By Sunil K Parameswaran
The requirement of a lack of arbitrage opportunities is at the core of most theories and models in modern finance. By precluding the possibility of arbitrage, researchers have been able to get path breaking results.
No arbitrage conditions can usually be derived with less stringent requirements, than major financial models. Take for instance the case of Put-Call Parity for European options on non-dividend paying stocks. It states that the difference between the call premium and the put premium, for options on the same stock and with the same exercise price and time till expiration, will be equal to the difference between the prevailing stock price and the present value of the exercise price.
This is a far more basic result than well known option pricing models such as the Binomial Model and the Black-Scholes Model. While these models make more stringent assumptions about the evolution of the stock price over time, they too rely on a no-arbitrage argument to derive the final results. Put-Call Parity, which requires only the absence of arbitrage, is consequently valid for all option pricing models, irrespective of the assumptions regarding the evolution of the price of the underlying stock.
For instance, a stock is trading at Rs 100 on the BSE and Rs 100.80 on the NSE. In principle, a trader can lift one phone and buy a million shares on the BSE and sell an equivalent amount immediately on the NSE by another call. Without factoring in transaction costs, he stands to make a cost-less risk-less profit of Rs 8,00,000. This is arbitrage.
Cost of transactions
In real life, traders encounter transactions costs such as bid-ask spreads and brokerage commissions. The issue is whether one can make a profit despite these costs. Also, both BSE and NSE have a T+2 settlement cycle. Hence to implement such a strategy, the trader must have prior access to adequate cash in his bank account, and enough shares in his Demat account.
A dealer who has enough resources in the form of both securities and cash, and who does not have to pay a commission to trade, may end up profiting from such opportunities, which typically last for fleeting moments.
Looking at an issue from a dealer’s and an arbitrageur’s perspective leads to the same conclusion. Assume a money market dealer is quoting the following rates for three-month and six-month loans, where the rates are quoted on a per annum basis.
3-M 5.22% – 5.40%
6-M 8.04% – 8.40%
The forward rate for a three-month contract should be such that the dealer makes a profit whether he borrows for three months and lends for six months or borrows for six months and lends for three months. Thus, the forward rate will have a lower bound of 10.5377% per annum, and an upper bound of 11.4308% per annum.
The logic is based on the argument that either way the dealer ought to make a profit. Now the dealer’s borrowing rate is the arbitrageur’s lending rate, while the dealer’s lending rate is the arbitrageur’s borrowing rate. Hence, if there’s no arbitrage profit, whether the arbitrageur borrows for three months and lends for six months, or the other way, we come to the same conclusion.