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The Black-Scholes-Merton model is a cornerstone in financial theory, providing a systematic approach to determining the fair value of an option contract. Beyond functioning as a form of financial insurance, an option’s price is influenced by a range of parameters.
In this module, we will concentrate on the core factors that have a large impact on option pricing—specifically, option type (call versus put), the price of the underlying asset, the strike price, volatility, and implied volatility.
While dividend expectations and the risk-free rate are important in broader applications of option pricing, they will be omitted from our discussion as they fall outside the scope of this course.
C = S * N(d1) - K * exp(-r * T) * N(d2)
d1 = [ln(S / K) + (r + σ² / 2) * T] / (σ * sqrt(T)) d2 = d1 - σ * sqrt(T)
An option’s classification as either a call or a put is fundamental to its valuation. A call option confers the right, but not the obligation, to purchase the underlying asset at a predetermined strike price, thereby gaining from upward price movements. Conversely, a put option allows the holder to sell the underlying asset at the strike price, benefiting from downward price movements. Consequently, an increase in the underlying asset’s price is beneficial to call options, while a decrease enhances the value of put options.
The current market price of the underlying asset plays a critical role in an option’s valuation. For call options, an increase in the asset’s price enhances the probability of the option expiring in the money, thereby raising its premium. In contrast, put options derive more value as the underlying price declines, increasing their intrinsic worth. Thus, the underlying price exerts opposite effects on calls and puts, making the concept of “moneyness” central to understanding option valuation. Moneyness refers to the relationship between an option’s strike price and the current price of the underlying asset, which determines whether the option holds intrinsic value at expiration. An option is considered in-the-money (ITM) if it would retain intrinsic value at expiration; for a call option, this means the underlying asset’s price is above the strike price, while for a put option, it occurs when the underlying asset’s price is below the strike price. In contrast, an option is deemed out-of-the-money (OTM) when it would have no intrinsic value at expiration—this is the case for call options when the underlying price is below the strike and for put options when the underlying price is above the strike. When the underlying asset’s price is exactly equal to the strike price, the option is classified as at-the-money (ATM).
The strike price is the predetermined price at which the underlying asset can be bought or sold within the confines of an option contract. For call options, when the strike price is lower relative to the market price of the underlying asset, the option becomes more likely to end in-the-money (ITM). A lower strike price increases the chance that the asset’s market price will exceed this strike price as the option approaches expiration, thereby increasing the intrinsic value of the option. In turn, for put options, when the strike price is higher compared to the underlying asset's price, the option has a greater potential for yielding a higher intrinsic value. A higher strike price enhances the probability that the asset's market price will fall below this threshold, allowing the put option holder to sell the asset for a better price than the current market value. In the broader context of options pricing, the strike price, in relation to the underlying price, directly affects the probability of the option expiring in-the-money (ITM). The closer the strike price is to the current market price, the higher the likelihood for both call and put options to have intrinsic value at expiration. As such, the strike price plays an essential role in determining the option's moneyness and the resulting probability of an option expiring profitably.
The strike price is the predetermined level at which the underlying asset may be transacted. For a call option, a lower strike price sets a lower threshold, meaning that the underlying asset’s market price is more likely to exceed the strike at expiration, thereby increasing the magnitude of any intrinsic value. Conversely, for a put option, a higher strike price establishes a higher threshold; if the market price falls below this level at expiration, the intrinsic value becomes greater. In both cases, the strike price, relative to the current market price of the underlying asset, is directly linked to the probability of the option expiring in-the-money, as it determines the conditions under which the option has intrinsic value.
Volatility, which measures the extent of fluctuation in the underlying asset’s price, is a critical determinant of an option’s premium. Higher volatility increases the range of possible price outcomes, thereby elevating the probability that an option—be it a call or a put—will finish in the money. This expanded range of potential outcomes raises the option’s value by reflecting the increased likelihood of achieving a significant payoff. It is essential to note that while volatility directly impacts the option’s premium, it is distinct from implied volatility, which is derived from the option’s market price.
Furthermore, anticipated events that may significantly move the underlying asset’s price—such as earnings announcements, regulatory decisions, or other major news—tend to increase expected volatility. As market participants adjust their outlook based on these events, the probability distribution of future price outcomes widens, leading to higher option premiums. Importantly, there is no explicit metric input within the Black-Scholes-Merton model to account for such event-driven volatility. Instead, the market’s expectations for future volatility are deduced from the option’s current market price, a measure known as implied volatility (discussed in the next paragraph).
Implied volatility is a metric that reflects the market’s expectations of future volatility, as inferred from the current option price. Unlike historical volatility, which is based on past price movements, implied volatility is obtained by inputting the observed market price into the Black-Scholes-Merton model. The resulting value is the volatility figure that aligns the theoretical price with the market price. An increase in the option’s market price will yield a higher implied volatility, thereby signaling that market participants expect more significant future fluctuations. In this way, implied volatility serves as a gauge of market sentiment rather than a direct driver of the option’s intrinsic premium.
Time to expiration is a fundamental factor in option pricing as it represents the remaining time until the option contract expires. A longer time to expiration generally increases the option's extrinsic or time value because there is more opportunity for the underlying asset's price to move favorably. This additional time allows for greater uncertainty and a higher probability of achieving a profitable outcome, thereby increasing the premium of the option. Conversely, as the expiration date approaches, the extrinsic value diminishes rapidly—a phenomenon known as time decay (captured by Theta). For example, an at-the-money option with several months until expiration will typically have a higher premium than an otherwise identical option with only a few days remaining, due to the greater uncertainty and opportunity provided by the longer time horizon.
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