9A.9 The term structure across weekly and monthly expiries

Motivation

Nifty has a weekly expiry every Tuesday and monthly, quarterly and longer series (M08 · 8.11). The implied vols across them form a term structure. Its shape carries the market's forecast of variance through time, and its kinks reveal events.

Total and forward variance With ATM implied vols $\sigma_1,\sigma_2$ for expiries $\tau_1<\tau_2$ , the total variances are $w_i=\sigma_i^2\tau_i$ and the forward variance between them is

\sigma_{1,2}^2=\frac{w_2-w_1}{\tau_2-\tau_1}. \tag{9A.12}

Proposition 9A.8calendar monotonicity

Suppose $r=q=0$ and the underlying is a $\mathbb Q$ -martingale. Then for each fixed strike the call price is non-decreasing in expiry, and the ATM (at $K=S_0$ ) total implied variance is non-decreasing in expiry. In particular $\sigma^2_{1,2}\ge0$ .

Proof

For $\tau_1<\tau_2$ and the convex function $f(s)=(s-K)^+$ , conditional Jensen and the martingale property give $\mathbb E[f(S_{\tau_2})\mid\mathcal F_{\tau_1}]\ge f(\mathbb E[S_{\tau_2}\mid\mathcal F_{\tau_1}])=f(S_{\tau_1})$ . Take expectations: $C(K,\tau_2)\ge C(K,\tau_1)$ . At $K=S_0$ the Black–Scholes price is $S_0[2N(\sqrt w/2)-1]$ , strictly increasing in $w$ , so $w(\tau_2)\ge w(\tau_1)$ .∎

With nonzero rates and dividends the same holds at fixed forward moneyness $K/F$ (Gatheral 2006, ch. 2). A violation in quoted IVs means either a data problem (stale prices, wrong forward, wrong time convention) or an arbitrage.

Real term structures (NSE F&O bhavcopies; ATM Black-76 IVs from closing prices with parity forwards; calendar days).

Date	1 day	7–8 days	13–14 days	21–22 days	28–29 days
5 Oct 2026 (Mon)	17.2% (6 Oct)	14.5% (13 Oct)	14.2% (19 Oct)	14.0% (27 Oct)	13.7% (3 Nov)
6 Oct 2026 (Tue)	—	12.2% (13 Oct)	12.6% (19 Oct)	12.6% (27 Oct)	12.5% (3 Nov)

On 5 Oct the curve was inverted (short vol above long), with an expiry-day option at 17.2%: an expiry-day IV is dominated by the final day's gamma trading and by the time convention. One day later, after a +1.0% index day, the curve was flat to slightly rising. On 6 Oct the total variances were $w=0.000283$ , $0.000561$ , $0.000906$ for 7, 13 and 21 days, so the forward vols were 13.0% (days 7–13) and 12.55% (days 13–21).

Time conventions Calendar time counts weekends; trading time does not. A Friday-to-Tuesday weekly option has 4 calendar days but 2 trading days. If weekend variance is close to zero, calendar-time IV understates the per-trading-day volatility by a factor $\sqrt{2/4}$ in that example. Use one convention for the whole term structure, and treat the comparison across a weekend with care: much of an apparent "term structure" in weeklies is the weekend.