Python Finance: Calculate In the money probability for an option

Ever wondered how likely it is for your option to finish in the money? Understanding this probability is crucial for traders and investors, and Python makes it easy to calculate using popular financial models. In this article, we’ll explore two approaches—the Black-Scholes formula and the binomial model—to estimate the in-the-money probability for options.

Black-Scholes Formula

def in_the_money_probability(option_type, strike_price, underlying_price, volatility, time_to_expiration):
  # Calculate d1 and d2
  d1 = (log(underlying_price / strike_price) + (volatility ** 2 / 2) * time_to_expiration) / (volatility * sqrt(time_to_expiration))
  d2 = d1 - volatility * sqrt(time_to_expiration)

  # Use the cumulative distribution function (CDF) of the standard normal distribution
  # to calculate the in the money probability
  if option_type == "call":
    in_the_money_probability = norm.cdf(d1)
  elif option_type == "put":
    in_the_money_probability = norm.cdf(-d2)

  return in_the_money_probability

In this function, option_type is either “call” or “put”, strike_price is the strike price of the option, underlying_price is the current price of the underlying asset, volatility is the volatility of the underlying asset, and time_to_expiration is the time until the option expires, measured in years.

This function applies the Black-Scholes formula to estimate the probability that an option will be in the money at expiration. The Black-Scholes model assumes the underlying asset follows a log-normal distribution and that the option is European (exercisable only at expiration).

Keep in mind, this function is for educational purposes and may not be suitable for real-world trading. The Black-Scholes formula can be inaccurate for certain options, such as those with high skew or long expirations, so it should not be solely relied upon for trading decisions.

Binomial Model

To estimate the in-the-money probability using a binomial model, you first construct a binomial tree for the underlying asset. This involves dividing the time to expiration into discrete intervals (such as days, weeks, or months) and simulating possible price movements at each step.