Contract

$

Spot price (S)

$

Strike (K)

Volatility (σ)

Annualised, 0.20 = 20%

Time to maturity (T)

Years

Risk-free rate (r)

Annualised, 0.05 = 5%

Dividend yield (q)

Annualised, continuous

Call

Put

Try this

0 of 3 done

Round trip

Use the model price as the market price and solve.

What to look for: The solver returns exactly the volatility you started from in a few Newton steps. Implied volatility inverts the pricing formula.

The vega problem

Set the strike to 150 with 0.1 years to maturity and a market price just above the lower bound.

What to look for: The solver needs more iterations or falls back to bisection. Vega is tiny far from the money, so Newton's method has almost no slope to work with.

Bend the smile

Generate the sample smile, then raise one strike's price by ten percent and solve all.

What to look for: That strike's implied volatility jumps up while its neighbours stay put. A single expensive option shows up as a bump in the smile.

Market price

$

Observed option price

No-arbitrage range: 4.8771 to 100.0000

Implied volatility

20.00%

σ = 0.200000

Iterations

3

converged

Check: price at solution

10.450600

Vega there: 37.5240

Newton steps

Step	σ tried	Price	Error
0	0.261958	12.788743	+2.338143
1	0.200234	10.459350	+0.008750
2	0.200000	10.450600	+0.000000
3	0.200000	10.450600	+0.000000

Price against volatility

The curve is monotone, so the root is unique. Its slope is Vega, which is what Newton's method divides by. The dashed line is the market price.

Smile explorer

Enter market prices at several strikes and see the implied volatility at each. The sample smile is synthetic: prices generated with volatility rising away from the money.

Strike	Market price	Implied vol
		32.98%
		26.33%
		22.25%
		20.93%
		20.00%
		19.41%
		19.12%
		19.26%
		20.19%