Historical Volatility, Implied Volatility and the Smile

Understand Volatility: Historical, Implied and Smile

Key rules to remember

Log return

R_i = ln(S_i ÷ S_(i-1))

Use continuously compounded returns, because GBM implies they are normal. Do not use simple percentage returns unless the question says so.

Sample variance of returns

s² = Σ(R_i − R̄)² ÷ (n − 1)

n is the number of returns, which is one fewer than the number of prices. Divide by n − 1 unless the question gives a different convention.

Annualised historical volatility

σ̂ = s × √m

m is the number of return periods per year: 12 for monthly, 52 for weekly, about 252 for daily (trading days). Use the m the question specifies.

Distribution of log return under GBM

ln(S_t ÷ S_0) ~ N((μ − σ²/2)t, σ²t)

The variance of the log return is σ²t, which justifies the √ time scaling.

Approximate standard error of a volatility estimate

se(σ̂) ≈ σ̂ ÷ √(2(n − 1))

Holds approximately when returns are independent and normal. It shows that more observations give a more precise estimate.

Black-Scholes call price

C = S_0 N(d1) − K e^(−rT) N(d2), d1 = [ln(S_0 ÷ K) + (r + σ²/2)T] ÷ (σ√T), d2 = d1 − σ√T

For a European call on a non-dividend-paying share, with constant r and σ. The put is P = K e^(−rT) N(−d2) − S_0 N(−d1).

Vega

Vega = ∂C/∂σ = S_0 φ(d1) √T

φ is the standard normal density. Vega is positive for calls and puts and is the same for both with the same strike and maturity. It is used as the slope in Newton-Raphson.

Newton-Raphson for implied volatility

σ_(n+1) = σ_n − (C_BS(σ_n) − C_market) ÷ vega(σ_n)

Start from a sensible guess such as 20%. Repeat until the model price matches the market price to the accuracy required.

How to solve Volatility: Historical, Implied and Smile questions

1. 1Identify what is asked: a historical estimate from data, an implied value from a price, or a discussion of the smile versus the model.
2. 2For historical volatility, convert each pair of consecutive prices to a log return ln(S_i ÷ S_(i-1)). Count the returns carefully.
3. 3Find the mean return, then the sample variance with divisor n − 1, then the standard deviation s for one period.
4. 4Annualise by multiplying s by √m, where m is the number of periods per year. State the unit of the answer, for example per year.
5. 5For implied volatility, write the Black-Scholes price as a function of σ. Choose a starting σ, compute d1, d2 and the price, and compare with the market price.
6. 6Adjust σ using vega (Newton-Raphson step) or bracket the answer by trying two values and interpolating. Check that the final price matches the market price.
7. 7For smile questions, state the Black-Scholes assumption (constant σ, lognormal returns), describe the observed pattern, and give the reasons (fat tails, skewness, jumps, stochastic volatility), linking each to the effect on option prices.
8. 8State your assumptions and conclusion clearly, for example no dividends, European exercise and constant interest rate.

Quickest way: Quick estimate and one-step implied volatility

Common mistakes in Volatility: Historical, Implied and Smile

• Dividing by n instead of n − 1 when computing the sample variance.

  Students remember the population formula from school and forget that the question uses a sample.

  Fix: Use n − 1 for an estimate from a sample of returns unless told otherwise, and write the divisor in your working.

• Using the number of prices as n, not the number of returns.

  Four prices look like n = 4, but they give only three returns.

  Fix: Count returns, not prices. The number of returns is one fewer than the number of prices.

• Annualising with the wrong factor, or multiplying by m instead of √m.

  Students mix up scaling the variance (by m) and scaling the standard deviation (by √m).

  Fix: Variance scales with time, so volatility scales with the square root of time. Multiply s by √m, with m matching the data frequency.

• Treating implied volatility as a forecast of the true future volatility.

  It is a forward-looking number, so it is tempting to read it as a prediction.

  Fix: Say that it is the volatility implied by the market price under the Black-Scholes model. It reflects expectations and risk premia, and it depends on the model being used.

• Claiming the smile shows that Black-Scholes is mathematically wrong.

  The word 'inconsistent' is misread as an error in the formula.

  Fix: Say that the formula is correct given its assumptions, and the smile shows that the assumptions (constant σ, lognormal returns) do not match market behaviour.

• Using simple returns (S_i − S_(i-1)) ÷ S_(i-1) when the question is within the Black-Scholes framework.

  Simple returns feel more natural.

  Fix: Use ln(S_i ÷ S_(i-1)). The model is built on normally distributed log returns.

Worked examples