Types of limits

Left and Right Limits:

Imagine you're walking on a path towards a bridge, but the bridge has a gap in the middle. As you approach the gap from the left side, you notice the ground's elevation. If you were to approach the same gap from the right side, you might observe a different elevation. The "left limit" is what you observe as you approach the gap from the left, and the "right limit" is what you observe as you approach from the right.

Mathematically:

The left-hand limit of a function ( f ) as ( x ) approaches a value ( c ) is given by:

$\lim_{{x \to c^-}} f(x)$

The right-hand limit is:

$\lim_{{x \to c^+}} f(x)$

If both these limits are the same and equal to ( f(c) ), then the function ( f ) is continuous at ( x = c ). If they are different, there's a discontinuity at ( x = c ).

Gamma and Delta Effects for Options - With Left and Right Limits:

1. Gamma (Γ): Gamma measures how much the Delta (Δ) of an option changes for a $1 move in the underlying asset. It's essentially the "acceleration" of the option's price.

2. Delta (Δ): Delta measures how much the option price changes given a $1 move in the underlying asset. It ranges between 0 and 1 for call options and -1 and 0 for put options.

Scenario - Pin Risk at Expiry:

Let's consider an option that's very close to being at-the-money (ATM) on its expiry day. This is where market makers face what's known as "pin risk."

Left and Right Limit in this Context:

As the underlying asset's price ( x ) approaches the strike price ( c ) of the option from the left (i.e., the asset price is slightly less than the strike), the Delta ( f(x) ) might be close to 0.5 for a call option (indicating a 50% chance it'll end up in-the-money).

However, as ( x ) approaches ( c ) from the right (i.e., the asset price is slightly more than the strike), even though the Delta might still be close to 0.5, the Gamma could be very high. This means small changes in the asset price can result in rapid changes in the option's Delta.

For market makers who are delta-hedging their positions, this can be problematic. As the underlying asset's price oscillates around the strike price, they may need to frequently adjust their hedges due to the high Gamma, leading to potential losses.

In trading, understanding these "limits" and the effects of Gamma and Delta, especially around expiry, is crucial for options traders and market makers. It guides their hedging strategies and risk management practices, ensuring they aren't caught off guard by sudden price movements.