then its expected value at time , denoted by is To avoid arbitrage opportunity, there should be no cost to enter a swap contract, meaning that is zero. Thus, the value of fair variance strike is simply expressed by which remains to be calculated either by finding its closed-form formula or utilizing numerical methods, like Monte Carlo methods. For the former version, according to the argument contributed by Rujivan and Rakwongwan (2021), once define and for , then the fair variance strike can be derived into the form of This solution subjects to arbitrary forms of and as long as they have continuous trajectories on , making it versatile to utilize in more general interpretation of the Black–Scholes models. Uses Many traders find variance swaps interesting or useful for their purity. An alternative way of speculating on volatility is with an option, but if one only has interest in volatility risk, this strategy will require constant delta hedging, so that direction risk of the underlying security is approximately removed. What is more, a replicating portfolio of a variance swap would require an entire strip of options, which would be very costly to execute. Finally, one might often find the need to be regularly rolling this entire strip of options so that it remains centered on the current price of the underlying security. The advantage of variance swaps is that they provide pure exposure to the volatility of the underlying price, as opposed to call and put options which may carry directional risk (delta). The profit and loss from a variance swap depends directly on the difference between realized and implied volatility. Another aspect that some speculators may find interesting is that the quoted strike is determined by the implied volatility smile in the options market, whereas the ultimate payout will be based upon actual realized variance. Historically, implied variance has been above realized variance, a phenomenon known as the variance risk premium, creating an opportunity for volatility arbitrage, in this case known as the rolling short variance trade. For the same reason, these swaps can be used to hedge options on realized variance. Related instruments Closely related strategies include straddle, volatility swap, correlation swap, gamma swap, conditional variance swap, corridor variance swap, forward-start variance swap, option on realized variance and correlation trading. References Derivatives (finance) Swaps (finance) Mathematical finance Financial economics Banking