Appendix - Local Volatility
Derivatives-based Portfolio Solutions


 

While Black-Scholes is the most popular method for pricing vanilla equity derivatives, exotic equity derivatives (and ITM American options) usually require a more sophisticated model. The most popular model after Black-Scholes is a local volatility model as it is the only completely consistent volatility model (Strictly speaking, this is true only for deterministic models. However, as the expected volatility of non-deterministic models has to give identical results to a local volatility model to be completely consistent, they can be considered to be a ‘noisy’ version of a local volatility model.)


LOCAL VOLATILITY IS INSTANTANEOUS VOLATILITY OF UNDERLYING
Instantaneous volatility is the volatility of an underlying at any given local point, which we shall call the local volatility. We shall assume the local volatility is fixed and has a normal negative skew (higher volatility for lower spot prices). There are many paths from spot to strike and, depending on which path is taken, they will determine how volatile the underlying is during the life of the option


BLACK-SCHOLES VOLATILITY IS AVERAGE OF LOCAL VOLATILITIES
It is possible to calculate the local (or instantaneous) volatility surface from the Black-Scholes implied volatility surface. This is possible as the Black-Scholes implied volatility of an option is the average of all the paths between spot (ie, zero maturity ATM strike) and the maturity and strike of the option. A reasonable approximation is the average of all local volatilities on a direct straight-line path between spot and strike. For a normal relatively flat skew, this is simply the average of two values, the ATM local volatility and the strike local volatility.

Black-Scholes skew is half local volatility skew as it is the average
If the local volatility surface has a 22% implied at the 90% strike, and 20% implied at the ATM strike, then the Black-Scholes implied volatility for the 90% strike is 21% (average of 22% and 20%). As ATM implieds are identical for both local and Black-Scholes implied volatility, this means that 90%-100% skew is 2% for local volatility but 1% for Black-Scholes. Local volatility skew is therefore twice the Black-Scholes skew.

ATM volatility is the same for both Black-Scholes and local volatility
For ATM implieds, the local volatility at the strike is equal to ATM, hence the average of the two identical numbers is simply equal to the ATM implied. For this reason, Black-Scholes ATM implied is equal to local volatility ATM implied.


LOCAL VOL IS THE ONLY COMPLETE CONSISTENT VOL MODEL
A local volatility model is complete (it allows hedging based only on the underlying asset) and consistent (does not contain a contradiction). It is often used to calculate exotic option implied volatilities to ensure the prices for these exotics are consistent with the values of observed vanilla options and hence prevent arbitrage. A local volatility model is the only complete consistent volatility model; a constant Black-Scholes volatility model (constant implied volatility for all strikes and expiries) can be considered to be a special case of a static local volatility model (where the local volatilities are fixed and constant for all strikes and expiries).
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