Arbitrage-Free Price Formula

The arbitrage-free price formula is a fundamental concept in financial mathematics that ensures no riskless profit opportunities exist through arbitrage. To understand this, we first need to grasp the basic principles of arbitrage and how pricing models prevent it. The core idea is that if a certain asset or financial instrument is mispriced relative to a benchmark or market condition, arbitrageurs would exploit this discrepancy to make riskless profits. This exploitation would continue until the price adjustments eliminate the arbitrage opportunity. The arbitrage-free price formula mathematically represents this equilibrium state where no such opportunities exist.

To illustrate the concept, consider a simplified scenario involving a stock and a derivative, such as a call option. Suppose the stock is trading at $100, and there’s a call option on this stock with a strike price of $105, expiring in one year. If the market is efficient and arbitrage-free, the price of this call option should align with the underlying stock price in a way that reflects the cost of holding the stock and the risk-free interest rate.

We use the put-call parity theorem to establish the relationship between the prices of European call and put options. The theorem states: $C-P=S-K{e}^{-rT}$C−P=S−Ke−rT where:

• $C$C is the price of the call option,
• $P$P is the price of the put option,
• $S$S is the current stock price,
• $K$K is the strike price of the options,
• $r$r is the risk-free interest rate,
• $T$T is the time to maturity.

For the market to be arbitrage-free, this relationship must hold. If it does not, then an arbitrage opportunity exists. For instance, if the actual market price of the call option is significantly higher or lower than what the formula predicts, arbitrageurs could exploit this discrepancy by buying or selling the options and stock in a combination that guarantees a profit without risk.

Another essential component of the arbitrage-free price formula is the no-arbitrage condition applied to bond pricing. In this case, the price of a bond must be such that it prevents arbitrage opportunities through borrowing and lending at the risk-free rate. For example, if a bond with a face value of $1000 and an annual coupon payment is mispriced, investors could either exploit the price difference or arbitrage by constructing portfolios that lock in guaranteed profits.

A Real-World Example: Imagine an investor observes two bonds: Bond A and Bond B. Bond A offers a 5% annual coupon payment and matures in 5 years, while Bond B has a similar coupon rate but matures in 10 years. If Bond B’s price is not aligned with its present value calculated using the 5% discount rate, an arbitrage opportunity arises. The investor could exploit this by buying the underpriced bond and selling the overpriced one, making a risk-free profit as the prices adjust.

Mathematical Derivation: To derive the arbitrage-free price of a derivative, we use the concept of risk-neutral pricing. The basic idea is to compute the expected payoff of the derivative under the risk-neutral measure and then discount this expected payoff to present value using the risk-free rate. For a call option, this is computed as: $C=e^{-rT}\cdot E\left[\right(S_T-K{)}^{+}]$C=e−rT⋅E[(ST​−K)+] where $E$E denotes the expectation under the risk-neutral measure, $S_T$ST​ is the stock price at maturity, and $(S_T-K{)}^{+}$(ST​−K)+ represents the payoff of the call option if it is in-the-money.

This formula ensures that the price of the option reflects the expected payoff discounted to the present value, preventing arbitrage opportunities through mispricing.

In Summary: The arbitrage-free price formula plays a crucial role in financial markets by ensuring that prices of financial instruments are consistent with the principles of no riskless profit opportunities. This concept is essential for the stability and efficiency of financial markets. By applying the no-arbitrage conditions and pricing models such as put-call parity and risk-neutral valuation, we can maintain equilibrium and prevent arbitrage.