Question

Let \(A(0)=100, A(1)=112\) and \(S(0)=34\) dollars. Is it possible to find an arbitrage opportunity if the forward price of stock is \(F=38.60\) dollars with delivery date \(1 ?\)

Short answer

Yes, an arbitrage opportunity exists as the forward price 38.60 differs from the no-arbitrage price 38.08.

Step-by-step solution

- Define Arbitrage

Arbitrage is the opportunity to profit from price differences in different markets without any risk. If the forward price, stock price, and bond price do not align properly, arbitrage may be possible.

- Calculate No-Arbitrage Forward Price

Using the given bond prices, calculate the no-arbitrage forward price. The formula for the no-arbitrage forward price is: \ \[ F_0 = S_0 \times \frac{A(1)}{A(0)} \] With \[ S_0 = 34 \] dollars, \[ A(0) = 100 \] dollars and \[ A(1) = 112 \] dollars, the calculation will be: \[ F_0 = 34 \times \frac{112}{100} = 38.08 \] dollars.

- Compare with Given Forward Price

Compare the calculated no-arbitrage forward price (\[ F_0 = 38.08 \]) with the given forward price (\[ F = 38.60 \]). If they are not equal, an arbitrage opportunity exists.

- Determine Arbitrage Opportunity

Since \[ 38.60 eq 38.08 \], there is a discrepancy. Hence, an arbitrage opportunity exists. One can achieve risk-free profit by entering into suitable positions in the bond, stock and forward markets.

Key concepts

no-arbitrage forward price

The no-arbitrage forward price is a crucial concept in financial markets. It ensures that the price of a forward contract is set in such a way that no arbitrage opportunities exist. In simple terms, it ensures that there are no easy profits without any risk. To calculate the no-arbitrage forward price, we use both the current stock price (S_0) and the bond prices (A(0) and A(1)). The formula for this is:

\[F_0 = S_0 \times \frac{A(1)}{A(0)}\]
For the given exercise, where S_0 = 34 dollars, A(0) = 100 dollars, and A(1) = 112 dollars, we calculate:

\[F_0 = 34 \times \frac{112}{100} = 38.08dollars\].
If the actual forward price deviates from this calculation, we can identify potential arbitrage opportunities.

stock price

The stock price represents the current value of a company's shares traded in the market. In the context of the given exercise, the initial stock price (S(0)) is provided as 34 dollars. The stock price is a key input when calculating the no-arbitrage forward price. It helps to determine the fair value of a forward contract on the stock.

The stock price fluctuates based on market conditions, company performance, and investor sentiment. Accurate valuation models and pricing mechanisms help in determining the fair stock price to ensure there are no arbitrage opportunities.

bond price

Bond prices play a pivotal role in financial calculations. They represent the value of debt securities issued by entities to raise capital. In this exercise, we have two bond prices: A(0) = 100 dollars at time 0 and A(1) = 112 dollars at time 1. These prices are used to discount the expected payoff and help in determining the no-arbitrage forward price.

The bond price at A(0) is the present value, while the bond price at A(1) represents its value at maturity or at a future date. By using these prices, we ensure that the forward contract on a stock is priced correctly to prevent arbitrage opportunities.

financial engineering

Financial engineering utilizes mathematical techniques to solve financial problems. It involves constructing financial strategies and instruments to achieve desired outcomes in terms of pricing, managing risk, and maximizing profits while ensuring no arbitrage opportunities exist. This field blends concepts from finance and mathematics to develop tools for decision-making.

In the context of this exercise, we used financial engineering principles to calculate the no-arbitrage forward price. We ensured that the stock and bond prices aligned correctly, making it possible to identify any discrepancies or arbitrage opportunities. Effective financial engineering practices help maintain market efficiency and fair pricing.