Subject 2. An arbitrage opportunity

Recall that the basic put-call parity equation is: c0 + X/(1 + r)T (fiduciary call )= p0 + S0 (protective put). Violations of put-call parity occur when one side of the equation is not equal to the other.

  • An arbitrageur can buy the lower-priced side and simultaneously sell the higher-priced side, thereby making a profit on the price difference. Since the fiduciary call and protective put have the same payoff, the arbitrageur's positions will perfectly offset at expiration.

  • As more and more arbitrageurs perform these transactions, the price of the lower-priced portfolio will increase and the price of the higher-priced portfolio will decrease, until put-call parity is restored.

Situation: Consider the following example involving call options with an exercise price of $100 expiring in half a year (T = 0.5). The risk-free rate is 10 percent. The call is priced at $7.5, and the put is priced at $4.25. The underlying price is $99.

Analysis: The left side of the put-call parity equation is c0 + X / (1 + r)T = 7.5 + 100/(1.10)0.5 = 102.85. The right side is p0 + S0 = 4.25 + 99 = 103.25. This means the protective put is overpriced.

Our strategy: We sell the protective put. This means we sell the put and sell short the underlying. Doing so will generate a cash inflow of $103.25. We buy fiduciary call, paying out $102.85, netting a cash inflow of $0.4. At expiration, if the price of the underlying is above 100:

  • the bond matures, paying $100.
  • use the $100 to exercise call, receiving the underlying.
  • deliver the underlying to cover the short sale.
  • the put expires with no value.
  • net effect: no money in or out.

What would you do if the price of the underlying is below 100 at expiration?

So we receive $0.4 up front and do not have to pay anything out. The position is perfectly hedged and represents an arbitrage profit. The combined effects of other investors performing this transaction will result in the value of the protective put going down and/or the value of the covered call going up until the two strategies are equivalent in value.

Practice Question 1

With two call options available, you decide to construct a bull-call spread. The first option has an exercise price of $30 at a premium of $2, and the second call has an exercise price of $40 with a premium of $0.50. If at expiration, the underlying asset price closes at $45, what is the most that you can profit from this trade?

A. $8.00.
B. $8.50.
C. $11.50.
Correct Answer: B

In a bullish spread strategy, the investor purchases the option with the lower exercise price and sells the option with the higher exercise price. That translates into paying a premium of $2 and receiving a premium of $0.50, in other words, the strategy involved a net premium payment of $1.50. At expiration, the value of the long call will be $15($45-$30) and the value of the short call will be $5($45-$40), resulting in a net payoff of $10($15-$5). However, to get net profit, we deduct net premium from net payoff to give us a figure of $8.50 ($10 - $1.50).