12.6 Asymmetric payoffs: convexity and options as geopolitical hedges

Motivation

Geopolitical payoffs are asymmetric: a deal moves Brent down 10–15%, an escalation can move it up 40% or more in weeks (2026). A linear position (futures) has the same exposure to both. An option buys exposure to one tail at a known maximum cost. Traders who survive geopolitical regimes think in payoff shapes, not point forecasts.

12.6.1Convexity and uncertainty#

Convexity and Jensen If a payoff $f(S_T)$ is convex, then $\mathbb E[f(S_T)]\ge f(\mathbb E[S_T])$ . Spreading the distribution while keeping the mean fixed, which is what a rise in uncertainty does, raises the expected payoff. A long option is long convexity, so it gains from an increase in uncertainty itself, not only from the direction of the move. This is why implied volatility rises when GPR rises: India VIX rose 30.3% on 24 Feb 2022 and 65.6% on 7 Apr 2025¹.

12.6.2Hedging through a scenario tree#

Worked example 12.3hedging a Nifty position through a scenario tree

Position. A trader is long one Nifty futures lot (65 units, the lot size since January 2026 contracts²) at 24,536.6.
Hedge. A three-month 23,000 put. The premium is ₹238.1 per unit, a Black–Scholes price at an assumed 18% implied volatility, 5.5% rate and 1.2% dividend yield (M08). It is not a market quote.
Scenarios at expiry. A deal moves the Nifty +6%, a stalemate 0% and escalation −15%. These are assumptions.

Scenario	$S_T$	Futures P&L (₹)	Futures + put P&L (₹)
Deal	26,008.8	+95,693	+80,216
Stalemate	24,536.6	0	−15,477
Escalation	20,856.1	−2,39,232	−1,15,356

Probability-weighted outcomes:

Probabilities	EV unhedged	EV hedged	sd unhedged	sd hedged
Prior $(0.5,0.3,0.2)$	0	+12,394	1,26,590	76,138
After news 1 $(0.217,0.391,0.391)$	−72,810	−33,757	1,38,147	74,566
After news 2 $(0.426,0.479,0.096)$	+17,815	+15,681	95,178	62,298

Three lessons, each general:

The hedge has positive EV under the trader's prior because the trader's 20% probability of a 15% fall is fatter than what an 18% implied volatility prices. Option prices embed risk-neutral probabilities. If your physical probability of the tail exceeds the risk-neutral one, the option is cheap to you. Usually it is the other way round: the variance risk premium (M09) means puts are on average overpriced. A hedge is then insurance with negative EV, bought to cut variance and ruin risk (M03).
The hedge cuts the standard deviation by about 40% under the prior and caps the loss at $(24{,}536.6-23{,}000+238.1)\times65=\text{₹}1{,}15{,}356$ however bad escalation gets. Without it the loss is unbounded in principle.
The value of the hedge depends on the news state. After news 2 the hedge costs EV, but it still cuts the sd by a third. Whether to keep it is a risk-budget question, not a forecast.

12.6.3Instruments for an Indian trader#

Practical instruments for an Indian trader

Nifty and Bank Nifty index options, for equity downside.
MCX crude options, which are options on MCX crude futures; an ITM option devolves into the future at expiry, needing full futures margin (M10) [S].
USDINR options on NSE and BSE, for rupee depreciation.
Gold ETFs or MCX gold, though gold's sign in a war depends on real yields (March 2026).

Each hedge has basis risk: the instrument may not move with your actual exposure. Each also has timing risk: options decay (theta, M08) while you wait.