Question

Consider the standard Black-Scholes model. Fix the time of maturity \(T\) and consider the following \(T\) claim \(\chi\). $$ \chi= \begin{cases}K & \text { if } S(T) \leq A \\ K+A-S(T) & \text { if } AK+A\end{cases} $$ \(8.28\) This contract can be replicated using a portfolio, consisting solely of bonds, stock and European call options, which is constant over time. Determine this portfolio as well as the arbitrage free price of the contract.

Short answer

Answer: The arbitrage-free price of the contract 𝜒 can be found by calculating the cost of the replicating portfolio, which consists of a combination of bonds, stocks, and European call options with weights w_B, w_C, and w_C', determined by solving a system of linear equations: $$\pi_\chi = w_B P(B)+ w_C P(C(K)) + w_C'P(C(K+A))$$

Step-by-step solution

Analyze the Payoff Structure of \(\chi\)

The payoff \(\chi\) is given by three different cases: - When \(S(T) \leq A\), the payoff is \(K\). - When \(A < S(T) < K+A\), the payoff is \(K+A-S(T)\). - When \(S(T) > K+A\), the payoff is \(0\).

Identify the Required Components of the Portfolio

We will need stock, bond, and European call option that expire at time \(T\) to replicate the contract. Let's denote the European call option with a strike price of \(X\) as \(C(X)\).

Calculate the Payoff for Each Component

To identify the combination of stock, bond and call options needed to replicate the payoff, we need to first express the payoff of each component at the maturity time \(T\): - Stock's payoff: \(S(T)\). - Bond's payoff: \(B(T)\), which is the present value of the bond multiplied by the interest rate. - Call option's payoffs: \(C(K)=\max(0, S(T)-K)\), \(C(K+A)= \max(0, S(T)-(K+A))\).

Replicate the Contract's Payoff

We need to find a combination of bonds, stocks, and European call options that result in the given payoff \(\chi\). Let's use \(w_B\), \(w_C\), and \(w_C'\) to represent the weight (number of units) of the bond, stocks and call options in the replicating portfolio, respectively. In the case when \(S(T) \leq A\): $$K = w_B B(T) - w_C C(K) - w_C' C(K+A)$$ In the case when \(A < S(T) < K+A\): $$K+A-S(T) = w_B B(T) + w_C (S(T)-K) - w_C' C(K+A)$$ In the case when \(S(T) > K+A\): $$0 = w_B B(T) + w_C (S(T)-K)+ w_C' (S(T)-(K+A))$$

Solve the System of Equations

We can form a system of linear equations with the three cases: $$ \begin{cases} K = w_B B(T) - w_C C(K) - w_C' C(K+A) \\ K+A-S(T) = w_B B(T) + w_C (S(T)-K) - w_C' C(K+A) \\ 0 = w_B B(T) + w_C (S(T)-K)+ w_C' (S(T)-(K+A)) \end{cases} $$ Solving this system of equations yields the weights of bonds, stocks, and call options, \(w_B\), \(w_C\), and \(w_C'\), which create the replicating portfolio.

Determine the Arbitrage-Free Price of the Contract

The price of the \(\chi\) contract can be found by calculating the cost of the replicating portfolio. To get the cost, we multiply the respective weights by the price of each component and sum them up: $$\pi_\chi = w_B P(B)+ w_C P(C(K)) + w_C'P(C(K+A))$$ This yields the arbitrage-free price of the contract \(\chi\).

Key concepts

Black-Scholes model

The Black-Scholes model is a cornerstone of modern financial theory used for valuing options. Developed by Fischer Black and Myron Scholes, it provides a mathematical framework for pricing European call and put options. The model assumes that the price of the underlying asset, typically a stock, follows a stochastic process known as geometric Brownian motion. This means that the prices are unpredictable but follow a certain statistical distribution.

Key assumptions of the Black-Scholes model include:

• The market is efficient, meaning no arbitrage opportunities exist.
• The underlying asset's returns are normally distributed.
• There are no transaction costs, and trading occurs continuously.
• The risk-free interest rate is constant over the option's life.

The formula derived from this model allows for calculation of the option's current price using variables such as the current stock price, strike price, volatility of the stock returns, time to expiration, and the risk-free rate. The simplicity yet effectiveness of the Black-Scholes model makes it a widely-adopted tool in finance for determining arbitrage-free prices in the options market.

European call options

A European call option is a type of option that gives the holder the right, but not the obligation, to purchase a specific stock at a predetermined price, known as the strike price, on a specific expiration date. Unlike American options, European options can only be exercised at maturity.

European call options are a critical component of the Black-Scholes model, as they impact the portfolio needed to replicate a particular option contract, like the one given in the original exercise. The value of a European call option derives from several factors:

• The current price of the underlying stock or asset.
• The strike price of the option.
• The time to maturity of the option.
• The volatility of the stock's returns.
• The risk-free interest rate over the life of the option.

When constructing a portfolio that replicates a specific contract, it is crucial to include European call options with appropriate strike prices. These options, combined with bonds and stocks, enable the creation of a portfolio that mimics the desired payoff structure, thus effectively neutralizing any potential for arbitrage.

Arbitrage-free pricing

Arbitrage-free pricing is a fundamental concept in financial markets that ensures no risk-free profits can be made. In a market that is arbitrage-free, identical assets or portfolios must have identical prices. Otherwise, traders could exploit price differences to earn guaranteed profits without risk.

In the context of option pricing, arbitrage-free pricing involves calculating the cost of a replicating portfolio that mimics the payoff of the option using a combination of:

• Bonds, for their interest-yielding nature.
• Stocks, representing the underlying asset's direct price exposure.
• European call options, providing leverage based on expected price movements.

The goal is to ensure the portfolio perfectly offsets all risk and achieves the same payoff as the contractual claim. By solving a set of linear equations, as shown in the solution, investors can determine the exact combination of assets needed to replicate the contract. The total cost of constructing this portfolio then determines the arbitrage-free price of the option or structured financial product, ensuring the market remains in equilibrium and free from arbitrage.