The cornerstone of the Black-Scholes model is the possibility of hedging of any option by creating a portfolio of shares and zero coupon bonds. The Black-Scholes formula is supposed to show the weights of the components composing the portfolio. Creating such a portfolio leads to profit by continuously changing the weights of the portfolio components, if the price of the option differs from the price of underlying share or risk-free bond. This process is called dynamic hedging, and the underlying portfolio is referred to as a dynamic-replicating portfolio.
Although an attractive approach in theory, this strategy has the following drawbacks:
1) High transaction costs due to the necessity of frequent correction of the weights to maintain the neutrality of the portfolio.
2) Such process implies the continuity of price, often impossible in financial markets. Markets often experience substantial jumps, known as price gaps.
For these reasons, models based on static replication have been developed. Here, an option is hedged by a portfolio of other options with different strike prices and different expiration dates. The portfolio can be kept with the original weights until expiration. The option being replicated is called the target option. The portfolio designed to copy it is the static-replicating portfolio; for short, we will call it basis. Static arbitrage has the following advantages:
1) With the basis, the target option can be hedged without using the dynamic arbitrage, eliminating the drawbacks of dynamic hedging mentioned above.
2) The basis price can approximate the price of the target option, and if the prices differ, it creates the opportunity to arbitrage between the target option and its basis.
3) Virtually any target option can be created, even one not traded on the market, with a particular strike price and expiration.
For perfect static hedging of a target option, an infinite number of standard options are needed. However, there are practical methods of hedging with the limited number of basis options available. These methods were developed in “Static options replication,” Journal of Derivatives, Summer 1995, by E. Derman, D. Ergener and I. Kani, and a 2002 working paper by P. Carr and L. Wu titled “Static hedging of standard options.” In these methods, the target option has a longer expiration, while the basis consists of options with shorter durations.
Putting time on your side
There is another solution, however, one that allows for the creation of various combinations of portfolios where both target and basis can have either longer or closer expiration dates. In addition, it is relatively easy to calculate parameters of such a portfolio, which makes this method suitable for market players and options traders. This method allows the creation of various portfolio combinations consisting of a target option and different replicating portfolios.
The drawback of static arbitrage is that replication between target and basis is incomplete because of hedging with a limited number of basis options. However, all methods have the same issue when it comes to real-life trading.
Our goal is as follows: given a standard option with a particular strike and expiration, we would like to obtain the portfolio of other standard options with behavior similar to the original (target) option. Research suggests a formula for such a portfolio, where the target option has longer expiration and the basis options are closer in expiration (see “Two timing"). Model testing determined the correlation between behavior of such an option portfolio and the target option. It considered not only Brownian movement as market behavior, but also Merton’s model assuming random jumps within the life of an option having a Poisson distribution. This research showed that static hedging in the case of such jumps provides better protection than dynamic hedging.
However, research describes only methods where the target option has the longer expiration. In our case, we can use target options with any expiration length relative to the basis. The relative ease of calculation allows the creation of various combinations of target and bases, making this method practical and useful.