Real World Examples

1. Impact of News on Stock Price:

Scenario: Imagine a company that has been rumored to launch a groundbreaking product. As the official announcement date approaches, the stock price might rise in anticipation.

Limit Concept: As the time ( x ) approaches the announcement date ( c ), the stock price ( f(x) ) approaches some value ( L ). This doesn't necessarily mean the stock will hit ( L ), but it gets closer and closer as the announcement nears.

2. Interest Rates and Bond Prices:

Scenario: Generally, as interest rates rise, bond prices fall, and vice versa.

Limit Concept: If we were to model the bond price as a function of interest rates, as the interest rate ( x ) approaches zero (or even negative in some cases), the bond price ( f(x) ) might approach a particular high value ( L ). In real-world scenarios, as central banks approach zero interest rate policies, the behavior of bond prices can be analyzed using limits.

3. Market Saturation and Product Sales:

Scenario: A company introduces a new product. Initially, sales skyrocket. But as more and more people buy it, the pool of potential new buyers decreases.

Limit Concept: As the time ( x ) since product launch increases, the number of new sales ( f(x) ) might approach a limit ( L ) representing market saturation. Even if everyone hasn't bought the product, sales growth slows down and tends towards a steady state.

4. Volatility and Stock Price Impact:

Scenario: On certain days, especially during major news events or economic releases, market volatility can spike.

Limit Concept: As volatility ( x ) approaches very high values, the predictability (or stability) of a stock's price movement ( f(x) ) might approach a limit ( L ) of being very unpredictable. During extreme volatility, traditional models might break down, and the stock's behavior becomes less and less predictable.

5. Trading Volume and Price Movement:

Scenario: In some cases, especially with less liquid stocks, a significant increase in trading volume can lead to drastic price changes.

Limit Concept: As trading volume ( x ) approaches unusually high values for the stock, the price change ( f(x) ) might approach some limit ( L ). This could be a sharp rise or drop in price due to the sudden influx of buying or selling.

6. Expiry Day Context:

The expiry day is the last day on which a futures or options contract is valid. Once the contract expires, the holder is required to settle, either by taking/giving delivery of the underlying asset or by settling in cash, depending on the market. This makes the expiry day especially volatile as traders rush to close, roll over, or settle their positions.

Limit Concept with Expiry Day:

Let's consider the time of day as our input ( x ) and the stock or index price movement ( f(x) ) as our output. As ( x ) approaches the market close on expiry day, ( f(x) ) can exhibit extreme volatility.

Scenario:

As the last half of the trading day progresses on the expiry day, especially the last hour or even the last few minutes, there's often a flurry of activity. Traders who don't wish to roll over their positions to the next cycle or don't want to take/give delivery will look to close their positions. This can lead to rapid price changes.

Limit Concept in this Scenario:

As the time ( x ) since the start of the trading day approaches the end of the trading day (let's say ( x = 3:30 ) PM for a market that closes at 4 PM), the price movement ( f(x) ) can become more erratic or volatile, approaching a state of unpredictability ( L ).

Trading Implications:

1. Liquidity Crunch: On expiry days, especially in the last half, there might be a liquidity crunch. Some contracts that are deep in-the-money or deep out-of-the-money might not have many willing buyers or sellers. This can lead to significant price jumps.

2. Gamma and Delta Effects for Options: For options traders, the effects of gamma (the rate of change of delta) become pronounced on expiry days, especially for at-the-money options. This can lead to significant price swings in the underlying stock or index as options market makers hedge their positions.

3. Unpredictability: Due to the combined actions of various market participants, predicting the exact movement becomes challenging. While some traders might observe patterns or use algorithms to gain an edge, the general consensus is that the last half of expiry day can be very unpredictable.

In summary, using the concept of limits, we can understand that as the expiry time approaches, the potential for unexpected and rapid price movements increases, approaching a state of high volatility or unpredictability. This understanding is crucial for traders to manage risk and make informed decisions on expiry days.

In each of these scenarios, the idea is that as one variable (the input) approaches a particular value, another variable (the output) tends towards a certain behavior or value. Understanding these behaviors through the lens of limits can give traders insights into potential market movements and risks.