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