ABSTRACT
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The prices of the options and futures of a stock both reflect the market’s expectation of future trends of the stock’s price. Their prices normally align with each other within a limited window. When they do not, arbitrage opportunities arise: an investor who spots the misalignment will be able to buy (sell) options on one hand, and sell (buy) futures on the other and make risk-free profits. In this chapter, we focus on put-call-futures parity arbitrage opportunities. The upper bound of a futures bid price, denoted by Fbt , is given by
Fbte −r a(T−t) ≤ Cat − Pbt + Xer b(T−t) + TC . (15.0.1)
Here, T is the expiration date and t is today, i.e., T − t is the remaining time to maturity; Cat is the option’s call premium at the ask; Pbt is the option’s put premium at the bid; X is the exercise price for the option; TC is the transaction cost; r a is the interest rate on the borrowing to finance the futures; and r b is the interest rate to lend. If equation (15.0.1) is violated, then the arbitrageur will be able to make a risk-free profit equal to
Fbte r a(T−t) − [Cat Pbt + Xer b(T−t) + TC] > 0. (15.0.2)
When equation (15.0.1) is violated, a short arbitrage profit can be realized by shorting futures and then protecting it by a synthetic long futures position by (a) buying a call option, (b) shorting a futures option, and (c) borrowing the present discounted value of the futures price and lending the same for the exercise price. Historical data suggest that option and futures prices on the LIFFE market (London) occasionally do not satisfy equation (15.0.1). In the LIFFE tick trade data from January 1991 to June 1998, we identified 8073 profitable short arbitrage and 7410 profitable long arbitrage opportunities when no transaction cost is considered. If we assume a transaction cost of £60 per put-call-futures arbitrage operation, then 2345 (or 29%) of the 8037 triplets would still be profitable. The profits in equation (15.0.2) are those that accrue if the arbitrageur could have obtained as quoted the trade prices recorded at these points in time. In reality, due to delay, the arbitrageur may not be able to obtain the quoted prices. Therefore, an arbitrageur may not be able to exploit all the profitable arbitrage opportunities, especially if it reacts passively. Besides, price misalignments are corrected rapidly by the market, so reacting ahead of the others is crucial to securing the risk-free profits. Therefore, the challenge is not only to spot such opportunities, but to discover them ahead of other arbitrageurs. This motivated us to turn our attention to our previous work on forecasting. EDDIE is a genetic programming tool for forecasting. A specialization of EDDIE, which we called EDDIE-ARB, was implemented for forecasting arbitrage opportunities. EDDIE uses constraints to focus its search in promising areas of the space. The task that we gave EDDIE-ARB was to predict arbitrage opportunities five minutes ahead of time.
