The Black-Scholes formula and delta hedging

Exam FAM – Fundamentals of Actuarial Mathematics · SOA (Society of Actuaries) · 6 min read

Welcome to the World of Black-Scholes!

Hello there! Today, we are diving into one of the most famous topics in all of finance: The Black-Scholes Formula and Delta Hedging. If you have ever wondered how professional traders decide what an option is worth or how they protect themselves from losing money when stock prices wiggle, you are in the right place.

Don't worry if these formulas look like "alphabet soup" at first glance. We are going to break them down piece by piece. By the end of these notes, you’ll see that the Black-Scholes formula is really just a way of balancing what you get against what you pay.

1. The Black-Scholes Formula: The Big Picture

The Black-Scholes model is used to calculate the "fair price" of European-style options. It assumes that stock prices follow a specific mathematical path (geometric Brownian motion) and that the market is "frictionless" (no taxes or transaction costs).

To find the price of a Call or Put, we need six ingredients:

$S_0$: Current Stock Price
$K$: Strike Price
$\sigma$ (Sigma): Volatility (how much the stock jumps around)
$r$: Risk-free interest rate
$T$: Time to expiration (in years)
$\delta$ (Delta): Dividend yield of the stock

The Formulas

The price of a Call option ($C$) is:
$C = S_0 e^{-\delta T} N(d_1) - K e^{-rT} N(d_2)$

The price of a Put option ($P$) is:
$P = K e^{-rT} N(-d_2) - S_0 e^{-\delta T} N(-d_1)$

Quick Review Box:
Notice the structure: It is (Value of Stock Part) minus (Cost of Cash Part) for a Call. For a Put, it's the reverse! $N(x)$ represents the standard normal cumulative distribution function—basically, the probability that a value is less than $x$ on a bell curve.

Calculating $d_1$ and $d_2$

These are the "engines" of the formula. You calculate $d_1$ first, then use it to find $d_2$:

$d_1 = \frac{\ln(S_0/K) + (r - \delta + 0.5\sigma^2)T}{\sigma\sqrt{T}}$

$d_2 = d_1 - \sigma\sqrt{T}$

Memory Aid: Think of $d_1$ as the "Stock side" and $d_2$ as the "Strike side." They help determine how likely it is the option will finish "in the money."

Key Takeaway: The Black-Scholes formula looks intimidating, but it is just a weighted difference between the current stock price and the present value of the strike price.

2. Understanding $N(d_1)$ and $N(d_2)$

It helps to have an intuitive sense of what these variables represent so you aren't just memorizing symbols.

$N(d_2)$: Under the "risk-neutral" world, this is the probability that the option will be exercised (i.e., that the stock price will be above the strike price at expiration).
$N(d_1)$: This is related to the Delta of the option (which we will cover next). It tells us how much the option price changes when the stock price changes.

Example: If $N(d_2) = 0.70$, there is roughly a 70% chance the option will end up "in the money."

3. Delta ($\Delta$): The Hedge Ratio

Delta is the most important "Greek" for Exam FAM. It measures the sensitivity of the option price to a change in the underlying stock price.

The Formulas for Delta:
For a Call: $\Delta_{Call} = e^{-\delta T} N(d_1)$
For a Put: $\Delta_{Put} = -e^{-\delta T} N(-d_1) = e^{-\delta T} [N(d_1) - 1]$

Analogy: The Steering Wheel

Imagine you are driving a car. The stock price is the gas pedal, and the option price is the speed. Delta tells you how much your speed increases for every inch you push the pedal.

A Call Delta is positive (0 to 1). If the stock goes up, the call price goes up.
A Put Delta is negative (-1 to 0). If the stock goes up, the put price goes down.

Did you know? If a stock pays no dividends ($\delta = 0$), the Call Delta is simply $N(d_1)$. This is a very common scenario on the exam!

Key Takeaway: Delta tells you how many shares of stock behave like one option. If Delta is 0.6, the option value moves like 0.6 shares of stock.

4. Delta Hedging: Staying Neutral

Market makers (banks and trading firms) don't want to gamble on whether a stock goes up or down. They want to earn fees while staying "neutral." They do this through Delta Hedging.

A Delta-Neutral Portfolio is one where the total Delta equals zero. This means if the stock price moves a tiny bit, the value of the portfolio doesn't change at all.

How to Create a Delta-Neutral Position

If you Sell (Write) options, you have "negative" exposure if it's a call, or "positive" exposure if it's a put. You must buy or sell shares of the underlying stock to cancel this out.

Step-by-Step Process:
1. Calculate the Delta of a single option ($\Delta$).
2. Multiply by the number of options you have.
3. Buy or sell shares to make the sum zero.
Shares needed = -(Number of Options $\times$ $\Delta$)

Real-World Example:
Suppose you sell 100 Call options, and each has a $\Delta$ of 0.60.
Your total Delta from the options is $100 \times (-0.60) = -60$.
To be "Delta Neutral," you must buy 60 shares of the stock. Now, if the stock goes up \$1, you lose \$60 on the options but gain \$60 on the shares. You are perfectly balanced!

Common Mistake to Avoid:
Don't forget the sign! If you sell a Call, you are "short" delta, so you must buy stock. If you sell a Put (which has a negative delta), you are "short" a negative, which is positive—meaning you have to sell stock to hedge.

Key Takeaway: Delta hedging is the art of balancing your "Option Delta" with "Stock Delta" so the total is zero.

5. The Costs of Hedging

Delta hedging isn't free! In the Black-Scholes world, we assume you can re-hedge constantly. In reality, every time the stock price moves, the Delta ($N(d_1)$) changes. This means you have to buy or sell more stock to stay neutral. This process is called dynamic hedging.

Quick Review: The Inputs and Delta
What happens to Call Delta ($\Delta_{Call}$) when these go up?
- Stock Price ($S$) $\uparrow$: Delta $\uparrow$
- Time to Expiration ($T$) $\uparrow$: Delta moves toward 0.5 (for at-the-money options)
- Volatility ($\sigma$) $\uparrow$: Delta moves toward 0.5

Summary Checklist for the Exam

Before you move on, make sure you can:

Identify all 6 inputs ($S, K, \sigma, r, T, \delta$).
Calculate $d_1$ and $d_2$ accurately (watch your parentheses in the calculator!).
Compute the Call and Put prices using the Black-Scholes formula.
Find the Delta for both Calls and Puts.
Determine how many shares are needed to make a portfolio Delta-neutral.
Explain why a Delta-neutral portfolio needs to be "rebalanced" over time.

Don't worry if this seems tricky at first! The math is the hardest part. Once you practice the $d_1$ calculation five or ten times, it will become muscle memory. You've got this!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What are the six essential inputs required to calculate an option price using the Black-Scholes formula?

The six inputs are the current stock price ($S_0$), strike price ($K$), volatility ($\sigma$), risk-free interest rate ($r$), time to expiration in years ($T$), and dividend yield ($\delta$).

What is the Black-Scholes formula for a European call option?

$C = S_0 e^{-\delta T} N(d_1) - K e^{-rT} N(d_2)$

What is the Black-Scholes formula for a European put option?

$P = K e^{-rT} N(-d_2) - S_0 e^{-\delta T} N(-d_1)$

How are the variables $d_1$ and $d_2$ calculated in the Black-Scholes model?

$d_1 = \frac{\ln(S_0/K) + (r - \delta + 0.5\sigma^2)T}{\sigma\sqrt{T}}$ and $d_2 = d_1 - \sigma\sqrt{T}$.

In the Black-Scholes model, what is the risk-neutral interpretation of $N(d_2)$?

$N(d_2)$ is the probability that the option will be in the money (exercised) at expiration.

What does the Delta ($\Delta$) of an option represent?

Delta measures the sensitivity of the option's price to a change in the underlying stock price.

What are the formulas for the Delta of European call and put options?

$\Delta_{Call} = e^{-\delta T} N(d_1)$ and $\Delta_{Put} = -e^{-\delta T} N(-d_1)$.

What is a delta-neutral portfolio?

A portfolio with a total Delta of zero, making its value insensitive to small changes in the stock price.

How do you calculate the number of shares needed to delta-hedge a position of options?

$\text{Shares needed} = -(\text{Number of Options} \times \Delta)$

What is dynamic hedging and why is it necessary?

Dynamic hedging is the process of periodically rebalancing a delta-neutral portfolio as the option's Delta changes over time and with stock price movements.

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