Appreciate different stochastic models for security prices and how and when they can be applied

Q: Why do actuaries use stochastic models for security prices?

Stochastic models are used to quantify risk, price complex financial derivatives (like options), and provide a framework for simulating various future financial scenarios.

Q: How does the Binomial Model represent asset price changes over a single time step?

The Binomial Model assumes that in one time step, a stock price $ S $ can only move to one of two values: an "up" state ($ uS $) or a "down" state ($ dS $).

Q: What is the no-arbitrage condition for the parameters in a Binomial model?

To avoid risk-free profit opportunities (arbitrage), the risk-free growth factor must sit between the down and up factors: \( d

Q: What is the Stochastic Differential Equation (SDE) for Geometric Brownian Motion (GBM)?

The SDE for GBM is given by: $ dS_t = \mu S_t dt + \sigma S_t dW_t $.

Q: In the GBM formula, what do the terms "Drift" and "Diffusion" represent?

Drift ($ \mu S_t dt $) represents the predictable average trend of the return, while Diffusion ($ \sigma S_t dW_t $) represents the random volatility or "noise."

Q: Why is it beneficial that GBM assumes prices follow a lognormal distribution?

A lognormal distribution ensures that security prices can never be negative, which matches the real-world characteristic of limited liability assets.

Q: How do Binomial and GBM models differ regarding time and complexity?

Binomial models are discrete (move in steps) and arithmetic-based, while GBM is continuous (moves smoothly) and requires stochastic calculus.

Q: Which model is preferred for valuing American Options and why?

The Binomial Model is preferred because it can easily account for the possibility of exercising the option at any point (node) before the expiry date.

Q: What are the two major real-world limitations of standard Binomial and GBM models?

The two major limitations are that volatility is not constant (it tends to cluster) and they do not account for jump risk (sudden, large price gaps).

CM2 – Economic Modelling · IFoA (Institute and Faculty of Actuaries) · 5 min read

Welcome to Stochastic Models for Security Prices!

Hi there! Welcome to one of the most exciting parts of the CM2 curriculum. If you’ve ever looked at a stock market chart and thought, "That looks completely random," you’re already thinking like an actuary! In this chapter, we’ll learn how to take that "randomness" and turn it into mathematical models. These models help us value assets and price complex financial products.

Don't worry if the term "stochastic" sounds intimidating—it’s just a fancy math word for "random." We are simply looking at different ways to describe how security prices move over time. Let's dive in!

1. Why Do We Need Stochastic Models?

In a perfect world, we might know exactly how much a stock will be worth tomorrow. But in the real world, prices are influenced by millions of unpredictable events. We use stochastic models because:

1. They help us quantify risk (how much could we lose?).
2. They allow us to price derivatives (like options).
3. They provide a framework for simulating future scenarios.

2. The Binomial Model (The Discrete-Time Approach)

Think of the Binomial Model as a game of "choose your own adventure." At every step in time, the price can only do two things: go Up or go Down.

How it Works

Suppose the current price of a stock is $ S $. In the next time step:

1. The price goes up to $ uS $ (where $ u > 1 $).
2. The price goes down to $ dS $ (where $ d < 1 $).

Key Assumptions of the Binomial Model:

1. No Arbitrage: You can't make a risk-free profit higher than the risk-free rate.
2. Constant Returns: The factors $ u $ and $ d $ stay the same at every step.
3. Discrete Time: Changes only happen at specific intervals (e.g., every day or every month).

Why Use It?

It is incredibly simple to visualize using a "tree" diagram. It’s also very flexible—you can use it to price "American Options" (options that can be exercised at any time), which is much harder to do with continuous models.

Quick Tip: Remember that for the model to be realistic and avoid arbitrage, the risk-free growth factor $ e^{rt} $ must sit between the down factor and the up factor: $ d < e^{rt} < u $.

Summary: The Binomial model is like a digital clock—it moves in "ticks." It’s simple, visual, and great for American-style options.

3. Geometric Brownian Motion (The Continuous-Time Approach)

If the Binomial model is a digital clock, Geometric Brownian Motion (GBM) is like an analog clock—the hands move smoothly and constantly. This is the model used in the famous Black-Scholes-Merton framework.

The Formula (Don't Panic!)

The price process $ S_t $ is often written as a Stochastic Differential Equation (SDE):
$ dS_t = \mu S_t dt + \sigma S_t dW_t $

Let’s break this down into "human English":
- $ \mu S_t dt $: This is the Drift. It represents the "average" trend or the expected return. It's the predictable part of the movement.
- $ \sigma S_t dW_t $: This is the Diffusion. This is the "noise" or volatility. $ W_t $ is a Wiener Process (or Brownian Motion), which provides the randomness.

Key Properties of GBM:

1. Lognormal Distribution: While the returns are normally distributed, the prices themselves follow a lognormal distribution. This is great because it means prices can never be negative (a stock can't be worth -\$10!).
\n2. Constant Volatility: The model assumes $ \sigma $ (volatility) never changes.
\n3. Independence: What happens today doesn't tell us anything about what will happen tomorrow (the "Efficient Market Hypothesis").

\n\n

Did you know? We call it "Geometric" because the percentage change is modeled, not just the absolute dollar change. This ensures that a 10% move on a \$100 stock feels the same as a 10% move on a \$10 stock.

Key Takeaway: GBM is the "gold standard" for theoretical finance. It assumes prices move smoothly and stay positive, but it assumes volatility is a boring, constant number.

4. Comparing the Models

Students often ask: "Which model is better?" The answer depends on what you are trying to do!

1. Time: Binomial is discrete (steps); GBM is continuous (smooth).
2. Complexity: Binomial is arithmetic-based; GBM requires calculus (Ito’s Lemma).
3. Application: Use Binomial for American options; use GBM/Black-Scholes for European options.

Common Mistake: Thinking these models are perfectly accurate. In real life, market prices often have "fat tails" (extreme events happen more often than the models predict) and "volatility smiles" (volatility isn't constant).

5. When and How to Apply These Models

As an actuary, you need to choose the right tool for the job.

Use the Binomial Model when:

- You need to value an option that can be exercised before the expiry date (American Options).
- You want to explain the valuation process to a non-technical client (the tree diagram is very intuitive).
- The underlying asset has complex features that change at specific dates (like dividends).

Use Geometric Brownian Motion (GBM) when:

- You are dealing with European Options (exercise only at the end).
- You need a fast formula (Black-Scholes) rather than building a massive tree.
- You are performing high-level theoretical modeling or risk calculations (like Value at Risk).

Analogy Time: Choosing a model is like choosing a map. A Binomial Model is like a "Turn-by-Turn" list of directions—it tells you exactly what happens at every intersection. GBM is like a satellite image—it gives you the big, smooth picture of the whole landscape.

6. Limitations to Remember

Even though these models are brilliant, they aren't perfect. For your exam, keep these shortcomings in mind:

1. Volatility is NOT constant: In the real world, volatility "clusters" (if it's crazy today, it'll likely be crazy tomorrow).
2. Jump Risk: Real prices can "jump" (e.g., after a CEO resigns). Neither model handles sudden gaps in price very well.
3. Transaction Costs: These models usually assume trading is free. We know that's not true!

Quick Review Box

- Binomial Model: Discrete steps, Up/Down movements, good for American options.
- GBM: Continuous movement, Lognormal prices, constant volatility, core of Black-Scholes.
- Drift ($ \mu $): The average "trend" of the price.
- Volatility ($ \sigma $): The "wiggliness" or risk of the price.
- Key Constraint: Prices cannot be negative in these models!

Don't worry if the math feels heavy at first. Focus on the behavior of the models. Once you understand how the "tree" grows or how the "smooth line" drifts, the formulas will start to make much more sense!

Quick check

Can you answer these now?

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

Why do actuaries use stochastic models for security prices?

Stochastic models are used to quantify risk, price complex financial derivatives (like options), and provide a framework for simulating various future financial scenarios.

How does the Binomial Model represent asset price changes over a single time step?

The Binomial Model assumes that in one time step, a stock price $ S $ can only move to one of two values: an "up" state ($ uS $) or a "down" state ($ dS $).

What is the no-arbitrage condition for the parameters in a Binomial model?

To avoid risk-free profit opportunities (arbitrage), the risk-free growth factor must sit between the down and up factors: $ d < e^{rt} < u $.

What is the Stochastic Differential Equation (SDE) for Geometric Brownian Motion (GBM)?

The SDE for GBM is given by: $ dS_t = \mu S_t dt + \sigma S_t dW_t $.

In the GBM formula, what do the terms "Drift" and "Diffusion" represent?

Drift ($ \mu S_t dt $) represents the predictable average trend of the return, while Diffusion ($ \sigma S_t dW_t $) represents the random volatility or "noise."

Why is it beneficial that GBM assumes prices follow a lognormal distribution?

A lognormal distribution ensures that security prices can never be negative, which matches the real-world characteristic of limited liability assets.

How do Binomial and GBM models differ regarding time and complexity?

Binomial models are discrete (move in steps) and arithmetic-based, while GBM is continuous (moves smoothly) and requires stochastic calculus.

Which model is preferred for valuing American Options and why?

The Binomial Model is preferred because it can easily account for the possibility of exercising the option at any point (node) before the expiry date.

What are the two major real-world limitations of standard Binomial and GBM models?

The two major limitations are that volatility is not constant (it tends to cluster) and they do not account for jump risk (sudden, large price gaps).

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