Calculate No Arbitrage Cost for Put Option

Understanding the no arbitrage cost for a put option is essential for investors and traders looking to maximize their returns while minimizing risk. This guide explains what the no arbitrage cost is, how to calculate it, and how to interpret the results.

What is No Arbitrage Cost for Put Option?

The no arbitrage cost for a put option refers to the minimum cost that must be paid to enter into a put option contract without creating an arbitrage opportunity. In financial markets, arbitrage occurs when investors can profit from price differences without taking on additional risk.

For a put option, the no arbitrage cost is determined by the relationship between the option price, the underlying asset price, and the strike price. If the calculated cost is higher than the market price of the put option, it indicates a potential arbitrage opportunity.

How to Calculate No Arbitrage Cost

Calculating the no arbitrage cost for a put option involves several key variables:

Underlying asset price (S): The current market price of the asset
Strike price (K): The price at which the put option can be exercised
Risk-free interest rate (r): The rate of return on risk-free investments
Time to expiration (T): The remaining time until the option expires
Volatility (σ): The expected volatility of the underlying asset

The calculation involves using the Black-Scholes model, which provides a theoretical estimate of the put option price. The no arbitrage cost is then derived from this model.

The Formula

The no arbitrage cost for a put option is calculated using the Black-Scholes formula for put options:

Put Option Price = K * e^(-rT) * N(-d2) - S * N(-d1)

Where:

d1 = [ln(S/K) + (r + σ²/2)T] / (σ√T)

d2 = d1 - σ√T

N(x) = Cumulative distribution function of the standard normal distribution

The no arbitrage cost is essentially the put option price calculated using this formula. If the market price of the put option is higher than this calculated price, it suggests an arbitrage opportunity exists.

Worked Example

Let's calculate the no arbitrage cost for a put option with the following parameters:

Underlying asset price (S): $50
Strike price (K): $55
Risk-free interest rate (r): 5% (0.05)
Time to expiration (T): 0.5 years
Volatility (σ): 20% (0.20)

Using the Black-Scholes formula, we calculate the put option price to be approximately $4.25. This is the no arbitrage cost for this put option.

Interpreting the Result

The calculated no arbitrage cost provides several important insights:

Market Efficiency: If the market price of the put option is higher than the calculated no arbitrage cost, it suggests the market is overpricing the option.
Arbitrage Opportunity: If the market price is lower than the calculated cost, it indicates a potential arbitrage opportunity.
Risk Assessment: The difference between the market price and the no arbitrage cost helps assess the risk of the option.

Understanding the no arbitrage cost helps investors make informed decisions about buying or selling put options, ensuring they are not overpaying for the option or missing out on arbitrage opportunities.

FAQ

What is the difference between no arbitrage cost and option premium?

The no arbitrage cost is the theoretical minimum price for a put option, calculated using the Black-Scholes model. The option premium is the actual price at which the option is traded in the market. The difference between these two prices can indicate market inefficiencies or arbitrage opportunities.

How does volatility affect the no arbitrage cost?

Higher volatility generally increases the no arbitrage cost of a put option. This is because higher volatility increases the chance that the underlying asset price will move significantly, making the put option more valuable.

Can the no arbitrage cost be negative?

No, the no arbitrage cost cannot be negative. The Black-Scholes formula ensures that the calculated put option price is always non-negative, as negative values would not make economic sense in this context.