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Have you ever noticed how life, business, and markets share one thing in common — uncertainty?
Finance doesn’t ignore that; it embraces it.
John Hull’s classic work on derivatives explains how we can actually model uncertainty — and that’s what this post is about. Let’s walk through it in plain words 👇
🌧️ 1. Stochastic Process – Life as a Sequence of Uncertainties
Imagine you’re walking home, and at every intersection, you toss a coin to decide whether to go left or right.
You don’t know exactly where you’ll end up, but you can describe the probability of reaching different places.
That’s what a stochastic process is — it describes how something (like a stock price, temperature, or your location) evolves through time with uncertainty.
🔁 2. Markov Process – “Memoryless” Future
A Markov process says: the future only cares about the present, not the past.
Think of Google Maps’ live traffic — when predicting how long your trip will take, it uses current traffic, not the route you took to get there.
The road behind you doesn’t matter — only the situation right now.
Stock prices behave a bit like that: today’s price already reflects everything known from the past. What happens next depends only on the present state.
🍃 3. Wiener Process – Pure Random Drift
A Wiener process is like a drunk person walking in a park: each step is random, and there’s no direction — sometimes left, sometimes right, with small unpredictable changes.
On average, they stay where they started (no drift), but over time, their position becomes more uncertain.
So if you track them for longer, the range of where they might be spreads out — that’s the growing variance.
🌊 4. Generalized Wiener Process – Random Drift with a Trend
Now imagine our drunk walker is on a gentle slope.
They still stumble randomly, but gravity pulls them slightly downhill — that’s the drift.
And maybe some days they’re more unstable than others — that’s the variance rate.
This is closer to how stock prices move: they wander unpredictably, but with an overall upward drift (the average return).
🔄 5. Ito Process – When Conditions Change Over Time
A Wiener process assumes drift and volatility are constant.
An Ito process says they can change — depending on time or on where you are.
Imagine a boat drifting on the sea:
- When near the shore, the waves are calmer (less volatility).
- In the open ocean, the waves get rougher.
- The current may change direction with the wind (time-dependent drift).
So the randomness is still there, but its intensity and direction vary with circumstances.
🎲 6. Monte Carlo Simulation – Playing Out Many Possible Futures
You can imagine the future price of a stock as millions of possible paths, each depending on random daily movements.
To understand what’s likely, we simulate these paths many times — like playing out a thousand versions of tomorrow to see the range of outcomes.
It’s like testing how many ways your weekend trip could go wrong or right depending on the weather.
🧮 7. Ito’s Lemma – How Uncertainty Transfers Between Variables
Ito’s Lemma helps us understand how randomness in one variable (like a stock price) affects another variable that depends on it (like the value of a stock option).
Think of it like wind and a kite:
- The wind (stock price movement) is random.
- The kite (option value) reacts to it in a more complex way — rising, falling, curving.
Ito’s Lemma tells us exactly how that randomness transfers from wind to kite.
📈 8. Geometric Brownian Motion – How Stock Prices “Grow Randomly”
Stock prices follow what’s called geometric Brownian motion — randomness combined with a tendency to grow.
Think of a tree in the wind:
- It grows taller over time (the drift — average return).
- But it also sways unpredictably (the random shocks).
After a while, no two trees (or stock price paths) look the same, but all have grown on average.
This idea is the foundation of the Black-Scholes model, which prices options by assuming stock prices evolve with this kind of “random growth.”
Adapted from: Options, Futures & Other Derivatives, John C. Hull