Advanced Securities Pricing with Monte Carlo Simulations

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Securities Pricing

There are a variety of methods used to price securities. I would argue that this topic is as much an art as it is a science. This article discusses the advantages of using simulations (mainly in the form of stochastic inputs) for pricing financial instruments. If you are unfamiliar with securities pricing I would suggest this progression of articles…

  • A gentle introduction to stochastic processes

Geometric Brownian Motion

  • The primary stochastic process used in the Black-Scholes model

Stochastic Integrals

  • The evaluation of stochastic integrals for deriving the Black-Scholes model

Deriving the Black-Scholes Model

  • Deriving the infamous…

Python and Black-Scholes Pricing for Dynamic Hedges

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Option Portfolios

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…

Researching Algorithmic Trading Signals

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Quantitative Research

In my previous articles, I have discussed at length the development of algorithmic trading systems in a variety of programming languages. Solutions to the intricacies of these systems (architecture, connection, persistence, trade execution, reporting, etc…) can all be found in my previous articles. I have yet to provide a specific example of a trading signal. It should be quite obvious why one may be unable to find academic (or any) literature on this particular topic: it’s lucrative. Trading signals are difficult to research and after identification, there are few reasons to make the findings public. The aim of this article…

A Fast Track Guide to Downloading Twitter Data using Python

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Twitter data is widely used in the world of natural language processing. This article is meant to give you everything you need to get started downloading and working with Twitter data right now.

Twitter’s Developer API

To use Tweepy you need to apply for developer access to receive the necessary keys and authentication tokens.

After applying for, and being granted developer access head over to the developer portal and create a new app.

Algorithmic Trading System Design Patterns

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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.

Establishing a Controller

EClient and EWrapper Classes

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

A guide to the notion of securities pricing with code

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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. …

Explanation, Consequences, Papers, Resources, and References

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Equity Portfolio Replication

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…

Quick and concise explanation and computation with code

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Risk Management

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…

Dynamic Greek Hedging in Option Portfolios

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Financial Derivatives: Options

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.

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

The development of modern options pricing

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A Brief History

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…

Roman Paolucci

Quantitative Finance, Mathematics, Artificial Intelligence and Computer Science

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