# Fugit (Option Greek)

Fugit is a concept in options trading and risk management, particularly for American and Bermudan options.

It provides information on the timing of option exercise and helps in hedging strategies.

It’s considered a less traditional Greek.

## Key Takeaways – Fugit (Option Greek)

- Fugit estimates the expected time to exercise for American and Bermudan options, which can be used for accurate hedging and risk management.

- It helps traders anticipate early exercise scenarios.
- Lower fugit values indicate higher likelihood of early exercise.

- Early exercise is an important consideration for options on dividend-paying stocks or in high interest rate environments.
- Fugit enables more precise delta hedging by adjusting for expected exercise timing, potentially improving hedge effectiveness and capital efficiency in options trading.
- We give a fugit trade example.

## Definition

Fugit, also known as “forward delta” or “expected time to exercise,” represents the expected time until an American or Bermudan option is likely to be exercised.

It’s expressed in years and provides a probabilistic estimate of when the option holder might choose to exercise their right.

## Significance in Options Trading

Fugit helps understand the behavior of early-exercise options and their potential impact on portfolio management.

It helps traders and risk managers make more informed decisions about hedging and pricing strategies.

## Calculation & Interpretation of Fugit

### Calculation Methods

Calculating fugit typically involves complex mathematical models and simulations.

The most common approaches include:

- Monte Carlo simulations
- Binomial tree models
- Finite difference methods

These techniques consider various factors such as the option’s underlying asset price, strike price, volatility, interest rates, and dividend yield to estimate the expected time to exercise.

### Interpretation of Fugit Values

- A fugit value close to zero indicates that the option is likely to be exercised soon or is already deep in-the-money.
- A fugit value close to the option’s time to expiration suggests that early exercise is unlikely, and the option will probably be held until maturity.
- Values between these extremes helps understand the probability distribution of exercise times.

## Applications in Risk Management

### Hedging Strategies

Fugit is useful for developing effective hedging strategies for American and Bermudan options.

(American options can be exercised at any time before expiration. Bermudan options can be exercised on specific dates before expiration.)

One can represent the flows of an American swaption as the flows of a swap starting at the fugit multiplied by delta.

This approach allows for a more accurate estimation of potential cash flows and risk exposures.

### Sensitivity Analysis

By incorporating fugit into sensitivity calculations, risk managers can gain a more thorough understanding of how changes in various market factors might affect the option’s behavior and value.

This includes:

#### Delta hedging

Adjusting for the expected timing of exercise can lead to more precise delta hedging strategies.

#### Gamma risk

Fugit can help in estimating the rate of change in delta over time (gamma), considering the probability of early exercise.

#### Vega sensitivity

Understanding the expected time to exercise helps in managing volatility risk more effectively.

## Fugit in Different Option Types

### American Options

For American options, which can be exercised at any time before expiration, fugit provides important information about the likelihood and timing of early exercise.

This is particularly important for options on dividend-paying stocks or for understanding how exercise becomes more likely after earnings releases.

### Bermudan Options

Bermudan options, which can be exercised on predetermined dates before expiration, also benefit from fugit analysis.

The fugit helps in assessing the probability of exercise at each possible exercise date.

### Comparison with European Options

While fugit doesn’t apply to European options (which can only be exercised at expiration), understanding the concept can still be useful when comparing different option types and strategies.

## Factors Influencing Fugit

Several factors can impact the fugit of an option:

### Moneyness

- Deep in-the-money options tend to have lower fugit values, as they’re more likely to be exercised early due to the intrinsic value (they tend to behave like the underlying asset).
- At-the-money or out-of-the-money options generally have higher fugit values, closer to their time to expiration.

### Volatility

Higher volatility can increase fugit values, as it may be beneficial to hold onto the option longer to capture potential upside.

### Interest Rates

In high interest rate environments, fugit values for call options may decrease, as early exercise becomes more attractive due to the opportunity cost of holding the option.

### Dividends & Corporate Events

For options on dividend-paying stocks, upcoming dividends can significantly impact fugit, potentially leading to lower values for call options as early exercise becomes more likely.

Earnings releases are also likely to move a stock’s value, which can make in-the-money options likely to exercise after.

## Fugit Trade Example

Let’s do an example.

This example will focus on using fugit to hedge an American swaption.

### Purpose of the Trade

The purpose of this trade is to hedge the interest rate risk of a long position in an American swaption.

By using fugit, the trader can more accurately estimate the expected cash flows and adjust their hedge accordingly.

This can potentially improve the effectiveness of their risk management strategy.

### Step-by-Step Process

#### Step 1: Analyze the American Swaption

Let’s assume we have a 5-year American swaption on a 10-year swap with the following characteristics:

**Notional amount**: $100 million**Option type**: Call (right to enter a pay-fixed swap)**Strike rate**: 3%**Current 10-year swap rate**: 3.5%**Implied volatility**: 20%**Option premium**: $5 million

#### Step 2: Calculate Fugit

Using an appropriate model (e.g., Monte Carlo simulation or binomial tree), we calculate the fugit of the swaption.

Let’s assume our calculation yields:

**Fugit**= 2.3 years

This means that, on average, we expect the swaption to be exercised in 2.3 years.

#### Step 3: Calculate Delta

We also need to calculate the option’s delta.

For this example, let’s assume:

**Delta**= 0.65

This means that for a 1 basis point change in the underlying swap rate, the option’s value changes by 0.65 basis points of the notional.

#### Step 4: Construct the Hedge

Based on the fugit and delta, we can construct a hedge that approximates the swaption’s behavior:

- Enter a forward-starting swap that begins in 2.3 years (the fugit)
- The notional of this swap should be: $100 million * 0.65 = $65 million

So, we enter a forward-starting swap with the following characteristics:

**Start date**: 2.3 years from now**Maturity**: 12.3 years from now (to match the total 10-year tenor of the underlying swap plus the 2.3-year forward start)**Notional =**$65 million**Direction**: Receive fixed (to offset the long call swaption)

#### Step 5: Calculate Hedge Effectiveness

To assess the effectiveness of this hedge, we can compare the DV01 (dollar value of a basis point) of the swaption to the DV01 of the hedging instrument.

Let’s assume:

**Swaption DV01 =**$50,000 per basis point**Forward swap DV01 =**$48,750 per basis point

**Hedge ratio** = $48,750 / $50,000 = 0.975 or 97.5%

This indicates that the hedge covers 97.5% of the swaption’s interest rate risk, which is quite effective.

### Math Behind the Hedge

The key mathematical concept here is that we’re approximating the American swaption’s behavior with a forward-starting swap.

The start date of this swap is determined by the fugit, and its notional is adjusted by the delta.

The expected cash flows of the swaption can be represented as:

**E[Swaption Cash Flows]**≈ Swap Cash Flows starting at Fugit * Delta

This approximation allows you to create a more accurate hedge than simply using the full tenor of the underlying swap.

### Benefits of Using Fugit

#### More accurate timing

Using fugit accounts for the expected early exercise of the American swaption, rather than assuming it will be held to maturity.

#### Improved hedge effectiveness

The fugit-based hedge more closely matches the swaption’s behavior, potentially reducing basis risk.

#### Capital efficiency

By hedging based on the expected exercise time rather than the full option tenor, you may be able to use capital more efficiently.

#### Better risk management

This approach provides a more nuanced understanding of the swaption’s risk profile, allowing for more precise risk management.

In practice, traders and risk managers would monitor and adjust this hedge as market conditions change – i.e., by recalculating fugit and delta as needed to maintain hedge effectiveness.

## Fugit Limitations & Considerations

### Model Dependency

Fugit calculations are highly dependent on the underlying model and assumptions.

Different models may produce varying results.

Careful interpretation and model risk management are required.

### Market Dynamics

Changes in market variables can affect the accuracy of fugit estimates.

Regular recalculation and monitoring are necessary to maintain effective risk management strategies.

### Behavioral Factors

Fugit assumes rational exercise behavior, which may not always align with actual trader/investor decisions.

Factors related to sentiment, transaction costs, and individual trader/investor constraints can lead to deviations from model predictions.

## Conclusion

Fugit helps understand the expected timing of option exercise, which can enable more sophisticated hedging strategies, improve risk assessment, and better inform trading decisions.

As with all financial models, use fugit in conjunction with other analytics and to maintain awareness of its limitations.

When applied judiciously, fugit can improve the understanding and management of options portfolios.

It can contribute to more effective risk control and potentially improve trading and investment outcomes.