## 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 uses the Black-Scholes formula to calculate the probability that an option will be in the money at expiration. The Black-Scholes formula assumes that the underlying asset follows a log-normal distribution and that the option is European (i.e. it can only be exercised at expiration).

Note that this function is just an example, and it may not be suitable for use in a real-world trading scenario. In particular, the Black-Scholes formula is known to be inaccurate for certain types of options (e.g. options with a high degree of skew or options with a long time to expiration), and it should not be relied on for making trading decisions.

## Binomial

To calculate the in the money probability for an option using a binomial model, you would first need to construct a binomial tree for the underlying asset. This involves breaking the time until expiration into a number of discrete time intervals (e.g. days, weeks, or months) and then simulating the possible price movements of the underlying asset at each interval.

Once the binomial tree has been constructed, you can use it to calculate the in the money probability for the option at each interval. To do this, you would need to consider all of the possible paths the underlying asset could take from the current time until expiration, and then calculate the probability that the option will be in the money at expiration for each path.

For example, suppose you have a call option with a strike price of $100 and an underlying asset that is currently trading at $110. If the binomial tree shows that the underlying asset could either go up by 10% to $121 or go down by 10% to $99 at the next time interval, then the in the money probability for the option at that interval would be the probability that the underlying asset ends up at $121 or higher, which could be calculated using the probability distribution of the underlying asset.

Once you have calculated the in the money probability for the option at each interval in the binomial tree, you can use these probabilities to estimate the overall in the money probability for the option at expiration. This would typically involve taking the average of the probabilities at each interval, weighting them by the length of the interval, or using some other method to combine the probabilities into a single estimate.

Overall, calculating the in the money probability for an option using a binomial model is a complex process that involves constructing a binomial tree for the underlying asset and then using the tree to calculate the probability that the option will be in the money at expiration. It can provide a more accurate estimate of the in the money probability than the Black-Scholes formula, but it is also more computationally intensive and may not be suitable for all situations.

```
def in_the_money_probability(option_type, strike_price, underlying_price, volatility, time_to_expiration, steps):
# Construct a binomial tree for the underlying asset
tree = construct_binomial_tree(underlying_price, volatility, time_to_expiration, steps)
# Calculate the in the money probability for the option at each interval in the tree
in_the_money_probabilities = []
for i in range(steps):
in_the_money_probability = 0
for j in range(i + 1):
if option_type == "call":
if tree[i][j] >= strike_price:
in_the_money_probability += tree[i][j]
elif option_type == "put":
if tree[i][j] <= strike_price:
in_the_money_probability += tree[i][j]
in_the_money_probabilities.append(in_the_money_probability)
# Calculate the overall in the money probability for the option at expiration
# by combining the probabilities at each interval
overall_in_the_money_probability = 0
for i in range(steps):
overall_in_the_money_probability += in_the_money_probabilities[i] * (time_to_expiration / steps)
return overall_in_the_money_probability
```

to calculate the value of the cumulative distribution function (CDF) of the standard normal distribution for a given value:

```
def norm_cdf(x):
return (1 + erf(x / sqrt(2))) / 2
```

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, `time_to_expiration`

is the time until the option expires, measured in years, and `steps`

is the number of intervals to use in the binomial tree.

This function first constructs a binomial tree for the underlying asset using the `construct_binomial_tree`

function, which is not shown in this code. Then, it uses the tree to calculate the in the money probability for the option at each interval in the tree. Finally, it combines the probabilities at each interval to calculate the overall in the money probability for the option at expiration.

Note that this function is just an example, and it may not be suitable for use in a real-world trading scenario. In particular, this function does not account for the possibility of early exercise for American options, and it should not be relied on for making trading decisions.