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Quantitative Finance Blog

Updated: Oct 1, 2023

Just like my machine learning blog this will document any projects, interesting topics I learn about, and just general path towards me (hopefully!) becoming an expert in quantitative finance. This first blog post will mostly be maths free, but later ones will go deep into topics like Black-Scholes, binomial pricing, Monte Carlo methods, and even machine learning/AI. Most of this blog is taken from Sheldon Natenberg's book "Option Volatility and Pricing"

 

Part 1: Types of Financial Instruments and their uses


When you start learning about finance, it can be difficult to keep up with all the securities and derivatives: options, futures, options on futures, stocks, indexes, swaps, the list goes on. I thought it would be helpful to put all of them into a big list, with intuitive explanations on what they are, how they work, and what function they serve in trading/investing. I've collected some of the basic instruments and their possible derivatives below:


Image 1: Basic Financial Instruments


Futures


It might seem counter-intuitive to start by looking at futures, but this is how I learned it. The structure of this section, and all sections in this blog will be as follows: we first explain the concept, go through some basic examples and different types of the instrument. Then we'll explain the maths behind the pricing of the instrument, and then we will go over the financial strategies that involve the instrument.


what is a future?


A future(technically a forward if it isn't on an exchange) is an agreement to buy a product in the future for a price agreed on by both parties now. Futures are useful for when the buyer knows he/she/they will want the product at some point in the future but don't want it now.


A situation where this could be useful is energy trading: companies know they will need energy in the future to be able to operate, but they don't want to have to store all the energy(Gas, Electricity etc) until that point, so they just buy energy futures instead, because it streamlines the whole process(Another situation where futures are useful is trading and hedging strategies, but we will get to those later).


Pricing Futures


note that in this post we don't go into detail on the maths of pricing futures, which can get extremely complex, but we just go over the logic and give a simple example.


You might be thinking "how can we agree on a price now if I'm buying it in the future", or "why don't we just agree on the price when I pay". And my response to that would be twofold:


1) The market for futures has existed for many centuries, and (is thought to have) developed because of the convenience and security that it gives. Admittedly I'm not an expert on this topic but essentially, for several reasons to do with operations, transport etc, its easier and safer for business to deal certain things where you know you're going to buy, or you have someone who you know will want to buy the thing you're selling. I'm sure you could write a very long book about the history and details of this but to understand how to make money off this stuff you don't need to know all that.


&, more importantly:


2) Futures can play an important role in a trading/hedging strategy, meaning that if we buy/sell the right amount, use the money in the right ways and properly know our way round all the figures and equations, we can take advantage of opportunities in the market! The specifics of this will be explained later but for now just know that there are many financial instruments that exist purely so that people like you and me can trade them and make money off it. This isn't to say they are "useless" in the real world, but it is to say that no business is buying a swaption on a future on an option for an index rate exchange that expires in 5 years time. I have no idea if this example is actually real but my point stands; the market has evolved past its initial roots of 17th century mercantile proto-capitalism.


Now that we understand that, we need to introduce a foundational equation when it comes to pricing futures. Before we do this we need to run through some basic definitions:


- The "Underlying" is the asset that we agree to buy/sell at some point in the future.


- The "exercise/expiration date" is the date at which we agree to either buy or sell the underlying.


- The "Spot price" is the price you would have to pay to buy the underlying right now.


- The "Future price" is the price we agree to pay for the underlying on the exercise date.


- The "Benefits/Costs" of buying now refer to the positive/negative cash flows that the buyer of the underlying will incur across the length of the future if they buy right now. Importantly this does not include the spot price of the underlying. It only includes the money made/lost as a result of owning the underlying until the exercise date.


Now that we know those;



Pretty much all the complicated maths that we will get into in a bit comes down to this equation, and in particular calculating the benefits/costs of buying now.


The logic of this equation is that if I'm going to buy the underlying on the exercise date, say in a weeks time, I am going to miss out on the benefits of buying it now, so we reduce the future price because of that, but I also dodge the costs of buying now, so I get that added on to the price. You might think, "well how do we know what the benefits and costs will be?", and therein lies the game; trying to predict what these will be in the future. This is where the maths comes in; we try to build models and analyse data to make a guess on what these will be, and then price futures(and other instruments) accordingly.


example 1: A simple future price calculation


consider the situation where we want to buy a plot of land a year into the future, and the price of the land right now (spot price) is £100,000. Say



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