Lesson 5.2: Delta (Δ): The "Speed" of an Option

In its simplest form, Delta is just the "speed" of your option relative to the stock.

The Intuitive Meaning

"If the stock price moves up by $1, how much will my option price move?"

• If $\Delta = 0.5$ (an "at-the-money" option): A $1 increase in S causes a $0.50 increase in V.
• If $\Delta = 1.0$ (a "deep-in-the-money" option): A $1 increase in S causes a $1.00 increase in V. The option now moves 1-for-1 with the stock.
• If $\Delta = 0.0$ (a "far-out-of-the-money" option): A $1 increase in S causes a $0.00 increase in V. The option is "dead" and doesn't react to small stock moves.

Delta is the slope of the option's price curve. It is your instantaneous risk exposure to the underlying stock.

This should sound incredibly familiar. In Module 4, we performed a "magic trick" to build a risk-free portfolio.

• We built a portfolio $\Pi = -V + \Delta_{\text{shares}} S_t$.
• We found its change $d\Pi$ and looked at the "Random Bin":
$$\text{Random Bin} = \left[ - \sigma S_t \frac{\partial V}{\partial S} + \Delta_{\text{shares}} (\sigma S_t) \right] dW_t$$
• We "killed" the risk by setting this to 0 and solving for $\Delta_{\text{shares}}$.
• The "magic number" of shares we needed was: $\Delta_{\text{shares}} = \frac{\partial V}{\partial S}$.

So, how do we calculate this magic number $\Delta$? We would have to take the derivative of the massive Black-Scholes formula from Lesson 5.1:

This looks like a nightmare. It requires the product rule on the first term ($S_t \cdot N(d_1)$) and the chain rule on the second term.

However, after a page of complex algebra (which we can skip), all the messy terms magically cancel each other out. The solution simplifies to the most elegant "wow" moment in finance.

That's it. The "speed" of the option is simply $N(d_1)$, the first "probability calculator" term that was already sitting in the price formula all along.

This gives Delta a new, powerful, intuitive meaning.

This is where we combine all the theory. A professional trader thinks of Delta in two ways at the same time:

Example 1: The Risk Manager (Delta as a "Hedge Ratio")

• The Situation: You are a trader. A client calls and sells you 100 call options.
• The Risk: You are now "short 100 calls." You need to find your risk. You look at your screen and see the option's Delta is $N(d_1) = 0.60$.
• Calculate Your Risk: Your portfolio's Delta is:
  $$(-100 \text{ options}) \times (0.60 \text{ Delta/option}) = -60 \text{ Delta}$$
• Physical Meaning: You are now "short 60 shares of stock." If the stock price goes up $1, your options lose $60, and your portfolio loses $60. You are exposed to risk.
• The Action (Delta-Hedging): To make your portfolio "delta-neutral" (risk-free), you must add +60 Delta to your book. The stock itself has a Delta of 1. You immediately buy 60 shares of the stock.
• Final Position:
  • Option Position: -60 Delta
  • Stock Position: +60 Delta
  • Net Portfolio Delta: 0
  You are now "delta-hedged." You are immune to small, instantaneous moves in the stock price. This is what a market-maker does all day.

Example 2: The Speculator (Delta as a "Probability")

• The Situation: You are a speculator. You don't want to hedge; you want to *bet*. You are trying to decide which option to buy.
• The Formula: $\Delta = N(d_1)$
• The Intuition: $N(d_1)$ is a "cumulative normal distribution." Its value is *always* between 0 and 1. This makes it *look just like a probability*.

While $N(d_2)$ (from Lesson 5.1) is the *actual* risk-neutral probability the option expires in-the-money, traders use $N(d_1)$ as a quick, real-time approximation.

• If an option's Delta is 0.20 ($N(d_1) = 0.20$): Traders think, "This is a cheap, out-of-the-money longshot. The market is giving it a ~20% chance of paying off."
• If an option's Delta is 0.50 ($N(d_1) = 0.50$): "This is a coin-flip, at-the-money option. A ~50% chance."
• If an option's Delta is 0.90 ($N(d_1) = 0.90$): "This is a deep-in-the-money, safe bet. A ~90% chance of paying off."

This intuition also tells you how the option will "act":

• A 0.20-Delta option will "act like" 20 shares of stock.
• A 0.90-Delta option will "act like" 90 shares of stock.

This tells you both the *risk* ($\Delta = 0.90$) and the *probability* ($N(d_1) = 90\%$) are high. Delta is the single number that connects all these ideas.