Module: OptionLab::BinomialTree

Defined in:
lib/option_lab/binomial_tree.rb

Overview

Implementation of the Cox-Ross-Rubinstein (CRR) binomial tree model for American and European options pricing

Class Method Summary collapse

Class Method Details

.get_greeks(option_type, s0, x, r, volatility, years_to_maturity, n_steps = 100, is_american = true, dividend_yield = 0.0) ⇒ Hash

Calculate option Greeks using the CRR model and finite difference methods

Parameters:

  • option_type (String) —

    'call' or 'put'

  • s0 (Float) —

    Spot price

  • x (Float) —

    Strike price

  • r (Float) —

    Risk-free interest rate

  • volatility (Float) —

    Volatility

  • years_to_maturity (Float) —

    Time to maturity in years

  • n_steps (Integer) (defaults to: 100) —

    Number of time steps

  • is_american (Boolean) (defaults to: true) —

    True for American options, false for European

  • dividend_yield (Float) (defaults to: 0.0) —

    Continuous dividend yield

Returns:

  • (Hash) —

    Option Greeks (delta, gamma, theta, vega, rho)



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# File 'lib/option_lab/binomial_tree.rb', line 190

def get_greeks(option_type, s0, x, r, volatility, years_to_maturity, n_steps = 100, is_american = true, dividend_yield = 0.0)
  # Small increment for finite difference calculation
  h_s = s0 * 0.001    # For Delta and Gamma
  h_t = 1.0 / 365     # For Theta (1 day)
  h_v = 0.001         # For Vega
  h_r = 0.0001        # For Rho

  # Base price
  price = price_option(option_type, s0, x, r, volatility, years_to_maturity, n_steps, is_american, dividend_yield)

  # Delta: ∂V/∂S
  price_up = price_option(option_type, s0 + h_s, x, r, volatility, years_to_maturity, n_steps, is_american, dividend_yield)
  price_down = price_option(option_type, s0 - h_s, x, r, volatility, years_to_maturity, n_steps, is_american, dividend_yield)
  delta = (price_up - price_down) / (2 * h_s)

  # Gamma: ∂²V/∂S²
  gamma = (price_up - 2 * price + price_down) / (h_s * h_s)

  # Theta: -∂V/∂t
  price_t_down = if years_to_maturity - h_t > 0
    price_option(option_type, s0, x, r, volatility, years_to_maturity - h_t, n_steps, is_american, dividend_yield)
  else
    option_payoff(option_type, s0, x)
  end
  theta = -(price_t_down - price) / h_t

  # Vega: ∂V/∂σ
  price_v_up = price_option(option_type, s0, x, r, volatility + h_v, years_to_maturity, n_steps, is_american, dividend_yield)
  vega = (price_v_up - price) / h_v

  # Rho: ∂V/∂r
  price_r_up = price_option(option_type, s0, x, r + h_r, volatility, years_to_maturity, n_steps, is_american, dividend_yield)
  rho = (price_r_up - price) / h_r

  # Return Greeks
  {
    delta: delta,
    gamma: gamma,
    theta: theta,
    vega: vega,
    rho: rho,
  }
end

.get_tree(option_type, s0, x, r, volatility, years_to_maturity, n_steps = 15, is_american = true, dividend_yield = 0.0) ⇒ Hash

Get a full binomial tree as a structured output

Parameters:

  • option_type (String) —

    'call' or 'put'

  • s0 (Float) —

    Spot price

  • x (Float) —

    Strike price

  • r (Float) —

    Risk-free interest rate

  • volatility (Float) —

    Volatility

  • years_to_maturity (Float) —

    Time to maturity in years

  • n_steps (Integer) (defaults to: 15) —

    Number of time steps

  • is_american (Boolean) (defaults to: true) —

    True for American options, false for European

  • dividend_yield (Float) (defaults to: 0.0) —

    Continuous dividend yield

Returns:

  • (Hash) —

    Tree structure with stock prices and option values



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# File 'lib/option_lab/binomial_tree.rb', line 102

def get_tree(option_type, s0, x, r, volatility, years_to_maturity, n_steps = 15, is_american = true, dividend_yield = 0.0)
  # Calculate time step
  dt = years_to_maturity / n_steps.to_f

  # Calculate up and down factors
  u = Math.exp(volatility * Math.sqrt(dt))
  d = 1.0 / u

  # Calculate risk-neutral probability
  effective_r = r - dividend_yield
  p = (Math.exp(effective_r * dt) - d) / (u - d)

  # Calculate discount factor
  discount = Math.exp(-r * dt)

  # Initialize price tree
  stock_prices = Array.new(n_steps + 1) { Array.new(n_steps + 1, 0.0) }
  option_values = Array.new(n_steps + 1) { Array.new(n_steps + 1, 0.0) }
  exercise_flags = Array.new(n_steps + 1) { Array.new(n_steps + 1, false) }

  # Fill stock price tree
  (0..n_steps).each do |i|
    (0..i).each do |j|
      stock_prices[i][j] = s0 * (u**(i - j)) * (d**j)
    end
  end

  # Calculate option values at expiration (i = n_steps)
  (0..n_steps).each do |j|
    option_values[n_steps][j] = option_payoff(option_type, stock_prices[n_steps][j], x)
  end

  # Work backwards through the tree
  (n_steps - 1).downto(0) do |i|
    (0..i).each do |j|
      # Expected value (European option value)
      expected_value = discount * (p * option_values[i + 1][j] + (1 - p) * option_values[i + 1][j + 1])

      if is_american
        # For American options, compare with immediate exercise value
        exercise_value = option_payoff(option_type, stock_prices[i][j], x)

        if exercise_value > expected_value
          option_values[i][j] = exercise_value
          exercise_flags[i][j] = true
        else
          option_values[i][j] = expected_value
        end
      else
        # For European options, use expected value
        option_values[i][j] = expected_value
      end
    end
  end

  # Return tree structure
  {
    stock_prices: stock_prices,
    option_values: option_values,
    exercise_flags: exercise_flags,
    parameters: {
      option_type: option_type,
      spot_price: s0,
      strike_price: x,
      risk_free_rate: r,
      volatility: volatility,
      time_to_maturity: years_to_maturity,
      steps: n_steps,
      is_american: is_american,
      dividend_yield: dividend_yield,
      up_factor: u,
      down_factor: d,
      risk_neutral_probability: p,
    },
  }
end

.option_payoff(option_type, stock_price, strike) ⇒ Float

Calculate option payoff at expiration

Parameters:

  • option_type (String) —

    'call' or 'put'

  • stock_price (Float) —

    Stock price

  • strike (Float) —

    Strike price

Returns:

  • (Float) —

    Option payoff



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# File 'lib/option_lab/binomial_tree.rb', line 81

def option_payoff(option_type, stock_price, strike)
  if option_type == 'call'
    [stock_price - strike, 0.0].max
  elsif option_type == 'put'
    [strike - stock_price, 0.0].max
  else
    raise ArgumentError, "Option type must be either 'call' or 'put'!"
  end
end

.price_option(option_type, s0, x, r, volatility, years_to_maturity, n_steps = 100, is_american = true, dividend_yield = 0.0) ⇒ Float

Price an option using the Cox-Ross-Rubinstein binomial tree model

Parameters:

  • option_type (String) —

    'call' or 'put'

  • s0 (Float) —

    Spot price

  • x (Float) —

    Strike price

  • r (Float) —

    Risk-free interest rate

  • volatility (Float) —

    Volatility

  • years_to_maturity (Float) —

    Time to maturity in years

  • n_steps (Integer) (defaults to: 100) —

    Number of time steps

  • is_american (Boolean) (defaults to: true) —

    True for American options, false for European

  • dividend_yield (Float) (defaults to: 0.0) —

    Continuous dividend yield

Returns:

  • (Float) —

    Option price



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# File 'lib/option_lab/binomial_tree.rb', line 24

def price_option(option_type, s0, x, r, volatility, years_to_maturity, n_steps = 100, is_american = true, dividend_yield = 0.0)
  # Calculate time step
  dt = years_to_maturity / n_steps.to_f

  # Calculate up and down factors
  u = Math.exp(volatility * Math.sqrt(dt))
  d = 1.0 / u

  # Calculate risk-neutral probability
  effective_r = r - dividend_yield
  p = (Math.exp(effective_r * dt) - d) / (u - d)

  # Calculate discount factor
  discount = Math.exp(-r * dt)

  # Initialize price tree
  stock_prices = Array.new(n_steps + 1) { Array.new(n_steps + 1, 0.0) }
  option_values = Array.new(n_steps + 1) { Array.new(n_steps + 1, 0.0) }

  # Fill stock price tree
  (0..n_steps).each do |i|
    (0..i).each do |j|
      stock_prices[i][j] = s0 * (u**(i - j)) * (d**j)
    end
  end

  # Calculate option values at expiration (i = n_steps)
  (0..n_steps).each do |j|
    option_values[n_steps][j] = option_payoff(option_type, stock_prices[n_steps][j], x)
  end

  # Work backwards through the tree
  (n_steps - 1).downto(0) do |i|
    (0..i).each do |j|
      # Expected value (European option value)
      expected_value = discount * (p * option_values[i + 1][j] + (1 - p) * option_values[i + 1][j + 1])

      if is_american
        # For American options, compare with immediate exercise value
        exercise_value = option_payoff(option_type, stock_prices[i][j], x)
        option_values[i][j] = [expected_value, exercise_value].max
      else
        # For European options, use expected value
        option_values[i][j] = expected_value
      end
    end
  end

  # Root node contains the option price
  option_values[0][0]
end