Editing Variance swap (section)

===Analytically pricing variance swaps with discrete-sampling===

One might find discrete-sampling of the realized variance, says, <math>\sigma^2_{\text{realized}}</math> as defined earlier, more practical in valuing the variance strike since, in reality, we are only able to observe the underlying price discretely in time. This is even more persuasive since there is an assertion that <math>\sigma^2_{\text{realized}}</math> converges in probability to the actual one as the number of price's observation increases. <ref>{{cite journal |last1=Barndorff-Nielsen |first1=Ole E. |last2=Shephard |first2=Neil |author-link=Ole Barndorff-Nielsen|author-link2=Neil Shephard  |date=May 2002 |title=Econometric analysis of realised volatility and its use in estimating stochastic volatility models|pages=253–280 |doi=10.1111/1467-9868.00336|journal=Journal of the Royal Statistical Society, Series B  |volume=64 |issue=2 |s2cid=122716443 |doi-access=free }}</ref>

Suppose that in the risk-neutral world with a martingale measure <math>\mathbb{Q}</math>, the underlying asset price <math>S=(S_t)_{0\leq t \leq T}</math> solves the following SDE:

: <math>\frac{dS_t}{S_t}=r(t) \, dt+\sigma(t) \, dW_t, \;\; S_0>0</math>

where:
*<math>T</math> imposes the swap contract expiry date,
*<math>r(t)\in\mathbb{R}</math> is (time-dependent) risk-free interest rate,
*<math>\sigma(t)>0</math> is (time-dependent) price volatility, and
*<math>W=(W_t)_{0\leq t \leq T}</math> is a Brownian motion under the filtered probability space <math>(\Omega,\mathcal{F},\mathbb{F},\mathbb{Q})</math> where <math>\mathbb{F}=(\mathcal{F}_t)_{0\leq t \leq T}</math> is the natural filtration of <math>W</math>.

Given as defined above by <math> (\sigma^2_{\text{realized}} - \sigma^2_{\text{strike}})\times N_{\text{var}} </math> the payoff at expiry of variance swaps, then its expected value at time <math>t_0</math>, denoted by <math>V_{t_0}</math> is 

: <math>V_{t_0}=e^{\int^T_{t_0}r(s)ds}\mathbb{E}^{\mathbb{Q}}[\sigma^2_{\text{realized}} - \sigma^2_{\text{strike}} \mid \mathcal{F}_ {t_0}] \times N_{\text{var}}.</math>

To avoid arbitrage opportunity, there should be no cost to enter a swap contract, meaning that <math>V_{t_0}</math> is zero. Thus, the value of fair variance strike is simply expressed by

: <math>\sigma^2_{\text{strike}}=\mathbb{E}^{\mathbb{Q}}[\sigma^2_{\text{realized}} \mid \mathcal{F}_{t_0}],</math>

which remains to be calculated either by finding its closed-form formula or utilizing numerical methods, like Monte Carlo methods.