Problem 7 - A Course in Financial Calculus

Question

If a stock price is \(S\) just before a dividend \(D\) is paid, what is its value immediately after the payment? Suppose that a stock pays dividends at discrete times, \(T_{0}, T_{1}, \ldots, T_{n}\). Show that it can be optimal to exercise an American call on such a stock prior to expiry.

Step-by-Step Solution

Answer

The stock price immediately after dividend payment is \( S - D \). Exercising an American call option can be optimal if it allows capturing intrinsic value higher than post-dividend stock price.

Step 1: Understanding the Problem

We need to determine the stock price immediately after a dividend is paid. The initial stock price is given as \( S \), and the dividend paid is \( D \). Additionally, we need to demonstrate why it might be optimal to exercise an American call option before expiry due to dividend payments.

Step 2: Calculating Stock Price Post-Dividend

A dividend payment reduces the stock price by the amount of the dividend. Thus, if the stock price is \( S \) before the dividend, the price immediately after the dividend of \( D \) is paid will be \( S - D \). This new stock price reflects the reduction caused by the dividend distribution.

Step 3: Exploring American Call Option Exercise

American call options can be exercised at any time before expiry. Typically, these options are not exercised early as it is usually beneficial to hold them and avoid the opportunity cost of exercising. However, if the stock pays a dividend, the stock price is expected to drop by the amount of the dividend. Exercising the option just before a dividend payment can be beneficial if the option's intrinsic value (\( S - K \), where \( K \) is the strike price) exceeds the expected post-dividend stock price, \( S - D \).

Step 4: Justification for Early Exercise

Consider the scenario where \( S - K > S - D \). Rearranging gives \( K < D \). In this case, the intrinsic value before the dividend payment is higher than holding the option beyond the dividend, arguing for early exercise to capture the current higher value and prevent loss in value post-dividend. This makes early exercise optimal.

Key Concepts

Dividend Impact

Dividends are periodic payments made by companies to their shareholders as a distribution of profits. When a dividend is paid out, the stock's price is adjusted downward. This is because the payout reduces the company's retained earnings. For instance, if a stock is priced at \( S \) and a dividend \( D \) is paid, the new stock price becomes \( S - D \). The reduction equals the dividend amount and reflects the transfer of value from the company to the shareholder.

• The immediate post-dividend price adjustment might affect stockholders' decisions, especially concerning options.
• Understanding this reduction is crucial for investors holding American call options, as it directly impacts their decision on when to exercise the option.

Early Exercise

An American call option provides the flexibility to exercise any time before the option’s expiration date. While traditionally it's practiced to avoid early exercise (due to the loss of potential time value), there are scenarios when doing so is advantageous, particularly with impending dividend payments.

• Early exercise might be optimal if a dividend payment is expected since the stock price will drop by the dividend amount.
• If the intrinsic value of the option (\( S - K \)), just before the dividend exceeds the anticipated value post-dividend (\( S - D \)), it is beneficial to exercise the option early.
• This ensures that the option holder captures a higher value now instead of a reduced value later.

Intrinsic Value

The intrinsic value of a call option is a measure of the profit an option holder would gain if the option is exercised immediately. It’s calculated as \( S - K \), where \( S \) is the stock price and \( K \) is the option's strike price. This value is essential in deciding whether to hold or exercise an option.

• If \( S - K > S - D \), this indicates that the intrinsic value before the dividend is greater than the expected value after the dividend.
• An option with a higher intrinsic value just before a dividend announcement might prompt early exercise to maximize profits and avoid potential losses from a post-dividend stock price drop.
• The intrinsic value thus plays a critical role in considering early exercise in light of approaching dividends.