Question

European call and put options with strike price $$\$ 24$$ and exercise date in six months are trading at $$\$ 5.09$$ and $$\$ 7.78$$. The price of the underlying stock is $$\$ 20.37$$ and the interest rate is $$7.48 \% .$$ Find an arbitrage opportunity.

Short answer

Formula for put-call parity is violated: 28.20 ≠ 28.15. Arbitrage opportunity exists.

Step-by-step solution

Identify option prices and conditions

Recognize that the European call option has a strike price of \(24 and costs \)5.09, while the European put option has a strike price of \(24 and costs \)7.78. The underlying stock price is $20.37 and the interest rate is 7.48%.

Calculate the future value of the stock price

Since the interest rate is 7.48% annually, find the six-month equivalent rate by halving it: \( r = 0.0748 / 2 = 0.0374 \). Then, calculate the future value of the underlying stock price: \( S = 20.37 \times e^{0.0374} \).

Compute Put-Call Parity

Use put-call parity to check for arbitrage opportunities. Put-call parity is given by: \( C + PV(K) = P + S \).Here, \( C = 5.09 \), \( P = 7.78 \), \( K = 24 \) and \( PV(K) = 24 \times e^{-0.0374} \).

Calculate Present Value of the Strike Price

Calculate the present value of the strike price \( K \): \( PV(K) = 24 \times e^{-0.0374} = 23.11 \).

Check for Arbitrage Opportunity

Substitute the evaluated values into the put-call parity formula: \( 5.09 + 23.11 = 7.78 + 20.37 \).Left-hand Side: \( 5.09 + 23.11 = 28.20 \) Right-hand Side: \( 7.78 + 20.37 = 28.15 \).Since both sides are unequal, an arbitrage opportunity exists. Buy call option and stock while selling put option.

Key concepts

European call options

A European call option gives the holder the right, but not the obligation, to buy an underlying asset at a predetermined strike price on a specific future date, which is the expiration date. This type of option cannot be exercised before the expiration date.
The main characteristics of a European call option include:

• **Strike Price**: The price at which the option holder can purchase the underlying asset.
• **Premium**: The price paid for purchasing the call option, which is \(5.09 in this case.
• **Expiration Date**: The future date when the option can be exercised, here it is six months from now.
• **Underlying Asset**: The asset that can be bought, with the current price of \)20.37.

This option becomes profitable if the price of the underlying asset rises above the strike price plus the premium. For instance, if the underlying stock rises significantly above \(24, the holder can buy the stock at \)24 and sell it at the market price, making a profit after considering the premium paid.

European put options

A European put option gives the holder the right, but not the obligation, to sell an underlying asset at a predetermined strike price on the expiration date. Like the call option, the put option cannot be exercised before the expiration date.
The key features of a European put option include:

• **Strike Price**: The price at which the option holder can sell the underlying asset.
• **Premium**: The price paid for purchasing the put option, which is \(7.78 in this case.
• **Expiration Date**: The future date when the option can be exercised, in this scenario, it is six months from now.
• **Underlying Asset**: The asset that can be sold, with the current price of \)20.37.

This option becomes profitable if the price of the underlying asset falls below the strike price minus the premium. For example, if the underlying stock price drops significantly below \(24, the holder can buy the stock at the market price and sell it at \)24, thus locking in a profit after considering the premium paid.

Put-Call Parity

Put-Call Parity is a fundamental principle that defines a specific relationship between the price of European call options and European put options with the same strike price and expiration date. The formula for Put-Call Parity is:
\[ C + PV(K) = P + S \ \]

where:

• **C**: The price of the European call option, which is \(5.09.
• **PV(K)**: The present value of the strike price (which can be calculated as the strike price discounted back to the present using the risk-free interest rate). Here, it is \)23.11.
• **P**: The price of the European put option, which is \(7.78.
• **S**: The current price of the underlying stock, which is \)20.37.

The formula effectively tells us that the value of holding a long call option and discounted cash equivalent of the strike price is equal to holding a long put option and the stock itself.
In this example, substituting the given values into the formula:
\[ 5.09 + 23.11 = 7.78 + 20.37 \ \]
The left-hand side (\[5.09 + 23.11 = 28.20 \ \]) and the right-hand side (\[7.78 + 20.37 = 28.15 \ \]) are not equal. This indicates an arbitrage opportunity, showing a slight mispricing in the options market.

Arbitrage opportunity

Arbitrage involves taking advantage of price differences in different markets or forms to make a risk-free profit. In the context of European options, arbitrage opportunities arise when the Put-Call Parity does not hold.
To exploit the arbitrage opportunity identified in the previous section, you can follow these steps:

• **Buy** the underpriced asset (here, the call option and the stock).
• **Sell** the overpriced asset (here, the put option).

Using the given example, since:
\[ 5.09 + 23.11 > 7.78 + 20.37 \ \]

You would:

• **Buy** the call option at \(5.09.
• **Buy** the stock at \)20.37.
• **Sell** the put option at \(7.78.

This results in a cost of \)5.09 + \(20.37 = \)25.46 to buy the call and the stock.
By selling the put option, you receive $7.78.
So the net cost is: \[25.46 - 7.78 = 17.68 \ \]
This should equal the value derived from holding the options and stock until expiration, leading to a guaranteed and risk-free profit due to the mispricing found in the Put-Call Parity.