Types

Types of Derivatives

The concepts of left and right derivatives arise from the need to understand how a function behaves precisely at a given point from different directions. This is particularly important for functions that have points of discontinuity, sharp turns, or cusps.

What are Left and Right Derivatives?

Left Derivative

This is the derivative of a function as you approach a point from the left side (or from values smaller than the point). It's formally represented as:

f'_{-}(a) = \lim_{{x \to a^-}} \frac{f(x) - f(a)}{x-a}

Right Derivative

This is the derivative of the function as you approach the point from the right side (or from values larger than the point). It's denoted as:

f'_{+}(a) = \lim_{{x \to a^+}} \frac{f(x) - f(a)}{x-a}

Why are they Important?

Discontinuous Functions: For functions that are not continuous at a point, the left and right derivatives can be different. Evaluating both can give insights into the nature of the discontinuity.
Sharp Turns or Cusps: Some functions might be continuous but have a sharp turn or cusp at a point (like the function (y = |x|) at (x=0)). At such points, the function doesn't have a well-defined tangent, and the left and right derivatives can differ.
Checking Differentiability: A function is differentiable at a point only if it's continuous at that point and the left and right derivatives at that point exist and are equal. If they differ, the function isn't differentiable at that point.

ELI5 Explanation:

Imagine you're hiking on a mountain trail. You're approaching a sharp cliff:

Left Derivative: As you come closer to the edge of the cliff from the left side, you notice the steepness of the trail. This steepness, as you approach the cliff from the left, is like the left derivative.
Right Derivative: After reaching the edge, you continue your hike by descending down the other side of the cliff. The steepness of the trail as you start your descent from the edge is like the right derivative.

If the steepness as you approach the cliff (left derivative) is different from the steepness as you start descending (right derivative), then there's a sharp turn or cusp at the cliff edge.

In trading or finance, understanding such "sharp turns" or sudden changes in data or trends is crucial. For example, if a stock price graph has a sharp turn, left and right derivatives can give insights into how the stock behaved right before and after a significant event, like a product launch or a regulatory announcement.

Real World Examples

1. Trading - Stock Price Analysis:

Imagine a major tech company is about to release its quarterly earnings report. Traders are keenly watching the stock price.

Left Derivative: This represents the stock's momentum just before the earnings release. It's like gauging the general sentiment or trend of the stock as you approach the release moment from the left (before the announcement).
Right Derivative: This represents the stock's immediate reaction after the announcement. It's the initial momentum or trend right after the earnings are made public.

If the left and right derivatives are very different, it indicates a sharp turn in the stock's trajectory due to the announcement.

2. Algo Trading - Trading Bots:

Algo trading often involves bots that automatically buy or sell based on certain triggers or patterns in the data.

Left Derivative: The bot analyzes the momentum of a cryptocurrency just before a major regulatory announcement, determining the trend leading up to that moment.
Right Derivative: After the announcement, the bot immediately gauges the market's reaction.

If there's a significant discrepancy between the left and right derivatives, the bot might interpret this as a major market event and could be programmed to make specific trades in response.

3. Expiry Trading:

As an option nears its expiry, its price can be sensitive to underlying asset price changes.

Left Derivative: This measures the option's price momentum leading up to a significant market event, like a Federal Reserve interest rate decision.
Right Derivative: This gauges the immediate effect on the option's price right after the interest rate announcement.

A divergence between these derivatives indicates a strong market reaction to the news, which traders can use to make last-minute decisions before option expiry.

4. OTM Option Price Explosion:

Out-of-the-money (OTM) options can have explosive price changes due to sudden market movements.

Left Derivative: Before a surprise merger announcement between two companies, the left derivative measures the momentum of an OTM option of one of those companies.
Right Derivative: Right after the merger news breaks, the right derivative captures the immediate skyrocketing price of the previously OTM option.

A stark contrast between these derivatives signals the sudden and unexpected market event, which can be a goldmine for traders holding such options.

In all these scenarios, the idea is that by examining the behavior of a financial instrument just before and right after a significant event (using left and right derivatives), traders and algorithms can gain insights into the market's reaction, helping them make more informed decisions.

The concepts of second, third, and higher-order derivatives arise when we differentiate a function multiple times. Let's dive into the basics and then provide real-life examples from the finance world.

Second Derivative

This is the derivative of the first derivative. It provides insight into the curvature or concavity of the function.

Positive Second Derivative: Indicates the function is concave upwards.
Negative Second Derivative: Indicates the function is concave downwards.

Third Derivative

This is the derivative of the second derivative. It provides insights into the rate of change of the curvature.

Higher-Order Derivatives

You can keep differentiating a function as many times as possible until you can't differentiate it any further (i.e., the derivative is zero or undefined). In practice, higher-order derivatives beyond the third or fourth are less commonly used because they often become too abstract or complicated.

Real-life Examples:

Trading/Algo Trading:

Second Derivative: When analyzing stock price movements, the first derivative provides the momentum or velocity of the price change. The second derivative can give traders an idea of how this momentum is changing. If a stock's price is rising but its second derivative is negative, it might indicate that the stock is losing upward momentum and could soon start to decline.
Third Derivative: It indicates how quickly the acceleration (second derivative) of a stock's price is changing. It's less commonly used but could help sophisticated algorithms anticipate inflection points in price movements.

Hedge Funds:

Second Derivative: When assessing risk, hedge funds might look at the volatility of an asset. If the second derivative of the price movement is high, it indicates that volatility is increasing, signaling higher risk.

Expiry Days:

Second Derivative: The gamma of an option is a second derivative. It measures the rate of change of delta (first derivative) with respect to changes in the underlying asset. As expiry approaches, gamma can become very high for at-the-money options, indicating a higher sensitivity to price changes in the underlying asset.

OTM Explosions:

Second Derivative: For OTM options, a high positive second derivative might signal that the option is becoming more sensitive to price changes in the underlying asset, indicating a higher chance of the option going in-the-money.

In essence, while the first derivative provides insights into the rate of change, the second and higher-order derivatives offer deeper insights into the nuances of these changes, helping traders and funds make more informed decisions.

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