Implied Volatility
Introduction to Financial Options / Lecture 06
Last time: volatility \longrightarrow price.
Today: price \longrightarrow volatility.
From volatility to price, and back
Introduction to Financial Options / Lecture 06
A trader’s description
Introduction to Financial Options / Lecture 06
The wrong number
in the wrong formula
to get the right price.
Trading options is trading volatility
Introduction to Financial Options / Lecture 06
Lecture 02 More uncertainty raises the value of calls and puts.
Lecture 03 A stock trade transforms calls into puts. Direction and optionality can be separated.
But option premiums depend on too many variables at once to compare directly.
\boxed{C=C_{\mathrm{BS}}(S_0,K,T,r;\sigma)}
Same strike, same implied volatility
Introduction to Financial Options / Lecture 06
Same stock S_0=\$100, same three-month expiry, same strike K=\$110. One call, one put.
At every volatility, C-P=S_0-K=-\$10. If market prices obey parity, the call and put must imply the same volatility.
European options, zero rates and dividends. The graph uses single prices, not bid-ask quotes.
Change only the strike
Introduction to Financial Options / Lecture 06
Same stock S_0=\$100, same three-month expiry. Only the call strike changes.
Near zero volatility, the $90 and $110 curves flatten. This is zero vega, sensitivity to volatility. At 30% vol, their call deltas are about 78 and 29.
All three comparisons: European calls, zero rates, no dividends.
Change only the expiration
Introduction to Financial Options / Lecture 06
Same out-of-the-money call, S_0=\$100 and K=\$110. Only the time to expiry changes.
At the same volatility, the longer-dated option costs more. The shorter-dated call stays near zero until volatility becomes large enough to make reaching the strike plausible.
Change only the stock price
Introduction to Financial Options / Lecture 06
Each call is at the money with three months to expiry. Stock price and strike scale together.
Even with identical volatility and moneyness, an option on a $250 stock has a larger dollar premium than one on a $50 stock.
We need a common measuring stick
Introduction to Financial Options / Lecture 06
Volatility is that stick. It lets us compare options with different prices, strikes, expirations, and underlyings.
For a fixed contract, Black-Scholes price rises monotonically with volatility. So we can run the formula backward, from a market price to one implied volatility.
Schematic. Each contract has its own curve. The rising shape is what makes inversion possible.
Read the same markets in volatility space
Introduction to Financial Options / Lecture 06
Illustrative European option boards. Zero rates and dividends.
Visa is centered near 20% and Mastercard near 21%. Wider in-the-money spreads and rounded dollar quotes add small differences to the displayed implied vols.
Pause
There’s just one little problem
Introduction to Financial Options / Lecture 06
Black-Scholes gives us the call price when we choose a volatility.
C_{\mathrm{BS}}(\sigma)=S_0N(d_1)-KN(d_2)
Plug in a volatility. Calculate the price.
There is no similar closed-form formula for this inverse. We have to find the volatility numerically.
Newton’s method finds the implied volatility
Introduction to Financial Options / Lecture 06
S_0=\$100, K=\$120, T=3 months, r=0. The market call price is \$0.50. Start at 45\% volatility.
At each guess, follow the curve’s tangent to the market-price line. Its crossing gives the next volatility.
Can every price have an implied volatility?
Introduction to Financial Options / Lecture 06
No. Newton’s method needs a price that the model can actually reach.
\underbrace{\max(S_0-Ke^{-rT},0)}_{\sigma=0}\ \leq\ C_{\mathrm{market}}\ <\ \underbrace{S_0}_{\sigma\to\infty}
Too low: below the zero-volatility price. No model volatility fits.
Too high: at or above the stock price. No finite model volatility fits.
European call, no dividends, nonnegative rates. At the lower bound, \sigma=0; at the upper bound, volatility only approaches infinity. Check executable bid and ask prices before calling a quote an arbitrage.
A flat price curve makes the answer fragile
Introduction to Financial Options / Lecture 06
Newton follows a tangent. What if the price curve is almost flat?
Steep price curve
A small quote change gives a small change in implied volatility.
Flat price curve
The same quote change gives a much larger change in implied volatility.
Near a flat part of the curve, a small bid-ask spread or tick-size change can move the fitted volatility a lot. The tangent in Newton’s method is nearly horizontal, so its next guess may jump far away.
The fitted volatility absorbs input mistakes
Introduction to Financial Options / Lecture 06
The market supplies the price. We supply the stock price, rates, dividends, and contract rules.
Suppose a dividend is due before expiry. At a fixed volatility, the call is worth less because the stock will drop when the dividend is paid.
Forget the dividend when fitting the market call price. Your model starts too high, so you must lower volatility to match the quote.
An American option also needs an early-exercise model, such as the Lecture 04 binomial tree. The same market price can imply a different volatility under a different pricing model.
Implied is not the same as expected
Introduction to Financial Options / Lecture 06
Both are expressed in annualized volatility units, but they answer different questions.
Forecast volatility
How much we expect the stock to move in the future. Compare this forecast with what actually happens later.
Implied volatility
The input that makes a chosen model reproduce today’s option quote. It summarizes the price, not just a prediction of future moves.
These can differ because traders may pay extra for protection, because supply and demand affect quotes, or because the pricing model is imperfect.
Your turn: read the quote
Introduction to Financial Options / Lecture 06
Return to the Visa one-month board. Stock price is 250. Focus on the 250 strike.
- If you buy the call, which price do you pay? Which volatility quote corresponds to that price?
- Why is the midpoint of these volatility quotes not necessarily executable?
- If an expected dividend was omitted from the model, which way would the fitted call volatility move after correcting the input?
Answers: Pay the call ask and use its ask implied volatility. You cannot assume someone will trade at the midpoint. Correcting for the dividend lowers the model call price at fixed volatility, so the fitted call volatility rises.
Looking forward
Introduction to Financial Options / Lecture 06
Black-Scholes tells us there is one volatility. If the world used that price model, every strike would imply the same volatility.
In fact, market quotes show something very different. The implied volatility changes with strike.
Stylized prices for one expiry. Lecture 09 will study volatility smiles and skews.
Summary
Introduction to Financial Options / Lecture 06
- Implied volatility is the model input that reproduces an observed option price.
- For a valid European vanilla price, the price-versus-volatility relationship lets us find it numerically.
- Bid and ask premiums become bid and ask implied volatilities. Flat price curves make those quotes fragile.
- Implied volatility depends on model inputs and exercise rules. It is not a direct forecast of future volatility.
- Different options can imply different Black-Scholes volatilities.
Next: how does an option’s stock exposure change as the stock moves?