How to forecast stock volatility?

Forecast Volatility : Forecasting the fluctuation in the stock market (prices) . It is important to the option traders, a higher fluctuation in the stock price means higher option price and hence higher profit.

There are many ways to forecast a stock volatility.

1. Simple way to forecast a stock volatility is to assuming its historical volatility. Calculate the volatility as std ( standard deviation) of the stock returns. Now calculate the std of daily returns and then scale it. Now if you want it for N days then just multiply the daily volatility by square root of N. This is based on Einstein's Equation ( called the T1/2 rule ) to find the distance travelled by a a particle in Brownian motion .

Note: But this is not true as volatility not remain same.

2. GARCH model ( Generally Autoregressive Conditional Heteroskedasticity) . This model assume that future volatility will depend on past volatility. Let me explain what is heteroscedasticity?
Let a structural model can be represented by

where ut ~ N(0, ).

If the varianace of erros is constant var(ut) = -> homoscedasticity
and if the variance of error is not constant -> heteroscedasticity

So, heteroscedasticity => standard error estimates could be wrong.

and in finance we never assume that the variance is constant, so we need a model to assume such and one such example is GARCH. But going into detail of GARCH , let me tell you about ARCH and problems with ARCH.

ARCH : In simple sense we can say it is nothing but today's conditional variance is a weighted average of past squared returns.
σ_t­ ² = α₀+α₁ε_t-1+….+α_pε_t-p ² for all i α_i >=0 and ε_{t ~ N(}0,σ_t­ ² <sub>) ---(i)

The above model is called ARCH(p).
Note: ARCH model are not often used in financial market
why ?</sub>

1. How to decide on p?
2. The required value of p may be very large
3. Non-negativity constraints might be violated.

GARCH(1,1) model has only three parameters in the conditional variance equation compared to p+1 for ARCH(P) model. However GARCH(1,1) ~ ARCH (infinity) . Please read any material to see its prove.
In general GARCH(p,q) model adds q autoregression term with ARCH(p) specification and the conditional variance equal to
σ_t­ ² = α₀+α₁ε_t-1+….+α_pε_t-p ² ₊ β₁σ _t-1² + … +β_qσ_t-p for all i α_{i ,}β_i>=0 --(ii)
Now the GARCH(1,1) model adds just one tagged error square & one autoregression term using the stand notation for GARCH constraint ω and error coefficient β
GARCH(1,1) Model :
σ_t­ ² = ω/1-β + α ( ε_{t-1 +} β ε_{t-1 +} β² ε_t-2+ … ) ~ ARCH ( infinity)

The size of α and β determine the short-term dyamics of resulting volatility time series.
Larger β => stocks to conditional variance takes long time to die & so volatility is persistant,