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Suppose I have the formula for computing $\mathbb E^P\big[\int_0^T v\,dt\big]$ for the variance process $v$ in the real world measure $P$. Can I set it to the VIX$^2$ price and solve for the variance risk premium? My concern is that VIX is a traded asset and its risk premium is zero and the equality does not hold. However, would it be correct to argue that VIX$^2$ is not traded and thus the procedure is correct?

I will put the above question in specific terms. Assume the variance $v$ undergoes the process $$dv = a\,dt+b\,dB$$ where $B$ is the standard Brownian motion. The transformation between the real-world and risk-neutral measures is $a_P = a_Q-\lambda_v b$ where $\lambda_v$ is the market price of variance risk, and subscripts $Q$ and $P$ denotes the risk-neutral and real-world measures, respectively. We than set $$f[a_Q,b]:=\frac1T\mathbb E^Q\Big[\int_0^T d\langle\ln S\rangle_t\Big]$$ where $f[u,v]$ denotes a function $f$ of functions $u$ and $v$.

From the vanilla option market, I calibrate the functions $a_Q$ and $b$. Now I set $f[a_Q-\lambda_vb,b]=$VIX$^2$ where VIX denotes the price TRADED on the market of a fresh start VIX of maturity $T$, to compute $\lambda_v$.

Is this correct? My rationale is that although $VIX$ is traded and thus evaluated in the risk-neutral measure $Q$, VIX$^2$ is not and therefore is evaluated in the real-world measure $P$.

Edit: I now think this is wrong because the variance swap is traded and thus should be valued in the risk-neutral measure $Q$ and we cannot obtain the risk premium this way. The correct way to estimate this is to set $$f[a_P,b]=\frac1T\int_0^T d\langle\ln S\rangle_t$$ for the realized $S_t$ to solve for $\lambda_v$.

As a matter of fact, if we have a mean reversion form for the variance process, for example, $a=-\kappa(v-v_\infty)$, which generates an asymptotic stationary variance $v_\infty$ at long time, we can simply set $$v_\infty[a_P,b]=\lim_{T\to\infty}\frac1T\int_0^T d\langle\ln S\rangle_t$$ where $S$ is the realized stock price, to compute $\lambda_v$.

Hans
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  • "My concern is that VIX is a traded asset" -- no? It's given by a formula that is a function of traded asset prices, but it is not traded itself. – user217285 Aug 01 '18 at 02:52
  • @Nitin: I agree with your last sentence. But what do you mean by the question mark following your "no"? Also, what is your conclusion? Do you agree with my rationale as stated in my last sentence that my setup is correct? – Hans Aug 01 '18 at 06:02
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    It's not entirely clear to me what you're trying to achieve. What is the variance process $(v_t)_{t \geq 0}$ which you are talking about? Is this in the framework of a particular model? What do you define as the variance risk premium? – Quantuple Aug 01 '18 at 07:55
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    So assuming $VIX^2$ represents the fair strike of a fresh-start, idealised (i.e. continuous returns sampling) variance swap maturing in 30 days then indeed: $VIX^2 = \frac{1}{T} \Bbb{E}^\Bbb{Q} \left[ \int_0^T d\langle \ln S \rangle_t \right]$ with $T=30/DCC$ years. And the idea would be to write that: $VRP = VIX^2 - \frac{1}{T} \Bbb{E}^P \left[ \int_0^T d\langle \ln S \rangle_t \right] = \frac{1}{T} \int_0^T \left( \Bbb{E}^Q[v_t] - \Bbb{E}^P[v_t] \right) dt$ assuming that $d\ln S_t = \cdot dt + v_t dW_t$. Did I understand that well? – Quantuple Aug 01 '18 at 12:50
  • @Quantuple: Your understanding is mostly correct but with some difference to my intended argument. I have edited my question to let it stand on a more specific ground. Please review. Thank you. – Hans Aug 01 '18 at 17:22
  • @Quantuple: I now think my rationale is wrong and wrote as much at the end of in my question. But please scrutinize my argument anyway. – Hans Aug 01 '18 at 18:35
  • Yes or edits make better sense. It's possible to do that easily if you're willing to make an assumption on the "form" of the market price of volatility risk like Heston did in his original paper. Have you had a look at this: https://quant.stackexchange.com/questions/37722/pricing-vix-futures/37805#37805 – Quantuple Aug 03 '18 at 08:06
  • @Quantuple: I do have a specific form for the stochastic volatility SDE as well as the market price of volatility risk. I read your answer you linked to which, as you say, shows the specific form of the risk premium. However, as I have realized and as your answer implies, the final express is in the risk neutral measure and thus does not give a means to compute the risk premium. This has to be done in the real world measure. I am edited further my question to compute the volatility risk premium via the long time asymptotics for a mean reverting variance. – Hans Aug 04 '18 at 16:52
  • Yes, I've already seen this technique used e.g. here: Dimitroff G., Szimayer A., Wagner A., Quanto Option Pricing in the Parsimonious Heston Model. – Quantuple Aug 06 '18 at 06:58
  • @Quantuple: Thank you for the reference. I will prove the ergodicity for my setting later. – Hans Aug 06 '18 at 19:59
  • @Quantuple: Would you like to take a look at a similar question regarding the interest rate risk premium https://quant.stackexchange.com/q/41286/6686? – Hans Aug 16 '18 at 01:26

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