Carlos Blanco is managing director at Ascend Analytics and faculty at the Oxford Princeton Programme, while Alessandro Mauro is risk officer at MKS and faculty at the Master in Commodity Trading, University of Geneva.
In this article, we introduce a framework to perform profit and loss (P&L) attribution for portfolios that contain non-linear market exposures. In a linear portfolio, the P&L is fully described by changes in volumes and market prices. The effect of the changes on the P&L are linear in the sense that a change of 10% in price will determine a change of 10% in P&L. Changes will then be explained by linear P&L attribution drivers such as new deals, deal amendments and deal realisation, to name a few.1 The majority of physical deals as well as forwards, futures and swaps financial instruments have a linear P&L.
Non-linear portfolios, additionally to their linear counterparts, include optionality elements that are either explicit or implicit. Non-linear P&L is found in portfolios containing financial options or physical deals that contractually embed optionality. One or both parties of the agreement have the option to exercise a predefined contractual right, normally based on profitability assessments. The presence of those optionalities determines a more complex definition of the P&L, which will not be dependent exclusively on volumes and prices.
In fact, in addition to volume and price sensitivities, the P&L in portfolios that contain optionality also depends on non-linear price effects, as well as changes in market volatility, time decay and correlation changes in the case of spread options. These additional factors will be crucial in the assessment of P&L attribution for non-linear portfolios.
A P&L attribution framework for non-linear exposures should answer some critical questions such as:
- What are the main risk drivers of the book?
- Are there any large long or short directional bets?
- What is the convexity or ‘gamma’ of the portfolio?
- Are we primarily long or short volatility?
- How does time decay impact the value of our books?
- Is correlation a large driver of P&L variability?
P&L attribution and Taylor series expansion
A common way of decomposing the P&L of a portfolio containing options is by taking the individual instrument sensitivities, more commonly known as ‘Greeks’, and applying them to the respective changes in market factors such as prices, volatilities and interest rates.
The decomposition is based on the Taylor Series Expansion formula:2
The P&L contribution of each of the main drivers is calculated by combining the Greeks and the changes in each of the risk drivers:
- linear price exposure contribution. Calculated by multiplying delta times the difference between the price at time t and t–1;
- non-linear price exposure contribution. Because gamma is a second derivative, it is calculated by multiplying gamma times the underlying change squared and then divided by two;
- volatility contribution. Calculated by multiplying vega times the difference in implied (model) volatility between t and t–1;
- time decay contribution. Calculated by multiplying theta times changes in time to maturity. One business day is commonly used; and
- interest rate exposure contribution. Calculated by multiplying rho times the difference in interest rates between t and t–1.
Additionally, the P&L decomposition needs to take into account the P&L effect of portfolio adjustments that were not taken into account for the calculation of the original portfolio Greeks. Those include new open positions, intraday trades, and closing existing positions.
An alternative way to represent the Taylor series expansion in terms of the Greeks and the underlying risk drivers is the following:
The exact derivation of the sensitivity parameters in the previous equations depends on the theoretical pricing model chosen for the calculation of the non-linear portfolio value.
For European options on futures or forwards using Black-76, the Greeks can be derived using closed-form solutions while for most other instruments valued with approximation, trees of Monte Carlo Simulation, it is common to apply finite difference to calculate the option price sensitivity for small changes in the underlying drivers.
Taylor series expansion for spread options
The Taylor series expansion for a two-asset non-linear exposure is given by:
Where the first four terms are the deltas and gammas for assets one and two of the spread, the fifth term is the cross-gamma, which is the second order derivative of price with respect to changes in both of the underlying assets. The sixth and seventh terms are the vegas for each of the two underlying assets and the eighth term is eta, which measures the sensitivity of the spread options versus changes in the correlation coefficient between the two assets.3
Sample P&L attribution reports
If the risk system and corresponding models are similar to those used for the P&L decomposition, the theoretic P&L from the Taylor series expansion should usually be close to the actual P&L generated from the risk system. The risk management policy can define some bands of tolerance for the ‘error term’, which is the difference between the proxy P&L and the actual system-generated P&L. For example, if the error term is larger than 10% in absolute value, risk groups could investigate and report on the reasons for the discrepancies. Some sources of model error may be the absence of higher-order Greeks such as cross-order sensitivities.
These, however, are typically small contributors to a portfolio of commodity derivatives, and their introduction in the P&L analysis framework might provide more nuisance than value. As we mentioned previously, the P&L attribution also needs to take into account the P&L effect of new trades that were not included in the calculation of the original Greeks, such as intraday trades and new portfolio positions.
Application of P&L decomposition for non-linear books
Table A shows the P&L decomposition of the portfolios of four trading desks as well as the overall portfolio. The model P&L is the sum of the delta, gamma, vega and theta columns, as well as the impact of new trades. The actual P&L is the accepted P&L for reporting purposes. The error term is the difference between actual P&L and model P&L. The last column is the error term as a percentage of the actual P&L.
In addition to serving as a tool to benchmark mark-to-model values, P&L decomposition can shed an important light on the sources of risk of the different positions.
The report shows that the trading book made a profit of $777,277 on that particular date. The main contributors were the North America oil desk with a gain of $445,000 followed by Europe Gas and Power with a gain of $282,539. We can also see that all books, except North America Oil, lost value because of time decay as communicated by the theta
While the overall P&L can be explained within the bands of tolerance, the Europe Oil desk might warrant further investigation as the error term is close to 10%. Depending on the results of the analysis, higher-order Greeks may be included in the report to explain the P&L dynamics of that book using sensitivities.
The P&L attribution analysis can answer the question of how the different trading desks made or lost money. Figure 1 presents a graphic display of the P&L sensitivity to each respective risk factor.
Starting with the delta, or directional price exposure contribution, we can see that the factor contribution of 148,082 for the Europe Gas and Power desks represent more than 50% of the overall P&L for the period. In contrast, we can see that the Europe Oil and North America Oil portfolios have a substantially lower contribution from the delta factor.
The gamma contribution shows the non-linear price exposures that are not explained by delta. Because the term is calculated by multiplying half of gamma times the price change squared, we can identify whether each desk is long or short gamma by looking at whether the gamma impact is positive or negative. We can see that the Europe Oil desk has a negative gamma exposure, which means they are short optionality. The long or short optionality may be due to bought or sold options as well as asset and contractual exposures or operational flexibility in their operations such as destination or sourcing options.
The vega contribution is a function of the directional volatility bets, as well the changes in market-implied volatility over the evaluation period. Gamma and vega of trading portfolios often have the same sign (positive for long optionality, negative for short optionality). As the vega contributions is negative for the Europe Oil desk but positive for the North America Oil desk, it is likely that implied volatility was higher for the period.
The time decay contribution measured by theta shows that the North America Gas and Power desk is long short-term options with significant theta, which has resulted in a loss over the period due to the lower value of those options as time passes by.
Risk equivalent positions for contracts, assets and structured products
In order to conduct P&L attribution for physical assets, long-term contracts and structured products, a common approach is to represent them as ‘risk equivalent positions’. The main idea is to summarise the profitability of those assets and contracts as a set of exposures to market prices, volatilities and any other variables that determine their value. For example, natural gas pipelines can be represented as a strip of locational basis options, while a natural gas-fired power plant asset can be represented as a strip of spark spread options.4
Tracking the cumulative results and the evolution of the contribution of the different risk factors that drive P&L can also prove to be a powerful tool in the risk manager’s arsenal
Further applications of Greek-based P&L analysis
A one-day snapshot of P&L decomposition is helpful, but it is not likely to provide the full picture of portfolio risk exposures on its own. Tracking the cumulative results and the evolution of the contribution of the different risk factors that drive P&L can also prove to be a powerful tool in the risk manager’s arsenal.
A cumulative analysis for the period under consideration or a time-series analysis of the risk drivers can prove very valuable in determining changes in trends and providing a more comprehensive footprint of the type of risks taken by each trading group as well as significant shifts in the risk profile.
Another application is the breakdown of the P&L for non-linear books into intrinsic and extrinsic value changes. The intrinsic value calculations, which are based on the automatic exercise of the options based on current market conditions at any point in time, are just dependent on the forward curves and therefore involve considerably less model risk. The extrinsic value changes may be a function of changes on unknown parameters such as volatilities, correlations or mean reversion rates, and therefore risk managers can gain an intuitive understanding of the importance of model parameter assumption in changes in the book value.
Option sensitivities and VAR models
Option portfolios often contain strategies that react in a highly non-linear fashion against changes in the underlying market. As a result, those books may appear to have relatively low risk according to standard value-at-risk (VAR) or sensitivity-analysis models, but often contain hidden non-linear exposures such as gamma and vega bets. For example, certain option strategies can be ‘delta-neutral’ in the sense that they are insensitive to small positive or negative changes in the underlying market. However, a large market change can result in large gains or losses.
In the two articles of this series, we have shown that an attribution framework can provide valuable insights into the behaviour and dynamics of key financial metrics such as P&L and, overall, improve the company’s business intelligence capabilities.
Non-linear exposures such as financial options, assets and physical contracts with embedded optionality are common in energy and commodity portfolios. P&L attribution tools are an effective way of benchmarking mark-to-model values and explaining the key sources of risk and return on those portfolios.
For non-linear exposures, the P&L attribution analysis can provide valuable insights into the value and risk of different desks and strategies and uncover the main exposures to the various risk drivers such as price, volatility and time decay.
1 The reader may refer to Blanco, C, Mauro, A, “A profit and loss attribution framework for physical and financial energy portfolios” , Energy Risk, October 2016.
2 Note that in the equation, we are assuming that the underlying quantity of the contract is constant. The change in contractual exposures is normally a P&L attribution factor, as shown in the first article of these series.
3 For readers interested in the closed-form solutions to calculate the Greeks for spread option with the Kirk pricing model, we recommend “Analytic Approximations for Spread Options” C. Alexander and A. Venkatramanan. ICMA Centre Discussion Papers in Finance DP2009–06.
4 The reader may find an application of this analysis in Mauro, A, Sgarioto, R, “An Evaluation of Italian Electricity Generation Assets Using a Spark Spread Option Model”. Available at SSRN: https://ssrn.com/abstract=1461493
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