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Option on realized variance
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<h2 id="definitions">Definitions</h2> <p>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 {\displaystyle n+1} sampling points of the underlying prices, says S t 0 , S t 2 , … , S t n {\displaystyle S_{t_{0}},S_{t_{2}},\dots ,S_{t_{n}}} observed at time t i {\displaystyle t_{i}} where 0 ≤ t i − 1 < t i ≤ T {\displaystyle 0\leq t_{i-1}<t_{i}\leq T} for all i ∈ { 1 , … , n } {\displaystyle i\in \{1,\dots ,n\}} , then the realized variance denoted by R V d {\displaystyle RV_{d}} is valued of the form </p> R V d := A n ∑ i = 1 n ln 2 ( S t i S t i − 1 ) {\displaystyle RV_{d}:={\frac {A}{n}}\sum _{i=1}^{n}\ln ^{2}{\Big (}{\frac {S_{t_{i}}}{S_{t_{i-1}}}}{\Big )}} <p>where </p> <ul><li> A {\displaystyle A} is an annualised factor normally selected to be A = 252 {\displaystyle A=252} if the price is monitored daily, or A = 52 {\displaystyle A=52} or A = 12 {\displaystyle A=12} in the case of weekly or monthly observation, respectively and</li> <li> T {\displaystyle T} is the options expiry date which is equal to the number n / A . {\displaystyle n/{A}.} </li></ul> <p>If one puts </p> <ul><li> K var C {\displaystyle K_{\text{var}}^{C}} to be a variance strike and</li> <li> L {\displaystyle L} be a notional amount converting the payoffs into a unit amount of money, say, e.g., USD or GBP,</li></ul> <p>then payoffs at expiry for the call and put options written on R V d {\displaystyle RV_{d}} (or just variance call and put) are </p> ( R V d − K var C ) + × L {\displaystyle (RV_{d}-K_{\text{var}}^{C})^{+}\times L} <p>and </p> ( K var C − R V d ) + × L {\displaystyle (K_{\text{var}}^{C}-RV_{d})^{+}\times L} <p>respectively. </p><p>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 σ ( t ) {\displaystyle \sigma (t)} determines the instantaneous volatility of the price process, then </p> R V c := 1 T ∫ 0 T σ 2 ( s ) d s {\displaystyle RV_{c}:={\frac {1}{T}}\int _{0}^{T}\sigma ^{2}(s)ds} <p>defines the continuous-sampling annualized realized variance which is also proved to be the limit in the probability of the discrete form<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> i.e. </p> lim n → ∞ R V d = lim n → ∞ A n ∑ i = 1 n ln 2 ( S t i S t i − 1 ) = 1 T ∫ 0 T σ 2 ( s ) d s = R V c {\displaystyle \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}} . <p>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. </p> <h2 id="pricing-and-valuation">Pricing and valuation</h2> <p>Suppose that under a risk-neutral measure Q {\displaystyle \mathbb {Q} } the underlying asset price S = ( S t ) 0 ≤ t ≤ T {\displaystyle S=(S_{t})_{0\leq t\leq T}} solves the time-varying <a href="/facts/Black%E2%80%93Scholes/TPPG5YWQ">Black–Scholes</a> model as follows: </p> d S t S t = r ( t ) d t + σ ( t ) d W t , S 0 > 0 {\displaystyle {\frac {dS_{t}}{S_{t}}}=r(t)\,dt+\sigma (t)\,dW_{t},\;\;S_{0}>0} <p>where: </p> <ul><li> r ( t ) ∈ R {\displaystyle r(t)\in \mathbb {R} } is (time varying) risk-free interest rate,</li> <li> σ ( t ) > 0 {\displaystyle \sigma (t)>0} is (time varying) price volatility, and</li> <li> W = ( W t ) 0 ≤ t ≤ T {\displaystyle W=(W_{t})_{0\leq t\leq T}} is a Brownian motion under the filtered probability space ( Ω , F , F , Q ) {\displaystyle (\Omega ,{\mathcal {F}},\mathbb {F} ,\mathbb {Q} )} where F = ( F t ) 0 ≤ t ≤ T {\displaystyle \mathbb {F} =({\mathcal {F}}_{t})_{0\leq t\leq T}} is the natural filtration of W {\displaystyle W} .</li></ul> <p>ฺBy this setting, in the case of variance call, its fair price at time t 0 {\displaystyle t_{0}} denoted by C t 0 var {\displaystyle C_{t_{0}}^{\text{var}}} can be attained by the expected present value of its payoff function i.e. </p> C t 0 var := e − ∫ t 0 T r ( s ) d s E Q [ ( R V ( ⋅ ) − K var C ) + ∣ F t 0 ] , {\displaystyle C_{t_{0}}^{\operatorname {var} }:=e^{-\int _{t_{0}}^{T}r(s)\,ds}\operatorname {E} ^{\mathbb {Q} }[(RV_{(\cdot )}-K_{\operatorname {var} }^{C})^{+}\mid {\mathcal {F}}_{t_{0}}],} <p>where R V ( ⋅ ) = R V d {\displaystyle RV_{(\cdot )}=RV_{d}} for the discrete sampling while R V ( ⋅ ) = R V c {\displaystyle RV_{(\cdot )}=RV_{c}} for the continuous sampling. And by <a href="/facts/Put-call_parity/C3vpwfdG">put-call parity</a> we also get the put value once C t 0 var {\displaystyle C_{t_{0}}^{\text{var}}} is known. The solution can be approached analytically with the similar methodology to that of the <a href="/facts/Black%E2%80%93Scholes/TPPG5YWQ">Black–Scholes</a> derivation once the <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> of R V ( ⋅ ) {\displaystyle RV_{(\cdot )}} is perceived, or by means of some approximation schemes, like, the <a href="/facts/Monte_Carlo_method/AkHjY7jc">Monte Carlo method</a>. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Variance_swap/UbUUclLe">Variance swap</a></li> <li><a href="/facts/Volatility_swap/MxcfxvNt">Volatility swap</a></li> <li><a href="/facts/Volatility_(finance)/mCdus1MW">Volatility (finance)</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Barndorff-Nielsen, Ole E.; Shephard, Neil (May 2002). "Econometric analysis of realised volatility and its use in estimating stochastic volatility models". Journal of the Royal Statistical Society, Series B. 64 (2): 253–280. doi:10.1111/1467-9868.00336. S2CID 122716443. <a href="/wiki/Ole_Barndorff-Nielsen" target="_blank">/wiki/Ole_Barndorff-Nielsen</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>