## Mini Chapter Six

# Volatility and its smiles

## Implied vs Realised Vol

- Black Scholes model assumes a normal distribution for the returns of the underlying which implies lognormal distribution for its prices (refer to the explanation above). Hence if one plugs in a vol (as indicated by one’s proprietary model) to this formula it would return an option price that counterparties agree to trade i.e. the observed market price.

- This plugged vol number is the popularly referred Implied Vol (IV) that gets quoted by dealers, also called the Black Scholes Vol.

- While the BS model assumes a constant vol for the underlying asset during the life of an option – market’s way of adjusting for the fact that prices may not be lognormally distributed (i.e. vol is not constant) is to trade different strikes for the same maturity at different implied vols effectively changing the probability distribution of the underlying.

- This phenomenon of different implied vols at different strikes of an asset for the same tenor is referred to as the Volatility Smile. Depending on the demand and supply of calls vs puts for an underlying, the smile (generally understood to be symmetrical) can develop a skew on either side.

**Graph 14 – Diagram for a smile, skew, flat vol**

Source: Pandemonium.

- Equity options are generally known to have a downside skew i.e. larger demand for OTM puts to hedge against the downside. Seen from the lens of an option seller as market spirals lower, gamma on lower strikes increases which adds to their short (gamma) positions which leads to larger losses (realised vol goes up too) at the time of hedging. To compensate for these losses, sellers charge a higher IV to the buyers resulting/reinforcing the skew.

## Skew can sometimes be seen as a function of time

**Volatility Surface**– is a three dimensional snapshot of an asset’s Implied Volatility against the different strikes and time to expiration.

**Graph 15 – Snapshot of a Volatility Surface**

Source: Pandemonium.

The implied vol surface can be understood to have an equilibrium of its own such that the **relative steepness and or flatness of the skew can’t be arbitraged**. Given a non-zero price for going long optionality one can understand this equilibrium by dissecting the surface as, a) different strikes for the same expiry (call spreads) b) same strike for different expiries (calendar spread) c) equidistant strikes for the same expiry (butterfly spreads). Therefore a vol surface to be free of arbitrage must meet the following conditions:

- Option premium for call spreads for all maturities must be positive, and the ratio of the premium to the distance between strikes would be less than or equal to 1. Recall that max payoff for a call spread is the distance between the strikes.
- Option premium for Calendar spreads across time horizons must be positive.
- Option premium for all butterfly spreads must be positive.
- Any interpolation or extrapolation of implied vols in a surface must adhere to the conditions above.
- Also the same conditions would be met if we were to replace calls with puts because of the Put Call parity.

- Standard Volatility measurement – measured by variance is computed as:

where T is the number of business days in the tenor of the option, 𝜎 is the standard deviation.

- Daily vol can therefore be annualised by multiplying the daily 𝜎 by √252, and conversely annualised can be converted to daily by dividing the annualised 𝜎 by √252, where 252 is the number business days in a year.

**Realised Volatility**– is the observed volatility or what’s been realised in terms of vol in a past time period, often used for a sanity check on implied vols (adjusting for market moving events). Relative cheapness of an option at a certain strike is also assessed by comparing its realised vs implied vols at that strike. Typically options with lower implied vols (vs realised) are purchased in the hope of making money on it if implied catches up with realised.