d 1 right-parenthesis"/> is the value of the standard normal cumulative distribution function at (1.13)
where σ is the volatility of the stock. If the equations seem gross, it's because they are.
Again, the purpose of this section is not to describe the underlying mechanics of the Black‐Scholes model in detail. Rather, Equations (1.10) through (1.14) are included to emphasize three important points.
1. There is inherent uncertainty in the price of stock. Stock price movements are also assumed to be independent of one another and log‐normally distributed.12
2. An estimate for the fair price of an option can be calculated according to the price of the stock, the volatility of the stock, the risk‐free rate, the duration of the contract, and the strike price.
3. The volatility of a stock, which plays an important role in estimating the risk of an asset and the valuation of an option, cannot be directly observed. This suggests that the “true risk” of an instrument can never be exactly known. Risk can only be approximated using a metric, such as historical volatility or the standard deviation of the historical returns over some timescale, typically matching the duration of the contract. Other than using a past‐looking metric, such as historical volatility to estimate the risk of an asset, one can also infer the risk of an asset from the price of its options.
As stated previously, the Black‐Scholes model only gives a theoretical estimate for the fair price of an option. Once the contract is traded on the options market, the price of the contract is often driven up or down depending on speculation and perceived risk. The deviation of an option's price from its theoretical value as a result of these external factors is indicative of implied volatility. When initially valuing an option, the historical volatility of the stock has been priced into the model. However, when the price of the option trades higher or lower than its theoretical value, this indicates that the perceived volatility of the underlying deviates from what is estimated by historical returns.
Implied volatility may be the most important metric in options trading. It is effectively a measure of the sentiment of risk for a given underlying according to the supply and demand for options contracts. For an example, suppose a non‐dividend‐paying stock currently trading at $100 per share has a historical 45‐day returns volatility of 20%. Suppose its call option with a 45‐day duration and a strike price of $105 is trading at $2 per share. Plugging these parameters into the Black‐Scholes model, this call option should theoretically be trading at $1 per share. However, demand for this position has increased the contract price significantly. For the model to return a call price of $2 per share, the volatility of this underlying would have to be 28% (assuming all else is constant). Therefore, although the historical volatility of the underlying is only 20%, the perceived risk of that underlying (i.e., the implied volatility) is actually 28%.
To conclude, the primary purpose of this section was not to dive into the math of the Black‐Scholes. These concepts were, instead, introduced to justify the following axioms that are foundational to this book:
● Profits cannot be made without risk.
● Stock log returns have inherent uncertainty and are assumed to follow a normal distribution.
● Stock price movements are independent across time (i.e., future price changes are independent of past price changes, requiring a minimum of the weak EMH).
● Options can theoretically be priced fairly based on the price of the stock, the volatility of the stock, the risk‐free rate, the duration of the contract, and the strike price.
● The volatility of an asset cannot be directly observed, only estimated using metrics like historical volatility or implied volatility.
The Greeks
Other than implied volatility, the Greeks are the most relevant metrics derived from the Black‐Scholes model. The Greeks are a set of risk measures, and each describes the sensitivity of an option's price with respect to changes in some variable. The most essential Greeks for options traders are delta
Delta
(1.15)
where V is the price of the option (a call or a put) and S is the price of the underlying stock, noting that ∂ is the partial derivative. The value of delta ranges from –1 to 1, and the sign of delta depends on the type of position:
● Long stock: Δ is 1.
● Long call and short put: Δ is between 0 and 1.
● Long put and short call: Δ is between –1 and 0.
For example, the price of a long call option with a delta of 0.50 (denoted 50Δ because that is the total Δ for a one lot, or 100 shares of underlying) will increase by approximately $0.50 per share when the price of the underlying increases by $1. This makes intuitive sense because a long stock, a long call, and a short put are all bullish strategies, meaning they will profit when the underlying price increases. Similarly, because long puts and short calls are bearish, they will take a loss when the underlying price increases.
Delta has a sign and magnitude, so it is a measure of the degree of directional risk of a position. The sign of delta indicates the direction of the risk, and the magnitude of delta indicates the severity of exposure. The larger the magnitude of delta, the larger the profit and loss potential of the contract. This is because positions with larger deltas are closer to/deeper ITM
Order refers to the number of mathematical derivatives taken on the price of the option. Delta has a single derivative of
