Option on realized variance
In finance, an option on realized variance (or variance option) is a type of variance derivatives which is the derivative securities on which the payoff depends on the annualized realized variance of the return of a specified underlying asset, e.g. stock index, bond, exchange rate, etc. Another liquidated security of the same type is variance swap, which is, in other words, the futures contract on realized variance. With a similar notion to the vanilla options, variance options give the owner a right but without obligation to buy or sell the realized variance in exchange with some  agreed price (variance strike) sometime in the future (expiry date), except that risk exposure is solely subjected to the price's variance itself. This property gains interest among traders since they can use it as an instrument to speculate the future movement of the asset volatility to, for example, delta-hedge a portfolio, without taking a directional risk of possessing the underlying asset. Definitions. In practice, the annualized realized variance is defined by the sum of the square of discrete-sampling log-return of the specified underlying asset. In the words, if there are formula_1 sampling points of the underlying  prices, says formula_2 observed at time formula_3 where formula_4 for all formula_5, then the realized variance denoted by formula_6 is valued of the form where If one puts then payoffs at expiry for the call and put options written on formula_6 (or just variance call and put) are and respectively. Note that the annualized realized variance can also be defined through continuous sampling, which resulted in the quadratic variation of the underlying price. That is, if we suppose that formula_19 determines the instantaneous volatility of the price process, then defines the continuous-sampling annualized realized variance which is also proved  to be the limit in the probability of the discrete form i.e. However, this approach is only adopted to approximate the discrete one since the contracts involving realized variance are practically quoted in terms of the discrete sampling. Pricing and valuation. Suppose that under a risk-neutral measure formula_22 the underlying asset price formula_23 solves the time-varying Black–Scholes model as follows: where: ฺBy this setting, in the case of variance call, its fair price at time formula_31 denoted by formula_32 can be attained by the expected present value of its payoff function i.e. where formula_34 for the discrete  sampling while formula_35 for the continuous sampling. And by put-call parity we also get the put value once formula_32 is known. The solution can be approached analytically with the similar methodology to that of the Black–Scholes derivation once the probability density function of formula_37 is perceived, or by means of some approximation schemes, like, the Monte Carlo method. Analytically pricing variance options with discrete sampling with time-varying risk-free interest rate and constant price volatility. Owning to Rujivan and Rakwongwan (2021), in the case of discrete sampling realized variance with time-varying risk-free interest rate formula_38  and constant price volatility formula_39, by exploiting the noncentral chi-square distribution of formula_6, the analytical pricing solution for variance call can be derived by the following argument. First, if we define then the variance call price is of the following closed-form formula for formula_47 indexed the cases formula_48 and formula_49, respectively, where Although the formula gives an exact solution to the variance options, it is only functional with deterministic interest rate and constant price volatility. And for the more inclusive models where either of the parameters evolves stochastically e.g. the Heston model, there is still no closed-form  formula uncovered, leaving an intriguing issue for the further contribution of derivatives pricing.