Option Arbitrage trades are performed to earn small profits with less or zero risk. It is a process of buying and selling an equivalent commodity in two different markets. Options arbitrage can be done through put-call parities. A call gives you the rights to purchase and put gives you the rights to sell.

## What is a Put-Call parity?

• It defines the connection between the worth of put options and call options of an equivalent class, with an equal underlying asset, strike price, and expiration date.
• It also suggests that holding a little put and long call of an equivalent class will deliver an equal return. That is because you are holding one forwarding contract on an identical underlying asset, with an equivalent expiration date and forward price adequate to the Options’ strike price.
• If the costs of the put and call options diverge so that this type of relationship does not hold, an arbitrage opportunity exists, because the traders can theoretically earn a risk-free profit
• The equation expressing put-call parity is c + Xe-rt = p+S

Where,

c= Call price

X = Strike Price

p = Put price

S= Initial price of underlying

r= Rate of interest

t= Time of expiry

e-rt = Discounting factor

If this equality is violated there is an arbitrage opportunity

## Arbitrage Strategy through an Example:

• Suppose Stock ABC is trading at Rupees 90.
• Construct two portfolios portfolio A and portfolio B. In portfolio A, we buy two things a call option and a zero-coupon bond.
• Within the protocol parity equation, the left-hand side represents portfolio A, and therefore the right hand side represents B.
• With a strike price of Rs 100, we buy the decision option on portfolio A priced at Rs 8 and a zero-coupon bond-trading activity at Rs 88.56, which can be Rs 100 at the choice expiry time.
• In portfolio B, we buy two options put options and stock. We buy the put options for an equivalent underlying price Rs 100 but priced at Rs 12.

The worth of portfolio A consisting of a call option and 0 bond coupon option is worth:

c + Xe-rt= 8 + 100*e-0.07*0.5= 8 + 88.56 = Rs 96.56

p+S = 12 + 90= Rs 102

Here we will see that portfolio A < portfolio B, there is an arbitrage opportunity and that we can take advantage of the discrepancy within the price of the two portfolios, which essentially should be equal.

We buy the cheaper portfolio that’s portfolio A and sell the portfolio B (expensive) to form a harmless profit of Rs. 5.44

There are different scenarios at Time of options expiry will affect this strategy

### Scenario 1: Time expiration affect this strategy

• The stock price drops to 0 at expiration.
• The call option that we bought in portfolio A expires worthless of any money, and the put option which we had sold in Portfolio B is now worth Rs. 100
• The put buyer exercises the option, and now we must pay Rs 100 to the put buyer. As we have Rs. 100 value bond which we can use to pay the put-option buyer.
• That how our net profit remains the same at Rs 5.44

### Scenario 2:

• Assume that the stock price has reached Rs. 200 at expiration.
• The call option is in the money, which is worth Rs 100 and we have a bond that is also worth Rs. 100. The put option expires worthless. But we had sold the stock in portfolio B, which now costs us Rs. 200.
• So, to cover the short position of Rs. 200, we can use Rs. 100 from the call option and Rs 100 from the bond so that our profit remains the same at Rs. 5.44.

You will be able to cover all the positions and make a profit of Rs 5.44. We buy the cheaper portfolio and sell the expensive one and make a profit of the difference between the prices of both the portfolios.

## Option Arbitrage Opportunities:

### The Synthetic Position:

• These methods involve what is referred to as synthetic positions. The fundamental positions in underlying stocks or its choices have a synthetic equivalent.
• That means risk profile (Possible Profit & Loss) of any position, are often precisely duplicated with different, but, a lot of complicated methods.
• The basic rule for making synthetics is that the strike worth and expiration date, of the calls and puts, should be identical.
• For making synthetics, with each the underlying stock and its choices, the number of shares of stock should equal the number of shares described by the options.
• To illustrate a synthetic strategy, think about a reasonably comfortable possibility position: the long decision. Once you get a decision, your loss is restricted to the premium paid, whereas the potential gain is unlimited.
• Now, think about the synchronic purchase of an extended put and 100 shares of the underlying stock. Once again, your loss is restricted to the premium paid for the put, and your profit potential is unlimited if the stock worth goes up.

The potential profit and loss of a long-put/long-stock position are sort of just like owning a call option with an equivalent strike and expiration. That’s why a long-put/long/stock position is commonly referred to as a ‘synthetic long call’.

The only distinction between them is the dividend that is paid throughout the holding amount of the trade. The owner of the stock would receive that extra quantity; however, the owner of a long call option would not.

Options arbitrage trades are considered profitable with no risk only if prices have moved out of alignment, and the put-call parity is being violated. If you place these trades when prices are not out of alignment, you may incur a loss.