Volatility formula reference

Every formula used anywhere on this site, extracted from the concept pages at build time, with every symbol defined exactly once.

Quick answer: The volatility formula reference lists every formula used anywhere on VolatilityGyan, pulled directly from the concept pages at build time so it cannot drift from them, together with a single definition for every symbol and an explicit statement of the conventions — 252 trading days for annualising volatility, 365 calendar days for time to expiry.

Every formula used anywhere on VolatilityGyan, extracted directly from the concept pages at build time. If a formula changes on its page it changes here, because both read the same data. 68 formulas, 189 distinct symbols.

The formulas

Concept	Formula	Units	Section
What is Volatility?	σ_annual = σ_daily × √252	Annualised %	Core
Historical Volatility	HV = √( (1 / (n − 1)) × Σ (r_t − r̄)² ) × √252, with r_t = ln(P_t / P_{t−1})	Annualised %	Core
Realized Volatility	σ_Parkinson = √( (1 / (4 ln 2)) × mean( ln(H_t / L_t)² ) ) × √252	Annualised %	Core
Implied Volatility	Find σ such that: BS(S, K, T, r, σ) = P_market	Annualised %	Core
Expected Volatility	σ²_t = λ·σ²_{t−1} + (1 − λ)·r²_{t−1}, λ ≈ 0.94	Annualised %	Core
Forward Volatility	σ_fwd = √( (σ₂²·T₂ − σ₁²·T₁) / (T₂ − T₁) )	Annualised %	Core
Annualized Volatility	σ_annual = σ_period × √N	Annualised %	Core
Intraday Volatility	σ_annual = σ_bucket × √(B × 252)	Annualised % or per-bucket	Core
How IV is Calculated	Find σ such that: BS(S, K, T, r, σ) = P_market	Annualised %	Implied
Why IV Changes	σ_implied = BS⁻¹(P_market), where P_market clears Demand(σ) = Supply(σ)	Annualised %	Implied
IV Expansion	Expansion: σ_t stays elevated for t = 1 … N with N large; Spike: σ_t → σ_baseline within weeks	Annualised %	Implied
IV Crush	Crush loss ≈ ν × (σ_before − σ_after)	Annualised %	Implied
Event Volatility	σ²_total × T = σ²_diffusive × (T − 1/365) + σ²_event × (1/365)	Annualised %	Implied
Earnings Volatility	Implied move (%) ≈ Straddle_ATM / S	Annualised %	Implied
Volatility Smile	σ(K) = σ_ATM + c · [ln(K / S)]²	Annualised %	Implied
Volatility Skew	σ(K) = σ_ATM − β · ln(K / S) + c · [ln(K / S)]²	Volatility points	Implied
Volatility Surface	w(k, T) = σ(k, T)² · T, with ∂w/∂T ≥ 0	Annualised %	Implied
Sticky Strike	Δ_adj = Δ_BS + ν · (∂σ/∂S)	Regime description	Implied
Sticky Delta	Δ_adj = Δ_BS + ν · (∂σ/∂S)	Regime description	Implied
IV Rank	IVR = (IV_today − IV_low) / (IV_high − IV_low) × 100	0–100	Metrics
IV Percentile	IVP = (D_below / N) × 100	0–100	Metrics
HV vs IV	spread = IV_T − HV_T	Volatility points	Metrics
Volatility Risk Premium	VRP = E[σ_implied] − E[σ_realised]	Volatility points	Metrics
Expected Move	EM(1σ) = S × σ × √(days ÷ 365)	Points or ₹	Metrics
Standard Deviation	σ = √( Σ(x_i − x̄)² ÷ (n − 1) )	Same units as the data	Metrics
Variance	σ² = ( Σ(x_i − x̄)² ÷ (n − 1) )	%² (or fraction²)	Metrics
Realized Variance	RV = Σ_{t=1}^{n} r_t²	%² (or fraction²)	Metrics
What is Term Structure?	σ_ATM(T) plotted against T subject to σ²(T)·T non-decreasing in T	Annualised %	Term
Contango	σ_ATM(T_far) is greater than σ_ATM(T_near) → curve in contango	Annualised %	Term
Backwardation	σ_ATM(T_near) is greater than σ_ATM(T_far) → curve inverted (backwardation)	Annualised %	Term
Calendar Structure	σ_fwd = √( (σ₂²·T₂ − σ₁²·T₁) / (T₂ − T₁) )	Volatility points	Term
Expiry Structure	ExpectedMove_expiry = S × σ_expiry × √(D / 365)	Annualised %	Term
Volatility Curve	RV_n = √( (252 / n) × Σ r_t² )	Annualised %	Term
Event Premium	σ²_total × T = σ²_diffusive × (T − t_e) + σ²_event × t_e	Volatility points or one-day move %	Term
Rolling Volatility	RV_n(t) = √( (252 / n) × Σ r_{t−i}² ), i = 0 … n−1	Annualised %	Term
India VIX	σ² = (2/T)·Σ (ΔK_i / K_i²)·e^{rT}·Q(K_i) − (1/T)·(F/K_0 − 1)²	Annualised %	Indices
CBOE VIX	σ² = (2/T)·Σ (ΔK_i / K_i²)·e^{rT}·Q(K_i) − (1/T)·(F/K_0 − 1)²	Annualised %	Indices
VVIX	VVIX = 100 × √( (2/T)·Σ (ΔK_i / K_i²)·e^{rT}·Q(K_i) − (1/T)·(F/K_0 − 1)² )	Annualised %	Indices
VIX Futures	F(t) = θ + (V_0 − θ)·e^{−k·t}	Index points	Indices
VIX Options	C = (F − K) if F is above K at expiry, else 0	Index points	Indices
VIX Term Structure	TS = VIX / VIX3M	Index points	Indices
Fear Index	σ_daily = VIX / √252	Index points	Indices
Market Sentiment	RR = σ_25put − σ_25call	Qualitative + index points	Indices
Long Volatility	V ≈ intrinsic + extrinsic; dV ≈ ν·dσ + ½·Γ·(dS)² + Θ·dt	₹ per unit	Strategies
Short Volatility	V_short ≈ premium − option value; dV_short ≈ −ν·dσ − ½·Γ·(dS)² − Θ_paid	₹ per unit	Strategies
Long Vega	ν = S · φ(d₁) · √T / 100; ν(120d) / ν(7d) ≈ √(120/7) ≈ 3.9	₹ per 1 IV point	Strategies
Short Vega	dV_short ≈ −ν·dσ − ½·vomma·(dσ)²; ν becomes more negative as σ rises	₹ per 1 IV point	Strategies
Gamma Scalping	hedged P&L over a step ≈ ½·Γ·S²·(σ_realised² − σ_implied²)·dt; per-hedge harvest ≈ ½·Γ·(ΔS)²	₹ per unit	Strategies
Delta Hedging	Position delta = Δ_option × contracts × lot; hedge = −(position delta) units of NIFTY futures	Delta / units of underlying	Strategies
Volatility Arbitrage	P&L ≈ ∫ ½ · Γ · S² · (σ²_realised − σ²_implied) · dt	Volatility points	Strategies
Dispersion Trading	σ_index ≈ σ_avg × √(ρ + (1 − ρ)/n)	Volatility points / correlation	Strategies
Calendar Trading Concepts	σ_forward = √( (σ²_far · T_far − σ²_near · T_near) / (T_far − T_near) )	Volatility points	Strategies
IV and Option Premium	Premium = Intrinsic + TimeValue, where Intrinsic = max(S − K, 0) for a call and TimeValue = BS(S, K, T, r, σ) − Intrinsic	₹	Options
IV and Theta	θ ≈ −½ · Γ · S² · σ² (per year)	₹ per calendar day	Options
IV and Vega	ν = ( S · φ(d₁) · √T ) / 100, d₁ = [ ln(S/K) + (r + σ²/2)·T ] / ( σ·√T )	₹ per 1 IV point	Options
IV Before Expiry	Premium_ATM ≈ S · σ · √(T / 2π), so ∂σ/∂Price = 1 / vega, and vega ∝ √T → 0	Annualised %	Options
IV After Expiry	σ_contract = σ(K, T_expiry); σ_30day = interpolate( σ_near · w, σ_far · (1 − w) ) to hold T = 30 days	Annualised %	Options
IV Around Events	σ²_total · T = σ²_diff · (T − t_e) + σ²_event · t_e, t_e = 1/365	Annualised %	Options
IV Around RBI Policy	σ²_event = σ²_total · D − σ²_diff · (D − 1), t_e = 1/365	Annualised %	Options
IV Around the Union Budget	σ²_event = σ²_total · D − σ²_diff · (D − 1), t_e = 1/365	Annualised %	Options
IV Around Election Results	σ²_event = σ²_total · D − σ²_diff · (D − 1), t_e = 1/365	Annualised %	Options
Low Volatility Markets	σ_ann = σ_daily × √252 · 'low volatility' ⇔ σ_ann under ~11%	Annualised %	Regimes
High Volatility Markets	Risk_new / Risk_old = σ_new / σ_old · e.g. 26 / 10 = 2.6	Annualised %	Regimes
Trending Markets	VR(q) = Var(r_q) / ( q × Var(r_1) )	Annualised %	Regimes
Range-bound Markets	VR(q) = Var(r_q) / ( q × Var(r_1) ) · range ⇒ VR(q) under 1	Annualised %	Regimes
Crisis Volatility	σ²_p = Σ wᵢ² σᵢ² + Σᵢ≠ⱼ wᵢ wⱼ ρᵢⱼ σᵢ σⱼ → as ρᵢⱼ → 1, σ_p → Σ wᵢ σᵢ	Annualised %	Regimes
Mean Reversion in Volatility	dσ = κ(θ − σ) dt + ξ√σ dW	Annualised %	Regimes
Volatility Clustering	σ²_t = ω + α·r²_{t−1} + β·σ²_{t−1}	Annualised %	Regimes

Every symbol, defined once

Notation across derivatives literature is not standardised. This is the notation VolatilityGyan uses, consistently, on every page.

Symbol	Meaning
1/365	The fraction of a year occupied by the single event day, in calendar-day convention.
252	Approximate number of trading days in an Indian calendar year — the annualisation factor for volatility built on daily returns.
365	Calendar days in a year — the denominator that scales an annual volatility down to the option's actual horizon.
4 ln 2	The scaling constant 4 × ln 2 ≈ 2.7726; its reciprocal ≈ 0.3607 makes the mean squared log-range an unbiased variance estimate under the model's assumptions.
B	The number of buckets in one trading session. For 5-minute buckets in a 375-minute NSE session, B = 75.
BS(·)	The Black–Scholes–Merton price of a European option, given all five inputs.
BS⁻¹(·)	The implied-volatility inversion: the market price mapped back to the volatility that reproduces it.
C	Value of the VIX call at expiry, in index points. A put replaces the payoff with (K − F) if positive.
D	Calendar days from today to that expiry. Calendar, not trading — the option decays over weekends even though the spot does not move.
D_below	The number of days in the window on which implied volatility closed strictly below today's reading.
Demand(σ)	Quantity of the option buyers want at each implied-volatility level — driven by hedging need, event fear, and positioning.
EM(1σ)	The one-standard-deviation expected move, in points or rupees, either side of spot.
E[σ_implied]	The expected implied volatility priced into options at the start of each period — the forward-looking price, known when the position is opened.
E[σ_realised]	The expected volatility that subsequently realises over the period — the outcome, knowable only at the end. This is the number that does not exist yet when the trade is placed.
ExpectedMove_expiry	The one-standard-deviation move, in points of the underlying, that this particular expiry is pricing over its own life.
F	The forward NIFTY level for the expiry, implied from the put-call parity strike where call and put prices are closest.
F(t)	Fair value of the VIX future maturing t months from today, in index points.
HV	Historical volatility — the annualised figure, expressed as a percentage (0.135 = 13.5%).
HV_T	Realised (historical) volatility over a horizon comparable to T, annualised on 252 trading days. Backward-looking — a measurement of a period that has finished.
H_t	The intraday high price on day t.
IVP	IV Percentile — the output, 0 to 100. The share of the window's days that were calmer than today.
IVR	IV Rank — the output, a number from 0 to 100 (some platforms report 0 to 1). It is a position, not a probability.
IV_T	Implied volatility of an option with tenor T (e.g. the 30-day at-the-money IV), annualised. Forward-looking — a price for the period ahead.
IV_high	The highest implied volatility observed in the lookback window. One panic spike sets this and holds it for the whole window.
IV_low	The lowest implied volatility observed in the lookback window (52 weeks by convention). One calm day can set this.
IV_today	The current implied volatility, conventionally the at-the-money near-month value or India VIX for NIFTY.
Intrinsic	Value if exercised immediately: max(S − K, 0) for a call, max(K − S, 0) for a put. Immune to implied volatility.
K	Strike price of the option contract.
K_0	The first strike at or below the forward F — the boundary between the puts and the calls in the strip.
K_i	The i-th option strike used in the strip.
L_t	The intraday low price on day t.
N	The number of periods in one year. 252 trading days, 52 weeks, or 12 months, depending on the sampling frequency of σ_period.
P&L	Accumulated profit and loss of the delta-hedged option over its life, per unit of the underlying.
P_market	The option's observed market price. On an illiquid strike, use the bid-ask midpoint; a single traded print can be stale.
P_t	The closing price on day t; P_{t−1} is the previous close. Only closing prices enter — the day's high and low are ignored.
Premium	The option's total market price, in rupees per unit of the underlying.
Premium_ATM	Time value of an at-the-money option, in rupees. Collapses toward zero as expiry nears because it is proportional to √T.
Q(K_i)	The price of the out-of-the-money option at strike K_i — the bid-ask MIDPOINT, never the last traded price, which can be stale on a far strike.
RR	25-delta risk reversal, in volatility points. Larger positive values mean a steeper downside skew — more crowded demand for protection.
RV	Realized variance — the accumulated squared return over the period, in squared units (%² or fraction²).
RV_n	Realised volatility measured over a window of n trading days, annualised as a decimal (0.14 = 14%).
RV_n(t)	Rolling realised volatility as of day t over an n-day window, annualised as a decimal (0.14 = 14%).
Risk_new	The position's daily risk (e.g. one-standard-deviation profit-and-loss) in the new, higher-volatility regime.
Risk_old	The position's daily risk when it was opened, in the calmer regime it was sized for.
S	Spot price of the underlying — 24,000 for NIFTY in every example on this site.
Straddle_ATM	Combined price of the at-the-money call and put on the nearest expiry after results, just before the announcement (in rupees).
Supply(σ)	Quantity sellers will provide at each level — driven by risk appetite, existing inventory, and hedging cost.
T	Time to expiry in years, measured as calendar days ÷ 365 (interest accrues on weekends, even though prices do not move).
TS	Term-structure ratio, dimensionless. Below 1 is upward-sloping and calm; above 1 is inverted and stressed.
T_expiry	Time to that contract's fixed expiry. When it reaches zero the contract, and its implied volatility, cease to exist.
T_far	Time to expiry of the far contract in years, calendar days ÷ 365. T_far is greater than T_near.
T_near	Time to expiry of the near contract in years, calendar days ÷ 365.
TimeValue	Premium minus intrinsic value — the part IV prices. Always at least zero for a European option that cannot be exercised early into a loss.
T₁	Time to the nearer date, in years (calendar days ÷ 365) or in days — used consistently with T₂.
T₂	Time to the farther date, in the same unit as T₁. T₂ − T₁ is the length of the forward window.
V	Value of the long-volatility position per unit of the underlying, in ₹.
VIX	The 30-day constant-maturity volatility-index level — the headline number.
VIX3M	The 3-month constant-maturity volatility-index level (the 91-day point).
VR(q)	The variance ratio at horizon q — the diagnostic statistic. Equals 1 under independence, exceeds 1 for a trend, falls under 1 for mean reversion.
VRP	The volatility risk premium in volatility points. Small and positive on average; the sign in any single period is unknown in advance.
VVIX	The reported index — the annualised 30-day expected volatility of the VIX, in percent.
V_0	Today's spot VIX level — 12.4 in the calm example, 27 in the stressed example.
V_short	Value of the short-volatility position to the seller, per unit, in ₹.
Var(r_1)	Variance of the one-period (daily) return of the series.
Var(r_q)	Variance of the q-period (multi-day) return of the series.
c	Curvature (convexity) of the smile. A larger c means steeper wings and a market pricing fatter tails. On the site's reference NIFTY chart c is about 2.6.
contracts	Number of option contracts held; negative if the position is short.
dS	Change in the underlying's price over the step, in points.
dV	Change in the position's value over a small time step.
dV_short	Change in the seller's position value over a small step.
dW	The increment of a Wiener process (standard Brownian motion) — the random, mean-zero shock over dt.
days	Calendar days to expiry, counted from now to the expiry date inclusive of weekends.
dt	The time step, measured in days.
dσ	Change in implied volatility, in percentage points.
d₁	The Black–Scholes d₁ term: (ln(S/K) + (r + σ²/2)·T) / (σ·√T). It carries the strike, rate and volatility dependence.
e	The base of the natural logarithm, ≈ 2.71828; e^{−k·t} is the fraction of today's gap to the mean that remains at expiry.
extrinsic	Time value: everything in the premium above intrinsic. This is the part that decays to zero at expiry.
hedged P&L	The profit or loss of the delta-hedged position over the step, in ₹ per unit — the residual after the delta hedge has stripped out direction.
intrinsic	The in-the-money amount that would be realised on immediate expiry — zero for an at-the-money option.
k	Log-moneyness, ln(K/S): 0 at the money, negative for puts below spot, positive for calls above.
ln	The natural logarithm. Log-moneyness ln(K/S) is used rather than K − S so that a strike and its mirror image (K/S and S/K) sit equal distances from the centre.
ln(H_t / L_t)	The log-range: the natural logarithm of the day's high divided by its low — the estimator's raw signal.
lot	Contract multiplier — 75 for NIFTY, 30 for BANKNIFTY — converting a per-unit delta into units of the underlying.
m	The one-standard-deviation event-day move the option market is implying, as a fraction of spot: m = σ_event · √(1/365).
mean(·)	The average over all days in the window.
n	The window length: the number of daily returns used, e.g. 10, 20 or 60. This choice determines the answer.
option value	The current cost to buy the sold options back. The seller profits as this falls toward zero and loses as it rises.
position delta	Total directional exposure of the option position, in units of the underlying, before hedging.
premium	The premium collected on entry — about ₹708 for the 30-day NIFTY at-the-money straddle. This is the maximum possible profit.
q	The aggregation horizon in periods — e.g. q = 5 tests weekly against daily returns.
r	Risk-free interest rate, taken as 6.5% as an Indian rupee proxy. Dividend yield is zero for NIFTY and BANKNIFTY, which are price indices.
r_t	The close-to-close log return on day t: the natural log of that day's close divided by the previous close.
r_{t−i}	The log return i days before t, ln(P_{t−i} / P_{t−i−1}), where P is the closing level.
r²_{t−1}	The previous day's squared return (log return of the underlying), the newest piece of information entering the forecast.
r̄	The mean of the r_t over the window; often taken as zero for short windows.
spread	The HV-vs-IV difference in volatility points. Usually small and positive; turns negative when realised volatility overtakes implied, which is when short-vol positions lose.
t	Days elapsed since the rise began.
t_e	The length of the event in years — one calendar day, so t_e = 1/365 ≈ 0.00274.
vega	The premium's sensitivity to a one-point change in σ, proportional to √T. As it collapses, the implied volatility extracted from the price becomes hypersensitive to noise.
vomma	The convexity of vega in volatility (∂ν/∂σ). A short-vega position is typically short vomma, so the convex term adds to the loss as σ rises.
w	Interpolation weight, chosen so the blended tenor equals 30 days. It rolls smoothly toward zero on the near contract as that contract approaches expiry, which is what removes the roll discontinuity.
w(k, T)	Total implied variance at log-moneyness k and expiry T — the natural, arbitrage-friendly coordinate for the surface.
wᵢ	Weight of constituent i in the portfolio; the weights sum to one.
wⱼ	Weight of a different constituent j; the double sum runs over all pairs i ≠ j.
x_i	The i-th observation. For volatility, this is the log return of day i, not the price.
x̄	The arithmetic mean of the observations. In the zero-mean finance estimator this is set to 0.
½	The one-half factor from the second-order (Taylor) term of the option's value in the underlying — the curvature contribution.
Γ	Gamma — the rate at which delta changes as spot moves. Positive for a long option, so the ½·Γ·(dS)² term is always a gain from movement.
ΔK_i	The strike interval around K_i — half the distance between the strike above and the strike below it (at the ends, the distance to the single neighbour).
ΔS	The move in the underlying since the last re-hedge, in points. The per-hedge harvest grows with its square.
Δ_BS	The Black–Scholes delta, computed at the option's own implied volatility. It is the whole answer only when ∂σ/∂S is zero.
Δ_adj	The regime-adjusted (minimum-variance) delta — the hedge ratio that actually minimises the variance of the hedged position.
Δ_option	The option's delta per unit of underlying — the sensitivity of its price to a ₹1 move in spot, between 0 and 1 for a call and 0 and −1 for a put.
Θ	Theta — the value lost per calendar day with everything else held still. Negative for a long option; about −₹12 per day on this straddle.
Θ_paid	Theta — the daily decay the buyer pays and the seller collects; the seller's rent for underwriting the position, about ₹12 per day here.
Σ	Summation over the n days in the window.
Σ r_t²	The sum of squared daily log returns across the n days in the window. The zero-mean assumption means we do not subtract a sample mean.
Σ r_{t−i}²	The sum of squared daily log returns across the n days in the trailing window; the zero-mean assumption means no sample mean is subtracted.
Σ_{t=1}^{n}	Summation over every period t from 1 to n.
α	The reaction coefficient (alpha): how strongly a large squared return yesterday raises today's variance. This term is clustering made explicit.
β	Skew slope — the strength of the downward tilt. Larger β means a steeper skew and a richer put wing. It rises in selloffs.
θ	The long-run mean of the VIX toward which futures price — taken as ≈ 17.5 in the examples on this site.
κ	Speed of reversion — how hard volatility is pulled toward the mean. Larger κ means faster reversion and a shorter half-life.
λ	The decay factor, between 0 and 1. RiskMetrics fixes it at 0.94 for daily data; a higher λ means a longer, smoother memory and a slower reaction to new moves.
ν	Vega — rupee change in the option's price per 1 percentage-point change in implied volatility. Largest for at-the-money options with time left.
ξ	Volatility of volatility — the size of the random shocks that push σ around. Often written as the Greek letter xi.
ρ	The average pairwise correlation between the constituents' returns — the single variable the dispersion trade is really exposed to. Low in calm markets, rising toward one in a crash.
ρᵢⱼ	Pairwise correlation between the returns of constituents i and j. Low in calm markets, close to one in a crisis.
ρ₁	First-order autocorrelation of returns — the correlation between one day's return and the next. Positive in a trend, negative in a range.
σ	Implied volatility — the unknown being solved for. Annualised, expressed as a decimal (0.147 = 14.7%).
σ(K)	Implied volatility of the option struck at K, expressed as an annualised decimal (0.135 = 13.5%).
σ(k, T)	Implied volatility at log-moneyness k and time to expiry T, as an annualised decimal. This is the height of the surface at that point.
σ_25call	Implied volatility of the 25-delta out-of-the-money call — the price of upside participation.
σ_25put	Implied volatility of the 25-delta out-of-the-money put — the price of downside protection.
σ_30day	A constant-maturity implied volatility, engineered to describe a fixed 30-day horizon regardless of the contract calendar.
σ_ATM	The at-the-money implied volatility — the minimum of the smile, the floor from which both wings rise.
σ_ATM(T)	At-the-money annualised implied volatility for the expiry that is T years away, expressed as a decimal (0.129 = 12.9%). This is the y-axis of the curve.
σ_ATM(T_far)	At-the-money implied volatility of the further expiry (0.146 = 14.6% at 90 days). Contango means this exceeds the near value.
σ_ATM(T_near)	At-the-money implied volatility of the nearer expiry, annualised as a decimal (0.117 = 11.7% at 7 days on the calm curve).
σ_Parkinson	The Parkinson estimate of realized volatility, annualised, as a percentage.
σ_after	Implied volatility immediately after the event, once the uncertainty is removed — typically back near the pre-build baseline.
σ_ann	Annualised realised volatility, the standard deviation of returns scaled to a one-year horizon and quoted as a percentage.
σ_annual	Annualised volatility — the standard, quotable figure, expressed as a percentage (0.13 = 13%).
σ_avg	The average volatility of the individual constituents, assumed equal across names in this simplified form; taken as 24% in the worked case.
σ_baseline	The prevailing implied-volatility level before the rise began — the level a spike returns toward.
σ_before	Implied volatility just before the event resolves, inflated by the event premium (annualised decimal or percentage points to match ν).
σ_bucket	The volatility (standard deviation of returns) measured over a single intraday bucket, such as one 5-minute interval.
σ_contract	Implied volatility of one specific option contract — a fixed strike and expiry. Exists only while the contract does; ceases at expiry.
σ_daily	Daily volatility — the sample standard deviation of one day's log returns, expressed as a decimal (0.0082 = 0.82%).
σ_diff	Calm base implied volatility, 12.2%.
σ_diffusive	The ordinary annualised volatility the market realises on a normal, non-event day — estimated from an expiry that does not contain the event, or from a nearby non-event baseline.
σ_event	The annualised volatility contributed by the single event day — the unknown being solved for.
σ_expiry	The at-the-money implied volatility of that specific listed expiry, annualised, as a decimal (0.135 = 13.5%).
σ_far	The implied volatility of the far-dated leg; 13.6% in the worked case.
σ_forward	The implied volatility of the period between the near and far expiries — the quantity a calendar spread is really long or short.
σ_fwd	The forward volatility — annualised — for the window running from T1 to T2. The unknown being solved for.
σ_implied	The implied volatility that changes — an annualised decimal read off the clearing option price.
σ_index	The volatility realised (or implied) by the index itself. For NIFTY with the site's numbers this is about 13.5% when ρ = 0.30.
σ_near	The implied volatility of the near-dated leg; 12.5% in the worked case.
σ_new	Annualised realised volatility of the current regime — about 26% in the stressed example.
σ_old	Annualised realised volatility when the position was opened — about 10% in the calm example.
σ_p	Volatility (standard deviation) of the whole portfolio or index.
σ_period	The volatility measured over one period: the standard deviation of that period's returns (0.0085 = 0.85% for a NIFTY day).
σ_realised	Realised volatility — the volatility the underlying actually delivers over the step, annualised as a decimal.
σ_t	Implied volatility at day t after the initial rise, annualised decimal.
σ_total	Total annualised implied volatility of the event-containing expiry, read off its at-the-money option (decimal).
σ²	The variance — the average squared deviation of returns from their mean, in squared units (%² or fraction²).
σ²(T)·T	Total (undiscounted) variance to expiry T — the quantity that must be non-decreasing. It is the variance the market is pricing over the whole life of the option, not the annualised rate.
σ²_diff	Diffusive (ordinary-day) variance — the square of the calm base implied volatility away from the event (0.122² here).
σ²_event	The variance contributed by the single event day, the quantity being solved for.
σ²_implied	The variance implied by the volatility paid when the option was bought or received when it was sold — the implied volatility squared.
σ²_realised	The variance the underlying actually realises over the interval — the realised volatility squared, annualised on 252 trading days.
σ²_t	The variance forecast for day t — the quantity being produced. Its square root, annualised by multiplying by √252, is the expected volatility.
σ²_total	Total implied variance the option is pricing — the square of the quoted at-the-money implied volatility on the eve (0.188² here).
σ²_{t−1}	The previous day's variance forecast, carried forward. This is what gives EWMA its memory.
σᵢ	Volatility of constituent i.
σ₁	Spot implied volatility to the nearer date T1, annualised as a decimal (0.129 = 12.9%).
σ₂	Spot implied volatility to the farther date T2, annualised as a decimal (0.139 = 13.9%).
σⱼ	Volatility of constituent j.
φ(d₁)	The standard-normal probability density evaluated at d₁; largest at the money, which is why vega peaks at the strike.
ω	The constant baseline (the Greek letter omega), tied to the long-run variance by ω = (1 − α − β) × long-run variance. Keeps the process from decaying to zero.
∂w/∂T	Rate of change of total implied variance with expiry at fixed k. It must be non-negative everywhere — that is the no-calendar-arbitrage constraint.
∂σ/∂Price	How much the extracted implied volatility moves for a one-rupee change in the option's price. Equal to one over vega, so it explodes as vega collapses.
∂σ/∂S	The rate at which the option's implied volatility changes as spot moves. This single term is what a regime assumes. Sticky strike sets it to zero for a fixed strike.
√	Square root. Variance grows linearly with time under independence, so volatility, its square root, grows with the square root of time.
√(B × 252)	The scaling factor from one bucket to one year — the square root of the total number of buckets in a year. For B = 75 it is √18900 ≈ 137.5.
√252	The annualisation factor: volatility scales with the square root of time, and there are approximately 252 trading days in a year, so √252 ≈ 15.87.
√N	The square root of N — the scaling factor. √252 ≈ 15.87, √52 ≈ 7.21, √12 ≈ 3.46. This is the factor volatility scales by, not N itself.
√T	Square root of time to expiry in years. Vega grows with √T, so a 180-day option carries several times the vega of a 7-day one at the same strike.

Conventions

Volatility is annualised and quoted in percent unless stated otherwise. Annualisation uses 252 trading days, not 365 calendar days, because markets do not move when they are shut.
Time to expiry T in a pricing formula is in years, measured in calendar days ÷ 365, because interest accrues on weekends even though prices do not move.
That inconsistency — 252 for volatility, 365 for discounting — is deliberate and standard. It is also the single most common source of a mismatched number when people check our arithmetic against their own.
Vega is quoted per one percentage point of implied volatility. Theta is quoted per calendar day.
The risk-free rate is taken as 6.5% as an Indian proxy, and the dividend yield as zero, which is correct for NIFTY and BANKNIFTY index options.
All figures are per unit of the underlying. Multiply by the lot size (NIFTY 75, BANKNIFTY 30 at the time of writing) for rupee figures per lot.

Frequently asked questions

Why does this site use 252 days for volatility and 365 for time to expiry?

Because a closed market does not move, so volatility should be annualised on trading days; but interest accrues on weekends, so discounting should use calendar days. The two conventions contradict each other, both are universal, and the mismatch is the most common reason a reader's spreadsheet disagrees with a broker's screen.

Are these formulas the same ones the calculators use?

Yes, by construction. Every diagram on this site and every calculator on it call the same volcharts.js engine, and this page is generated from the same concept data that those pages render. A formula cannot appear here in one form and be applied elsewhere in another.

What convention does this site use for vega and theta?

Vega is quoted per one percentage point of implied volatility. Theta is quoted per calendar day. Both are per unit of the underlying — multiply by the lot size for rupee figures per lot.

Why is the risk-free rate 6.5%?

It is a proxy for Indian rupee rates, used consistently so that every figure on this site is internally comparable. The rate matters little for short-dated options and materially for long-dated ones.

Is the dividend yield really zero?

For NIFTY and BANKNIFTY, yes — they are price indices and their options are European and cash-settled, so a zero dividend yield is correct. For a single stock with an ex-dividend date before expiry it is wrong, and the single-stock pages say so.

Which standard deviation estimator does this site use?

The sample estimator with an n−1 denominator, which is what a spreadsheet's STDEV() gives. Some desks prefer a zero-mean estimator dividing by n; the difference is well under one percent over a typical window.

Last reviewed 10 July 2026. Educational content only — not investment advice.