# Multi-Asset Greeks

In the context of financial derivatives, Greeks are essential risk measures that help traders and analysts understand how various factors affect the value of an option or other derivative instrument.

While single-asset Greeks are well-known and widely used, multi-asset Greeks are used when dealing with derivatives that depend on two or more underlying assets.

Multi-asset Greeks extend the concept of traditional Greeks to account for the complex interactions between multiple underlyings.

These measures provide a more thorough and accurate view of risk exposure and sensitivity in multi-asset derivatives.

This allows for more accurate pricing, hedging, and risk management strategies.

## Key Takeaways – Multi-Asset Greeks

Correlation Delta (Cega)– Measures the sensitivity of a derivative’s value to changes in the correlation between underlying assets. Helps in managing cross-asset risk.Cross Gamma– Quantifies the rate of change of delta in one underlying due to changes in another underlying’s level.Cross Vanna– Cross Vanna tracks the rate of change of vega in one underlying due to changes in another’s level.Cross Volga– Cross Volga measures vega sensitivity to another’s volatility, enhancing portfolio risk management.

## Correlation Delta (Cega)

### Definition

Correlation delta, also known as cega, measures the sensitivity of a derivative’s value to changes in the correlation between its underlying assets.

### Importance

In multi-asset derivatives, the correlation between underlyings can impact the instrument’s value.

Cega helps traders understand and manage this risk – particularly in products like basket options or spread options.

For example, assets within the same asset class tend to be positively correlated.

### Application

#### Risk management

Traders use cega to measure and hedge correlation risk in their portfolios.

#### Pricing

Accurate correlation estimates are important for proper pricing of multi-asset derivatives.

Correlations are not stable, so traders may not use backward-looking correlation measures but rather base their analysis on the intrinsic drivers of the assets.

### Example Correlation Delta (Cega) Trade

A trader buys a basket option on a group of technology stocks, including Apple, Microsoft, and Google.

The trader expects the correlation between these stocks to decrease due to diverging business strategies (an example of looking at correlations in a forward-looking way rather than unstable coefficients looking backward).

They use cega to measure the option’s sensitivity to correlation changes and implement a hedge by taking a short position in correlation swaps on the same stock pairs.

If correlations decrease as expected, the gain from the correlation swaps offsets the potential loss in the basket option’s value.

## Cross Gamma

### Definition

Cross gamma measures the rate of change of delta in one underlying asset with respect to a change in the level of another underlying asset.

### Importance

Cross gamma captures the second-order effects of price movements across different assets.

This helps understand how the hedging requirements for one asset change as the price of another asset moves.

### Application

**Dynamic hedging**– Cross gamma helps in adjusting hedges when multiple asset prices change simultaneously.**Risk assessment**– It provides a more complete picture of portfolio risk when dealing with multiple correlated assets.

### Example Cross Gamma Trade

A trader holds a spread option on the price difference between WTI crude oil and Brent crude oil.

They notice that when WTI prices move significantly, the delta of their position with respect to Brent also changes rapidly.

To manage this risk, the trader uses the cross gamma to dynamically adjust their delta hedges in both WTI and Brent futures.

When WTI prices spike, they quickly rebalance their Brent hedge to maintain a delta-neutral position.

## Cross Vanna

### Definition

Cross vanna measures the rate of change of vega in one underlying asset due to a change in the level of another underlying asset.

Alternatively, it can be viewed as the rate of change of delta in the second underlying due to a change in the volatility of the first underlying.

### Importance

Cross vanna captures the relationship between price levels and volatility across different assets, important for understanding the behavior of multi-asset options.

### Application

**Volatility trading**– Cross vanna is important for traders focusing on volatility-based strategies across multiple assets.**Risk management**– It helps in understanding how changes in one asset’s price can affect the volatility exposure to another asset.

### Example Cross Vanna Trade

An options trader has a position in EUR/USD currency options.

They observe that when the S&P 500 index rises sharply, the volatility of EUR/USD tends to decrease.

Using cross vanna, they quantify this relationship and implement a strategy that involves buying S&P 500 call options while simultaneously adjusting their EUR/USD option position.

This allows them to profit from both the equity market rally and the corresponding change in currency volatility.

**Related**: Vanna-Volga Pricing

## Cross Volga

### Definition

Cross volga measures the rate of change of vega in one underlying asset with respect to a change in the volatility of another underlying asset.

### Importance

Cross volga helps understand how volatility changes in one asset affect the volatility sensitivity of another asset.

It’s used for managing risk in multi-asset volatility-dependent instruments.

### Application

**Volatility of volatility risk**– Cross volga is essential for traders dealing with products sensitive to changes in the volatility of multiple assets.**Portfolio management**– It helps in understanding and managing the complex volatility relationships in multi-asset portfolios.

### Example Cross Volga Trade

A volatility trader focuses on the relationship between implied volatilities of gold and silver.

They use cross volga to measure how changes in gold’s implied volatility affect their vega exposure to silver.

When they anticipate an increase in gold’s volatility, they buy gold options and simultaneously sell silver options.

The cross volga helps them size these positions to maintain a balanced volatility exposure across both metals, which in turn allows them to profit from the relative changes in volatility between the two assets.

## Implications and Applications of Multi-Asset Greeks

### Enhanced Risk Management

Multi-asset Greeks give a more thorough framework for risk management in complex derivative portfolios.

By accounting for cross-asset effects, traders and risk managers can:

- Identify hidden risks that may not be apparent when looking at single-asset Greeks alone.
- Develop more effective hedging strategies that account for interrelationships between assets.
- Improve stress testing and scenario analysis by incorporating multi-dimensional risk factors.

### Improved Pricing Accuracy

The use of multi-asset Greeks leads to more accurate pricing of complex derivatives by:

- Incorporating correlation effects that impact the value of multi-asset options.
- Accounting for cross-asset volatility dynamics that influence option premiums.
- Providing a more nuanced understanding of how changes in one asset can affect the valuation of derivatives tied to multiple assets.

### More Sophisticated Trading Strategies

Traders can leverage multi-asset Greeks to develop and implement more sophisticated trading strategies to better capture effects that other traders ignore in their models:

**Correlation trading**– Using cega to profit from changes in asset correlations.**Cross-asset volatility strategies**– Exploiting opportunities identified through cross vanna and cross volga.**Dynamic delta-hedging**– Using cross gamma to improve hedging effectiveness across multiple assets.

### Portfolio Optimization

Multi-asset Greeks enable more effective portfolio optimization by:

- Providing a clearer picture of overall portfolio sensitivity to various market factors.
- Allowing for better diversification by understanding cross-asset effects.
- Facilitating more accurate risk-return analysis in multi-asset portfolios.

## Challenges in Implementing Multi-Asset Greeks

### Computational Complexity

Multi-asset Greeks aren’t favorable to discretionary trading styles.

Calculating multi-asset Greeks can be computationally intensive, especially for large portfolios or complex derivatives.

It will generally require:

- Numerical methods and algorithms.
- Significant computational resources and optimized software.
- Approximation techniques to balance accuracy with computational efficiency.

### Data Requirements

Accurate estimation of multi-asset Greeks requires extensive and high-quality data:

- Historical price and volatility data for all relevant assets.
- Reliable correlation estimates
- These can be challenging to obtain, especially in stressed market conditions where liquidity isn’t available.

- Up-to-date market information to be sure the Greeks reflect current market dynamics.

### Model Risk

The calculation of multi-asset Greeks relies on models that may have limitations:

- Assumptions about asset price dynamics and correlations may not always hold in real markets.
- Model calibration can be challenging, especially for less liquid or more exotic multi-asset derivatives.
- The choice of model can impact the calculated Greek values, requiring careful model selection and validation.

## Conclusion

Multi-asset Greeks are used in financial risk management and derivative pricing.

These measures provide a more thorough and nuanced understanding of risk exposures in multi-asset derivatives and portfolios by capturing the complex interactions between multiple underlying assets.

The implementation of multi-asset Greeks presents challenges in terms of computational complexity, data requirements, and model risk.

But their benefits in enhancing risk management, improving pricing accuracy, and enabling sophisticated trading strategies make them popular concepts for modern financial practitioners.