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Equities that have a very straightforward exposure to idiosyncratic and systematic risk. Options, on the other hand, have exposure to not only the underlying asset, but also interest rates, time, and volatility. These exposures are inputs to the Black-Scholes option pricing model(see Deriving the Black-Scholes Model). Since these inputs affect the value of the…

Efficiency and reusability is critical, especially when building systems that trade multiple securities. I have discussed development using Interactive Broker’s Java and Python API in the past:

I would suggest reading those articles first if you are unfamiliar with Interactive Broker’s API. In this article, I want to further the notion of algorithmic trading by exhibiting a system design pattern that allows for ease of extension to multiple securities.

The EWrapper class is responsible for receiving data from the server that is requested by the EClient class. More intuitively, we can think…

The majority of the time in undergraduate quantitative finance coursework is spent on pricing different securities. Initially, there is a heavy emphasis on the time value of money, and analyzing annuities and perpetuities. Afterward, the focus shifts to learning, analyzing, and pricing equity derivatives including forwards, futures, options, and swaps. Once the groundwork is laid for pricing derivatives, the remainder of coursework is spent researching and becoming a specialist in a market sector or security type. In this article, I aim to breakdown the underlying theme that determines the fair price for all of these securities in the market. …

The notion of a replicating portfolio first appears in the argument for the Black-Scholes model (1973). In order to bind an option’s price to their model under a no-arbitrage assumption, they develop an offsetting equity position replicating the opposite value of the option at expiration. Holding such a portfolio is therefore risk-neutral, consequently, the portfolio must earn an appropriate risk-free rate. However, Black-Scholes use geometric Brownian motion to model the underlying asset price. Infinite variation is implied by this stochastic process meaning overtime the initial equity value will stray linearly where the option value strays non-linearly. This creates a hedging…

There are a variety of metrics and tools that a portfolio manager can use to understand their exposure to risk, from economic indicators to Monte Carlo simulations. In this article, we will take a look at one of the most important metrics when analyzing an investment portfolio: *Value at Risk (VaR)*. There are several assumptions that are made when computing a portfolio’s VaR. However, in practice, it is possible to relax these assumptions — consequently, the computations will be much more complex. It is very helpful to have software that can automatically compute VaR so as a portfolio manager you…

Option contracts give the buyer the right but not the obligation to purchase (call option) or sell (put option) shares in an underlying asset at a predetermined price.

How and what changes an option price in the market? Economics tells us that the market will find an equilibrium price given supply and demand, and thanks to the Black-Scholes model (see Deriving the Black-Scholes Model) we can explain the market’s pricing by five key inputs.

**S —**The price of the underlying asset at time t**X —**The strike price for the option contract**r —**The rate of interest…

Remarkably, options trading can be traced back to 332 B.C. where there is an account of Thales of Miltetus, an astronomer, philosopher and mathematician purchased the rights to an oil harvest — making a fortune. The next most notable account of options trading was a period in the Dutch Golden Age known as Tulip Mania. In 1636 these contracts were used to speculate the rising prices of tulips until prices collapsed in 1637 which is often recognized as the first speculative bubble. This continued in London during the early 18th century where options trading was given its own organized market…

AI has turned into a buzzword everyone in the tech industry throws around, leaving the uninitiated mystified and in the dark. This article is meant to be an introduction to artificial intelligence and machine learning for those who may be unfamiliar with what it actually is, how it works, and what it can do.

Let's get started…

The realm of artificial intelligence is vast, however, there are certain subsections that narrow down its applications. Let’s start by introducing what is meant when we hear words like AI, ML, and Data Science. Whenever words like this are thrown around, generally, they…

After the derivation of the Black-Scholes model, the discussion is open to its place in pricing vanilla options. The market dictates the interest-rate and option price for a particular strike and expiration by the laws of supply and demand. The only parameter not readily available as an input in the Black-Scholes equation is volatility. Since every parameter except volatility is available and dictated by the market, the inverse of the Black-Scholes equation allows us to find the volatility expected by market participants: implied volatility. …

The world of AI is as exciting as it is misunderstood. Buzz words like “Machine Learning” and “Artificial Intelligence” end up skewing not only the general understanding of their capabilities but also key differences between their functionality against other models. In this article, I want to discuss the key differences between a linear regression model and a standard feed-forward neural network. To do this, I will be using the same dataset (which can be found here: https://archive.ics.uci.edu/ml/datasets/Energy+efficiency) for each model and compare the differences in architecture and outcome in Python.

We are looking at the Energy Efficiency dataset from UCI…

Quantitative Finance, Mathematics, Artificial Intelligence and Computer Science