What is Arbitrage?
Arbitrage is making a sure profit in excess of the risk-free rate of return. In the language of quantitative finance we can say that an arbitrage opportunity is a portfolio of zero value today which is of positive value in the future with positive probability, and of negative value in the future with zero probability.
The assumption that there are no arbitrage opportunities in the market is fundamental to classical finance theory. This idea is popularly known as 'there's no such thing as a free lunch.'
An at-the-money European call option with a strike of $100 and an expiration of six months is worth $8. A European put with the same strike and expiration is worth $6. There are no dividends on the stock and a six-month zero-coupon bond with a principal of $100 is worth $97.
Buy the call and a bond, sell the put and the stock, which will bring in$ - 8 - 97 + 6 + 100 = $1. At expiration this portfolio will be worthless regardless of the final price of the stock. You will make a profit of $1 with no risk. This is arbitrage. It is an example of the violation of put-call parity.
The principle of no arbitrage is one of the foundations of classical finance theory. In derivatives theory it is assumed during the derivation of the binomial model option-pricing algorithm and in the Black-Scholes model. In these cases it is rather more complicated than the simple example given above. In the above example we set up a portfolio that gave us an immediate profit, and that portfolio did not have to be touched until expiration. This is a case of a static arbitrage. Another special feature of the above example is that it does not rely on any assumptions about how the stock price behaves. So the example is that of model-independent arbitrage. However, when deriving the famous option-pricing models we rely on a dynamic strategy, called delta hedging, in which a portfolio consisting of an option and stock is constantly adjusted by purchase or sale of stock in a very specific manner.
Now we can see that there are several types of arbitrage that we can think of. Here is a list and description of the most important.
• A static arbitrage is an arbitrage that does not require rebalancing of positions
• A dynamic arbitrage is an arbitrage that 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 is an arbitrage which 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 all apparent arbitrage opportunities can be exploited in practice. If you see such an opportunity in quoted prices on a screen in front of you then you are likely to find that when you try to take advantage of them they just evaporate. Here are several reasons for this.
• Quoted prices are wrong or not tradable
• 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
References and Further Reading
Merton, RC 1973 Theory of rational option pricing. Bell Journal of Economics and Management Science 4 141-183
Wilmott, P 2007 Paul Wilmott Introduces Quantitative Finance, second edition. John Wiley & Sons Ltd