option trading -Basics

In this article, short summary of Black, Scholes and Merton model, ways to measure and forecast volatility and dynamics of implied volatility surface are discussed.

BSM Model: Delta hedged portfolio consists of a call option and ∆ units of underlying short stock or future. With passage of time this portfolio will be rebalanced to make it delta neutral.

Option prices change due to passage of time, change in underlying and change in volatility. Of three components, change in underlying impacts the option price most.

From Taylor’s rule, we get a relation between volatility, theta and gamma.

Option trading using quoted prices and estimate of option volatility can have P/L effects in the following fashion.

GAMMA profits:

(½)*S2*Γ*(σ2- σ2implied)

Vega Profits:

Vega*(σ- σimplied)

Assumptions:

1) Underlying is a trade able asset
2) Underlying pays no dividends
3) Can short the underlying in any size
4) Interest rates are constant
5) Volatility is constant
6) Underlying changes continuously


Defining and measuring volatility

Volatility is defined as square root of variance. Variance is measured as


Unbiased estimate

To get unbiased estimate of volatility needs correction factor. This correction factor depends on the assumption of the underlying process follows particular distribution (Normal distribution).



There are other estimators that can be used to measure volatility. But due to simplicity and well understood sampling properties make this estimator most desirable.


Forecasting volatility:

Volatility is a mean reverting process
Volatility of volatility is positively related to level.

In making an estimate of volatility, we need to understand primarily what events are being included and what are getting excluded. Exponentially weighted moving average model does a good job in giving higher weights to recent events and lower weighting to past events.


In the above equation most recent return values are given weightings. Λ values used generally range between 0.9 and 0.99. One draw back of exponential moving averages is that they do not address the mean reversion nature of volatility. High volatility regimes follow calm and low volatility ranges.

GARCH (generalized auto-regressive conditional heteroskedasticity) models address the above issue of mean reversion. These models are developed by Engle-Bollerslev. But this model is not Holy Grail for forecasting volatility.
This equation is specification for GARCH (1,1) Model.



This model does good job in some situations. One undesirable feature for this model is that it needs estimation of calibration parameters. These estimates are not persistent and some times they tend to be highly unstable.

Volatility cones: Forecasting volatility involves in coming up with point estimates for the volatility. When we need a range of volatilities then volatility cones will come in aid for such analysis. This analysis was initially developed by Bughardt. In this analysis, what we try to find is chart a series of 10, 20, 40, 100, 300 days volatility for securities for non-overlapping periods. This will give a context to today’s volatility.

Implied volatility dynamics: Implied volatilities for a particular stock at different strikes and maturities form a 3D surface. This volatility surface will have different shapes and contours. From time to time market changes cause the shape to undergo changes. PCA (principal component analysis) when applied to yield curve data it provides insights to level, slope and curvature changes in the yield curve. Similarly when applied to implied volatilities as a deviation from ATM (at the money volatilities) we get similar factors that explain variation.

Level Dynamics: VIX Index published by CBOE.



Level of VIX has 3 regimes, less than 20, above 20 and below 40 and above 40.
Volatility of the index is positively related to the level.
There are more large ups compared to downs.
It is mean reverting and settles with new level at each regime.

ATM volatility level for contracts maturing from front month to last month will have embedded event volatility. In other words contract maturing after event will see a sharp drop in volatility.

Smile dynamics: Good understanding of volatility smile will give us a handle to spot best strike to trade. Volatility smile is a phenomena where OTM/ITM strikes trade at different volatilities compared to ATM. These volatilities can be higher or lower compared to ATM vol subjected to market conditions.

1) retail investors buy OTM strike options (akin to lottery ticket)
2) Large funds who buy downside protection and writing covered calls


Skewness and kurtosis are 3rd and 4th moments for a distribution. These two elements additionally required to specify a particular distribution. Jarrow and Rudd (1982) have made first attempt to include these two elements into option pricing. Corrado and Su (1996) have provided a better solution to estimate these parameters.

The European call price is given by



This equation will be solved for ATMVOL, SKEW and Kurtosis.