Implied volatility

Of the inputs to an option pricing model — spot, strike, time to expiry, interest rate and volatility — every one is observable except the last. Volatility is invisible.

One way to estimate it is to look backwards: take the past closing prices and compute the standard deviation of the price changes. That is historical volatility, and it is easy to calculate and often beside the point.

Implied volatility runs the model in reverse instead. Take the option's actual traded price as given, hold every other input fixed, and solve for the volatility number that produces it. What comes out is not a measurement of the past. It is the consensus of every participant in that option about how much the underlying will move over the remaining life of the contract.

IV is quoted as an annualised percentage and is, in the workbook's phrasing, "a proxy of market risk", expressed in standard deviations over a horizon.

Three behaviours to know:

• IV does not predict direction. High IV means a large expected swing — it could be up or down. IV is a statement about magnitude only.
• IV and option prices move together. When implied volatility rises, premiums rise, other things equal. So a rise in IV after a trade is placed is good for the option owner and bad for the writer; a fall is good for the writer.
• IV generally rises in bearish markets and falls in bullish ones, because falling equity markets are the ones investors find risky and undesirable.

That produces the standard trading rule the workbook states directly: when IV is high, option traders sell options; when IV is low, they buy them. And it explains why writing options into an event is not the free money it looks like — the writer has already been paid for the event in the premium.

Black-Scholes(Spot, Strike, Time, Rate, σ) = Market premium
                                    solve for σ  →  Implied volatility

Annualised IV converted to an expected move over a shorter horizon:

One standard deviation move = Spot × IV × √(days ÷ 365)

Solving for IV. The index is at 17,562. A call with a strike of 17,500 and 9 days to expiry is trading at Rs 95, and the interest rate is 6%. Spot, strike, time and rate are all known; the premium is known. The only unknown left in the model is volatility — so plug the rest in and solve for the σ that returns Rs 95. That number is the implied volatility.

What IV is actually telling you. A pharmaceutical stock at Rs 1,200 has been range-bound for a year at a historical volatility of 15%. A court verdict lands in 7 days, and option premiums now imply 80%.

At 15% IV:  1,200 × 0.15 × √(7 ÷ 365) = 1,200 × 0.15 × 0.1385 = Rs 25
At 80% IV:  1,200 × 0.80 × √(7 ÷ 365) = 1,200 × 0.80 × 0.1385 = Rs 133

The market is pricing a one-standard-deviation move of Rs 133 over the week instead of Rs 25 — a swing of over 11% either way, and it has no view on which way.

Why selling into that is not free money. Suppose you write the Rs 1,200 call for a premium of Rs 90 because 80% IV looks absurd against 15% history. The verdict is favourable and the stock jumps to Rs 1,270.

Payoff owed = 1,270 − 1,200 = Rs 70
Premium kept              = Rs 90
Net                       = +Rs 20 per unit

You survived a 5.8% overnight gap and still made money — precisely because the 80% IV had already priced the verdict in. Had you written the same option at 15% IV for a premium of about Rs 17, the same jump would have cost you Rs 53 a unit. The premium was the margin of safety, and IV is how it got there.

Chapter 16.9 (Implied volatility of an option) is the source of the 17,562 / 17,500 / Rs 95 example and the pharmaceutical court-case illustration; Chapter 21.7 repeats the treatment for interest rate options and adds the three-way classification — historical, forecasted and implied volatility. Expect a definitional question distinguishing implied from historical volatility, one on the direction of the IV-premium relationship, and one on why traders sell when IV is high.

• Implied volatility is derived from the option price; historical volatility is derived from the underlying's past prices. They answer different questions and rarely agree.
• IV says nothing about direction. High IV means a big move is expected, not an up move.
• Rising IV helps the buyer and hurts the writer. Candidates often reverse this because they associate volatility with risk and risk with loss.
• IV rises in bearish markets, not bullish ones. The workbook is explicit.
• Vega, not IV, is the sensitivity. Vega measures how much the premium moves for a 1% change in IV; IV is the input itself.
• Without a traded option there is no implied volatility. It is extracted from a market price, so an underlying with no options market simply has none.