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