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Chapter 17: Problem 6

"For a dividend-paying stock, the tree for the stock price does not recombine; but the tree for the stock price less the present value of future dividends does recombine" Explain this statement.

Short Answer

Expert verified
The tree for stock price less future dividends recombines because adjusting for present value of dividends aligns the paths.

Step by step solution

01

Understanding Stock Price Trees

In a stock price tree, nodes represent potential future prices of a stock. Each node branches into potential outcomes in subsequent time intervals, reflecting the possible price movements of the stock.
02

Non-recombining Trees and Dividends

Dividends, which are cash payments to shareholders, can cause stock price trees to be non-recombining. This is because future dividends affect each projected price, creating unique paths and nodes that do not merge or recombine.
03

Recombining Tree Concept

A recombining tree has branches that merge, reducing the number of nodes needed as time progresses. This happens when the tree is built using only the stock price forecasts excluding dividends, because similar future states exist after adjusting for dividends.
04

Adjusting for Present Value of Dividends

By calculating the present value (PV) of future dividends and subtracting this from future stock prices, it transforms the stock price tree into a recombining format. This is because excluding dividends aligns similar price states that naturally merge over time.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Stock Price Trees
Understanding stock price trees is like exploring the possible future paths a stock's price can take over time. Think of it as mapping out a journey where each stop is a potential price level that could be reached. In a stock price tree:
  • Each node represents a possible price at a specific point in time.
  • Branches extending from a node signify possible price increases or decreases in the next time step.
  • The tree grows with time, expanding as more potential outcomes are mapped.
These trees are essential for visualizing and calculating different scenarios, allowing investors to anticipate how stock prices might evolve.
Non-recombining Trees
When dividends come into play, stock price trees can become non-recombining. This means that different paths in the tree don鈥檛 merge back together, making each trajectory unique. Here's why:
  • Dividends are cash payouts to shareholders, reducing the company's cash reserve and potentially its stock price.
  • Future dividend payments alter the projected price of a stock at each node, introducing unique branches that don't converge back with others.
  • This often results in a larger and more complex tree with more distinct paths.
Each unique path must be analyzed separately because the impact of dividend payments leads to an increasingly different outcome compared to paths without dividends.
Present Value of Dividends
The present value of dividends is a concept that helps simplify the stock price tree by eliminating the complexity introduced by future dividends. Essentially, it鈥檚 the current value of dividends you expect to receive in the future, calculated by considering time and a discount rate. Calculating present value (PV):
  • Calculate dividends expected at future times.
  • Discount these to their present value using an appropriate discount rate, typically reflecting the time value of money.
  • The formula is generally \( PV = \frac{D}{(1 + r)^n} \) where \( D \) is the dividend, \( r \) the discount rate, and \( n \) the number of periods until the dividend is received.
Subtracting this present value from stock prices in the tree helps align potentially overlapping nodes, paving the way for a recombining tree structure.
Recombining Tree Concept
The recombining tree concept simplifies the tree model by allowing branches to merge back into shared nodes. This occurs when future stock price scenarios become identical after adjusting stock prices by subtracting the present value of expected dividends. In a recombining tree:
  • Branches that would otherwise remain separate combine into common nodes if they lead to similar future price levels.
  • This reduces computational complexity, as fewer unique nodes need to be considered over time.
  • It is a more efficient way to forecast future stock prices, especially for dividend-paying stocks when dividends are adjusted for.
Recombining trees are particularly useful because they streamline the analysis, providing clearer outcomes and faster calculations.

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Most popular questions from this chapter

Which of the following can be estimated for an American option by constructing a single binomial tree: delta, gamma, vega, theta, rho?

Calculate the price of a 3 -month American put option on a non-dividend-paying stock when the stock price is \(\$ 60,\) the strike price is \(\$ 60,\) the risk- free interest rate is \(10 \%\) per annum, and the volatility is \(45 \%\) per annum. Use a binomial tree with a time interval of 1 month.

How can the control variate approach improve the estimate of the delta of an American option when the tree approach is used?

A 2-month American put option on a stock index has an exercise price of 480 . The current level of the index is 484, the risk-free interest rate is \(10 \%\) per annum, the dividend yield on the index is \(3 \%\) per annum, and the volatility of the index is \(25 \%\) per annum. Divide the life of the option into four half-month periods and use the tree approach to estimate the value of the option.

The current value of the British pound is $$\$ 1.60$$ and the volatility of the pound/dollar exchange rate is $$15 \%$$ per annum. An American call option has an exercise price of $$\$ 1.62$$ and a time to maturity of 1 year. The risk-free rates of interest in the United States and the United Kingdom are \(6 \%\) per annum and \(9 \%\) per annum, respectively. Use the explicit finite difference method to value the option. Consider exchange rates at intervals of \(0.20\) between \(0.80\) and \(2.40\) and time intervals of 3 months.
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