More than you ever wanted to know about Volatility Swaps
The note titled,” More than you ever wanted to know about Volatility Swaps”, written by Derman, Demeterfi, Kamal and Zhou, is a fantastic fifty page write up highlighting many aspects of valuing a variance swap and a volatility swap. I love the structure followed in the note. Instead of heading right in to the math behind valuation, the paper gives starts off by giving a superb intuition into the need for variance swap and how does one go about pricing a variance swap with nothing more than common sense. In this blog post, I will summarize some of the points from the note.
What’s the need for variance swap or volatility swap?
If an investor wants to take a long volatility or short volatility position, exposure to an option is one of the common ways. However this has a problem as the option position gives the trader exposure to direction of the stock as well as the volatility. What if the trader wants to trade forward volatility? A delta hedged option removes the exposure to the stock direction to a certain extent but not completely. There is a clear need for a trader to directly trade volatility. That’s where variance swaps and volatility swaps come in to prominence. Volatility swaps are forward contracts on annualized volatility. These swaps have several characteristics that make trading attractive. Even though option market participants talk about volatility, it is the variance that has a more fundamental theoretical significance. This is so because the correct way to value a swap is to value the portfolio that replicates it, and the swap that can be replicated most reliably is a variance swap.
What’s the intuition behind pricing a variance swap?
A trader looking to have an exposure on pure volatility needs a portfolio that is sensitive to changing volatility. If he/she takes an exposure in to a single option, then as the spot moves away from the strike price of the contract, the option loses its vega and hence is no longer responds to the changing volatility. Ok,so a single option is not enough. Hence the trader has to have a portfolio of options with different strikes. Does equal exposure to all the strikes makes a portfolio immune to stock price movement? Using some visuals the paper makes an intuitive argument that the options must be weighed inversely proportion to the strike squared. This fact is also proved mathematically in the appendix. However one need to go through the math to appreciate the fact that a set of options weighted in a certain manner makes a portfolio immune to stock price movement. The authors then show that an exposure to a set of options with varying strikes resembles a log contract, an exotic option that isn’t traded in the market. Having established the connection between variance swap payoff and log contract payoff, the note moves on to the actual math.
What’s the math behind obtaining the fair value of a variance swap?
Well, the math is not as daunting as I expected. In just a few steps it is extremely clear that the difference between the SDE that drives a log Forward contract and a SDE of a certain exposure in the forward contract gives a pure exposure to the realized variance. To put it in simple words, a log contract hedged by a static replication involving forward contract gives the trader exposure to pure realized variance devoid of any stock movement contamination.
How to replicate the hedge in the real world ?
Once the expression for the expected value of a variance swap is derived, the rest of the details revolve around applying the fair value of the variance swap to the real world where one has to deal with the following aspects among many:

Finite set of strikes

Volatility Skew

Price Jumps
All the above aspects are discussed at great lengths. For volatility skew, the note also gives a closed price formula for the fair value, depending on whether one assumes skew to vary linearly with strike or black schools delta. The final section of the paper deals with pricing volatility swap from variance swap. Naïve pricing of volatility swap from variance swap will result in incorrect pricing. The note concludes saying that a healthy variance swaps market is needed to price and value volatility swaps and this entails requiring an arbitrage free stochastic evolution of volatility surface.
This paper is a great paper to learn many aspects of math finance. Pricing, valuation, replicating in the real world, dealing with real world hedging issues, etc. are all discussed in the context of a variance swap.