An equation relating the Black-Merton-Scholes Greeks vega, vanna, and volga is derived by making use of the generalised Hull-White formula. Given only three options the equation automatically gives a robust and accurate approximation for the volatility swap strike, the variance swap strike, and...

In this paper we give a model-free approximation for the price of forward starting volatility swaps. Moreover, we show that a self-financing and model-independent approximate hedge is achieved by dynamically trading zero vanna forward starting straddles with a skew adjusted notional. To the best...

The forward start dual volatility swap is introduced. It can be regarded as the analog for volatility of what the entropy contract is for variance. Under the risk neutral measure it is shown that the difference between the forward start volatility swap and its dual is approximately the...

We describe the steps required to approximately price volatility derivatives by making use of only three near the money vanilla index options per maturity. The only assumption we make is that the index process is a diffusion with a volatility process that does not depend on the index level. No...

The Alos decomposition formula is written down, but making use of Bachelier price formula instead of the Black-Scholes price formula. Due to the linearity of the Bachelier formula in volatility at the at-the-money strike, the decomposition formula gives an exact expression for the price of a...

It is well-known that in normal stochastic volatility models with zero correlation the fresh volatility swap price is exactly equal to the at-the-money implied volatility. To replicate a volatility swap, however, the price of a volatility swap at inception is insufficient. Its price throughout...

The main assumption of the vanna-volga method is the existence of a flat but stochastic implied volatility for the pricing of vanilla options. Even though `flat but stochastic' appears to be self-contradictory, it is to an extent justified on empirical grounds. In this short note we argue that...