Deriving Risk-Neutral Simple-Return Moments from Log Returns
Summary
The document asks how to connect risk-neutral moments of simple gross returns to moments of log returns under a Black–Scholes model. It applies Itô’s lemma to obtain the stock’s terminal value, then uses the moment-generating function of a normal variable to derive second- and third-moment expressions, substituting the risk-free rate for the drift under the risk-neutral measure and using implied variance as an estimate of integrated variance.
The author reports that the second-moment calculation is close to a result obtained by replicating the payoff with options, while the third moment differs substantially, and asks what is wrong. The document does not include a resolution, so the derivations should be treated as a question rather than a validated method. In particular, the stated VIX measure refers to a 30-day variance horizon, and the comparison depends on matching the payoff’s maturity and the assumptions used to infer risk-neutral moments from option prices.
Key ideas
- The question derives simple-return moments from normally distributed log returns in a Black–Scholes setting.
- The risk-neutral calculation substitutes the risk-free rate for the stock’s drift.
- The author uses implied integrated variance as an input to the return-moment formulas.
- The reported third-moment discrepancy remains unexplained in the document.
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# Risk-neutral Simple Return Moment Log-return Moment
# Risk-neutral Simple Return Moment Log-return Moment
I am trying to find a way to link Risk-neutral moment of simple return to risk-neutral moment of log-returns.
Specifically, by making the same standard assumptions of the Black-Scholes model with the stock having the SDE
$$\frac{d S_{t}}{S_{t}}=\mu d t+\sigma d W(t)$$
with $W(t)$ being a standard Brownian motion, $\mu$ and $\sigma$ being constant, I apply Ito's lemma and obtain
$$S_{T}=S_{t} e^{\mu(T-t)+\sigma(W(T)-W(t))}$$
Then, by knowing the VIX, which is the risk-neutral (integrated) variance over the next 30days
$$V I X^{2}=E_{t}^{Q}\left[\int_{t}^{t+30 d} \sigma^{2} d t\right]$$
I obtain the risk-neutral 2nd moment of the simple gross return as:
\begin{aligned} E_{t}^{Q}\left[\left(\frac{S_{T}}{S_{t}}\right)^{2}\right] &=E_{t}^{Q}\left[e^{2\left[\left(\mu-\frac{1}{2} \sigma^{2}\right)(T-t)+\sigma(W(T)-W(t))\right]}\right.\\ &=e^{2\left(r_{f}-\frac{1}{2} \sigma^{2}\right)(T-t)} E_{t}^{Q}\left[e^{Z}\right], \quad Z \sim N\left(0,4 \sigma^{2}(T-t)\right) \\ &=e^{2\left(r_{f}-\frac{1}{2} \sigma^{2}\right)(T-t)} e^{0+\frac{4}{2} \sigma^{2}(T-t)} \\ &=e^{2 r_{f}(T-t)+\sigma^{2}(T-t)} \end{aligned}
by using the property of expectation of the exponential of a normal random variable and with $\sigma^2(T-t)$ being my VIX estimate.
I tried using the same methodolody on the 3rd moment, which should yield:
\begin{aligned} E_{t}^{Q}\left[\left(\frac{S_{T}}{S_{t}}\right)^{3}\right] &=E_{t}^{Q}\left[e^{3\left[\left(\mu-\frac{1}{2} \sigma^{2}\right)(T-t)+\sigma(W(T)-W(t))\right]}\right.\\ &=e^{3\left(r_{f}-\frac{1}{2} \sigma^{2}\right)(T-t)} E_{t}^{Q}\left[e^{Z}\right], \quad Z \sim N\left(0,9 \sigma^{2}(T-t)\right) \\ &=e^{3\left(r_{f}-\frac{1}{2} \sigma^{2}\right)(T-t)} e^{0+\frac{9}{2} \sigma^{2}(T-t)} \\ &=e^{3 r_{f}(T-t)+3 \sigma^{2}(T-t)} \end{aligned}
I tried testing whether this approach would give the same result as the Breeden and Litzenberger formula of writing the contract $g(S_T) = (\frac{S_T}{S_t})^n$ in terms of options, and while for the 2nd moment it gives very similar result for the 3rd it gives totally different ones.
I was wondering why it's wrong.
Thanks a lot!Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.