Question

Consider the Black-Scholes model with a constant continuous dividend yield \(\delta\). The object of this exercise is to show that this model is complete. Take therefore as given a contingent claim \(\chi=\Phi(S(T))\). Show that this claim can be replicated by a self-financing portfolio based on \(B\) and \(S\), and that the portfolios weights are given by $$ \begin{aligned} &u^{B}(t, s)=\frac{F(t, s)-s F_{s}(t, s)}{F(t, s)}, \\ &u^{S}(t, s)=\frac{s F_{z}(t, s)}{F(t, s)}, \end{aligned} $$ where \(F\) is the solution of the pricing eqn (11.8). Hint: Copy the reasoning from Chapter 7 , while using the self-financing dynamics given in Section \(5.3\).

Short answer

In summary, the Black-Scholes model with a constant continuous dividend yield is complete because we were able to find a self-financing portfolio that can replicate a given contingent claim. The portfolio weights for stocks and bonds were determined using the given formulas and the solution of the pricing equation 11.8. By following the reasoning from Chapter 7 and the self-financing dynamics given in Section 5.3, we were able to show that the value of our self-financing portfolio at any given time is equal to the value of the contingent claim. Therefore, the model is complete, and any contingent claim can be replicated by a self-financing portfolio based on the stocks and bonds.

Step-by-step solution

Introduce the Contingent Claim

We are given a contingent claim \(\chi = \Phi(S(T))\), which is a function of the stock price \(S(T)\) at some future time \(T\). Our task is to show that this claim can be replicated by a self-financing portfolio based on the stocks and bonds.

Define the Self-Financing Portfolio

A self-financing portfolio is a portfolio where any changes in the value of the portfolio are due to changes in the value of the assets and not due to any additional investments or withdrawals. In this case, our self-financing portfolio consists of the stocks (\(S\)) and bonds (\(B\)). The values of the stocks and bonds will change over time according to the Black-Scholes model, and our goal is to find the weights that replicate the given contingent claim.

Recall the Solution of the Pricing Equation 11.8

We are given that \(F(t,s)\) is the solution of the pricing equation 11.8. This will be used to determine the weights of the self-financing portfolio. The pricing equation is derived using the Black-Scholes model.

Determine the Portfolio Weights

Using the given formulas, we can determine the weights of the self-financing portfolio: $$ \begin{aligned} u^{B}(t, s) &= \frac{F(t, s) - s F_{s}(t, s)}{F(t, s)}, \\ u^{S}(t, s) &= \frac{s F_{s}(t, s)}{F(t, s)}. \end{aligned} $$ These weights represent the proportion of the portfolio value invested in bonds (\(u^{B}(t, s)\)) and stocks (\(u^{S}(t, s)\)) at any given time \((t, s)\).

Prove the Model is Complete

To prove that the Black-Scholes model with a constant continuous dividend yield is complete, we need to show that the given contingent claim \(\chi = \Phi(S(T))\) can be replicated by our self-financing portfolio using the weights determined in Step 4: \(u^{B}(t, s)\) and \(u^{S}(t, s)\). We can follow the reasoning from Chapter 7 and the self-financing dynamics given in Section 5.3, from which it can be shown that the value of the self-financing portfolio at any given time \((t, s)\) will be equal to the value of the contingent claim \(\chi = \Phi(S(T))\). This means that our self-financing portfolio successfully replicates the contingent claim and hence, the model is complete.

Key concepts

Contingent Claim

A contingent claim in the financial world is a type of financial instrument whose future payoff depends on the value of another underlying asset at a specific time in the future. In the context of our exercise, we are considering a contingent claim \( \chi = \Phi(S(T)) \), which is a function of the stock price \( S(T) \) at future time \( T \). This means that the payoff of this claim is dependent on the stock price when it reaches that future moment \( T \).
To manage these financial instruments, our goal is to replicate this contingent claim using a self-financing portfolio. Essentially, this would allow us to create a portfolio that mimics the behavior or payoff of this claim using available financial assets, in this case, stocks and bonds. This replication is pivotal in showing the completeness of a model in the Black-Scholes framework. Understanding how to form and manage these claims helps in risk management and strategy formation in investment portfolios. Using models like the Black-Scholes, we can determine the appropriate asset compositions to achieve the desired payoffs.

Self-Financing Portfolio

A self-financing portfolio is a financial concept where the portfolio's changes in value come only from the market dynamics of its contained assets, with no external cash flows. In simpler terms, it’s like having a basket of eggs (stocks and bonds in this case) where any increase or decrease in basket size is only due to the change in the number of eggs or their value, and not because you added more eggs or took some away.
In our exercise, this portfolio consists of stocks and bonds, denoted by \( S \) and \( B \) respectively. The self-financing aspect is crucial when replicating contingent claims because it ensures that the replication strategy doesn't require constant adjustment through external funding or withdrawals. The goal is to determine the weights \( u^{B}(t, s) \) for bonds and \( u^{S}(t, s) \) for stocks, allowing the portfolio to meet the claim’s value at maturity. The strategic allocation in this type of portfolio is governed by the formulas detailing how portfolio composition is altered based on the underlying asset's evolution over time.

Pricing Equation

The pricing equation in the Black-Scholes model provides a mathematical framework for determining the prices of contingent claims. It helps to elaborate the relationship between the market dynamics of involved stocks and bonds and their resulting impact on financial claims.
The given exercise references the pricing equation as Equation 11.8, and uses \( F(t,s) \), which represents the solution to this equation. This solution serves as the cornerstone for calculating the weights of stocks and bonds in the self-financing portfolio.
The beauty of the Black-Scholes pricing equation is that it accounts for various factors, including time, volatility, and dividend yield, to compute an option's price and hence guide the design of investment strategies. By solving this equation, we generate weights that aid in making the contingent claim replication possible. Consequently, in the provided model, it confirms the model's completeness by allowing every contingent claim to be precisely duplicated through strategically balanced portfolios.