Arbitrage Theory in Continuous Time Explained


Arbitrage Theory in Continuous Time is a textbook, published by Oxford Finance, which seeks to address the mathematics that are used in financial sectors. At the same time, these mathematics principles are applied to basic economics while teaching core fundamentals of this learning discipline.

Written by Thomas Bjork, the goal of this work is to concentrate on the probabilistic theory of continuous arbitrage pricing of financial derivatives. This includes the fund separation theory and the optimal control theory. To assist with the learning process, there are several exercises to follow within the textbook.

In the third edition, Bjork includes information on the optimal stopping theory, optimal investment problems, and positive interest models. He then connects each principle to potential theories and discount factors that may influence the final results.

What Is Arbitrage Theory?

Arbitrage is the practice of taking advantage of a price difference that can be found in 2+ markets. By striking a deal to take advantage of that difference, it becomes possible to create a profit for oneself because of the difference in market pricing.

At the same time, arbitrage can also be a transaction that involves at least one state of positive cash flow without a negative cash flow at any state.

In essence, arbitrage is a risk-free profit.

In the world of finance, arbitrage theory is a general theory of asset pricing. It holds that the expected return of a specific asset can be modeled as a linear function of different indices and factors, from macro-economics to the theoretical. It applies when sensitivity to changes within each factor can be represented by a factor-specific coefficient.

That allows for a rate of return to be calculated by the model, making it possible to price assets correctly. By using the arbitrage theory, an asset price can be equalized to the expected end of period price that is implied by the model. If there is a diversion, then the theory suggests that arbitrage can bring it back into line.

Why Is the Arbitrage Theory Model Important?

Using the principles outlined by Bjork in his textbook, as well as the core concepts of the arbitrage theory, it becomes possible to identify assets that may have been mispriced. In this theory, assets become mispriced if the current price diverges from what the model predicts. That divergence can either make the price be too low or too high.

When the divergence creates a price that is too low, then the inference given is that the assets would have appreciated more than the implied model rate. That allows an individual to short sell the portfolio and then purchase the mispriced asset with the proceeds that are received. Then, at the end of the period, the mispriced asset can be sold so the portfolio can be bought back, allowing for the difference to be pocketed as profit.

When the divergence creates a price that is too high, then the inference give is that the assets would have appreciated less than the implied model rate. When this occurs, it allows the mispriced asset to be short sold, but then the portfolio is purchased with the proceeds. Then, at the end of the period in question, the portfolio is sold and the mispriced asset is bought back. That allows for the difference to be pocketed as profit.

What Are the Factors of the Arbitrage Theory

The arbitrage theory does not reveal the identity of its priced factors within its models. The factors are allowed to change their nature and number over time and even when evaluated within different economies. Bjork provides the foundations for the mathematics behind these factors within his textbook.

As a result of this, the issue becomes empirical, but there are certain guidelines to watch for to determine if the characteristic present qualify as potential factors.

  • Impact on asset pricing manifests in unexpected movement.
  • There should be a representation of influences that cannot be diversified.
  • Information on the variables is required to be accurate and timely.
  • Relationships should be justifiable on economic grounds.

There are certain macro-level events that can be factors when looking at the risks of returns when using the arbitrage theory. Surprises in inflation, GNP as an industrial production index, investor confidence, and shifts in the yield curve can all create significant alterations to the expected outcome.

Certain indices can be influential on the arbitrage model as well. Many of them are quite direct from a mathematical standpoint, including short-term interest rates, commodity pricing, currency exchange rates, and diversification within the stock index.

The Binomial Model and the Arbitrage Theory

One of the core components of Bjork’s work involves covering the binomial options pricing model. First introduced in 1979, it is a lattice-based model that examines varying price over time with the use of an underlying financial instrument. It is used to provide evidence that binomial options pricing models do not provide closed-form solutions.

Option valuation is created using this method through a 3-step process. One must first create the price tree, then calculate the option value at each note. Then there must be a sequential calculation of each option vale at each preceding node.

Each node in the binomial model represents a possible price of the underlying instrument at any given time. Valuation begins at the final nodes and then works backward to the first note on the tree. This makes it possible to more accurately calculate the mispricing that the arbitrage model attempts to identify.

Why Choose Arbitrage Theory in Continuous Time for Studying?

Bjork has created a textbook that is very organized. It is easy enough to understand for someone who is new to this theory or even to advanced mathematics. At the same time, there are clearly marked sections within the textbook where the intent is for the student to review and practice the information with a qualified instructor so the concepts can be fully understood.

When looking at the various theories, equations, and formulas, Bjork provides rigorous proofs throughout each chapter to provide evidence-based perspectives. Each proof is given a thorough explanation as to why it is important or how it supports the intuitive nature of the arbitrage theory.

If it is used consistently, using the same principles, proofs, and equations, then this textbook can provide readers with the chance to become predictive with market pricing. Assets that are mispriced can be quickly identified and potential profits created.

Thomas Bjork may not have created the arbitrage theory or the other areas of mathematical theory that are found in his textbook, but his explanations of them make sense to the average person. That is why it is a quality book to pick up for anyone who has an interest in the financial sector.