Calculation of Greeks for Cross-Asset Structured Derivatives
Calculation of Greeks for Cross-Asset Structured Derivatives
In the field of derivatives trading and risk management, Greeks are core tools for measuring the sensitivity of product prices to market parameters. For structured derivatives linked to different underlying assets (interest rates, exchange rates, equities, commodities), while the definitional logic of Greeks is similar—they all measure the impact of a unit change in a market parameter on the price—significant differences in the price scales, volatility characteristics, and market conventions of various asset classes necessitate establishing a standardized framework for calculation and reporting.
This article aims to provide a comprehensive guide to calculating Greeks, covering all aspects from basic definitions to advanced practices, with special emphasis on standardized methods across asset classes and strategies for handling complex scenarios.
I. Core Greeks
1.1 The Basic Greeks Family
- Delta (Δ): Impact of a change in the underlying asset price on the derivative's value.
- Gamma (Γ): Impact of a change in the underlying asset price on Delta (second-order sensitivity).
- Vega (ν): Impact of a change in implied volatility on value.
- Theta (θ): Impact of the passage of time on value.
- Rho (ρ): Impact of a change in the risk-free interest rate on value.
1.2 The Need for Standardization
Price scales vary dramatically across different asset classes:
- Interest Rates: Expressed in percentages (e.g., 5.00%), where a 1% change is too large.
- Equities: Expressed in absolute monetary value (e.g., $100), where a 1bp change ($0.01) is difficult to measure.
- Commodity Futures: Lack a directly tradable spot price; futures contracts serve as the underlying.
These differences require us to standardize the definition of a "unit change" to ensure that Greeks across asset classes are comparable and practical.
II. Cross-Asset Class Bump Standards: Industry Practices
The table below summarizes standardized practices in the Bump and Price method for various asset classes:
| Metric | Interest Rates/FX (IR/FX) | Equities/Indices (Equity) | Commodity Futures (Commodities) | Remarks |
|---|---|---|---|---|
| Delta Bump | 1 basis point (bp) | 1 currency unit or 1% | 1 currency unit | For commodities, the futures price is the underlying |
| Gamma Bump | Delta change after 1 bp | Delta change after 1 unit | Delta change after 1 unit | Must distinguish between Gamma(Unit) and Gamma(%) |
| Vega Bump | 1% change in volatility | 1% change in volatility | 1% change in volatility | Always uses percentage change |
| Theta Bump | 1 calendar day | 1 calendar day | 1 calendar day | The most consistent Greek |
| Rho Bump | 1 bp | 1 bp | 1 bp | Commodities require additional convenience yield Rho |
2.1 Interest Rate/FX Products: Basis Points (BP) Rule
For interest rate and foreign exchange products, the market customarily uses basis points (bp) as the unit of change because:
- Interest rates themselves are percentages; a 1% change is too large an amplitude.
- Market quotes and trades are typically conducted in bp.
Delta Calculation Example (Central Difference Method):
Benchmark Rate: 5.00%
V_up: Product value when the rate increases to 5.01%
V_down: Product value when the rate decreases to 4.99%
Delta ≈ (V_up - V_down) / 22.2 Equity/Index Products: Currency Units or Percentages
For equity and index products:
- Using 1 currency unit as the Bump Size is most intuitive.
- Some risk control systems require reporting Delta corresponding to a 1% change, necessitating subsequent conversion.
- Gamma requires special attention to the double standardization problem (see Part III for details).
2.3 The Specificity of Commodity Futures: Futures Contracts as Underlying
Commodity derivatives face unique challenges:
- No Spot Price: Commodity futures options directly use futures contracts as the underlying.
- Solution: Treat the futures price as the "underlying asset price" for bumping.
- Example: For WTI crude oil futures options, bump the futures price (e.g., from $80.00 to $81.00), not any spot price.
III. In-Depth Analysis of Key Concepts
3.1 The "Double Standardization" Problem of Gamma
Gamma is one of the most confusing Greeks in practice due to two different standardization methods:
A. Gamma (Unit) / Gamma ($)
- Definition: The change in Delta (in absolute monetary amount) after a 1 standard unit change in the underlying asset price.
- Calculation: Γ_unit = Δ(S+1) - Δ(S)
- Example: Stock price $100 → $101, Delta changes from $0.50 → $0.54, then Γ_unit = $0.04
B. Gamma (%) / Gamma (Per%)
- Definition: The change in Delta (as a percentage) after a 1% change in the underlying asset price.
- Calculation: Γ_% = Γ_unit × (Underlying Price / 100)
- Example: Same as above, Γ_% = $0.04 × ($100/100) = 0.04 (or 4%)
- Importance: Γ_% makes assets at different price levels comparable and is a more commonly used metric in risk management.
3.2 Convenience Yield Risk for Commodity Futures
In addition to the standard risk-free rate Rho, commodity derivatives have another key parameter:
Convenience Yield Rho (Φ)
- Definition: Impact of a 1bp change in the convenience yield on the option value.
- Importance: The convenience yield reflects fundamental factors such as storage costs and supply-demand relationships of the commodity.
- Calculation: Similar to standard Rho, but bumping the convenience yield instead of the risk-free rate.
3.3 Refined Decomposition of Volatility Risk
For complex structured products, a single Vega is often insufficient to describe volatility risk:
| Metric | Definition | Importance |
|---|---|---|
| Vega Buckets | Vega decomposed by volatility tenor | Managing volatility risk across tenors |
| Volga (Vega Gamma) | ∂Vega/∂Vol | Measures volatility convexity, key for volatility smile/skew |
| Vanna | ∂Delta/∂Vol = ∂Vega/∂Spot | Connects directional and volatility risks |
| Vomma | Second derivative of Vega | Risk management in high-volatility environments |
IV. Implementation Details of the Bump and Price Method
4.1 Central Difference Method: Best Practice
For most Greeks, using the central difference method is recommended over the forward difference:
Greek ≈ [V(parameter+ε) - V(parameter-ε)] / (2ε)Where ε is the Bump Size (1bp, 1 unit of currency, or 1%).
Advantages:
- Higher accuracy (error is O(ε²) rather than O(ε)).
- Reduces bias caused by model asymmetry.
4.2 Trade-offs in Choosing Bump Size
Although the industry has standard Bump Sizes, adjustments are needed in specific scenarios:
| Scenario | Recommended Bump Size | Reason |
|---|---|---|
| Standard Products | Standard size | Maintain comparability across assets/products |
| Highly Nonlinear Regions | Reduce Bump Size | Avoid estimation bias from overly large bumps |
| Barrier Options Near Barrier | 0.1-0.5× Standard | Price function changes sharply near barriers |
| Stress Testing | Increase Bump Size | Assess risk under extreme scenarios |
4.3 Model Consistency: Hidden Risk
The biggest source of risk in the Bump and Price method is model inconsistency:
Common Issues:
- Differences in Numerical Methods: Using different convergence thresholds or numerical methods for the baseline price and bumped prices.
- Recalibration of Implied Parameters: After bumping market parameters, other implied parameters (e.g., local volatility) are not adjusted accordingly.
- Random Number Generator: Monte Carlo simulations using different random number paths.
V. Higher-Order and Cross Greeks
For complex structured derivatives, first-order Greeks may not capture all risks:
5.1 Important Higher-Order Greeks
| Greek | Mathematical Expression | Risk Management Significance |
|---|---|---|
| Charm | ∂Delta/∂Time | Rate of Delta decay over time, key for barrier options |
| Speed | ∂Gamma/∂Spot | Sensitivity of Gamma to price changes |
| Color | ∂Gamma/∂Time | Rate of Gamma decay over time |
| Ultima | ∂Volga/∂Vol | Higher-order volatility sensitivity |
5.2 Cross Greeks
Measure interactive effects between different market parameters:
- Delta-Vanna: Impact of price changes on Vanna.
- Vega-Rho: Impact of volatility changes on interest rate risk.
- Cross-Asset Greeks: For multi-asset derivatives, measure the impact of Asset A's change on the sensitivity to Asset B.
Calculation Method: Double bumping, for example:
Cross Greek = [V(S+ε, σ+δ) - V(S+ε, σ-δ) - V(S-ε, σ+δ) + V(S-ε, σ-δ)] / (4εδ)VI. Asset Class-Specific Considerations
6.1 Interest Rate Derivatives: Curve Risk
Interest rate products face risks from the entire yield curve, requiring:
- Key Rate Deltas: Bump interest rates at various key tenors on the yield curve separately.
- Curve Shape Risk: Measure the impact of changes in curve slope and curvature.
- OIS-LIBOR Basis Risk: For products involving basis.
6.2 Foreign Exchange Derivatives: Dual Currency Risk
Foreign exchange options involve two currencies:
- Domestic Rate Rho and Foreign Rate Rho need to be calculated separately.
- Correlation Risk: For Quanto products, measure the impact of changes in the correlation between domestic interest rates and the exchange rate.
6.3 Equity Derivatives: Dividend Risk
In addition to standard Greeks, consider:
- Dividend Rho: Impact of changes in the expected dividend yield.
- Ex-Dividend Day Effect: For high-dividend stocks, Greeks change sharply near ex-dividend dates.
6.4 Commodity Derivatives: Rollover and Term Structure
- Rollover Rules: Define how to switch to the next active contract upon futures contract expiry.
- Term Structure Risk: Bump futures prices of different delivery months to measure commodity futures curve shape changes.
- Seasonality Impact: For seasonal commodities like agricultural products, consider the impact of seasonality on Greeks.