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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 , gamma , and theta .

      Delta is one of the most important and widely used Greeks. It is a first‐order13 Greek that measures the expected change in the option price given a $1 increase in the price of the underlying (assuming all other variables stay constant). The equation is as follows:

      (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 and more sensitive to changes in the underlying price. A contract with a delta of 1.0 (100Δ ) has maximal directional exposure and is maximally ITM. 100Δ options behave like the stock price, as a $1 increase in the underlying creates a $1 increase in the option's price per share. A contract with a delta of 0.0 has no directional exposure and is maximally OTM. A 50Δ contract is defined as having the ATM strike.14

      Because delta is a measure of directional exposure, it plays a large role when hedging directional risks. For instance, if a trader currently has a 50Δ position on and wants the position to be relatively insensitive to directional moves in the underlying, the trader could offset that exposure with the addition of 50 negative deltas (e.g., two 25Δ long puts). The composite position is called delta neutral.

      Gamma is a second‐order Greek and a measure of the expected change in the option delta given a $1 change in the underlying price. Gamma is mathematically represented as follows:

      (1.16)

      As with delta, the sign of gamma depends on the type of position:

      ● Long call and long put: .

      ● Short call and short put: .

      In other words, if there is a $1 increase in the underlying price, then the delta for all long positions will become more positive, and the delta for all short positions will become more negative. This makes intuitive sense because a $1 increase in the underlying pushes long calls further ITM, increasing the directional exposure of the contract, and it pushes long puts further OTM, decreasing the inverse directional exposure of the contract and bringing the negative delta closer to zero. The magnitude of gamma is highest for ATM positions and lower for ITM and OTM positions, meaning that delta is most sensitive to underlying price movements at –50Δ and 50Δ .

      Awareness of gamma is critical when trading options, particularly when aiming for specific directional exposure. The delta of a contract is typically transient, so the gamma of a position gives a better indication of the long‐term directional exposure. Suppose traders wanted to construct a delta neutral position by pairing a short call (negative delta) with a short put (positive delta), and they are considering using 20Δ or 40Δ contracts (all other parameters identical). The 40Δ contracts are much closer to ATM (50Δ ) and have more profit potential than the 20Δ positions, but they also have significantly more gamma risk and are less likely to remain delta neutral in the long term. The optimal choice would then depend on how much risk traders are willing to accept and their profit goals. For traders with high profit goals and a large enough account to handle the large P/L swings and loss potential of the trade, the 40Δ contracts are more suitable.

      Theta is a first‐order Greek that measures the expected P/L changes resulting from the decay of the option's extrinsic value (the difference between the current market price for the option and the intrinsic

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