TQT Quant Talk 1

Source: https://sidvishwanath.com/tqt-2024/#/title-slide

Probability & Statistics in Quant Finance - Siddharth Vishwanath

Topological Data Analysis

Persistence diagrams to compare objects topologically
Comparing the distribution of points

Statistics & finance

Martin Baxter

A Tale of Two Probabilities

The forward probability

• Goal: Model the future
• Uses: Risk management, investing
• Reference: the “real world” probability
• Machinery: High-dimensional statistics, machine learning, etc.

The risk neutral probability

• Goal: Extrapolate the present
• Uses: Pricing, hedging
• Reference: the “risk neutral” probability
• Machinery: Ito calculus, Partial Differential Equations

What is the risk-neutral probability?

Consider the following setup:

• For time intervals
• At each time you have access to:
  • A stock with price
  • A bond (risk-free asset) with price
• From time to the obnd always gives you risk-free return $$B_{n+1}$

Portfolio

A portfolio is a collection of assets you own at any given time.

Note

• At each time , choose a value such that
• Your net value is maximized

European Call Option

A European Call option is a derivative where the payoff at time is
where is called the strike price
Can only be exercised at time of expiration, unlike American Call Options
At time suppose one of two thing can happen:

here, is the “real-world” probability


Solving for and





In other words… If you took the money and invested it all in bonds at time

Expected returns from the call option at time
Black Scholes
Here is the risk-neutral probability
When the stock price doesn’t just go up/down but can take a range of values

Then

• Stochastic differential equations
• Brownian Motion
• Girsanov’s theorem

Examples of cutting-edge DL models

• Transformers

Derivative Pricing

Use real world probability to model the stock price
Use the risk neutral probability to price the derivative
The price of the derivative is the expected value of the derivative at time T under the risk-neutral probability

Estimating $\mathbb{P}$

is a stochastic interest rate
Used to evaluate bond prices, create interest rate swaps, and underlies almost every other financial derivative
A common model for is to assume it follows a Vasicek model, i.e.

where is a Brownian motion

Let be the probability density function of at time . Then the likelihood of the data is

The statistical advantage

If you can do the math, you can

• Estimate using a dinosaur computer
• Use to make predictions about the future
• Quantify how much uncertainty you have in your predictions
• Quantify the effect that changing to + has on your predictions

21st Century Forecasting

If you have enough compute you can

• estimate theta hat using state of the art GPUs
• Use to make predictions about the future
• But it comes at the price of uncertainty quantification
• But you don’t have to worry about the math

Physics informed deep learning

Which is better?

It depends

Philosophically

• The first method is based on assumptions
  • Assumptions have consequences
• The second method is based on data
  • Garbage in, garbage out!
  • Signal to noise ratio - more noise than signal

Realistically

• First approach for mathematicians, second approach for computer scientists