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.
While strike prices for basis can be calculated using formulas from Carr and Wu’s research paper, we suggest that the quantity of each option constituting basis is calculated using the linear equations set shown below. The replication can be achieved by equating price and the strike with the prices and strikes of the basis options. The sum of factors in basis should equal 1.
Consider a direct problem where the option with longer expiration is replicated by three closer-to-expiration options as shown in “Three to one” (below).
Weights in this problem are determined with the following set of equations:
w1*P1 + w2*P2 + w3*P3 = Pt
w1*S1 + w2*S2 + w3*S3 = St
w1 + w2 + w3 = 1
Pt: is the price of target option
St: strike price
P1, P2, P3: prices of the first, second and third options in the basis
w1, w2, w3: weights (necessary quantities) of options in basis
By solving the simultaneous equations, we obtain the weights that suggest the necessary quantity of each option in the basis.
As a numeric illustration, assume that we need to replicate the target option expiring in 100 days having a strike of $100, volatility 20%, zero dividends and interest, with three basis options expiring in 70 days. According to the Black-Scholes model, for an underlying share quoted at $100, the price of the target call option is $4.17, and the difference in expirations between the target option and basis is 30 days.
For the purpose of the equation in “Two timing,” strike prices for the basis are 99.84, 90.38 and 110.28. Because in real life, strike prices are typically round numbers, let’s round them to 100, 90 and 110, respectively. Thus, their prices are 3.49, 10.46 and 0.64. Now, let’s determine the weights of each option in the portfolio necessary to make the entire basis equal 4.17. We enter the corresponding parameters into the equation above:
w1*3.49 + w2*10.46 + w3*0.64 = 4.17
w1*100 + w2*90 + w3*110 = 100
w1 + w2 + w3 = 1
By solving this equation, we can obtain the quantity for each option of the basis necessary to replicate the target option:
w1 = 0.669
w2 = 0.165
w3 = 0.165
The first table in “Priced to market” (below) includes the standard deviation and average for the positive difference between the target option and basis, consisting of 0.669 options with strike 100, 0.165 options with strike 90 and 0.165 options with strike 110, at various underlying share prices. The deviation of basis from the target option increases as the expiration of the closest options approaches. The biggest difference begins 10 days prior to expiration.
Now, consider the inverse problem, where the target price has the closest expiration date. For this case, we will use data from the previous numeric illustration, except modify the expiration target to 70 days and the basis options to 100 days. Now the equation set is as follows:
w1*4.17 + w2*10.82 + w3*1.08 = 3.49
w1*100 + w2*90 + w3*110 = 100
w1 + w2 + w3 = 1
The solutions of these equations are:
w1 = 1.384
w2 = - 0.192
w3 = - 0.192
Notable is that two factors have negative values, which suggest the sale of the corresponding options.
This basis is represented in the second table in “Priced to market,” (above) which includes the standard deviation and average of the target options for various expirations.
The deviation of behavior of the portfolio is slightly worse than in the previous case. Here, the deviation of the basis from the target will also increase if we consider the basis with the longer expiration. For that reason, using the closer expiration time between basis and target is recommended.
It is also possible to compose more complicated portfolios. One such alternative can be referred to as a “kite.” The essence of such a composition is that it consists of a central target option that is simultaneously the target for both the closest basis and the longer-expiration basis. “High flyer” (below) diagrams this example, consisting of three options.
In this article, we examined the benefits and construction of static-replicating portfolios. This is an effective method of replicating a plain-vanilla position. However, on the theoretical level, the technique does require an indefinite number of options to compose the basis. Even in such limited cases, though, static arbitrage can be quite useful for market makers and for those who have to hedge options already sold in anticipation of important economic events. An important area of additional research is how positions, including the target and the basis, react in anticipation of that news.
Dmitriy Taubman is a senior manager at JSC Option Management and Global Derivatives, III LP hedge fund. Gary Berg has a degree in mechanical engineering and is a partner and trader at Derivatives Research LLC.