In real life, volatility is changing every moment in an uncertain manner. This means that the underlying variable can change at any moment and can take to any value. Any variable which changes in an uncertain manner is said to be following a stochastic process.

The Generalised Autoregressive Conditional Heteroskedasticity GARCH ( 1 , 1 ) Model tries to capture three components and uses three parameters in estimating the current volatility-

Innovation (μ2n-1) i.e., actual previous volatility based on the continuous change in the variable.

Innovation (μi) = LN ( Si / Si-1 )

Where,

Si = current at the end of the day closing stock price

Si-1 = previous at the end of the day closing stock price

Persistence i.e., previous volatility estimate (σ2n-1), where current volatility estimate (σ2n) tends to cluster around the previous estimate of volatility (σ2n-1).

Mean reverting tendency i.e., volatility has a tendency of reverting to its long-term mean-variance. For suppose, if the previous volatility (σ2n-1) is more than far from its long-term variance (σ2nLt) or is less than far from its long-term variance, σ2n will revert to σ2nLt. This means volatility term-structure is downward sloping in case of σ2n is more than σ2nLt or upward sloping in case of σ2n is less than σ2nLt. To incorporate this feature, we have-

Current Volatility Estimate (σ2n) = [ Gamma (γ) * Long-term Variance (σ2nLt) ] + [ Beta (β) * Previous Volatility Estimate (σ2n-1) ] + [ Alpha (α) * Innovation (μn-1) ]

Where,

γ = 1 - Beta(β) - Alpha(α)

σ2nLt = long-term variance/volatility estimate

β = beta, estimated using Parameter Estimation Engine

σ2n-1 = previous volatility estimate

α = alpha, estimated using Parameter Estimation Engine

μ2n-1 = innovation i.e., actual previous volatility based on continuous change in the variable

Omega(ω) = Gamma (γ) * Long-term Variance (σ2nLt)

Parameter Gamma(γ) is the residual value i.e., 1 - Beta(β) - Alpha(α). Suppose, Gamma(γ) is 0.04. It means that 4% importance (weight) is given to its long-term variance/volatility.

Parameter Beta(β) is estimated using Parameter Estimation Engine either via Unbiased Estimator Approach or Maximum Likelihood Estimator Approach. Suppose, Beta(β) estimated using the Parameter Estimation Engines is 0.91. It means that 91% importance (weight) is given to Persistence.

Parameter Alpha(α) is estimated using Parameter Estimation Engine either via Unbiased Estimator Approach or Maximum Likelihood Estimator Approach. Suppose, Alpha(α) estimated using the Parameter Estimation Engines is 0.05. It means that 5% importance (weight) is given to Innovation.

If the series of the variable is misbehaved, Beta(β) will be low as it is obvious to give less importance to Persistence and more importance to Innovation. And if the series is well-behaved, Beta(β) will be high and therefore, more importance to Persistence and less importance to Innovation. Also, if the previous volatility (σ2n-1) is more than far from its long-term variance (σ2nLt) or is less than far from its long-term variance, σ2n will revert to σ2nLt.

### Let us understand GARCH ( 1 , 1 ) Model for Estimating Volatility using an illustration-

The volatility of a stock which is currently trading at $500 is estimated at 1.5%. Update the volatility estimate if the stock price at the end of the day closing happens to be $495. Long-term variance is estimated to be 0.000119.

Parameter Beta (β) is estimated to be 0.91 and Parameter Alpha (α) is estimated to be 0.05.

Previous volatility estimate (σn-1) = 1.5%

Innovation (μn-1) = LN ( Si / Si-1 ) = LN ( $495 / $500 ) = -1.005%

Long-term variance (σ2nLt) = 0.000119 i.e., volatility (σnLt) = √0.000119 = 0.010912 = 1.0912%

Beta (β) = 0.91

Alpha (α) = 0.05

Gamma (γ) = 1 - 0.91 - 0.05 = 0.04

Current Volatility Estimate (σ2n) = [ Gamma (γ) * Long-term Variance (σ2nLt) ] + [ Beta (β) * Previous Volatility Estimate (σ2n-1) ] + [ Alpha (α) * Innovation (μn-1) ]

Therefore,

σ2n = 0.04 * (1.0912)^2 + 0.91 * ( 1.5 )^2 + 0.05 * ( -1.005 )^2 = 2.1456%2

σn = √2.1456 = 1.4648%

Since, Innovation i.e., actual previous volatility based on the change in the variable of -1.005% (getting weightage of 0.05) is lower than the previous volatility estimate (σ2n-1) of 1.5% (with Beta (β) 0.91). Hence, the currently estimated volatility of 1.4648% is slightly less than the previous volatility estimate of 1.5%.

Let us understand GARCH ( 1 , 1 ) Model for Estimating Volatility using market data-

We want to estimate the daily volatility of a stock (IndusInd Bank) for the next day and to estimate that, we have 737 days closing stock price of the period 01-01-2017 to 31-12-2019.

We calculate the daily return of the stock [ LN ( Sn / Sn-1 ) ] as shown above- [ represented in column "Change" ]

Graph representing actual volatility of the stock over the period.

As we know, to calculate the volatility of the stock, we take the square root of the squared returns so as to incorporate both upside and downside fluctuation in the stock price. [ represented in column "Change^2" ]

Note: As per Efficient Market Hypothesis, it is very difficult to predict the stock price at the end of the next day. The best prediction would be the current stock price, such that the expected return (i.e., mean(X) of the current returns) would be 0. Hence, the square root of X and not X - X̅.

Finally, to calculate the Current Volatility Estimate (σ2n)-

Current Volatility Estimate (σ2n) = [ Gamma (γ) * Long-term Variance (σ2nLt) ] + [ Beta (β) * Previous Volatility Estimate (σ2n-1) ] + [ Alpha (α) * Innovation (μn-1) ]

Where, for Alpha(α) and Beta(β), Parameter Estimation Engine: Maximum Likelihood Estimator Approach which maximizes the joint probability of observing the data, can be used to identify the the Parameters: Alpha(α) = 0.945204540619286 and Beta(β) = 0.0547954593807137.

γ = 1 - Beta(β) - Alpha(α) = 0.00000000000000034

σ2nLt = long-term variance = 0.00034047 (i.e., Unconditional Variance)

Therefore,

σ2n = Gamma (γ) * Long-term Variance (σ2nLt) + Beta (β) * Previous Volatility Estimate (σ2n-1) + Alpha (α) * Innovation (μn-1)

σ2n = 0.00000000000000034 * 0.00034047 + ( 0.945204540619286 * 0.00034047 ) + ( 0.0547954593807137 * 0.000198041294580257 ) = 0.00033267%2

σn = √0.00033267 = 1.8239%

The current volatility estimated using the Unbiased Approach is 1.8452%. And under the GARCH (1,1) Approach, the current volatility estimated is 1.8239%- very close to the Unbiased Approach. We can conclude that GARCH (1,1) Approach done a very good job. Also note that the parameters, Alpha(α) and Beta(β), are estimated using Parameter Estimation Engine: Maximum Likelihood Estimator Approach which maximizes the joint probability of observing the data- also done a very good job in our estimation. The comment on the preciseness can only be done once it is backtested.

Drawbacks of GARCH ( 1 , 1 ) Model

GARCH ( 1 , 1 ) Model requires three parameters to estimate using the parameter estimation engine which may or may not be accurate together and can lead to greater noise, and therefore, estimating volatility may not be precise. Also, gamma(γ) can be negative (having the residual value after estimating beta(β) & alpha(α). This means previous volatility (σ2n-1) may not be reverting to its long-term variance (σ2nLt) i.e., mean fleeing rather than mean-reverting.

Greek Alphabets

lambda(λ), volatility(σ), omega(ω), innovation(μ), alpha(α), beta(β), gamma(γ)