In the world of finance, mathematics play an important role in modeling the behavior of the stock market and transactions that occur in the stock exchanges. Within the stock market, there are many ways to invest one’s money, with each type of investment having a certain amount of risk associated with it. One of the most common investments is called an option, a type of financial derivative. An option is a tradable contract that gives the buyer the right (but is not required) to purchase the stock at a fixed price before a certain date, at which point the contract expires. Options carry less risk with them than owning the underlying security (such as shares of Google stock), which is why there is usually a premium paid for the right to exercise an option.

The value of an option fluctuates based on multiple factors, but other than the intrinsic value, two major factors are the time value and the volatility of the underlying. The intrinsic value of an option is the difference in the price of the underlying stock when the option is bought versus the price when the option is exercised. The time value of an option describes the risk the person issuing the option is undertaking, as the larger the time interval in which the option can be exercised, the greater the chance the option will undergo a favorable change in price for the buyer. The value of the underlying security is constantly changing. The degree at which the value of the underlying security changes is called the volatility, and leads to greater risk.

**There are many financial models used to calculate the probability distribution of future security prices, but probably none more famous than the Black-Scholes-Merton model (BSM). The BSM model is widely used to estimate prices of European options because of its simplicity and closed form solution. In the BSM model, it is possible to compute a unique implied volatility from a given price of an option. Implied volatility is determined by the market price of the derivative contract itself, rather than the historical movement of the underlying stock price over a recent history. Implied volatility is a useful measure as it allows comparisons between different strike prices, expiration dates, and underlying assets based on a more consistent measurement. **