Computing Risk-Neutral Price Moments from Option Prices
Summary
The document explains how to recover moments of an asset’s risk-neutral terminal price distribution from call prices, assuming a zero discount rate. It relates the second strike derivative of call prices to the distribution’s density, then uses integration by parts to express the mean as the zero-strike call value and the second moment as twice the area under the call-price curve.
For functions such as the logarithm, the answer recommends splitting the calculation at a cutoff strike and using put prices below the cutoff and call prices above it. Integrating by parts twice then expresses the expectation through option prices, their derivatives, and the function’s second derivative. The author argues that suitable tail behavior keeps the boundary terms finite. The explanation is general rather than a worked numerical example, and finiteness depends on the distribution and tail conditions; the zero-rate assumption also simplifies the setup.
Key ideas
- The second strike derivative of a call price gives the risk-neutral terminal price density under the stated assumptions.
- The risk-neutral mean equals the zero-strike call price when the discount rate is zero.
- The second price moment can be written as twice the integral of call prices over strikes.
- For functions with problematic endpoint behavior, split the expectation into lower-strike puts and upper-strike calls.
- Two integrations by parts yield an option-price representation, subject to suitable tail behavior.
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# Computing moments of implied distribution
# Computing moments of implied distribution
For simplicity let's assume that the discount rate $r = 0$, then a price of a call option with strike $K$ and maturity $T$ on an asset with positive price can be computed as $C(K) = \Bbb E_Q(S_T-K)^+$. A known fact that this leads to $C''(K)$ being the density of the risk-neutral distribution $(S_T)_*Q$ of the asset price at maturity. I was playing with this fact to compute first two moments of this distribution for the price itself and its logarithm.
What I got is $\Bbb E_QS_T = C(0)$, that is the forward price, which makes perfect sense. Computations are pretty easy thanks to integration by parts: $$ \Bbb E_QS_T = \int_0^\infty KC''(K)\mathrm dK = \int_0^\infty K\mathrm dC'(K) = \int_0^\infty C'(K) \mathrm dK =C(0) $$ Then I got $\Bbb E_QS^2_T = 2\int_0^\infty C(K) dK$ using the same technique, which does not look natural to me, and in particular I don't see why would that always be greater than $C^2(0)$. Perhaps, monotonicity and concavity guarantees that.
Nevertheless, what I am having problem with is computing $\Bbb E_Q\log S_T$ and $\Bbb E_Q\log^2 S_T$ since if I do integration by parts there, on the very first step I get two terms that are infinite. I do however see $\frac1{K^2}$ starting to appear there, which afaik should be a part of the final result, however I was unable to get the explicit formulas. Can someone help here?
## Answer by Frido (score 3, accepted)
https://quant.stackexchange.com/a/76045
With calculating moments I'll assume you mean calculating $$ E[S_T^n] \enspace\text{for} \enspace n \in \mathbb N $$ Now in general \begin{align*} E[f(S_T)] &= \int_0^\infty f(K) C''(K) dK \\ &= \int_0^\alpha f(K) C''(K) dK + \int_\alpha^\infty f(K) C''(K) dK \\ &= \int_0^\alpha f(K) P''(K) dK + \int_\alpha^\infty f(K) C''(K) dK \\ \end{align*} because $P''(K) = C''(K)$ as you've already pointed out, and $\alpha$ is some cut-off point (eg ATM).
Integrating by parts twice gives \begin{align} E[f(S_T)] &= [f(K)P'(K)]_0^\alpha + [f(K)C'(K)]_\alpha^\infty \\ &\quad - [f'(K)P(K)]_0^\alpha - [f'(K)C(K)]_\alpha^\infty \\ &\quad + \int_0^\alpha f''(K) P(K) dK + \int_\alpha^\infty f''(K) C(K) dK \end{align}
In general, for example if $f(S_T) = S_T^2$ or $f(S_T) = \log S_T$ the terms $P'(K), P(K)$ decay faster than $f(K), f'(K)$ for $K \to 0$ and $C'(K), C(K)$ decay faster than $f(K), f'(K)$ for $K \to \infty$. Thus the integrals and the boundary terms will remain finite.
Let me know if this is answers your question, and if not pls let me know which part has not been clarified.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.