Option on realized variance

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 other words, if there are n+1 sampling points of the underlying prices, says S_{t_0},S_{t_2},\dots,S_{t_{n}} observed at time t_i where 0\leq t_{i-1} for all i\in \{1,\dots,n\}, then the realized variance denoted by RV_d is valued of the form :RV_d:=\frac{A}{n}\sum_{i=1}^{n}\ln^2\Big(\frac{S_{t_i}}{S_{t_{i-1}}}\Big) where • A is an annualised factor normally selected to be A=252 if the price is monitored daily, or A=52 or A=12 in the case of weekly or monthly observation, respectively and • T is the options expiry date which is equal to the number n/{A}. If one puts • K^C_{\text{var}} to be a variance strike and • L be a notional amount converting the payoffs into a unit amount of money, say, e.g., USD or GBP, then payoffs at expiry for the call and put options written on RV_d (or just variance call and put) are :(RV_d-K^C_{\text{var}})^+\times L and :(K^C_{\text{var}}-RV_d)^+\times L 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 \sigma(t) determines the instantaneous volatility of the price process, then :RV_{c}:= \frac{1}{T}\int_{0}^{T}\sigma^2(s)ds defines the continuous-sampling annualized realized variance which is also proved to be the limit in the probability of the discrete form i.e. :\lim_{n\to\infty}RV_d=\lim_{n\to\infty}\frac{A}{n}\sum_{i=1}^{n}\ln^2\Big(\frac{S_{t_i}}{S_{t_{i-1}}}\Big)=\frac{1}{T}\int_{0}^{T}\sigma^2(s)ds=RV_{c}. 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 \mathbb{Q} the underlying asset price S=(S_t)_{0\leq t \leq T} solves the time-varying Black–Scholes model as follows: : \frac{dS_t}{S_t}=r(t) \, dt+\sigma(t) \, dW_t, \;\; S_0>0 where: • r(t)\in\mathbb{R} is (time varying) risk-free interest rate, • \sigma(t)>0 is (time varying) price volatility, and • W=(W_t)_{0\leq t \leq T} is a Brownian motion under the filtered probability space (\Omega,\mathcal{F},\mathbb{F},\mathbb{Q}) where \mathbb{F}=(\mathcal{F}_t)_{0\leq t \leq T} is the natural filtration of W. ฺBy this setting, in the case of variance call, its fair price at time t_0 denoted by C_{t_0}^\text{var} can be attained by the expected present value of its payoff function i.e. :C_{t_0}^\operatorname{var}:=e^{-\int^T_{t_0} r(s) \, ds}\operatorname{E}^{\mathbb{Q}}[(RV_{(\cdot)}-K^C_{\operatorname{var}})^+\mid\mathcal{F}_{t_0}], where RV_{(\cdot)} = RV_d for the discrete sampling while RV_{(\cdot)} = RV_c for the continuous sampling. And by put-call parity we also get the put value once C_{t_0}^\text{var} 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 RV_{(\cdot)} is perceived, or by means of some approximation schemes, like, the Monte Carlo method. ==See also==