Module: Math::GreekCalculations

Included in:
Math::Greeks::Calculator::GreekCalculations
Defined in:
lib/greeks/calculations/iv.rb,
lib/greeks/calculations/rho.rb,
lib/greeks/calculations/vega.rb,
lib/greeks/calculations/delta.rb,
lib/greeks/calculations/gamma.rb,
lib/greeks/calculations/theta.rb,
lib/greeks/calculations/time_values.rb,
lib/greeks/calculations/normal_distribution.rb

Instance Method Summary collapse

Instance Method Details

#annualized_premium_value(opts) ⇒ Object

Annualized Premium The annualized premium is the value of the option divided by the strike price. You can use annualized premium to develop an intuitive understanding of how much the market is “paying” for a dollar of risk. For example, if a stock is trading at $50 and you sell a $50 strike 6 month call for $4, you are getting paid 8% in 6 months, or about 16% annualized, in exchange for being willing to buy at $50, the current price.



51
52
53
54
55
56
57
 
# File 'lib/greeks/calculations/time_values.rb', line 51

def annualized_premium_value(opts)
  return nil if opts[:option_price].nil?
  return nil if opts[:option_price] < 0
  opts.requires_fields(:option_price, :option_strike, :option_expires_pct_year)

  nil_or_gte0(100 * Math.log(1 + opts[:option_price] / opts[:option_strike]) / opts[:option_expires_pct_year])
end

#annualized_time_value(opts) ⇒ Object

Annualized Time Value The time value of the option divided by the strike price, then annualized. You can use annualized time value to develop an intuitive understanding of how much value the option market is adding to an in-the-money option beyond the intrinsic value. For example, if a stock is trading at $40 and a six month call on that stock with a strike price of $35 has an intrinsic value of $5 and a total value of $7, the time value ($2) divided by the strike is ($2/$40) = 5%. Annualizing that time value to a one year horizon on a continuously compounded basis yields 9.76% (2 × ln(1 + 0.05)).



67
68
69
70
71
72
 
# File 'lib/greeks/calculations/time_values.rb', line 67

def annualized_time_value(opts)
  return nil if opts[:time_value].nil? || opts[:time_value] < 0
  opts.requires_fields(:option_strike, :option_expires_pct_year, :time_value)
  
  nil_or_gte0(100 * Math.log(1.0 + opts[:time_value] / opts[:option_strike]) / opts[:option_expires_pct_year])
end

#break_even(opts) ⇒ Object

Chance of Breakeven The probability that a stock will be trading beyond the breakeven price as implied by the option price. Chance of Breakeven can be used to get a sense for the valuation of the option by comparing the markets’ estimate of Chance of Breakeven to estimates derived from your own fundamental research. If you believe the Chance of Breakeven is less than the probability that a stock will be beyond the breakeven price at option expiration, then you believe the option is undervalued, and visa versa.



81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
 
# File 'lib/greeks/calculations/time_values.rb', line 81

def break_even(opts)
  opts.requires_keys_are_present(:option_price, :iv)
  return nil if opts[:option_price].nil?
  return nil if opts[:option_price] < 0
  return nil if opts[:iv].nil?

  opts.requires_keys_are_not_nil(:option_type, :option_price, :option_strike, :option_expires_pct_year, :option_expires_pct_year_sqrt, :stock_price, :stock_dividend_rate_f, :federal_reserve_interest_rate_f, :iv)
  
  part1 = (opts[:federal_reserve_interest_rate_f] - opts[:stock_dividend_rate_f] - opts[:iv] * opts[:iv] / 2) * opts[:option_expires_pct_year]
  part2 = opts[:iv] * opts[:option_expires_pct_year_sqrt]
  
  case opts[:option_type]
  when :call
		return normal_distribution((Math.log(opts[:stock_price] / (opts[:option_strike] + opts[:option_price])) + part1) / part2)
  when :put
	return normal_distribution(-(Math.log(opts[:stock_price] / (opts[:option_strike] - opts[:option_price])) + part1) / part2)
  else
    raise ArgumentError, "Invalid option_type = #{opts[:option_type]}"
  end
end

#delta(opts) ⇒ Object

Delta A measurement of the change in the price of an option resulting from a change in the price of the underlying security. Delta is positive for calls and negative for puts. Delta can be calculated as the dollar change of the option that an investor can expect for a one-dollar change in the underlying security. For example, let’s say an option on a stock trading at $50 costs $1 and has a delta of $0.50 per dollar of underlying stock price change. If the stock price rises to $52, the price of the option will increase by $1 (the $2 price change times the $0.50 delta). After the stock price movement, the option will be worth $2 ($1 initial cost plus $1 delta). Delta can also be calculated as a percentage change in the option price for a one-percent change in the underlying security; this method of viewing the delta value is also known as “leverage.”



12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
 
# File 'lib/greeks/calculations/delta.rb', line 12

def delta(opts)
  opts.requires_keys_are_present(:iv)
  return nil if opts[:iv].nil?
  
  opts.requires_keys_are_not_nil(:option_type, :rate_vs_expires, :d1_normal_distribution, :iv)


  multiplier = case opts[:option_type]
  when :call
    1.0
  when :put
    -1.0
  else
    raise "Invalid option_type = #{opts[:option_type].inspect}"
  end
  
  multiplier * opts[:rate_vs_expires] * opts[:d1_normal_distribution]
end

#gamma(opts = {}) ⇒ Object

Gamma A measurement of the change in delta as the price of the underlying stock changes. As the underlying stock price changes, the delta of the option changes, too. Gamma indicates how quickly your exposure to the price movement of the underlying security changes as the price of the underlying security varies. For example, if you have a call with a strike of $50 and the stock price is $50, the delta likely will be approximately $0.50 for a one-dollar movement of the stock. At a stock price of $60, the delta will be greater, closer to $0.75. At a stock price of $40, the delta will be less, closer to $0.25. In this example, if the stock price changes from $50 to $60, then the delta will change from $0.50 to $0.75. The $10 change in stock price caused a $0.25 change in delta, so gamma is approximately $0.25/10, or $0.025, in this case.



11
12
13
14
15
16
17
18
 
# File 'lib/greeks/calculations/gamma.rb', line 11

def gamma(opts = {})
  opts.requires_keys_are_present(:iv)
  return nil if opts[:iv].nil?
  
  opts.requires_keys_are_not_nil(:stock_price, :option_expires_pct_year_sqrt, :nd1, :rate_vs_expires, :iv)
  
	opts[:nd1] * opts[:rate_vs_expires] / (opts[:stock_price] * opts[:iv] * opts[:option_expires_pct_year_sqrt])
end

#iv(opts) ⇒ Object 



3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
 
# File 'lib/greeks/calculations/iv.rb', line 3

def iv(opts)
  opts.requires_keys_are_present(:option_price)
  return nil if opts[:option_price].nil?
  return nil if opts[:option_price] <= 0
  
  opts.requires_keys_are_not_nil(:stock_price, :option_strike, :option_expires_pct_year, :option_expires_pct_year_sqrt, :federal_reserve_interest_rate_f, :stock_dividend_rate_f, :option_type, :option_price, :rate_vs_expires, :price_vs_rate_vs_expires, :strike_vs_fed_vs_expires, :price_ratio_log_less_rates)
  
  iv_calc(
    opts[:stock_price], 
    opts[:option_strike], 
    opts[:option_expires_pct_year], 
    opts[:option_expires_pct_year_sqrt],
    opts[:federal_reserve_interest_rate_f], 
    opts[:stock_dividend_rate_f], 
    opts[:option_type], 
    opts[:option_price],
    opts[:price_vs_rate_vs_expires],
    opts[:price_ratio_log_less_rates],
    opts[:strike_vs_fed_vs_expires]
  )
end

#iv_calc(stock_price, option_strike, option_expires_pct_year, option_expires_pct_year_sqrt, federal_reserve_interest_rate_f, stock_dividend_rate_f, option_type, option_price, price_vs_rate_vs_expires, price_ratio_log_less_rates, strike_vs_fed_vs_expires) ⇒ Object 



54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
 
# File 'lib/greeks/calculations/iv.rb', line 54

def iv_calc(stock_price, option_strike, option_expires_pct_year, option_expires_pct_year_sqrt, federal_reserve_interest_rate_f, stock_dividend_rate_f, option_type, option_price, price_vs_rate_vs_expires, price_ratio_log_less_rates, strike_vs_fed_vs_expires)
  # Contstant values for the calculations
	price_limit = [0.005, 0.01 * option_price].min

  # Lambda for short-hand calculations
  calc_option_price            = lambda { |volatility_guess| iv_option_price(stock_price, option_strike, option_expires_pct_year, option_expires_pct_year_sqrt, volatility_guess, federal_reserve_interest_rate_f, stock_dividend_rate_f, option_type, price_ratio_log_less_rates, price_vs_rate_vs_expires, strike_vs_fed_vs_expires) } 
  
  # Lambda for short-hand calculations
  calc_vega                    = lambda { |volatility_guess| iv_vega(stock_price, option_strike, option_expires_pct_year, option_expires_pct_year_sqrt, volatility_guess, federal_reserve_interest_rate_f, stock_dividend_rate_f, price_ratio_log_less_rates, price_vs_rate_vs_expires) }
  
  # Lambda for short-hand calculations
  calc_volatility_guess1       = lambda { |var_volatility_guess, var_option_price, var_vega| var_volatility_guess - (var_option_price - option_price) / var_vega }
  
  # Lambda for short-hand calculations
  is_terminal_volatility_guess = lambda { |var_option_price| ((option_price - var_option_price).abs < price_limit) }

  # Lambda for short-hand calculations
  cleanup_volatility_guess     = lambda { |volatility_guess| volatility_guess.nil? || volatility_guess <= 0 ? nil : volatility_guess.to_f }
  
	var_volatility_guess = iv_volatility_guess0(stock_price, option_strike, option_expires_pct_year, federal_reserve_interest_rate_f, stock_dividend_rate_f)
  var_volatility_guess = 0.1 if var_volatility_guess <= 0
	var_option_price     = calc_option_price.call(var_volatility_guess)

  if is_terminal_volatility_guess.call(var_option_price)
    return cleanup_volatility_guess.call(var_volatility_guess)
  end

	var_vega = calc_vega.call(var_volatility_guess)

	var_volatility_guess1 = calc_volatility_guess1.call(var_volatility_guess, var_option_price, var_vega)

	var_step = 1
  max_steps = 13
	while ((var_volatility_guess - var_volatility_guess1).abs > 0.0001 && var_step < max_steps)
		var_volatility_guess = var_volatility_guess1
		var_option_price = calc_option_price.call(var_volatility_guess)

		if is_terminal_volatility_guess.call(var_option_price)
      return cleanup_volatility_guess.call(var_volatility_guess)
    end

		var_vega = calc_vega.call(var_volatility_guess)

		var_volatility_guess1 = calc_volatility_guess1.call(var_volatility_guess, var_option_price, var_vega)
		if (var_volatility_guess1 < 0)
      return cleanup_volatility_guess.call(var_volatility_guess1)
    end

		var_step += 1
  end

	if (var_step < max_steps)
    return cleanup_volatility_guess.call(var_volatility_guess1)
  end

  var_option_price = calc_option_price.call(var_volatility_guess1)

	if is_terminal_volatility_guess.call(var_option_price)
    return cleanup_volatility_guess.call(var_volatility_guess1)
	else
		return nil
  end
end

#iv_option_price(stock_price, option_strike, option_expires_pct_year, option_expires_pct_year_sqrt, volatility_guess, federal_reserve_interest_rate_f, stock_dividend_rate_f, option_type, price_ratio_log_less_rates, price_vs_rate_vs_expires, strike_vs_fed_vs_expires) ⇒ Object 



33
34
35
36
37
38
39
40
41
42
43
44
45
 
# File 'lib/greeks/calculations/iv.rb', line 33

def iv_option_price(stock_price, option_strike, option_expires_pct_year, option_expires_pct_year_sqrt, volatility_guess, federal_reserve_interest_rate_f, stock_dividend_rate_f, option_type, price_ratio_log_less_rates, price_vs_rate_vs_expires, strike_vs_fed_vs_expires)
	var_d1                       = (price_ratio_log_less_rates + volatility_guess * volatility_guess * option_expires_pct_year / 2) / (volatility_guess * option_expires_pct_year_sqrt)
	var_d2                       = var_d1 - volatility_guess * option_expires_pct_year_sqrt
  
  case option_type
  when :call
		return price_vs_rate_vs_expires * normal_distribution(var_d1) - strike_vs_fed_vs_expires * normal_distribution(var_d2)
  when :put
		return strike_vs_fed_vs_expires * normal_distribution(-var_d2) - price_vs_rate_vs_expires * normal_distribution(-var_d1)
  else
    raise "Invalid option_type = #{option_type.inspect}"
  end
end

#iv_vega(stock_price, option_strike, option_expires_pct_year, option_expires_pct_year_sqrt, volatility_guess, federal_reserve_interest_rate_f, stock_dividend_rate_f, price_ratio_log_less_rates, price_vs_rate_vs_expires) ⇒ Object 



26
27
28
29
30
 
# File 'lib/greeks/calculations/iv.rb', line 26

def iv_vega(stock_price, option_strike, option_expires_pct_year, option_expires_pct_year_sqrt, volatility_guess, federal_reserve_interest_rate_f, stock_dividend_rate_f, price_ratio_log_less_rates, price_vs_rate_vs_expires)
	var_d1 = (price_ratio_log_less_rates + volatility_guess * volatility_guess * option_expires_pct_year / 2) / (volatility_guess * option_expires_pct_year_sqrt)
	var_nd = Math.exp(-var_d1 * var_d1 / 2) / Math::sqrt(2 * Math::PI)
	return price_vs_rate_vs_expires * option_expires_pct_year_sqrt * var_nd
end

#iv_volatility_guess0(stock_price, option_strike, option_expires_pct_year, federal_reserve_interest_rate_f, stock_dividend_rate_f) ⇒ Object 



48
49
50
51
 
# File 'lib/greeks/calculations/iv.rb', line 48

def iv_volatility_guess0(stock_price, option_strike, option_expires_pct_year, federal_reserve_interest_rate_f, stock_dividend_rate_f)
  Math.sqrt(
  (Math.log(stock_price / option_strike) + (federal_reserve_interest_rate_f - stock_dividend_rate_f) * option_expires_pct_year).abs * 2 / option_expires_pct_year)
end

#misc_d1(opts) ⇒ Object 



146
147
148
149
150
151
152
153
 
# File 'lib/greeks/calculations/time_values.rb', line 146

def misc_d1(opts)
  opts.requires_keys_are_present(:iv)
  return nil if opts[:iv].nil?

  opts.requires_keys_are_not_nil(:price_ratio_log_less_rates, :iv, :option_expires_pct_year, :option_expires_pct_year_sqrt)
  
  (opts[:price_ratio_log_less_rates] + opts[:iv] * opts[:iv] * opts[:option_expires_pct_year] / 2) / (opts[:iv] * opts[:option_expires_pct_year_sqrt])
end

#misc_d2(opts) ⇒ Object 



156
157
158
159
160
161
162
163
 
# File 'lib/greeks/calculations/time_values.rb', line 156

def misc_d2(opts)
  opts.requires_keys_are_present(:iv)
  return nil if opts[:iv].nil?

  opts.requires_keys_are_not_nil(:d1, :iv, :option_expires_pct_year_sqrt)
  
  opts[:d1] - opts[:iv] * opts[:option_expires_pct_year_sqrt]
end

#misc_d_normal_distribution(opts) ⇒ Object 



165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
 
# File 'lib/greeks/calculations/time_values.rb', line 165

def misc_d_normal_distribution(opts)
  opts.requires_keys_are_present(:d_value)
  return nil if opts[:d_value].nil?

  opts.requires_keys_are_not_nil(:option_type, :d_value)
  
  multiplier = case opts[:option_type]
  when :call
    1.0
  when :put
    -1.0
  else
    raise ArgumentError, "Invalid option_type = #{opts[:option_type]}"
  end
  
  normal_distribution(multiplier * opts[:d_value])
end

#misc_nd1(opts) ⇒ Object 



136
137
138
139
140
141
142
143
 
# File 'lib/greeks/calculations/time_values.rb', line 136

def misc_nd1(opts)
  opts.requires_keys_are_present(:d1)
  return nil if opts[:d1].nil?

  opts.requires_keys_are_not_nil(:d1)

  Math.exp(-0.5 * opts[:d1] * opts[:d1]) / Math.sqrt(2 * Math::PI)
end

#misc_price_ratio_log_less_rates(opts) ⇒ Object 



108
109
110
111
112
 
# File 'lib/greeks/calculations/time_values.rb', line 108

def misc_price_ratio_log_less_rates(opts)
  opts.requires_fields(:stock_price, :option_strike, :option_expires_pct_year, :federal_reserve_interest_rate_f, :stock_dividend_rate_f)
  
  Math.log(opts[:stock_price] / opts[:option_strike]) + (opts[:federal_reserve_interest_rate_f] - opts[:stock_dividend_rate_f]) * opts[:option_expires_pct_year]
end

#misc_price_vs_rate_vs_expires(opts) ⇒ Object 



122
123
124
125
126
 
# File 'lib/greeks/calculations/time_values.rb', line 122

def misc_price_vs_rate_vs_expires(opts)
  opts.requires_fields(:stock_price, :option_expires_pct_year, :stock_dividend_rate_f)
  
  opts[:stock_price] * misc_rate_vs_expires(opts)
end

#misc_rate_vs_expires(opts) ⇒ Object 



115
116
117
118
119
 
# File 'lib/greeks/calculations/time_values.rb', line 115

def misc_rate_vs_expires(opts)
  opts.requires_fields(:option_expires_pct_year, :stock_dividend_rate_f)
  
  Math.exp(opts[:option_expires_pct_year] * -opts[:stock_dividend_rate_f])
end

#misc_strike_vs_fed_vs_expires(opts) ⇒ Object 



129
130
131
132
133
 
# File 'lib/greeks/calculations/time_values.rb', line 129

def misc_strike_vs_fed_vs_expires(opts)
  opts.requires_fields(:option_strike, :option_expires_pct_year, :federal_reserve_interest_rate_f)
  
  opts[:option_strike] * Math.exp(opts[:option_expires_pct_year] * -opts[:federal_reserve_interest_rate_f])
end

#nil_or_gte0(value) ⇒ Object 



4
5
6
 
# File 'lib/greeks/calculations/time_values.rb', line 4

def nil_or_gte0(value)
  value.nil? || value.to_f < 0.0 ? nil : value
end

#normal_distribution(value) ⇒ Object

Moddeled after the Excel NORMSDIST function



4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
 
# File 'lib/greeks/calculations/normal_distribution.rb', line 4

def normal_distribution(value)
  p  =  0.2316419
  b1 =  0.319381530
  b2 = -0.356563782
  b3 =  1.781477937
  b4 = -1.821255978
  b5 =  1.330274429

  y = value.abs
  z = Math.exp(-y*y/2) / Math.sqrt(2 * Math::PI)
  t = 1 / ( 1 + p * y)
  cum = 1 - z * (b1*t + b2*t*t + b3*t*t*t + b4*t*t*t*t + b5*t*t*t*t*t)

  cum = 1 - cum if (value < 0)
  cum
end

#normal_distribution_gaussian(value) ⇒ Object

Normal distribution function (Gaussian bell curve)



23
24
25
 
# File 'lib/greeks/calculations/normal_distribution.rb', line 23

def normal_distribution_gaussian(value)
  Math.exp(-0.5 * value * value) / Math.sqrt(2 * Math::PI)
end

#premium_value(opts) ⇒ Object

Intrinsic Value The value that the option would pay if it were executed today. For example, if a stock is trading at $40, a call on that stock with a strike price of $35 would have $5 of intrinsic value ($40-$35) if it were exercised today. However, the call should actually be worth more than $5 to account for the value of the chance of any further appreciation until expiration, and the difference between the price and the intrinsic value would be the time value.



15
16
17
18
19
20
21
22
23
24
25
26
27
28
 
# File 'lib/greeks/calculations/time_values.rb', line 15

def premium_value(opts)
  opts.requires_fields(:option_type, :option_strike, :stock_price)
  
  value = case opts[:option_type]
  when :call
    [opts[:stock_price] - opts[:option_strike], 0].max
  when :put
    [opts[:option_strike] - opts[:stock_price], 0].max
  else
    raise ArgumentError, "Invalid option_type = #{opts[:option_type]}"
  end
  
  nil_or_gte0(value)
end

#rho(opts = {}) ⇒ Object

Rho The change in the value of an option for a change in the prevailing interest rate that matches the duration of the option, all else held equal. Generally rho is not a big driver of price changes for options, as interest rates tend to be relatively stable.



6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
 
# File 'lib/greeks/calculations/rho.rb', line 6

def rho(opts = {})
  opts.requires_keys_are_present(:iv)
  return nil if opts[:iv].nil?
  
  opts.requires_keys_are_not_nil(:option_type, :option_expires_pct_year, :strike_vs_fed_vs_expires, :d2_normal_distribution, :iv)

  multiplier = case opts[:option_type]
  when :call
    1.0
  when :put
    -1.0
  else
    raise "Invalid option_type = #{opts[:option_type].inspect}"
  end
  
  multiplier * opts[:option_expires_pct_year] * opts[:strike_vs_fed_vs_expires] * opts[:d2_normal_distribution] / 100
end

#theta(opts = {}) ⇒ Object 



3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
 
# File 'lib/greeks/calculations/theta.rb', line 3

def theta(opts = {})
  opts.requires_keys_are_present(:iv)
  return nil if opts[:iv].nil?
  
  opts.requires_keys_are_not_nil(:stock_dividend_rate_f, :federal_reserve_interest_rate_f, :option_type, :option_expires_pct_year_sqrt, :iv, :strike_vs_fed_vs_expires, :price_vs_rate_vs_expires, :nd1, :d1_normal_distribution, :d2_normal_distribution)
  
  part0 = opts[:price_vs_rate_vs_expires] * opts[:nd1] * opts[:iv]
  part1 = 2 * opts[:option_expires_pct_year_sqrt]
  part2 = opts[:stock_dividend_rate_f]           * opts[:price_vs_rate_vs_expires] * opts[:d1_normal_distribution]
  part3 = opts[:federal_reserve_interest_rate_f] * opts[:strike_vs_fed_vs_expires] * opts[:d2_normal_distribution]
  
  case opts[:option_type]
  when :call
    return (-part0 / part1 + part2 - part3) / 365
  when :put
    return (-part0 / part1 - part2 + part3) / 365
  else
    raise "Invalid option_type = #{opts[:option_type].inspect}"
  end
end

#time_value(opts) ⇒ Object

Time Value The value of an option that captures the chance of further appreciation before expiration. The value of an option can be broken down into intrinsic value, or the value of the option if it were exercised today, and time value, or the added value of the option over and above the intrinsic value. For example, if a stock is trading at $40 and a call with a strike price of $35 were trading for $7, the call would have a $5 intrinsic value ($40-$35) and a $2 time value ($7-$5). Time value will decay by expiration assuming the underlying security stays at the same price.



38
39
40
41
42
43
 
# File 'lib/greeks/calculations/time_values.rb', line 38

def time_value(opts)
  return nil if opts[:option_price].nil? || opts[:option_price] < 0
  return nil if opts[:premium_value].nil? || opts[:premium_value] < 0
  
  nil_or_gte0(opts[:option_price] - opts[:premium_value])
end

#vega(opts = {}) ⇒ Object

Vega The change in the price of an option for a change in the implied volatility of the option, all else held equal. In general, as the options market thinks it is more difficult to value a stock, implied volatility and therefore the price of the options will increase. For example, if an option is trading for $1, the implied volatility is 20%, and the vega is $0.05, then a one-percentage-point increase in implied volatility to 21% would correspond to an increase in the price of the option to $1.05. In percentage terms, the vega in this case would be ($0.05/$1.00)/(1 percentage point) = 5%.



9
10
11
12
13
14
15
16
 
# File 'lib/greeks/calculations/vega.rb', line 9

def vega(opts = {})
  opts.requires_keys_are_present(:iv)
  return nil if opts[:iv].nil?
  
  opts.requires_keys_are_not_nil(:price_vs_rate_vs_expires, :nd1, :option_expires_pct_year_sqrt, :iv)

  opts[:price_vs_rate_vs_expires] * opts[:option_expires_pct_year_sqrt] * opts[:nd1] / 100
end