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I am grateful to Frank de Jong for explaining this to me at a conference. 7 σ t,T | t−1 denotes a volatility forecast formulated at time t − 1 for volatility over the period from t to T . In pricing options, the required volatility parameter is the expected volatility over the life of the option. The pricing model relies on a riskless hedge to be followed through until the option reaches maturity. Therefore the required volatility input, or the implied volatility derived, is a cumulative volatility forecast over the option maturity and not a point forecast of volatility at option maturity.

For j period) is taken to be the sum of individual multi-step point forecasts sj=1 h T + j|T . These multi-step point forecasts are produced by recursive substitution and using the fact that εT2 +i|T = h T +i|T for i > 0 and εT2 +i|T = εT2 +i for T + i ≤ 0. g. ) and the forecast horizon. If returns are iid (independent and identically distributed, or strict white noise), then variance of returns over a long horizon can be derived as a simple multiple of single-period variance. 2. 7 Complication in relation to the choice of forecast horizon is partly due to volatility mean reversion.

1 Single-state historical volatility models The simplest historical price model is the random walk model, where the difference between consecutive period volatility is modelled as a random noise; σt = σt−1 + vt , Historical Volatility Models 33 So the best forecast for tomorrow’s volatility is today’s volatility: σ t+1 = σt , where σt alone is used as a forecast for σt+1 . In contrast, the historical average method makes a forecast based on the entire history 1 (σt + σt−1 + · · · + σ1 ) . t The simple moving average method below, σ t+1 = 1 (σt + σt−1 + · · · + σt−τ −1 ) , τ is similar to the historical average method, except that older information is discarded.