Volatility Cone and Smile

Volatility Cone and Smile

Volatility Cone and Smile: How It Works

What it does

The page answers two questions about one stock. First: how much has its price moved, and is the latest movement large or small against its own history? It measures the realised volatility of pasted daily closing prices over windows of 10 to 250 trading days and sets the latest value of each window beside the range that window has shown before. Drawn against the window length, the ranges form the volatility cone. Second, in Advanced mode: what volatility do quoted option prices imply at each strike? It turns each line of a pasted option chain into an implied volatility, draws the curve they form (the smile) and reports three numbers that summarise it. The page fetches no prices or quotes; everything is pasted or typed.

The method

Daily log return. With P_t the closing price of day t, the log return of the day is

u_t = ln(P_t / P_t−1)

Realised volatility of a window. A window of n trading days holds n returns (n + 1 prices). With ū their mean, the daily standard deviation and the yearly volatility are

s = √[ Σ(u_t − ū)² / (n − 1) ]     σ = s √252

The divisor n − 1 is the usual one for a standard deviation estimated from a sample. The factor √252 turns a daily figure into a yearly one: if daily returns are independent, variances add over the 252 trading days of a year, so the standard deviation grows with the square root of their number. This estimate is called close-to-close because it uses one closing price a day.

The cone. The window is slid through the history one day at a time, so N prices give N − n windows of n days. For each of n = 10, 20, 60, 120 and 250 the page sorts the windows' volatilities and reports the lowest, the 25 %, 50 % (median) and 75 % points, the highest, and the latest window's value. A point p of the way through m sorted values is read at position (m − 1)p, by a straight line between the two neighbouring values. An implied volatility typed into the form is drawn on the chart as a level line, and the table gives the share of past windows that lay below it.

Implied volatility (Advanced). The Black-Scholes model gives the value of a European option from the stock price S₀, the strike K, the time to expiry T in years (calendar days / 365), the interest rate r, the dividend yield q (both continuously compounded) and the volatility σ. The implied volatility of a quoted price is the σ at which the model value equals the quote, found by Newton steps kept inside a bracket (the routine of the Option Pricing tool, which also shows the formula). A solution exists only when the quote lies strictly between the bounds

call: max(S₀e^−qT − Ke^−rT, 0) < price < S₀e^−qT     put: max(Ke^−rT − S₀e^−qT, 0) < price < Ke^−rT

A line outside them is listed with the reason and left out.

Forward price and moneyness. F = S₀e^{(r − q)T} is the price for delivery at expiry. Moneyness is K / F; an option with K = F is at the money. The smile line uses one volatility per strike: the out-of-the-money option (the put when K < F, the call when K ≥ F), or the only option quoted there.

At-the-money level. σ_atm is the smile read at K = F by a straight line between the two quoted strikes on either side.

25-delta points. Delta is the model's change in the option value for a 1.00 rise in the stock: e^−qTN(d₁) for a call and −e^−qTN(−d₁) for a put, each computed at the line's own implied volatility. σ_25c is the implied volatility of the call with delta 0.25 and σ_25p that of the put with delta −0.25, each read by a straight line in delta between the two quoted lines that enclose it. Then

risk reversal RR = σ_25c − σ_25p     butterfly BF = (σ_25c + σ_25p) / 2 − σ_atm

The risk reversal measures the tilt of the smile and the butterfly its curvature. They are reported only when the quoted strikes enclose the points needed.

What it assumes

The prices are daily closes in date order, adjusted for splits and dividends (an unadjusted split looks like one enormous return). A year has 252 trading days. Daily returns are treated as independent when the daily figure is scaled to a year.

The option quotes belong to one expiry date and to the same moment as the stock price. The options are valued as European, with a constant interest rate and a continuous dividend yield. Listed U.S. stock options are American-style and stocks pay discrete dividends, so the implied volatilities of deep in-the-money lines, and of puts when rates are high, differ a little from a broker's figures.

Conventions differ between markets. Here at the money means K = F, delta is the spot delta without a premium adjustment, and the 25-delta points are read by straight lines between quoted strikes. Foreign-exchange desks, for example, quote these numbers with other conventions and will not match to the last digit.

When it misleads

The cone describes one past. Its lowest and highest lines are the extremes of the history pasted, not limits. A two-year history that happened to be calm gives a narrow cone, and the next month can fall outside it.

Overlapping windows. Neighbouring windows share all but one day. Five hundred prices hold only two separate 250-day windows, so the quartiles of the long windows rest on very little independent information, and the long end of the cone looks narrower than it should.

Implied against realised is not a like-for-like comparison. Implied volatility looks forward and carries compensation for risk; on average it has stood above the volatility later realised for stock indexes. Seeing it above or below the realised figure therefore does not by itself say whether an option's price is right.

One price a day. Close-to-close volatility ignores what happened inside the day and is a noisy estimate over short windows: a 10-day figure can change by a third when one large day enters or leaves it. Estimators that use the open, high and low (Parkinson, Garman-Klass) are less noisy; they are not computed here.

Quotes. Far out-of-the-money options cost a few cents, and one tick of price moves their implied volatility by several points. Wide bid-ask spreads, stale last-trade prices and quotes taken at a different time from the stock price all bend the wings of the smile.

Worked example

The default inputs are illustrative: 500 made-up daily prices from a random-number generator (seed 357) and a made-up option chain; neither is market data.

Cone. The last two prices are 101.59 and 100.00, so the latest return is ln(100.00 / 101.59) = −0.015775. The latest 20 returns add up to 0.031904, a mean of 0.001595. Their squared distances from the mean add up to 0.00314260, so s = √(0.00314260 / 19) = 0.012861 and σ = 0.012861 × √252 = 0.2042: a 20-day realised volatility of 20.4 % a year. The 480 windows of 20 days ran from 6.1 % to 30.9 % with a median of 14.5 %. The entered implied volatility of 22 % is above 90 % of them.

Smile (Advanced). With S₀ = 100, 60 days, r = 4 % and q = 1 %: T = 60 / 365 = 0.1644 and F = 100 e^{0.03 × 0.1644} = 100.49. The 70 call quoted at 29.50 is below its lower bound 100 e^{−0.01 × 0.1644} − 70 e^{−0.04 × 0.1644} = 99.84 − 69.54 = 30.29 and is left out. The 100 put (3.32) implies 22.11 % and the 105 call (1.68) implies 21.02 %, so σ_atm = 22.11 + (100.49 − 100) / 5 × (21.02 − 22.11) = 22.00 %. The 105 and 110 calls have deltas 0.318 and 0.145 and volatilities 21.02 and 20.28 %, giving σ_25c = 20.73 %; the 90 and 95 puts (deltas −0.131 and −0.262, volatilities 25.44 and 23.62 %) give σ_25p = 23.79 %. Risk reversal = 20.73 − 23.79 = −3.06 points; butterfly = (20.73 + 23.79) / 2 − 22.00 = 0.26 points.

Real data: daily price histories can be downloaded free from many exchanges' and data providers' websites as a spreadsheet; paste the closing-price column (or the date and close columns), oldest row first. Option chains are shown by brokers; paste strike, C or P and the mid price of each line.

A calculator for education and analysis, not investment advice.