9A.7 The straddle-implied move

Motivation

"The straddle is pricing a 300-point move" is the most common sentence on an options desk before an event. It has an exact meaning, and it is not "one standard deviation".

9A.7.1The ATM-forward straddle#

Proposition 9A.5ATM-forward straddle

Under Black-76 with forward $F$ , volatility $\sigma$ and $x=\sigma\sqrt\tau$ , the straddle struck at $K=F$ is

\Pi_{ATM}=2e^{-r\tau}F\left[2N\!\left(\tfrac x2\right)-1\right] \tag{9A.7}

and

\begin{aligned} \Pi_{ATM}&=e^{-r\tau}F\sigma\sqrt\tau\sqrt{\tfrac2\pi}\left(1-\frac{\sigma^2\tau}{24}+O(\sigma^4\tau^2)\right)\\ &\le e^{-r\tau}F\sigma\sqrt\tau\sqrt{\tfrac2\pi}\approx0.8\,e^{-r\tau}F\sigma\sqrt\tau. \end{aligned} \tag{9A.8}

Proof

At $K=F$ , $d_1=x/2$ and $d_2=-x/2$ , so the call is $e^{-r\tau}F[N(x/2)-N(-x/2)]=e^{-r\tau}F[2N(x/2)-1]$ . Parity at $K=F$ gives $C-P=0$ , so the straddle is twice the call: (9A.7). Write $2N(y)-1=2\int_0^y\varphi(u)\,du$ with $\varphi(u)=(2\pi)^{-1/2}(1-u^2/2+O(u^4))$ ; then $2N(y)-1=\sqrt{2/\pi}\,(y-y^3/6+O(y^5))$ . With $y=x/2$ : $2\big[2N(x/2)-1\big]=\sqrt{2/\pi}\,(x-x^3/24+O(x^5))$ , which is (9A.8). The bound: $\varphi(u)\le\varphi(0)$ , so $2N(y)-1\le2y\varphi(0)=y\sqrt{2/\pi}$ .∎

The relative error of the linear rule is $\sigma^2\tau/24$ : for a 7-day Nifty straddle at 13% it is $1.35\times10^{-5}$ . The approximation is exact for every practical expiry.

9A.7.2The expected absolute move#

Proposition 9A.6the straddle is the expected absolute move

(a) Under the pricing measure, $e^{r\tau}\Pi_{ATM}=\mathbb E^{\mathbb Q}|F_T-F|$ exactly. (b) If the move $S_T-S$ is normal with mean zero and standard deviation $S\sigma\sqrt\tau$ ,

\begin{gathered} \mathbb E|S_T-S|=S\sigma\sqrt\tau\sqrt{\tfrac2\pi},\\ \mathbb P\big(|S_T-S|<\mathbb E|S_T-S|\big)=2N\!\left(\sqrt{2/\pi}\right)-1=0.5751. \end{gathered} \tag{9A.9}

Proof

(a) The straddle pays $|F_T-K|$ at $T$ ; at $K=F$ its price is $e^{-r\tau}\mathbb E^{\mathbb Q}|F_T-F|$ by risk-neutral valuation (M08 · 8.7). (b) For $Z\sim N(0,1)$ , $\mathbb E|Z|=2\int_0^\infty z\varphi(z)dz=2\varphi(0)=\sqrt{2/\pi}$ . Then $\mathbb P(|Z|<\sqrt{2/\pi})=2N(\sqrt{2/\pi})-1$ .∎

So the straddle price is the market's mean absolute move, about 0.8 standard deviations, and in a normal world the index ends inside $\pm$ straddle about 57.5% of the time. The one-standard-deviation move is $\Pi/0.8\approx1.25\,\Pi$ . Two cautions: (a) is a statement under $\mathbb Q$ ; with a variance premium (M09 · 9.4) the real-world mean absolute move is smaller. And real moves are fat-tailed, so the probability of finishing inside the straddle is typically higher than 57.5% and the probability of a very large move also higher.

9A.7.3Nifty straddles in 2026#

Worked example 9A.3the eve of the RBI decision

On 6 Oct 2026 the Nifty 13 Oct expiry (7 days) had a parity forward of 22,747.10 and the 22,750 call and put closed at 151.15 and 154.05: straddle 305.20 (NSE F&O bhavcopy, 6 Oct 2026). The quick inversion $\sigma\approx\Pi e^{r\tau}/(0.798F\sqrt\tau)$ gives 12.155%; the exact Black-76 inversion with $r=5.25\%$ gives 12.154%. One standard deviation over the week is $22{,}776\times0.1215\sqrt{7/365}=383$ points; the expected absolute move is 306 points, the straddle itself.

Indian sample Over the 31 expiries of 9A.16 the day-before ATM straddle of the expiring Nifty series averaged 178.3 points and the realised absolute move to settlement averaged 149.6 points (ratio 0.84). The move was smaller than the straddle on 71% of days, and a buyer of every straddle at the closing prices lost 28.6 points on average before costs ( $t=-1.44$ ). The direction is what a variance premium predicts; the size of the sample cannot establish it.

See it live: the current ATM straddle price is on each F&O stock's page (https://oicharts.com/stocks/SYMBOL); divide by the spot price for the market's expected absolute move to expiry, and by 0.8 for one standard deviation (Proposition 9A.6).

9A.7.1The ATM-forward straddle#

9A.7.2The expected absolute move#

9A.7.3Nifty straddles in 2026#

The Nifty option chain from the NSE bhavcopy

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