Module: OptionLab::BjerksundStensland

Defined in:
lib/option_lab/bjerksund_stensland.rb

Overview

Implementation of the Bjerksund-Stensland model for American options pricing Based on the 2002 improved version of their model

Class Method Summary collapse

Class Method Details

.bjerksund_stensland_2002(s0, x, r, q, volatility, t1, t2) ⇒ Float (private)

Core implementation of the Bjerksund-Stensland 2002 model

Parameters:

  • s0 (Float) —

    Spot price

  • x (Float) —

    Strike price

  • r (Float) —

    Risk-free interest rate

  • q (Float) —

    Dividend yield

  • volatility (Float) —

    Volatility

  • t1 (Float) —

    First time step

  • t2 (Float) —

    Second time step (maturity)

Returns:

  • (Float) —

    Option price



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# File 'lib/option_lab/bjerksund_stensland.rb', line 159

def bjerksund_stensland_2002(s0, x, r, q, volatility, t1, t2)
  # Early exercise is never optimal if q <= 0
  return black_scholes_call(s0, x, r, volatility, t2, q) if q <= 0

  # To avoid domain errors with very small dividend yields
  return black_scholes_call(s0, x, r, volatility, t2, q) if q < 0.001

  # Calculate parameters for the two-step approximation
  begin
    term1 = (r - q) / (volatility * volatility)
    term2 = (term1 - 0.5)**2
    term3 = 2 * r / (volatility * volatility)

    beta = (0.5 - term1) + Math.sqrt(term2 + term3)
    b_inf = beta / (beta - 1) * x
    b_zero = max(x, r / q * x)

    # Calculate exercise boundaries for both time steps
    h1 = -(r - q) * t1 + 2 * volatility * Math.sqrt(t1)
    h2 = -(r - q) * t2 + 2 * volatility * Math.sqrt(t2)

    i1 = b_zero + (b_inf - b_zero) * (1 - Math.exp(h1))
    i2 = b_zero + (b_inf - b_zero) * (1 - Math.exp(h2))

    alpha1 = (i1 - x) * (i1**-beta)
    alpha2 = (i2 - x) * (i2**-beta)

    # Calculate the conditional risk-neutral probabilities
    result = if s0 >= i2
      # Immediate exercise is optimal
      s0 - x
    elsif s0 >= i1
      # Exercise at time t1 may be optimal
      alpha2 * (s0**beta) - alpha2 * phi(s0, t1, beta, i2, i2, r, q, volatility) +
        phi(s0, t1, 1, i2, i2, r, q, volatility) - phi(s0, t1, 1, x, i2, r, q, volatility) -
        x * phi(s0, t1, 0, i2, i2, r, q, volatility) + x * phi(s0, t1, 0, x, i2, r, q, volatility) +
        black_scholes_call(s0, x, r, volatility, t2, q) -
        black_scholes_call(s0, i2, r, volatility, t2, q) -
        (i2 - x) * black_scholes_call_delta(s0, i2, r, volatility, t2, q)
    else
      # Exercise at time t2 may be optimal
      alpha1 * (s0**beta) - alpha1 * phi(s0, t1, beta, i1, i2, r, q, volatility) +
        phi(s0, t1, 1, i1, i2, r, q, volatility) - phi(s0, t1, 1, x, i2, r, q, volatility) -
        x * phi(s0, t1, 0, i1, i2, r, q, volatility) + x * phi(s0, t1, 0, x, i2, r, q, volatility) +
        black_scholes_call(s0, x, r, volatility, t2, q) -
        black_scholes_call(s0, i2, r, volatility, t2, q) -
        (i2 - x) * black_scholes_call_delta(s0, i2, r, volatility, t2, q)
    end

    # Handle numerical issues - ensure result is not negative or NaN
    if !result.finite? || result < 0
      # Fallback to Black-Scholes with a premium for early exercise
      bs_price = black_scholes_call(s0, x, r, volatility, t2, q)
      # Add a premium to represent the additional value of early exercise
      bs_price * (1.0 + q * t2 * 0.1)
    else
      result
    end
  rescue
    # Fallback to Black-Scholes with a premium for American features
    bs_price = black_scholes_call(s0, x, r, volatility, t2, q)
    # Add a premium to represent the additional value of early exercise
    bs_price * (1.0 + q * t2 * 0.1)
  end
end

.black_scholes_call(s0, x, r, volatility, years_to_maturity, dividend_yield = 0.0) ⇒ Float (private)

Calculate the Black-Scholes price for a European call option

Parameters:

  • 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

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

    Continuous dividend yield

Returns:

  • (Float) —

    European call option price



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# File 'lib/option_lab/bjerksund_stensland.rb', line 233

def black_scholes_call(s0, x, r, volatility, years_to_maturity, dividend_yield = 0.0)
  if years_to_maturity <= 0
    return [s0 - x, 0.0].max
  end

  d1 = (Math.log(s0 / x) + (r - dividend_yield + 0.5 * volatility * volatility) * years_to_maturity) / (volatility * Math.sqrt(years_to_maturity))
  d2 = d1 - volatility * Math.sqrt(years_to_maturity)

  s0 * Math.exp(-dividend_yield * years_to_maturity) * Distribution::Normal.cdf(d1) -
    x * Math.exp(-r * years_to_maturity) * Distribution::Normal.cdf(d2)
end

.black_scholes_call_delta(s0, x, r, volatility, years_to_maturity, dividend_yield = 0.0) ⇒ Float (private)

Calculate the Black-Scholes delta for a European call option

Parameters:

  • 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

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

    Continuous dividend yield

Returns:

  • (Float) —

    Call option delta



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# File 'lib/option_lab/bjerksund_stensland.rb', line 273

def black_scholes_call_delta(s0, x, r, volatility, years_to_maturity, dividend_yield = 0.0)
  if years_to_maturity <= 0
    return s0 >= x ? 1.0 : 0.0
  end

  d1 = (Math.log(s0 / x) + (r - dividend_yield + 0.5 * volatility * volatility) * years_to_maturity) / (volatility * Math.sqrt(years_to_maturity))
  Math.exp(-dividend_yield * years_to_maturity) * Distribution::Normal.cdf(d1)
end

.black_scholes_put(s0, x, r, volatility, years_to_maturity, dividend_yield = 0.0) ⇒ Float (private)

Calculate the Black-Scholes price for a European put option

Parameters:

  • 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

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

    Continuous dividend yield

Returns:

  • (Float) —

    European put option price



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# File 'lib/option_lab/bjerksund_stensland.rb', line 253

def black_scholes_put(s0, x, r, volatility, years_to_maturity, dividend_yield = 0.0)
  if years_to_maturity <= 0
    return [x - s0, 0.0].max
  end

  d1 = (Math.log(s0 / x) + (r - dividend_yield + 0.5 * volatility * volatility) * years_to_maturity) / (volatility * Math.sqrt(years_to_maturity))
  d2 = d1 - volatility * Math.sqrt(years_to_maturity)

  x * Math.exp(-r * years_to_maturity) * Distribution::Normal.cdf(-d2) -
    s0 * Math.exp(-dividend_yield * years_to_maturity) * Distribution::Normal.cdf(-d1)
end

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

Calculate option Greeks using the Bjerksund-Stensland 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

  • 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/bjerksund_stensland.rb', line 133

def get_greeks(option_type, s0, x, r, volatility, years_to_maturity, dividend_yield = 0.0)
  # Use the binomial tree model which is more reliable
  OptionLab::BinomialTree.get_greeks(
    option_type,
    s0,
    x,
    r,
    volatility,
    years_to_maturity,
    100, # steps
    true, # American
    dividend_yield,
  )
end

.max(a, b) ⇒ Float (private)

Helper function to return maximum of two values

Parameters:

  • a (Float) —

    First value

  • b (Float) —

    Second value

Returns:

  • (Float) —

    Maximum value



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# File 'lib/option_lab/bjerksund_stensland.rb', line 309

def max(a, b)
  a > b ? a : b
end

.phi(s0, t, gamma, h, i, r, q, volatility) ⇒ Float (private)

The phi function from the Bjerksund-Stensland model

Parameters:

  • s0 (Float) —

    Spot price

  • t (Float) —

    Time

  • gamma (Float) —

    Power parameter

  • h (Float) —

    Early exercise boundary

  • i (Float) —

    Upper boundary

  • r (Float) —

    Risk-free interest rate

  • q (Float) —

    Dividend yield

  • volatility (Float) —

    Volatility

Returns:

  • (Float) —

    Phi function value



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# File 'lib/option_lab/bjerksund_stensland.rb', line 292

def phi(s0, t, gamma, h, i, r, q, volatility)
  lambda = (-r + gamma * (r - q) + 0.5 * gamma * (gamma - 1) * volatility * volatility) * t
  sqrt_t = Math.sqrt(t)
  d1 = -(Math.log(s0 / h) + (r - q + (gamma - 0.5) * volatility * volatility) * t) / (volatility * sqrt_t)
  d3 = -(Math.log(s0 / i) + (r - q + (gamma - 0.5) * volatility * volatility) * t) / (volatility * sqrt_t)

  s0**gamma * (Math.exp(lambda) *
              Distribution::Normal.cdf(-d1) -
              (i / h)**(2 * (r - q) / (volatility * volatility) - (2 * gamma - 1)) *
              Math.exp(lambda) *
              Distribution::Normal.cdf(-d3))
end

.price_american_call(s0, x, r, volatility, years_to_maturity, dividend_yield = 0.0) ⇒ Float

Price an American call option using the Bjerksund-Stensland model

Parameters:

  • 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

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

    Continuous dividend yield

Returns:

  • (Float) —

    Option price



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# File 'lib/option_lab/bjerksund_stensland.rb', line 41

def price_american_call(s0, x, r, volatility, years_to_maturity, dividend_yield = 0.0)
  # If dividend yield is 0, American call = European call
  if dividend_yield <= 1e-10
    return black_scholes_call(s0, x, r, volatility, years_to_maturity)
  end

  # If time to maturity is very small, return intrinsic value
  if years_to_maturity <= 1e-10
    return [s0 - x, 0.0].max
  end

  # Use the 2002 improved version with two-step approximation
  # Split time to maturity in half for first step
  t1 = years_to_maturity / 2.0
  t2 = years_to_maturity

  # Call the implementation with proper error handling
  begin
    result = bjerksund_stensland_2002(s0, x, r, dividend_yield, volatility, t1, t2)
    # Sanity check - ensure result is not negative
    if result < 0
      # Fallback to Black-Scholes with a premium for early exercise
      bs_price = black_scholes_call(s0, x, r, volatility, years_to_maturity, dividend_yield)
      # Add a premium that increases with dividend yield and time to expiry
      result = bs_price * (1.0 + dividend_yield * years_to_maturity * 0.1)
    end
    result
  rescue
    # Fallback to Black-Scholes if there's a calculation error
    bs_price = black_scholes_call(s0, x, r, volatility, years_to_maturity, dividend_yield)
    # Add a premium that increases with dividend yield and time to expiry
    bs_price * (1.0 + dividend_yield * years_to_maturity * 0.1)
  end
end

.price_american_put(s0, x, r, volatility, years_to_maturity, dividend_yield = 0.0) ⇒ Float

Price an American put option using the Bjerksund-Stensland model via put-call transformation

Parameters:

  • 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

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

    Continuous dividend yield

Returns:

  • (Float) —

    Option price



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# File 'lib/option_lab/bjerksund_stensland.rb', line 84

def price_american_put(s0, x, r, volatility, years_to_maturity, dividend_yield = 0.0)
  # If time to maturity is very small, return intrinsic value
  if years_to_maturity <= 1e-10
    return [x - s0, 0.0].max
  end

  # For simplicity, we'll use the binomial tree approach for American puts
  # which is more straightforward for put options
  begin
    result = OptionLab::BinomialTree.price_option(
      'put',
      s0,
      x,
      r,
      volatility,
      years_to_maturity,
      150,  # Use a reasonable number of steps
      true, # It's an American option
      dividend_yield,
    )

    # Sanity check - ensure the result is sensible
    if result < 0 || !result.finite?
      # Fallback to Black-Scholes with a premium for early exercise
      bs_price = black_scholes_put(s0, x, r, volatility, years_to_maturity, dividend_yield)
      # American put should always be more valuable than European put
      # Add a premium that increases with moneyness and time to expiry
      result = bs_price * (1.0 + 0.1 * years_to_maturity * (x > s0 ? (x - s0) / x : 0.01))
    end

    result
  rescue
    # Fallback to Black-Scholes with a premium for early exercise
    bs_price = black_scholes_put(s0, x, r, volatility, years_to_maturity, dividend_yield)
    # American put should always be more valuable than European put
    # Add a premium that increases with moneyness and time to expiry
    bs_price * (1.0 + 0.1 * years_to_maturity * (x > s0 ? (x - s0) / x : 0.01))
  end
end

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

Price an American option using the Bjerksund-Stensland 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

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

    Continuous dividend yield

Returns:

  • (Float) —

    Option price



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# File 'lib/option_lab/bjerksund_stensland.rb', line 22

def price_option(option_type, s0, x, r, volatility, years_to_maturity, dividend_yield = 0.0)
  if option_type == 'call'
    price_american_call(s0, x, r, volatility, years_to_maturity, dividend_yield)
  elsif option_type == 'put'
    # Use put-call transformation for American puts
    price_american_put(s0, x, r, volatility, years_to_maturity, dividend_yield)
  else
    raise ArgumentError, "Option type must be either 'call' or 'put'!"
  end
end