Local vs Stochastic vs Implied Volatilities

Derives and explains the connection and links between the three important concepts of volatilities (local, stochastic, and implied volatilities)-i.e., local volatility vs stochastic volatility, and local volatility vs Black Scholes’ implied volatility. Contents by timeline:

01:29- Explain relationship between local and stochastic vol using Gyongy’s lemma
08:30 - Explain relationship between local and Black Scholes’ implied vol
17:58 - Relationship between local and Black Scholes’ time-dependent implied vol
21:43 - Explain visually the relationship between local and implied volatilities
26:39 - Show that Implied vol is time-average of spatial-average of local vol
28:26 - sub-topic, Show dollar delta and dollar gamma satisfy Black Scholes PDE
32:15 - Summarise the relationship between the the stochastic, local, and implied volatilities

In terms of literature, this video covers the topics discussed in the last section of Chapter 1 (Local Variance as a Conditional Expectation of Instantaneous Variance), and the first part of Chapter 3 (Getting Implied Volatilities from Local Volatilities) of Jim Gatheral’s The Volatility Surface, A Practitioner’s Guide; though our spin, that’s the approach that we take and the details we include, is markedly different.