s0 * Math.exp(-y * years_to_maturity)

  # PDF of d1
  cdf_d1_prime = Math.exp(-0.5 * d1 * d1) / Math.sqrt(2.0 * Math::PI)

  s * cdf_d1_prime * Math.sqrt(years_to_maturity) / 100
end