Options Arbitrage Strategies - Example & How to Arbitrage Options

What is arbitrage?

Arbitrage is taking advantage of pricing differences of the same asset, but in a different market. In essence, arbitrage is a situation where a trader can profit from the imbalance of asset prices in different markets. The simplest form of arbitrage is purchasing an asset in the market where the price is lower and selling the asset in the market where the asset’s price is higher.

Warren Buffet, as a child used to buy a pack of 6 Coca-Cola’s bottles for 20 Cents and used to sell each bottle for 5 Cents to people in his neighborhood, earning 30 Cents in total for a pack and profiting 10 Cents per pack. Young Buffet saw the arbitrage opportunity that he could profit from. The difference in price of a pack against the price of an individual bottle that people were willing to pay. 

The most popular form of arbitrage is spot-to-futures trade, which is used across asset classes and markets. The most proven age-old arbitrage is taking advantage of mis-pricing in stocks between NSE and BSE, however, with deployment of algorithms these arbitrage opportunities are almost nonexistent for retail investors without access to algorithms.

Option arbitrage

This refers to buying and selling of options to take advantage of mis-pricing in premium or price of options. This kind of trades carry very low to zero risk and profit potential is also on the lower side.

Arbitrage opportunities in options arise on two fronts. Option arbitrage can either be initiated between two options or between options and an underlying asset. The former is based on the principle of put-call parity and the latter is based on divergence of intrinsic value and moneyness of options.

Put-call parity is an important concept that shows the prices of put, call and underlying asset must be consistent with one another. The put-call parity relationship shows that a portfolio consisting of long call option and short put option (Long synthetic futures position) would be equal to the short future contract with the same underlying asset, expiration and strike price. This can be explained using simplified equation;

Call premium + Strike price = Put premium + Future price

The LHS (left hand side) and RHS (right hand side) of this equation have to be equal as per put-call parity theory. Any imbalance resulting in one side getting heavier leads to arbitrage opportunity. Thus, by rearranging this equation by using algebraic manipulation, we get;

Long Call option + Short Put option + Short futures = Zero

The put-call parity equation states that if one of the asset prices deviates from the relationship, an arbitrage opportunity will arise. This allows traders to exploit the opportunity by buying the underpriced asset and selling the overpriced asset.

Illustration

Let’s consider a scenario where the Nifty50 index is currently trading at 17,815, this would make the strike of 17,800 the At-the-Money strike. The Nifty50 strike 17,800 call and put option are trading at premiums of ₹87 and ₹70 respectively.  

As per put call parity or arbitrage spread, the equation should result in zero, anything other than zero signifies existence of an arbitrage opportunity.

The buying of call and selling of put at the same strike price creates a long synthetic futures position.

Long call option + Short Put option = Long synthetic futures position

This is executed by buying Nifty50 Call option at strike price of 17,800 for ₹87 (premium paid) and selling Nifty50 Put option at strike price of 17,800 for ₹70 (premium received). The cost of creating this long synthetic spread is net of premium paid and received i.e. (₹87 - ₹70) = ₹17. This also implies that we are long on Nifty50 at 17,800 at cost of ₹17.

To make this riskless we have to assume a counter position to our long synthetic futures position, we do that by taking a short position on Nifty50 futures at 17,830.

The spread between short futures and long synthetic futures (long call, short put strike)
= (17,830 - 17,800) = ₹30

The net premium paid is ₹17 for long synthetic futures, therefore it is deducted from spread = (30 - 17) = ₹13. This spread signifies the existence of riskless arbitrage opportunity at the very initiation of trading positions and remain unchanged irrespective of market direction on expiry.

Farming these value in equation, we get;

Long Call option + Short Put option + Short futures = (-87 + 70 + 30) = ₹13

The non-zero value in the equation defines the existence and quantum of risk free arbitrage gain.

Let’s consider a few scenarios. 

Scenario 1: Nifty trades at 17,700 on expiry

The long call option position (bullish) expires worthless because it is Out-of-the-Money and we lose ₹87 paid upfront as premium. 

The short put option position (bullish) has an intrinsic value of 100 and the premium received of ₹70 is adjusted against intrinsic value (70 - 100) = -30. We lose only ₹ 30

The short futures position initiated at 17,830 expires in profit at 17,700 thus earning ₹130 (17,830 - 17,700)

Therefore, by substituting new values in put call parity (arbitrage) equation;

Long Call option + Short Put option + Short futures = - 87 - 30 + 130 = 13

The strategy results in risk free return of ₹650 (13 * 50)

Scenario 2: Nifty Trades at 17,800 on expiry

Both call and put options expire worthless. We lose the premium paid on call option of ₹87 and retain the premium received on put option of ₹70. 

The short futures position initiated at 17,830 expires with gains at 17,800 thus earning ₹30 (17,830 - 17,700)

Therefore, the by substituting new values in put call parity (arbitrage) equation;

Long Call option + Short Put option + Short futures = - 87 + 70 + 30 = ₹13

The strategy results in risk free return of ₹650 (₹13 * 50)

Scenario 3: Nifty trades at 18,000 on expiry 

The long call option position (bullish) fetches a big gain on account of rally in Nifty50. But, the 200 points gain in intrinsic value is adjusted against premium paid to long call position.

Profit = (18,000 - 17,800 - 87) = ₹113

The short put option position (bullish) expires worthless and retains the premium earned of ₹70

The short futures position initiated at 17,830 expires in loss at 18,000 thus losing ₹170 (17,830 - 18,000)

Therefore, the by substituting new values in put call parity (arbitrage) equation;

Long Call option + Short Put option + Short futures = 113 + 70 - 170 = ₹13

The strategy results in risk free return of ₹650 (₹13 * 50)

This signifies that irrespective of direction of markets, the arbitrage will fetch a 13 points difference which is risk free and consistent at all levels of market activity.

Other forms of option arbitrage

The option price is the composition of intrinsic and extrinsic values. The extrinsic value is based on time value. The time value in turn is derived from option Greeks; Delta, Gamma, Vega, Theta and Rho. Option Greeks are instrumental in measuring the sensitivity of option price to the change in the price of underlying security, time till expiration, volatility and interest rates.

Time value = option price - intrinsic value

The Delta and Vega values of the options pair (both call and put) can be calculated using the Black-Scholes option pricing formula, any deviation in these values to the actual traded price gives rise to an opportunity to set up arbitrage trade. Volatility arbitrage (or Vol% Arb) is a commonly employed option arbitrage strategy by traders, however since this is a level 2 derivative and advanced arbitrage strategy it is advisable to exercise extreme caution.

Box option spread and short box option spread are other four-legged option arbitrage strategies deployed by traders to earn risk free or low risk profits from arbitrage opportunities in NSE options.