# Introduction
## What is Stochastic Calculus?
Finance people hire math people to model price for them. Why? Because finance people make money in the markets, by buying low and selling high. The better they understand how price behaves, the more money they make.
But prices look random, and they often are. They move based on fear, greed, sentiment, the economy, underlying fundamentals - all sorts of things. Stochastic calculus is the branch of mathematics built to make sense of random-looking things. I'll teach you all you need to know about it to be interview-ready in 7 days.
## Why Should You Care ?
If you're interviewing for a quant / trading position on the trading floor of an investment bank / hedge fund / trading firm, you're likely to be asked about stochastic calculus. Why?
Because people on the trading floor actually use it. Hedge funds use it to make better bets. Investment banks use it to figure out a fair price of a product. Then they buy for slighly less and sell for slightly more to make a profit.
They also ask it becuase it's hard. They suffered through it in school. And they want to figure out if you're smart.
## What Will They Ask?
Since the topic is known to be hard, the questions tend not to be. Often you get asked for definitions of conceps, e.g.:
What is brownian motion? What are its properties?
What is a martingale? How do you check if a process is a martingale?
What is a filtration?
Then, you can be asked to apply the definitions to simple cases. There's a few tricks to solve most such problems:
Show that if B is a brownian motion and $s\leq t$ then $cor(B_s, B_t)=\sqrt{\frac{s}{t}}$
If B is a brownian motion, show that for any $c>0$ $X_t=c^{-1}B_{c^2t}$ is also a brownian motion.
If $B_t$ is a brownian motion, then show $B_t^2-t$ is a martingale.
Then, they might check your intuition around (seemingly) complicated concepts / theorems:
Explain the intuition behind Itô's lemma.
Explain Girsanov's theorem and why we need it.
Explain why martingales are important to model price?
Finally, you can expect questions on real world applications:
A drunk man walks 1m left or right with equal chance every second. He's at the 23rd meter of a 100m bridge. What's the chance he makes it to the end of the bridge? On average, in how many seconds does he reach either side of the bridge?
How would you price an option using Black Scholes?
## What's the Plan?
To make this less dry, let me quickly paint a real-life application I'll teach you about. Picture a top dog equity options trader at an investment bank. His desk buys and sells stock options to hedge funds hoping to make a bet. He takes home 1.5 million a year, because he tends to 'get it right'. He buys up options just before their price shoots up and sells them just before they plummit. Most mornings he comes into his office and scans the *implied volatility surface* to find mispricings. He looks for signs the market is scared or wrong - and bets against it. This likely means gibberish to you now, but it won't in 7 days. To understand what he's talking about, we will have covered all the topics you need to answer the above interview questions.
So how do we get from knowing nothing / very little about stochastic calculus to trading options and answering interview questions about it in 7 days?
Here's how:
- Day 1: Brownian motion - first attempt at modelling price
- Day 2: SDEs - a better attempt at modelling price and new toolkit
- Day 3: Stochastic Integrals - the new integral
- Day 4: Itô's lemma - the new derivative
- Day 5: Pricing financial products and Girsanov theorem
- Day 6: Martingales
- Day 7: Solving Black Scholes \& what Xavier was talking about