14.14 Scenario P&L: full revaluation versus the delta–gamma–vega expansion

Motivation

A stress test asks: if the Nifty gaps 8% down, implied volatility jumps 6 points and crude rises 12%, what is my P&L tonight? There are two ways to answer. Full revaluation reprices every instrument at the shocked inputs. The Greek expansion multiplies the shocks by sensitivities. The expansion is fast, additive across positions, and the language risk reports are written in. It is also wrong for large moves on option books, and you need to know in which direction.

14.14.1The second-order expansion#

Setup

The book's value is $V(x, t)$ , where $x = (S_1, \dots, S_n, \sigma_1, \dots, \sigma_m)$ collects underlying prices (index futures, stocks, MCX futures) and implied volatilities, and $t$ is calendar time. $V$ is twice continuously differentiable in a neighbourhood of the current point (true for European options before expiry). A scenario is a shock $(\Delta x, \Delta t)$ .

Proposition 14.22delta–gamma–vega–theta approximation

Write $\Delta_i = \partial V/\partial S_i$ , $\Gamma_{ij} = \partial^2V/\partial S_i\partial S_j$ , $\mathcal V_k = \partial V/\partial\sigma_k$ , $\Theta = \partial V/\partial t$ . Then

\Delta V = \Theta\,\Delta t + \sum_i \Delta_i\,\Delta S_i + \tfrac12\sum_{i,j}\Gamma_{ij}\,\Delta S_i\,\Delta S_j + \sum_k \mathcal V_k\,\Delta\sigma_k + R, \tag{14.13}

where $R$ collects the second-order terms in $(\Delta\sigma, \Delta t)$ and their cross terms with $\Delta S$ (vanna $\partial^2V/\partial S\partial\sigma$ , volga $\partial^2V/\partial\sigma^2$ , charm $\partial^2 V/\partial S\partial t$ ) and all third-order terms. In particular $R = O(\|(\Delta x, \Delta t)\|^2)$ for the dropped second-order terms plus $O(\|\Delta S\|^3)$ .

Proof

Taylor's theorem in several variables to second order at $(x, t)$ , with remainder in integral form, then separate the retained terms (all first-order terms and the second-order terms in $\Delta S$ only) from the rest.∎

The approximation keeps second order in price because gamma is the dominant curvature for short-dated index options. It keeps vega at first order only. Theta enters because a one-day scenario includes one day of decay.

Rupee Greeks Risk is reported per percentage move, not per index point, so that instruments with different price levels can be added. Writing $\Delta S_i = S_i r_i$ :

\begin{gathered} \Delta V \approx \Theta\,\Delta t + \sum_i D_i\,r_i + \tfrac12\sum_{i} G_i\,r_i^2 + \sum_k \mathcal V_k\,\Delta\sigma_k, \\ D_i = S_i\Delta_i,\qquad G_i = S_i^2\Gamma_{ii}, \end{gathered} \tag{14.14}

for a book whose options are each on a single underlying (no cross-gammas). Quoted per 1% ( $r_i = 0.01$ ): ₹ delta $D_i/100$ is the rupee P&L of a 1% move, and ₹ gamma $G_i/10^4$ is the change in ₹ delta for a 1% move; $\mathcal V_k/100$ is ₹ per volatility point.

Factor mapping Cash equities are mapped to the Nifty by beta: if $r_i = \beta_i r_N + \varepsilon_i$ , the book's Nifty ₹ delta is $D_N = \sum_i \beta_i D_i$ , and stock-option gammas map as $\beta_i^2G_i$ . The idiosyncratic terms $\varepsilon_i$ are dropped in an index stress and must be stressed separately (a single name can gap 20% on results; M06A). Betas themselves rise in crises (14.6), so the mapped delta understates stress exposure for low-beta names.

14.14.2When the quadratic fails, and in which direction#

Proposition 14.23exact error of the delta–gamma term

Fix volatility and time. For a single option position with value $v(F)$ and gamma $\Gamma(F)$ , the exact P&L of a move $h$ in the underlying is

v(F_0 + h) - v(F_0) = \Delta(F_0)\,h + \int_0^h (h - s)\,\Gamma(F_0 + s)\,ds, \tag{14.15}

so the error of the delta–gamma approximation is

E(h) = \int_0^h (h - s)\big[\Gamma(F_0 + s) - \Gamma(F_0)\big]\,ds .

Proof

Taylor's theorem with integral remainder: $v(F_0 + h) = v(F_0) + v'(F_0)h + \int_0^h (h - s)v''(F_0 + s)\,ds$ , and $\tfrac12\Gamma(F_0)h^2 = \int_0^h (h - s)\Gamma(F_0)\,ds$ .∎

The weight $(h - s)$ is non-negative along the path, so the sign of the error is the sign of the gamma change along the path. Black-76 gamma (M08 · 8.9) is a hump in $F$ that peaks near the strike. Two cases follow for a short option (whose position gamma is $-\Gamma$ ):

Move towards the strike (short out-of-the-money option, market moving against it): gamma grows along the path, the true loss is larger than the quadratic predicts.
Move through and far beyond the strike: gamma falls back towards zero, the option becomes delta-one, and the quadratic, whose curvature never stops, overstates the loss.

For a short Nifty 25500 call (23-Nov-2026 expiry, Nifty future 24,588, implied volatility 13.47%, 18 days), one lot of 65: a 4% rally costs ₹19,533 in full revaluation against ₹15,561 from delta–gamma, a 20% understatement; at 6% the understatement is 27% (₹40,669 against ₹29,531). For a short at-the-money straddle on the same expiry, a 10% fall costs ₹1,15,110 in full revaluation against ₹1,91,903 from delta–gamma. Neither error is small, and their signs differ. Only full revaluation is reliable for scenarios beyond about one daily standard deviation of the underlying for options within a few weeks of expiry.

Practitioner notes

SPAN is full revaluation on a grid. Exchange margin (M07; M09A) reprices the portfolio over a fixed set of price and volatility scenarios and charges the worst. That is why a short strangle's margin is not delta-based: the worst point is the edge of the scan range, where the Greek expansion is worst.
The volatility shock must be specified with the price shock. In a sell-off, index implied volatility rises and the put skew steepens (M09 · 9.5). A stress that shocks price and holds volatility fixed understates the loss on short puts twice over: no vega loss, and the wrong gamma.
Greeks are for monitoring, revaluation is for limits. Use ₹ delta, gamma, vega and theta to see the shape of the book every morning; use full-revaluation scenarios to decide whether the book is within its stress limit.