Learn implied vs historical volatility — Stock-Market notes

WHY do we even need two kinds of volatility?

Volatility is the single most important input to an option's price (via Black–Scholes). But "volatility" is ambiguous:

WHAT actually happened to the stock — measurable, backward-looking → Historical Volatility (HV).
WHAT the market expects to happen — inferred from live option prices, forward-looking → Implied Volatility (IV).

Historical Volatility — derive it from scratch

HOW we build it, step by step:

Step 1 — Daily log returns. Why logs? Because returns compound multiplicatively, and logs turn multiplication into addition (so they're additive over time and roughly normal). $r_t = \ln\!\left(\frac{S_t}{S_{t-1}}\right)$

Step 2 — Standard deviation of those returns. Volatility = spread of returns, so we measure how far returns scatter around their mean. $\sigma_{\text{daily}} = \sqrt{\frac{1}{N-1}\sum_{t=1}^{N}(r_t - \bar r)^2}$ Why $N-1$ ? Bessel's correction — we estimated the mean $\bar r$ from the sample, so we lose one degree of freedom.

Step 3 — Annualize. Why $\sqrt{252}$ ? Variance of independent returns adds over time, so variance scales with time $T$ , and standard deviation scales with $\sqrt{T}$ . There are ~252 trading days per year. $\boxed{\sigma_{\text{annual}} = \sigma_{\text{daily}} \times \sqrt{252}}$

Implied Volatility — where does it come from?

HOW: Black–Scholes gives price as a function $C = f(S, K, T, r, \sigma)$ . You know everything except $\sigma$ . So you invert: find the $\sigma$ such that $f(\ldots,\sigma) = \text{market price}$ .

$\text{Market Price} = f(S,K,T,r,\sigma_{IV}) \;\Longrightarrow\; \text{solve for } \sigma_{IV}$

Why can't we just rearrange the formula? Because $\sigma$ sits inside the cumulative-normal $N(d_1), N(d_2)$ — there is no closed-form inverse. We solve numerically (Newton–Raphson, using vega $\partial C/\partial\sigma$ as the slope).

Putting them side by side

Figure — Learn implied vs historical volatility

	Historical (HV)	Implied (IV)
Direction	Backward-looking	Forward-looking
Source	Actual past prices	Current option prices
Method	Std dev of log returns	Invert Black–Scholes
Meaning	What did happen	What market expects
Uses	Baseline / reality check	Pricing, timing trades

Worked examples

Worked example Example 1 — Compute HV

A stock's daily log returns over 5 days: $0.01, -0.02, 0.015, -0.005, 0.02$ . Step 1: mean $\bar r = (0.01-0.02+0.015-0.005+0.02)/5 = 0.004$ . Why? Need the centre to measure spread from. Step 2: deviations from the mean are $0.006, -0.024, 0.011, -0.009, 0.016$ . Square and sum: $(0.006)^2+(-0.024)^2+(0.011)^2+(-0.009)^2+(0.016)^2$ $= 0.000036+0.000576+0.000121+0.000081+0.000256 = 0.00107$ . Step 3: $\sigma_{\text{daily}}=\sqrt{0.00107/4}=\sqrt{0.0002675}=0.01636$ . Why $/4$ ? $N-1=4$ . Step 4: annualize $=0.01636\times\sqrt{252}=0.01636\times15.87=0.2597 \approx \mathbf{26.0\%}$ . Why $\sqrt{252}$ ? Variance adds over 252 days.

Common mistakes (steel-manned)

Recall Feynman: explain to a 12-year-old

Imagine a bouncy ball. Historical volatility is how high it has been bouncing when you watched it — you measured it. Implied volatility is how high everyone bets it will bounce next, based on how much they're paying to play the bouncing game. If people are paying a lot (high IV) but the ball has been bouncing gently (low HV), the game is overpriced — you might want to be the one selling tickets instead of buying.

Active recall

What does historical volatility measure?

The annualized standard deviation of past log returns — how much the stock actually moved.

What does implied volatility measure?

The market's expected future volatility, inferred by inverting Black–Scholes from the option's market price.

Why annualize by √252 not 252?

Variance scales with time, so standard deviation scales with the square root of time; ~252 trading days/year.

Why can't IV be found by algebra?

σ sits inside the N(d₁), N(d₂) normal terms in Black–Scholes; there's no closed-form inverse, so we solve numerically.

When are options 'expensive' by this framework?

When IV ≫ HV — the market prices in more movement than the stock historically shows.

Does high IV predict direction?

No — it predicts magnitude of movement (up or down), not direction.

Formula for HV daily std dev?

σ_daily = √( Σ(rₜ − r̄)² / (N−1) ), rₜ = ln(Sₜ/Sₜ₋₁).

Expected 4-day move if daily σ = 1.5%?

1.5% × √4 = 3.0%.

What is the VIX conceptually?

The market-implied 30-day volatility of the S&P 500 — a broad IV/'fear' gauge.

Connections

Black-Scholes Model — the engine that IV is inverted from
Vega — the Greek measuring sensitivity to volatility (slope used to solve for IV)
The Greeks — parent chapter
VIX Index — market-wide implied volatility
Log Returns and Random Walks — why volatility scales with √t
Straddles and Volatility Trading — how the IV vs HV gap is traded

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, volatility do type ki hoti hai aur dono ka matlab alag hai. Historical volatility (HV) matlab stock actually kitna hila past mein — hum uske purane prices ke log returns ka standard deviation nikaalte hain aur usko √252 se multiply karke annual bana dete hain. Ye reality hai, jo ho chuka hai. Yaad rakho: annualize karne ke liye √252 se multiply, sidha 252 se nahi — kyunki variance time ke saath add hota hai, isliye std dev sirf square-root of time se badhta hai.

Doosri taraf implied volatility (IV) hoti hai — ye market ki expectation hai ki aage kitna move aayega. Isko hum option ke live market price se nikaalte hain: Black–Scholes formula mein sab kuch pata hota hai (spot, strike, time, rate) sirf σ nahi. Toh hum ulta chalke woh σ dhoondte hain jo formula ka price market price ke barabar kar de. Wahi σ IV hai. IV algebra se nahi nikalti kyunki σ N(d1), N(d2) ke andar chhupa hota hai — isliye numerically (Newton method) solve karte hain.

Ab kyun important hai: agar IV bahut zyada hai HV se (jaise earnings se pehle), matlab options mehnge ho gaye hain — log darr ke maare zyada pay kar rahe hain. Tab option bechne ka bias banta hai. Agar IV kam hai HV se, options saste hain — tab kharidne ka bias. Bas ek golden habit: har option trade se pehle IV ko HV se compare karo. Aur ek cheez clear rakho — high IV ka matlab hai bada move aayega (upar ya neeche), direction ka koi guarantee nahi. IV magnitude batati hai, direction nahi.

Test yourself — The Greeks

Connections