Understand the principles of the Black-Scholes derivative-pricing model and its application

Q: What does the Black-Scholes PDE imply about the pricing of derivatives?

It implies risk-neutral pricing , meaning the price of the derivative does not depend on the expected growth rate of the stock, only on its volatility and the risk-free interest rate.

Q: What do the Greeks Gamma, Theta, Vega, and Rho represent?

Gamma: Sensitivity of Delta to stock price; Theta: Sensitivity to time (time decay); Vega: Sensitivity to volatility; Rho: Sensitivity to interest rates.

Q: What is the difference between Historical Volatility and Implied Volatility?

Historical volatility is calculated from past stock price data, while implied volatility is the volatility value that makes the Black-Scholes price equal the current market price.

CM2 – Economic Modelling · IFoA (Institute and Faculty of Actuaries) · 6 min read

Welcome to the World of Black-Scholes!

Hello! Today we are diving into one of the most famous topics in financial mathematics: the Black-Scholes Model. If you’ve ever wondered how traders decide the "fair price" for an option, this is the answer. It might look intimidating with all its Greek letters and long formulas, but don't worry! We are going to break it down step-by-step. By the end of these notes, you'll see it's just a recipe for finding a price based on a few key ingredients.

1. The Rules of the Game: Black-Scholes Assumptions

Before we use the model, we have to understand the "perfect world" it lives in. In the IFoA exams, you are often asked to list or explain these assumptions. Think of these as the ground rules for the model to work correctly.

The Main Assumptions:

1. The Stock Price Follows Geometric Brownian Motion: This is a fancy way of saying that the returns on the stock follow a Normal Distribution, and the stock price itself follows a Log-normal distribution. It also assumes volatility ($ \sigma $) is constant.
2. No Dividends: The basic model assumes the stock doesn't pay out any cash during the life of the option.
3. Constant Risk-Free Rate: The interest rate ($ r $) is known and stays the same.
4. No "Friction": There are no transaction costs, no taxes, and you can buy or sell any amount of stock (even tiny fractions).
5. No Arbitrage: There are no "free lunches." You cannot make a certain profit without taking any risk.
6. Continuous Trading: You can trade the stock at any micro-second you want.

Quick Review: Why do we care? Because if these assumptions aren't met in the real world (which they often aren't!), the model might give a price that is slightly "off."

2. The Black-Scholes Partial Differential Equation (PDE)

The Black-Scholes PDE is the "engine" under the hood. It describes how the value of an option ($ f $) changes over time ($ t $) in relation to the stock price ($ S $).

The equation looks like this:
$ \frac{\partial f}{\partial t} + rS \frac{\partial f}{\partial S} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 f}{\partial S^2} = rf $

Don't panic! You rarely have to solve this from scratch in the exam. What you need to know is what it means: It tells us that if we set up a "delta-hedged" portfolio (a mix of the option and the stock), the portfolio should earn the risk-free rate of return.

Key Takeaway: The PDE is a statement about risk-neutral pricing. It says the price of the derivative doesn't depend on how much the stock is expected to grow, but rather on its volatility and the risk-free rate.

3. The Black-Scholes Pricing Formulas

When we solve that scary PDE for a European call and put option, we get these famous formulas. These are your best friends for the CM2 exam!

The European Call Option Price ($ C_t $):

$ C_t = S_t \Phi(d_1) - K e^{-r(T-t)} \Phi(d_2) $

The European Put Option Price ($ P_t $):

$ P_t = K e^{-r(T-t)} \Phi(-d_2) - S_t \Phi(-d_1) $

What do these letters mean?
- $ S_t $: Current stock price
- $ K $: Exercise (Strike) price
- $ r $: Risk-free interest rate
- $ T-t $: Time remaining until expiry
- $ \Phi(d) $: The cumulative standard normal distribution (the "tables" value)
- $ \sigma $: Volatility of the stock price

Memory Aid: Think of the Call formula as: (What I expect to get) minus (What I expect to pay).
$ S_t \Phi(d_1) $ is roughly the expected value of the stock you receive, and $ K e^{-r(T-t)} \Phi(d_2) $ is the present value of the strike price you have to pay.

4. Calculating $ d_1 $ and $ d_2 $

To use the formulas above, you first need to calculate $ d_1 $ and $ d_2 $. This is where most students make calculator errors, so take it slow!

$ d_1 = \frac{\ln(S_t / K) + (r + \frac{1}{2} \sigma^2)(T-t)}{\sigma \sqrt{T-t}} $

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

Step-by-Step Process:
1. Calculate the log part: $ \ln(S/K) $.
2. Calculate the "drift" part: $ (r + 0.5\sigma^2) \times \text{time} $.
3. Add them and divide by the "volatility over time": $ \sigma \sqrt{\text{time}} $.
4. Once you have $ d_1 $, just subtract $ \sigma \sqrt{\text{time}} $ to get $ d_2 $.
5. Look up $ \Phi(d_1) $ and $ \Phi(d_2) $ in your Formula Tables.

Common Mistake: Make sure $ T-t $ is always in years. If the option expires in 6 months, use 0.5!

5. The "Greeks": Measuring Risk

The "Greeks" tell us how the option price changes when one of the "ingredients" changes. They are essential for risk management.

- Delta ($ \Delta $): How much the option price changes if the stock price moves by \$1. For a call, $ \Delta = \Phi(d_1) $.
- Gamma ($ \Gamma $): How much the Delta changes if the stock price moves. It measures the "stability" of your hedge.
- Theta ($ \theta $): The "time decay." Options lose value as they get closer to expiry.
- Vega ($ \nu $): How much the option price changes if volatility ($ \sigma $) changes.
- Rho ($ \rho $): How much the option price changes if interest rates change.

Did you know? Delta is also the number of shares you should hold to "hedge" (cancel out the risk of) the option you sold. This is called Delta Hedging.

6. Historical vs. Implied Volatility

Volatility is the only "secret ingredient" in the Black-Scholes formula because we can't see it directly in the future.

- Historical Volatility: Looking at the past. We calculate the standard deviation of past stock returns.
- Implied Volatility (IV): Looking at the market. We take the current market price of the option and "work backward" through the Black-Scholes formula to see what $ \sigma $ makes the formula equal the market price.

Key Takeaway: Implied volatility represents the market's expectation of future risk. If IV is high, options are expensive!

7. Summary and Quick Tips

Quick Review Box:
- Call Price: $ S\Phi(d_1) - Ke^{-rt}\Phi(d_2) $
- Put Price: Use Put-Call Parity or the Put formula.
- Delta: $ \Phi(d_1) $ for calls; $ \Phi(d_1) - 1 $ for puts.
- Lognormal: Prices are lognormal; returns are normal.

Encouragement: If the formulas look like a bowl of alphabet soup right now, don't worry! Practice calculating $ d_1 $ and $ d_2 $ five times with different numbers. Once you master the calculation of those two values, the rest of the chapter falls into place perfectly. 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 key assumptions of the Black-Scholes model regarding the underlying asset and market environment?

Key assumptions include: stock prices follow Geometric Brownian Motion (log-normal distribution), constant volatility (̑), no dividends, a constant risk-free rate (̑), no transaction costs/taxes, no arbitrage opportunities, and continuous trading.

What is the Black-Scholes Partial Differential Equation (PDE)?

The PDE is: $ \frac{\partial f}{\partial t} + rS \frac{\partial f}{\partial S} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 f}{\partial S^2} = rf $.

What does the Black-Scholes PDE imply about the pricing of derivatives?

It implies risk-neutral pricing, meaning the price of the derivative does not depend on the expected growth rate of the stock, only on its volatility and the risk-free interest rate.

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

$ C_t = S_t \Phi(d_1) - K e^{-r(T-t)} \Phi(d_2) $

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

$ P_t = K e^{-r(T-t)} \Phi(-d_2) - S_t \Phi(-d_1) $

How is the term $ d_1 $ calculated in the Black-Scholes pricing formula?

$ d_1 = \frac{\ln(S_t / K) + (r + \frac{1}{2} \sigma^2)(T-t)}{\sigma \sqrt{T-t}} $

How is $ d_2 $ related to $ d_1 $?

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

What is Delta ($ \Delta $) and how is it calculated for a call option?

Delta measures the sensitivity of the option price to changes in the underlying stock price; for a European call, $ \Delta = \Phi(d_1) $.

What do the Greeks Gamma, Theta, Vega, and Rho represent?

Gamma: Sensitivity of Delta to stock price; Theta: Sensitivity to time (time decay); Vega: Sensitivity to volatility; Rho: Sensitivity to interest rates.

What is the difference between Historical Volatility and Implied Volatility?

Historical volatility is calculated from past stock price data, while implied volatility is the volatility value that makes the Black-Scholes price equal the current market price.

Ready to test yourself?

Turn these notes into exam-style practice. Get unlimited AI questions on this topic with instant marking and explanations.

Practice This Topic

More CM2 – Economic Modelling chapters

* The content provided by thinka is generated by AI and may not always be accurate or up-to-date. Please use it as a supplementary resource and verify with official materials.

Put These Notes into Practice

Reading the notes is a great start. Now practise with unlimited AI-generated questions and get instant feedback. 100,000+ students are already improving their grades.

Start Practising Now View Pricing

Done reading? Test yourself with AI practice questions

Practice This Topic Now