# Implied Volatility Dynamics

Implied Volatility Delta

There is a natural order of market data speed:

- Spot levels change faster than ATM volatility

- ATM volatility changes faster than volatility skew

- Volatilities are more volatile than dividend forecasts.

Hedging performance can be improved by assuming a link between different market parameters.

Example: When calculating a price with a new spot, or computing th delta using a spot shift, one may assume that this move is accompanied by a volatility move in the opposite direction or a change in expected dividends in the same direction.

Thus, a delta hedge also hedges part of vega if stock and volatility are correlated.

If delta and vega are hedged separately one has to be careful not to double count vega exposure.

This section discusses pricing approaches assuming a spot move but no new volatility information.

Deterministic Smile Dynamics

Deterministic Smile Dynamics assumes that the IV surface depends on spot only.

Thus there is a function  XXX which denotes the IV surface observed at time t if spot level is St

There a 2 important special cases:

1. Sticky strike The dynamics of Vanilla option is thus described by the BS model, which is the only complete model with sticky strike dynamics.

Otherwise an arbitrage can be constructed by fixing 2 calls with IV: > The P&L of a delta hedged position over dt is given by: An arbitrage can be constructed by trading since the corresponding P&L has no gamma/theta, and is positive: The ATM volatility for equities around the ATM strike behaves as a sticky strike movement. In other words, the volatility associated with a given strike does not change when the spot moves.

2. Sticky delta refers to a dynamics of Vanilla options depending on moneyness and term: The only complete models with sticky delta dynamics are those which assume independent returns.

For currencies, behavior is sticky delta. In other words, the volatility associated with an option and given delta does not change when spot changes.

Volatility reacts slowly on spot moves. In quiet markets, volatility is quoted by strike and is updated much less frequently than spot --> Sticky strike.

When markets are volatile then IV will be updated more frequently and dynamics may resemble sticky delta.

Realistic models should exhibit stochastic IV dynamics, in the sense that the smile dynamics may allow for both sticky strike and sticky delta dynamics, as well as random changes between the two. To some extent local stochastic volatility models capture this behaviour.