Principles of Finance - Lecture 11

The analysis of investment opportunities that consider the value of flexibility and future decisions in a multi-period setting.

An assumption in ROA that the underlying asset being analyzed is not traded in a market, but rather its value is based on the project itself.

States that there should be no opportunity to make risk-free profits by exploiting the current market conditions.

An assumption in ROA that the market efficiently prices assets based on all available information.

A model used in ROA to simulate the evolution of a project's value over time while accounting for uncertainty and future decisions.

The expected value of the underlying asset at time t. It considers the probability of the asset going up (u) or down (d).

Represents the upward movement of the underlying asset during a time period.

Represents the downward movement of the underlying asset during a time period.

The risk-neutral probability of the underlying asset going up. It reflects the expected return on the asset in a risk-free environment.

The value of the underlying asset at time 0. It represents the starting point for the asset's price.

The present value of the project without flexibility. It represents the value of the underlying asset at time 0.

The value of the option at the terminal date (time T). It is determined by the payoff function of the option, which is based on the difference between the underlying asset price and the strike price.

The process of working backwards from the terminal date (time T) to the present (time 0) to determine the value of the option at each stage.

The value of an option at the end of its life (time T), determined by the underlying asset's price (ST) and the exercise price (K). For a call option, the value is the maximum of the difference between ST and K, or zero. For a put, it's the maximum of the difference between K and ST, or zero.

A portfolio constructed using the underlying asset and risk-free bonds that mimics the payoff of an option. It eliminates the risk of holding the option.

The probability used in option pricing that assumes the expected return of the underlying asset is equal to the risk-free rate. This simplifies valuation by eliminating risk premiums.

The number of shares of the underlying risky asset required in the replicating portfolio to match the option's payoff.

The amount of money invested in risk-free default-free bonds in the replicating portfolio.

The principle that states the price of an option should equal the price of its replicating portfolio. This eliminates arbitrage opportunities where risk-free profits could be made.

The formula used to calculate the risk-neutral probability (q), based on the risk-free rate, the time period, and the possible price movements of the underlying asset.

A tool used to value an option based on the concept of risk-neutral probabilities and replicating portfolios. It simplifies option pricing by assuming a risk-free world.

A real option gives the holder the right, but not the obligation, to take a particular action in the future. For example, an abandonment option allows the holder to abandon a project if it becomes unprofitable.

In real option analysis, we model the future value of a project using a binomial tree. Each node represents a possible future value, and the branches represent the possible upward or downward movements in value.

The risk-neutral probability (q) represents the likelihood of an upward movement in the value of a project in a binomial tree model, assuming that investors are risk-neutral.

A contraction option gives the holder the right to reduce the scale of a project, potentially leading to a decrease in value but also a decrease in costs.

An expansion option gives the holder the right to increase the scale of a project, potentially leading to an increase in value but also an increase in costs.

The value of an option at a given time (t) before the expiration date (T) is determined by the greater of the exercise value and the continuation value.

The payoff you get if you exercise the option immediately. For a call, it's the difference between the stock price and the strike price. For a put, it's the difference between the strike price and the stock price.

The value you'll get if you hold onto the option instead of exercising it immediately. It's calculated using the risk-free rate, the probability of different stock price movements, and the expected future values of the option.

The value of an option at the moment before its expiration date. It depends on the stock price (S), the strike price (K), and the probability (q) of the stock price going up.

In the context of an option, the value today (t=0) is calculated using the expected values of the option in the next period (Vu and Vd), the risk-free rate, and probabilities.

A type of option that allows the holder to exercise the option at any time before the expiration date.

The concept of real options applies to projects and investments, giving you the flexibility to adjust your strategy based on uncertain future events.

An example of a real option where you can choose to delay starting a project, allowing you to gather more information or wait for more favorable conditions.

The value of an option at time s = t, where t is not equal to the expiration date T, and j represents the possible states of the underlying asset at time s. The formula reflects the discounted expected value of the option in the subsequent period (t+1, s+1) considering the probabilities of upward and downward movements (q and 1-q, respectively) of the underlying asset.

The value of an option at time t = 1, considering the possible states of the underlying asset at time t = 2. The formula represents the discounted expected value of the option at the final period, taking into account the probabilities of upward and downward movements of the underlying asset.

The value of an option today (t=0) is calculated as the discounted expected value of the option at the next period (t=1), taking into account the probabilities of upward and downward movements of the underlying asset.

The value of an option at the expiration date (t=T) is determined by the payoff function, which depends on the type of option. For a call option, it's the maximum of the difference between the underlying asset price (ST) and the strike price (K) or zero. For a put option, it's the maximum of the difference between the strike price (K) and the underlying asset price (ST) or zero.

American options can be exercised at any time before the expiration date (T). To determine the optimal exercise strategy, it is necessary to compare the value of the option if exercised at each time point with its continuation value. The continuation value is the expected value of the option if it is held until the next period.