What is Arbitrage?
Arbitrage is making a sure profit in excess of the risk-free rate of return. Quantitative finance terminology defines an arbitrage opportunity as a portfolio with zero value now that has a positive value with a positive probability in the future and a negative value with a zero probability in the future.
Classical finance theory is predicated on the premise that there are no opportunities for this in the market. The phrase “there’s no such thing as a free lunch” describes this concept.
A Short Example to Understand
The value of an at-the-money European call option with a $100 strike and a six-month expiration period is $8. It is worth $6 to purchase a European put with the same strike and expiration. The stock has no dividends, and a $100 principle, six-month zero-coupon bond is worth $97.
Purchase the bond and call option, and sell the put and stock to generate revenue of $ − 8 − 97 + 6 + 100 = $1. Regardless of the stock’s final price, this portfolio will be worthless after expiration. You will profit $1 without taking any risks. It is arbitrage.
A Deeper Understanding
One of the foundations of classical finance theory is the idea of no arbitrage. It is presumpted in derivatives theory when the Black-Scholes model and the binomial model option pricing algorithm are derived. Therefore In these situations, it’s a little trickier than in the previous straightforward example.
In the example above, we created a portfolio that made us money right away and didn’t need to be touched until it expired. An example of a static method is this. The aforementioned example is unique in that it doesn’t make any assumptions about how the stock price will behave. Therefore, model-independent method serves as the illustration. However, in order to derive the well-known option-pricing models, we rely on a dynamic method known as delta hedging, in which a portfolio made up of stocks and options is continuously modified by the precise purchase or sale of stocks.
Types of Arbitrage
- A Static Arbitrage does not require rebalancing of positions
- A Dynamic Arbitrage requires trading instruments in the future, generally contingent on market states
- A Statistical Arbitrage is not an arbitrage but simply a likely profit in excess of the risk-free return (perhaps even suitably adjusted for risk taken) as predicted by past statistics
- Model-independent Arbitrage does not depend on any mathematical model of
financial instruments to work. For example, an exploitable violation of put-call parity or a violation of
the relationship between spot and forward prices, or between bonds and swaps. - Model-dependent Arbitrage does require a model. For example, options mispriced because of incorrect volatility estimate. To profit from the arbitrage you need to delta hedge and that requires a model.
Not every apparent opportunity for arbitrage can actually be taken advantage of. When you try to take advantage of an opportunity that you see in stated rates on a screen in front of you, you probably won’t find that it lasts long. Here are several reasons for this.
- Quoted prices are wrong or not tradeable
- Option and stock prices were not quoted synchronously
- There is a bid-offer spread you have not accounted for.
- Your model is wrong, or there is a risk factor you have not accounted for.
Related Readings
- Put-Call Parity: All You Need To Know
- Modelling Approaches in Quantitative Finance: All You Need To Know
- Central Limit Theorem In Finance: Power Your Portfolio Now
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