Stressed Scenarios and Model Risk

Setup

Why Scenarios and Model Risk Matter

Greeks are local sensitivity measures: they describe the P&L for infinitesimal changes in market inputs. They are insufficient for risk management because:

1. Real market moves are not infinitesimal. A 10% spot move in a day (as occurred during Black Monday 1987, Lehman 2008, or COVID March 2020) is far outside the regime where a first-order Taylor expansion is accurate.

2. Models are wrong. The price computed under a model depends on the model's assumptions. Any two models that agree on vanilla option prices can disagree on the price of a complex exotic. The spread between model prices is model risk, and it must be quantified and reserved against.

3. Historical extremes are not hypothetical. Regulatory capital (Basel III/IV, FRTB) requires firms to hold capital against both historical scenarios and internally defined stress tests. Understanding these scenarios is not optional — it determines the firm's capital base.

Notation

• $V_0$: current portfolio value.
• $\Delta V = V(\text{stressed}) - V_0$: P&L under a scenario (negative = loss).
• $\Delta S$, $\Delta\sigma$, $\Delta r$: moves in spot, implied vol, and rates under the scenario.
• Greeks: $\Delta$ (delta), $\Gamma$ (gamma), $\Theta$ (theta), $\nu$ (vega), Vanna, Volga — as defined in prior modules.

Historical Scenarios

Historical scenarios replay actual observed market moves, applied to the current portfolio. Key equity derivatives scenarios (indicative figures):

How they are applied: the observed percentage moves in each risk factor on the scenario date are applied to today's levels. The portfolio is repriced under the stressed market, and the P&L $\Delta V$ is computed. Multiple repricing models are used (base model plus alternatives) to assess model sensitivity.

Critique. Historical scenarios are limited to events that have occurred. They do not include scenarios that are plausible but unprecedented. The 2020 COVID scenario was not captured by any historical sample used in pre-2020 stress tests because the exact combination (a respiratory pandemic shutting down global economies) had not happened in the observable financial data record.

Hypothetical Scenarios

Hypothetical scenarios are designed by the risk function to probe specific vulnerabilities of the book, regardless of whether the scenario has historical precedent. They are organised by risk type:

Equity Scenarios

Parallel vol shift: add $+\Delta\sigma_0$ to all implied vols across all strikes and maturities. Measures the book's total vega exposure. Standard bump: $\Delta\sigma_0 = \pm 5\%$ or $\pm 10\%$.

Skew steepening: rotate the vol surface around the ATM point — increase OTM put vol by $+\delta$, decrease OTM call vol by $-\delta$, keep ATM unchanged. Measures vanna exposure. Standard bump: shift of $\pm 2\%$ per 10-delta unit.

Term structure twist: flatten or steepen the vol term structure — increase short-dated vol, decrease long-dated vol (or vice versa). Measures the position's sensitivity to the term structure slope.

Combined (stress test): apply simultaneous spot move and vol move. E.g., spot $-20\%$ simultaneously with implied vol $+25\%$. This is the most realistic scenario for equity options because spot and vol are correlated. The P&L is not simply the sum of the individual spot and vol P&Ls — the cross-Greek (vanna) term is significant.

Interest Rate Scenarios

Parallel shift: $\pm 100$bp move in all rates simultaneously. Measures DV01.

Steepener/flattener: long end rises, short end falls (steepener) or vice versa. Measures curve exposure (key-rate DV01 differences).

Inversion: very short rates rise, long rates fall — replicates rapid central bank tightening. Relevant for caps/floors (short-dated rate options).

Cross-Asset Scenarios

Correlation shock: equity-credit correlation changes — equities fall and credit spreads widen simultaneously. Relevant for convertibles, CLNs, and structured credit products.

Liquidity shock: bid-ask spreads widen by $3\times$. Measures the mark-to-market impact of a liquidity crisis.

P&L Attribution

The Taylor Decomposition

Daily P&L is attributed to each risk factor via a truncated Taylor expansion:

\delta V \approx \underbrace{\Delta \cdot \delta S}_{\text{delta P&L}} + \underbrace{\frac{1}{2}\Gamma \cdot (\delta S)^2}_{\text{gamma P&L}} + \underbrace{\Theta \cdot \delta t}_{\text{theta P&L}} + \underbrace{\nu \cdot \delta\sigma}_{\text{vega P&L}} + \underbrace{\mathrm{Vanna} \cdot \delta S \cdot \delta\sigma}_{\text{vanna P&L}} + \underbrace{\frac{1}{2}\mathrm{Volga} \cdot (\delta\sigma)^2}_{\text{volga P&L}} + \underbrace{\varrho \cdot \delta r}_{\text{rho P&L}} + R,

where $R$ is the residual — the unexplained P&L after all Greek contributions.

Residual Analysis

The residual $R = \delta V_{\mathrm{actual}} - \delta V_{\mathrm{Taylor}}$ captures:

1. Higher-order Taylor terms (cubic and above) that are non-negligible for large moves.
2. Model error: if the pricing model is incorrect, the Greeks it outputs are inconsistent with how the actual price responds to market moves.
3. Market-making spread and bid-ask crossing: the portfolio is repriced at mid, but actual P&L includes bid-ask costs.
4. New trades added during the period: these are not in the prior end-of-day Greeks.

A small, zero-mean residual is evidence that the model is capturing the book's risk correctly. A systematic residual (always positive after spot drops, for example) indicates a model bias — typically, a missing risk factor or an incorrect smile dynamics assumption.

Regulatory requirement (FRTB Internal Model Approval): under the Basel Fundamental Review of the Trading Book (FRTB), a trading desk's internal model must pass P&L attribution tests: the fraction of days where the residual exceeds the Greek-based P&L estimate cannot exceed prescribed thresholds. A desk that fails this test loses approval to use its internal model and must use the standardised approach (more conservative capital charge).

Model Risk Quantification

What Is Model Risk?

Model risk is the risk that the model used for pricing or risk management is incorrect, leading to mispriced trades, incorrect hedges, or under-reserved positions. It arises from:

1. Model choice: different model families (Black-Scholes, Heston, LV, rough vol) assign different prices to the same exotic.
2. Parameter uncertainty: even within a model, calibrated parameters are uncertain — the surface can be fitted by different parameter combinations with similar in-sample error but different extrapolated prices.
3. Model limitations: every model is wrong outside the instruments it was calibrated to. A Heston model calibrated to vanilla options mispricings barrier options, cliquets, and forward-starting options.

Reserve Calculation

The standard approach to quantifying model risk is the model spread:

$\mathrm{ModelRisk} = \max_{m \in \mathcal{M}} |V_m - V_{\mathrm{base}}|,$

where $\mathcal{M}$ is the set of approved alternative models and $V_{\mathrm{base}}$ is the base model price. The maximum deviation (or a conservative percentile thereof) is held as a model risk reserve — a contra-P&L charge that reduces the book's reported value.

In practice: for a vanilla book, the model spread between Black-Scholes and Heston is small (both calibrate to the smile). For a barrier book, the difference between local vol and stochastic vol can be 10–30% of the option premium — a significant reserve.

Model Validation

Model risk management requires:

1. Independent validation: the model used for pricing is validated by a team independent of the trading desk, using test cases (analytic benchmarks, exotic pricing comparisons, historical backtests).
2. Model inventory: a formal register of all models in production, with version history, validation status, and approved usage scope.
3. Limitation awareness: every model in production must have documented limitations — the conditions under which it should not be used and the estimated mispricing in those conditions.

Marking to Model vs. Marking to Market

• Level 1: positions priced from quoted market prices (no model required).
• Level 2: positions priced using observable market inputs via a model (e.g., vanilla options priced with the Black-Scholes formula and market-implied vols).
• Level 3: positions priced using a model with significant unobservable inputs (e.g., long-dated exotic options with model-dependent correlation parameters). Level 3 assets carry the highest model risk and are scrutinised by regulators.

VaR and Expected Shortfall

Definition

Value at Risk at confidence level $\alpha$ (typically $\alpha = 99\%$):

$\mathrm{VaR}_\alpha = \inf\{l > 0 : \mathbb{P}(\mathrm{Loss} > l) \leq 1 - \alpha\}.$

Interpreted as: the loss that is not exceeded with probability $\alpha$ over a given time horizon (typically 1 day or 10 days).

Expected Shortfall (ES) at confidence level $\alpha$:

$\mathrm{ES}_\alpha = \mathbb{E}[\mathrm{Loss} \mid \mathrm{Loss} > \mathrm{VaR}_\alpha] = \frac{1}{1-\alpha}\int_\alpha^1 \mathrm{VaR}_u\,du.$

ES is the expected loss in the $(1-\alpha)$ worst-case scenarios. ES is preferred to VaR under Basel III/IV because:

1. Coherent risk measure: ES satisfies subadditivity ($\mathrm{ES}(A + B) \leq \mathrm{ES}(A) + \mathrm{ES}(B)$), meaning diversification is always rewarded. VaR can violate subadditivity for non-elliptical distributions.
2. Tail sensitivity: ES captures the severity of losses beyond the threshold, not just whether a loss exceeds it. A portfolio with a 99% VaR of $10M can have wildly different tail behaviour — ES distinguishes between a $10.1M and a $100M average tail loss.

Greeks-Based VaR (Parametric VaR)

For a delta-gamma-vega approximation:

$\delta V \approx \Delta \cdot \delta S + \frac{1}{2}\Gamma \cdot (\delta S)^2 + \nu \cdot \delta\sigma,$

where $(\delta S, \delta\sigma)$ are jointly normal with covariance matrix $\Sigma$ estimated from historical data. The distribution of $\delta V$ is a non-central chi-squared (due to the gamma term) plus a normal (due to delta and vega). VaR is computed analytically or via moment-matching approximations (Cornish-Fisher expansion).

Limitations of parametric VaR: assumes normality of risk factors — underestimates tail risk. Fat tails, jump risk, and volatility clustering all cause parametric VaR to understate true tail losses.

Historical Simulation VaR

Apply the last $N$ days (typically 250–500) of observed market factor changes to today's portfolio, reprice fully, and take the empirical quantile:

$\mathrm{VaR}_\alpha^{\mathrm{HS}} = -\mathrm{Quantile}_{(1-\alpha)}(\{\delta V_1, \ldots, \delta V_N\}).$

Full repricing (not Greek approximation) is preferred for exotic books. Computational cost: $N \times$ (number of positions) full repricings per risk run.

Limitations

VaR horizon mismatch. A 10-day VaR is typically scaled from 1-day VaR as $\mathrm{VaR}_{10} = \sqrt{10} \cdot \mathrm{VaR}_1$. This scaling assumes i.i.d. returns — invalid during crises (serial correlation in large-loss events). The Basel III FRTB moves to stress-period VaR (calibrated during a 1-year stress period) to address this.

Model risk in VaR. VaR itself is model-dependent: the choice of historical window, the treatment of jumps, and the repricing model all affect the VaR estimate. A bank that underestimates tail risk in its VaR model holds insufficient capital.

Scenarios vs. probabilistic measures. Stress scenarios identify specific large moves; VaR/ES aggregate across all scenarios according to their probability. Neither is sufficient alone: VaR may miss the severity of specific crisis events (which may be rare); scenarios may miss the probability structure of joint moves.

Interview Angle

L1. What is Value at Risk? Why has Expected Shortfall replaced VaR as the primary regulatory risk measure under Basel III/IV?

VaR at confidence level $\alpha$ is the loss not exceeded with probability $\alpha$. ES is the expected loss given that loss exceeds VaR. ES is preferred because: (a) it is a coherent risk measure (subadditive — diversification always reduces risk); (b) it captures tail severity, not just whether a loss exceeds a threshold; (c) VaR can be non-subadditive for fat-tailed distributions, perversely penalising diversification.

L2. Write down the full second-order P&L attribution formula, including vanna and volga. For a scenario with spot $-15\%$ and implied vol $+20\%$, which terms dominate for (a) an ATM call, (b) an OTM put, (c) a knock-out barrier?

$\delta V \approx \Delta\delta S + \frac{1}{2}\Gamma(\delta S)^2 + \Theta\delta t + \nu\delta\sigma + \text{Vanna}\cdot\delta S\cdot\delta\sigma + \frac{1}{2}\text{Volga}\cdot(\delta\sigma)^2$.

(a) ATM call: delta dominates (−15% spot move); gamma is non-trivial (spot fell, gamma cash is positive); vega is large (vol up 20%, long vega position wins); vanna is small (ATM $d_2 \approx 0$); volga is also small (ATM volga is negative).

(b) OTM put: delta (now deep ITM after −15%: delta $\to -1$); gamma small (now deep ITM); vanna is significant (OTM put was long vanna — as vol rose, delta became more negative, consistent with positive vanna sign for short put); volga positive and substantial for a put that was initially OTM.

(c) Knock-out barrier: if spot approaching the barrier, delta P&L is large and discontinuous (the option is close to being knocked out). Vanna P&L is extreme — the knock-out delta changed sharply as spot moved. The simultaneous vol rise $\delta\sigma > 0$ increased the barrier-crossing probability, creating large negative vanna P&L for a long knock-out position. The gamma near the barrier is large negative (the knock-out has negative curvature near the barrier: the option value decreases as spot approaches the barrier). This combination — large negative gamma, extreme vanna, near-barrier proximity — is why barrier options require tight stop-loss limits and frequent rebalancing near the barrier.

L3. What is the distinction between Level 1, Level 2, and Level 3 fair value? How is a model risk reserve computed, and what determines its size for a barrier option book?

Levels refer to the observability of pricing inputs. Level 1: directly quoted prices (exchange-listed). Level 2: model with observable inputs (vanilla options with market-quoted implied vols). Level 3: model with significant unobservable inputs (long-dated exotics, illiquid barrier options).

Model risk reserve: $\mathrm{Reserve} = \max_{m \in \mathcal{M}}|V_m - V_{\mathrm{base}}|$, where $\mathcal{M}$ includes all approved alternative model specifications. For a barrier option book, the key alternative models are: local vol (Dupire), stochastic vol (Heston), SABR-LV, jump diffusion (Merton/Bates). The spread between LV and SV prices can be 10–30% of premium for short-dated near-barrier options, because LV and SV imply radically different smile dynamics (as discussed in the Derivatives Pricing course). The reserve is the maximum such spread, held as a contra-P&L charge. Its size grows with: (a) proximity of spot to the barrier, (b) short time to expiry, (c) magnitude of the smile (steeper skew → bigger model spread).