Outline the methods for modelling derivatives risk and the difficulties met in doing so.
The idea
Derivatives credit risk is modelled by simulating how the market value of a netting set may evolve. Risk factors are simulated with Monte Carlo over many dates, trades are revalued on each path, and netting and collateral terms are applied path by path. Exposure profiles then follow: expected exposure, expected positive exposure as the time average, and potential future exposure as a high percentile. In the next formula EE is expected exposure at each date, PD the marginal default probability for that period, DF the discount factor and LGD the loss given default, summed over the dates. Credit valuation adjustment, the price of counterparty default risk on the netting set, is then roughly: CVA = LGD × Σ (EE × PD × DF). The difficulties are heavy computation for large portfolios and long horizons. Calibration is hard, since CVA is priced under risk-neutral parameters while exposure limits use real-world ones. Wrong-way risk, collateral and close-out assumptions, and model risk add to the difficulty. Exam point: exposure must be aggregated per netting set on each path, not trade by trade.
Formulas
Credit valuation adjustment from an exposure profile
CVA = LGD × Σ (EEₜ × PDₜ × DFₜ)
- LGD
- loss given default, one minus the recovery rate
- EEₜ
- the expected exposure at date t
- PDₜ
- the marginal default probability for the period ending at t
- DFₜ
- the discount factor to t
Assumes exposure and default are independent; wrong-way risk breaks it. Use marginal, not cumulative, default probabilities.
Expected positive exposure
EPE = Σ EEₜ ÷ n
- EEₜ
- the expected exposure at date t, on equally spaced dates
- n
- the number of dates
A time average of the profile. The peak of the profile is the maximum expected exposure, not the EPE.
A practice question
Potential future exposure (PFE) is best defined as:
- The time average of expected exposure over the life of the portfolioWhy not: Describes expected positive exposure, not a high percentile.
- A high percentile of future exposure at a given date - correct
- The current replacement cost of the portfolioWhy not: Describes current exposure; PFE looks forward.
- The notional amount of the portfolio less the expected recoveryWhy not: Mixes notional and recovery; PFE is a measure of future mark-to-market value.
PFE is the worst-case future exposure at a confidence level such as 97.5 or 99 percent. It differs from expected exposure (the mean) and EPE (the time average).
A harder one
A portfolio's expected exposure profile at four equally spaced dates is USD 2 million, 4 million, 5 million and 3 million. The expected positive exposure (EPE), as the simple average of the profile, is:
- USD 5.0 millionWhy not: Reports the peak of the profile rather than its average.
- USD 3.0 millionWhy not: Reports the final date's expected exposure only.
- USD 2.8 millionWhy not: Adds today's zero exposure as a fifth point and averages over five dates.
- USD 3.5 million - correct
EPE = (2 + 4 + 5 + 3) / 4 = 3.5 million. The peak is the maximum expected exposure, not the average.
The hardest tier
A risk manager computes the PFE used for credit limits with historical, real-world parameters, while the xVA desk calculates CVA for pricing and hedging using market-implied parameters. Which statement about this practice is best?
- It is inconsistent: both numbers should use historical parameters because a price must reflect past behaviourWhy not: Pricing and hedging require market-implied parameters, so that CVA is consistent with traded instruments.
- Consistent: limits need real-world risk, while CVA as a price needs risk-neutral parameters - correct
- It is inconsistent: both should use market-implied parameters because limits must equal pricesWhy not: Limit monitoring aims at real-world outcomes, not at prices.
- It is consistent, but only because PFE uses risk-neutral default probabilities and CVA uses historical onesWhy not: Reverses the split between exposure and pricing measures.
Pricing and hedging CVA uses risk-neutral calibrations such as CDS spreads and implied volatilities, while PFE for limits and capital uses real-world calibrations. Differences in calibration are a standing modelling difficulty.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
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