8.7 Threshold Cointegration and Arbitrage
In this section, we focus on detecting arbitrage opportunities in index trading by using multivariate time series methods. We also demonstrate that simple univariate nonlinear models of Chapter 4. can be extended naturally to the multivariate case in conjunction with the idea of cointegration.
Our study considers the relationship between the price of the S&P 500 index futures and the price of the shares underlying the index on the cash market. Let ft, ℓ be the log price of the index futures at time t with maturity ℓ, and let st be the log price of the shares underlying the index on the cash market at time t. A version of the cost-of-carry model in the finance literature states
where rt, ℓ is the risk-free interest rate, qt, ℓ is the dividend yield with respect to the cash price at time t, and (ℓ − t) is the time to maturity of the futures contract; see Brenner and Kroner (1995), Dwyer, Locke, and Yu (1996), and the references therein.
The process of model (8.43) must be unit-root stationary; otherwise there exist persistent arbitrage opportunities. Here an arbitrage trading consists of simultaneously buying (short-selling) the security index and selling (buying) the index futures whenever the log prices diverge by more than the cost ...