Problem 12 - A Course in Financial Calculus

Question

Suppose that the value of a certain stock at time \(T\) is a random variable with distribution \(\mathbb{P}\). Note we are not assuming a binary model. An option written on this stock has payoff \(C\) at time \(T\). Consider a portfolio consisting of \(\phi\) units of the underlying and \(\psi\) units of bond, held until time \(T\), and write \(V_{0}\) for its value at time zero. Assuming that interest rates are zero, show that the extra cash required by the holder of this portfolio to meet the claim \(C\) at time \(T\) is $$ \Psi \triangleq C-V_{0}-\phi\left(S_{T}-S_{0}\right) $$ Find expressions for the values of \(V_{0}\) and \(\phi\) (in terms of \(\mathbb{E}\left[S_{T}\right], \mathbb{E}[C], \operatorname{var}\left[S_{T}\right]\) and \(\left.\operatorname{cov}\left(S_{T}, C\right)\right)\) that minimise $$ \mathbb{E}\left[\Psi^{2}\right] $$ and check that for these values \(\mathbb{E}[\Psi]=0\) Prove that for a binary model, any claim \(C\) depends linearly on \(S_{T}-S_{0} .\) Deduce that in this case we can find \(V_{0}\) and \(\phi\) such that \(\Psi=0\). When the model is not complete, the parameters that minimise \(\mathbb{E}\left[\Psi^{2}\right]\) correspond to finding the best linear approximation to \(C\) (based on \(S_{T}-S_{0}\) ). The corresponding value of the expectation is a measure of the intrinsic risk in the option.

Step-by-Step Solution

Answer

\(\phi = \frac{\text{cov}(S_T, C)}{\text{var}(S_T)}\) and \(V_0 = \mathbb{E}[C] - \phi \mathbb{E}[S_T-S_0]\). In a binary model, \(\Psi = 0\).

Step 1: Understanding the Task

We need to find expressions for \(V_0\) and \(\phi\) that minimize the expectation of the square of the extra cash required, \(\mathbb{E}[\Psi^2]\). We also need to check if \(\mathbb{E}[\Psi]=0\) for these values and show that \(C\) depends linearly on \(S_T-S_0\) in a binary model.

Step 2: Expression for Extra Cash

The extra cash required is given by \(\Psi = C - V_0 - \phi(S_T - S_0)\). Our goal is to minimize \(\mathbb{E}[\Psi^2]\).

Step 3: Objective Function

We rewrite \(\Psi\) as \(\Psi = C - V_0 - \phi S_T + \phi S_0\). Expanding and using expected values, we have: \(\mathbb{E}[\Psi^2] = \mathbb{E}[(C - V_0 - \phi(S_T - S_0))^2]\).

Step 4: Expanding the Expression

Expanding the square, we get \(\mathbb{E}[\Psi^2] = \mathbb{E}[C^2] - 2\mathbb{E}[C(V_0 + \phi(S_T-S_0))] + \mathbb{E}[(V_0 + \phi(S_T-S_0))^2]\).

Step 5: Calculating Optimal \(\phi\) and \(V_0\)

To minimize \(\mathbb{E}[\Psi^2]\), differentiate with respect to \(\phi\) and \(V_0\) and set the derivatives to zero. Solving these equations gives us: \(\phi = \frac{\operatorname{cov}(S_T, C)}{\operatorname{var}(S_T)}\) and \(V_0 = \mathbb{E}[C] - \phi \mathbb{E}[S_T-S_0]\).

Step 6: Checking Expected Value of \(\Psi\)

Verify that \(\mathbb{E}[\Psi] = \mathbb{E}[C - V_0 - \phi(S_T - S_0)] = 0\). Given \(V_0\) and \(\phi\) were derived to minimize the function under these constraints, \(\mathbb{E}[\Psi]\) evaluates to zero.

Step 7: Binary Model and Linear Dependency

In a binary model, if \(C\) only takes values based on two possible stock outcomes, \(C\) is a linear function of \(S_T - S_0\). In such cases, it's possible to perfectly hedge, meaning \(\Psi = 0\) for certain \(V_0\) and \(\phi\).

Step 8: Conclusion for Non-Complete Model

When the market model is not complete, \(\phi\) and \(V_0\) minimize \(\mathbb{E}[\Psi^2]\) and therefore provide the best linear approximation based on the stock's expected price change. Any remaining \(\mathbb{E}[\Psi^2]\) indicates intrinsic option risk.

Key Concepts

Options Pricing

Options pricing is fundamental in financial calculus. Generally, an option gives the holder the right, but not the obligation, to buy or sell an asset at a predetermined price within or at a specific time. The actual worth of an option is closely linked to the expected movements in the underlying asset's price. In this exercise, we focus on the payoff of an option, denoted as \( C \), at time \( T \) and derive an equation for the extra cash required to meet this payoff, denoting it as \( \Psi \). The value of the option contributes to a portfolio's balance when managing risks associated with the stock valuation \( S_T \). By determining the optimal values for \( V_0 \) and \( \phi \), which represent initial investment values in stocks and bonds, we can price options effectively using the expectation

• Expected value of the stock and payoff: \( \mathbb{E}[S_T] \), \( \mathbb{E}[C] \)
• Variability and correlation: \( \operatorname{var}[S_T] \), \( \operatorname{cov}(S_T, C) \)

Using parameters like covariance and variance helps identify the best investment strategy by calculating the most efficient distribution of resources across different assets.

Risk Minimization

Risk minimization in financial calculus involves strategies to reduce potential financial losses. The crux of this exercise is to minimize the expectation of the square of the extra cash required (\( \mathbb{E}[\Psi^2] \)). This is achieved through strategic allocation of \( \phi \) and \( V_0 \), which are critical components in defining our portfolio at time zero. By finding optimal expressions for these values, we can manage the risks that come with the uncertainty of future stock prices. To minimize risk effectively, we employ linear regression techniques to find the:

• Optimum number of units in the stock \( \phi = \frac{\operatorname{cov}(S_T, C)}{\operatorname{var}(S_T)} \)
• Optimum initial value of the portfolio \( V_0 = \mathbb{E}[C] - \phi \mathbb{E}[S_T-S_0] \)

These calculations ensure that the expected value of \( \Psi \) is zero, indicating that our approach has minimized the intrinsic financial risk by aligning closely with the stock price's expected shifts.

Linear Regression in Finance

In finance, linear regression is pivotal for predicting outcomes and making informed investment decisions. This exercise highlights the use of linear regression in determining the best financial strategies for options pricing and risk management. Whenever the market model isn't complete, or in a non-binary model where variances and covariances play significant roles, linear regression provides a reliable means to approximate the expected changes in stock prices. The essence of linear regression here is that it explains the relationship between the stock price change \((S_T - S_0)\) and the outcome \(C\). When perfect hedging isn't feasible, this mathematical approach helps find the values of \( \phi \) and \( V_0 \) that most accurately model the payoff based on historical price changes. Hence, linear regression provides:

• An analytical framework to minimize unwanted risks
• A way to find the best approximation of options intrinsic worth

In essence, linear regression empowers portfolio managers to make adjustments that address risks and optimize returns, making it an indispensable tool in the world of finance.