Volatility surface

Black-Scholes assumes one volatility. Markets do not agree: the volatility implied by option prices changes with strike, the smile, and with maturity, the term structure. Choose a shape for the "market", generate its quotes, edit any of them, and solve the whole grid back into implied volatilities.

Market shape

At-the-money volatility: 20%

Skew: -0.10

Curvature: 0.04

Term slope: +0.02 per year

Market

Spot (S)

Rate (r)

Yield (q)

Strikes run from 70% to 130% of the spot. Maturities are 0.1, 0.25, 0.5, 1 and 2 years. Each quote is the out-of-the-money option at that strike, a put below the forward and a call above it, which is how markets quote.

Try this

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Flat is Black-Scholes

Choose the flat preset and solve.

What to look for: Every cell shows the same volatility. This is the only surface Black-Scholes can describe with a single parameter.

Equity skew

Choose the equity skew preset and look at the smile chart.

What to look for: Volatility falls as the strike rises, and the short-dated smile is steeper than the two-year one. Puts far below the money are expensive insurance.

Edit one quote

Raise one quote's price by a few percent and solve again.

What to look for: That cell's implied volatility rises on the heatmap and the smile for that maturity gets a bump. Real surfaces are full of such wrinkles from supply and demand.

Quotes

Equity skew

Option quotes by maturity (rows) and strike (columns); each cell is an editable price
T \ K	70	80	90	100	110	120	130
0.1y	put	put	put	put	call	call	call
0.25y	put	put	put	put	call	call	call
0.5y	put	put	put	put	call	call	call
1y	put	put	put	put	call	call	call
2y	put	put	put	put	put	call	call

How to read it

If Black-Scholes described the market, every cell would show the same number. A smile or skew is the market's way of saying that big moves are more likely than the lognormal model allows, and a skew says which direction it fears more. Traders use the surface as a lookup table: to price an option at any strike and maturity, read off its implied volatility and put that into Black-Scholes.

The short-dated smile is steeper because a given price move is a bigger shock over a week than over two years. Models such as Heston or local volatility exist to reproduce these shapes with one consistent set of parameters.