How to Find No Arbitrage Price

Finding the no arbitrage price is fundamental in financial markets, ensuring that asset prices are aligned with the principles of no arbitrage. This concept prevents the possibility of risk-free profits from discrepancies in asset pricing. Here’s a detailed exploration of how to determine the no arbitrage price, using practical examples, theoretical foundations, and data analysis to illustrate the process.

Understanding No Arbitrage

No arbitrage is a principle in financial economics that asserts that it’s impossible to make a risk-free profit through discrepancies in pricing across different markets or financial instruments. If such discrepancies exist, they will be exploited by arbitrageurs until prices adjust to eliminate the opportunity. To find the no arbitrage price, one must ensure that the price of an asset or a portfolio of assets reflects its true value, accounting for the time value of money, risk, and other factors.

Theoretical Foundations

1. Arbitrage Pricing Theory (APT): APT suggests that the price of an asset is determined by its sensitivity to various risk factors, which should be accounted for in pricing models. This theory helps in finding the no arbitrage price by ensuring that prices are consistent with the risk factors affecting the asset.

2. Law of One Price (LOOP): According to LOOP, identical assets or portfolios should sell for the same price in different markets, assuming no transaction costs or barriers. This principle helps in finding the no arbitrage price by ensuring that there are no price discrepancies for identical assets across markets.

3. Put-Call Parity: This is a financial principle used in options pricing. It states that the price of a call option and a put option should be in equilibrium with the underlying asset price and the risk-free rate. Violations of put-call parity indicate arbitrage opportunities, and correcting these violations helps find the no arbitrage price.

Practical Methods to Determine No Arbitrage Price

1. Use of Option Pricing Models:

   • Black-Scholes Model: The Black-Scholes model calculates the theoretical price of options based on factors like the underlying asset price, strike price, time to expiration, risk-free rate, and volatility. Ensuring the price derived from this model matches market prices helps maintain no arbitrage conditions.

   • Binomial Model: The binomial model provides a discrete-time framework for option pricing, using a binomial tree to estimate the option price. By comparing the binomial model’s price with market prices, arbitrage opportunities can be identified and corrected.

2. Pricing of Bonds and Fixed Income Securities:

   • Yield to Maturity (YTM): YTM is used to calculate the no arbitrage price of bonds by discounting future cash flows at the risk-free rate. Ensuring that bond prices reflect their YTM helps maintain no arbitrage conditions.

   • Duration and Convexity: These measures help in assessing the sensitivity of bond prices to interest rate changes. By adjusting bond prices for duration and convexity, one can ensure that prices align with no arbitrage principles.

3. Portfolio Pricing:

   • Replication: Constructing a portfolio that replicates the payoff of a derivative or other financial instrument helps determine its no arbitrage price. If the cost of replicating the portfolio differs from the market price, arbitrage opportunities exist.

   • Hedging: Hedging involves creating a portfolio that neutralizes the risk of price changes. The cost of hedging helps in determining the no arbitrage price by ensuring that the price of the asset reflects its risk profile.

Data Analysis and Examples

To illustrate the concept of no arbitrage price, let’s consider a simple example using option pricing.

• Example: Call Option Pricing

  Suppose we have a European call option with the following parameters:

  • Underlying asset price: $100
  • Strike price: $95
  • Time to expiration: 1 year
  • Risk-free rate: 5%
  • Volatility: 20%

  Using the Black-Scholes model, the theoretical price of the call option can be calculated as follows:

  $C=S_0\cdot N\left(d_1\right)-K\cdot e^{-rT}\cdot N\left(d_2\right)$C=S0​⋅N(d1​)−K⋅e−rT⋅N(d2​)

  where:

  $d_1=\frac{\ln \left(S_0/K\right)+(r+{\sigma }^2/2)\cdot T}{\sigma \cdot \sqrt{T}}$d1​=σ⋅T​ln(S0​/K)+(r+σ2/2)⋅T​ $d_2=d_1-\sigma \cdot \sqrt{T}$d2​=d1​−σ⋅T​

  and $N\left(d\right)$N(d) is the cumulative distribution function of the standard normal distribution.

  Plugging in the values:

  $d_1=\frac{\ln \left(100/95\right)+(0.05+0.2^2/2)\cdot 1}{0.2\cdot \sqrt{1}}\approx 0.693$d1​=0.2⋅1​ln(100/95)+(0.05+0.22/2)⋅1​≈0.693 $d_2=0.693-0.2\approx 0.493$d2​=0.693−0.2≈0.493 $C=100\cdot N\left(0.693\right)-95\cdot e^{-0.05}\cdot N\left(0.493\right)\approx 10.45$C=100⋅N(0.693)−95⋅e−0.05⋅N(0.493)≈10.45

  If the market price of the call option differs significantly from $10.45, an arbitrage opportunity exists. Traders can exploit this by buying undervalued options and selling overvalued ones until prices converge.

Tables and Data Analysis

To enhance understanding, consider the following table comparing theoretical and market prices of various options:

Option Type	Underlying Price	Strike Price	Theoretical Price	Market Price	Arbitrage Opportunity
Call	$100	$95	$10.45	$12.00	Yes
Put	$100	$105	$8.30	$7.00	No
Call	$50	$45	$7.20	$6.50	Yes

Conclusion

Determining the no arbitrage price involves understanding key financial principles and applying them through various pricing models and methods. By ensuring that asset prices reflect their true value and correcting any discrepancies, one can prevent arbitrage opportunities and maintain market efficiency. Theoretical models, practical methods, and data analysis are crucial in this process, providing a comprehensive approach to finding no arbitrage prices.