Editing Variance swap (section)

==Pricing and valuation==
The variance swap may be hedged and hence priced using a portfolio of European [[call option|call]] and [[put option|put]] options with weights inversely proportional to the square of strike.<ref name="DDKZ99">{{cite web
  | last = Demeterfi, Derman, Kamal, Zou
  | title = More Than You Ever Wanted To Know About Volatility Swaps
  | publisher = Goldman Sachs Quantitative Strategies Research Notes
  | year = 1999
  | url = http://elis.sigmath.es.osaka-u.ac.jp/research/gs-volatility_swaps.pdf
  | archive-url = https://web.archive.org/web/20150906120858if_/http://elis.sigmath.es.osaka-u.ac.jp/research/gs-volatility_swaps.pdf
  | archive-date = 2015-09-06
  }}</ref><ref name="BSG05">{{cite web
  | last = Bossu, Strasser, Guichard
  | title = Just What You Need To Know About Variance Swaps
  | publisher = JPMorgan Equity Derivatives report
  | year = 2005
  | url = http://www.sbossu.com/docs/VarSwaps.pdf
  | archive-url = https://web.archive.org/web/20160304040051if_/http://www.sbossu.com/docs/VarSwaps.pdf
  | archive-date = 2016-03-04
  }}</ref>

Any [[volatility smile]] model which prices [[vanilla option]]s can therefore be used to price the variance swap. For example, using the [[Heston model]], a closed-form solution can be derived for the fair variance swap rate. Care must be taken with the behaviour of the smile model in the wings as this can have a disproportionate effect on the price.

We can derive the payoff of a variance swap using [[Ito's Lemma]]. We first assume that the underlying stock is described as follows:

: <math> \frac{dS_t}{S_{t}}\ = \mu \, dt + \sigma \, dZ_t </math>

Applying Ito's formula, we get:

: <math> d(\log S_t) = \left ( \mu - \frac{\sigma^2}{2}\ \right) \, dt + \sigma \, dZ_t </math>

: <math> \frac{dS_t}{S_t}\ - d(\log S_t) = \frac{\sigma^2}{2}\ dt </math>

Taking integrals, the total variance is:

:<math> \text{Variance} = \frac{1}{T}\ \int\limits_0^T \sigma^2 \, dt\ = \frac{2}{T}\ \left ( \int\limits_0^T \frac{dS_t}{S_t}\ \ - \ln \left ( \frac{S_T}{S_0}\ \right ) \right ) </math>

We can see that the total variance consists of a rebalanced hedge of <math> \frac{1}{S_{t}}\ </math> and short a log contract. <br>Using a [[static replication]] argument,<ref name="CM98">{{cite web
  | last = Carr, Madan
  | title = Towards a Theory of Volatility Trading
  | publisher = In "Volatility: New Estimation Techniques for Pricing Derivatives," R. Jarrow (ed.) RISK Publications, London
  | year = 1998
  | url = http://www.math.nyu.edu/research/carrp/papers/pdf/twrdsfig.pdf
  | archive-url = https://web.archive.org/web/20160418031433if_/http://www.math.nyu.edu/research/carrp/papers/pdf/twrdsfig.pdf
  | archive-date = 2016-04-18
  }}</ref> i.e., any twice continuously differentiable contract can be replicated using a bond, a future and infinitely many puts and calls, we can show that a short log contract position is equal to being short a futures contract and a collection of puts and calls:

: <math> -\ln \left ( \frac{S_T}{S^{*}}\ \right ) = -\frac{S_T-S^{*}}{S^{*}}\ + \int\limits_{K \le S^{*} } (K-S_T)^{+} \frac{dK}{K^2}\ + \int\limits_{K \ge S^{*} } (S_T-K)^{+} \frac{dK}{K^2}\ </math>

Taking expectations and setting the value of the variance swap equal to zero, we can rearrange the formula to solve for the fair variance swap strike:

: <math> K_\text{var} = \frac{2}{T}\ \left ( rT- \left (\frac{S_{0}}{S^{*}}\ e^{rT} -1 \right ) - \ln\left ( \frac{S^{*}}{S_0} \ \right ) + e^{rT}  \int\limits_0^{S^{*}} \frac{1}{K^2}\ P(K)\, dK + e^{rT}  \int\limits_{S^{*}}^\infty \frac{1}{K^2} C(K) \, dK  \right )</math>

where:

: <math> S_0 </math> is the initial price of the underlying security,
: <math> S^{*}>0 </math> is an arbitrary cutoff,
: <math> K </math> is the strike of the each option in the collection of options used.

Often the cutoff <math>S^{*}</math> is chosen to be the current forward price <math> S^{*} = F_0 = S_0e^{rT} </math>, in which case the fair variance swap strike can be written in the simpler form:

: <math> K_{var} = \frac{2e^{rT}}{T} \ \left (  \int\limits_0^{F_0} \frac{1}{K^2}\ P(K) \, dK +   \int\limits_{F_0}^\infty \frac{1}{K^2}\ C(K) \, dK \right )</math>

===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.