Mean Reversion and Option Pricing Under Risk-Neutral Valuation
Summary
The document explains why mean reversion in a stock’s historical price process does not, by itself, change its no-arbitrage option value. Under the risk-neutral measure, the stock’s drift becomes the short rate while its instantaneous volatility remains the same, so the Black–Scholes formula applies with that volatility, assuming the model’s other conditions hold.
The historical dynamics still matter when estimating volatility from observed returns. Mean reversion affects the relationship between return variability over a sampling interval and instantaneous volatility, so a simple standard deviation of returns may not provide the appropriate input. The document cites an example in which strong mean reversion makes estimated instantaneous volatility higher than historical annualized return variability. It does not derive that conversion or discuss broader model limitations, such as jumps or stochastic volatility; its conclusion is specific to the stated diffusion and replication framework.
Key ideas
- Under no-arbitrage pricing, the stock’s risk-neutral drift is the short rate.
- The document says the diffusion volatility is unchanged when moving from the historical to the risk-neutral measure.
- For the stated process, option value follows Black–Scholes if its volatility input is the instantaneous volatility.
- Historical mean reversion changes how volatility should be inferred from sampled returns.
- A standard deviation of returns alone may misstate instantaneous volatility when mean reversion is strong.
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# Options Pricing and Mean Reversion
# Options Pricing and Mean Reversion
I'm confused about the impact that a mean reverting stock price process has on the value of an option on it.
Several sources say that there is indeed an impact on the price of an option:
Option pricing and mean reversion
Lo and Wang (1995)
Yet, another source seems to say that mean reversion has no impact on the price of an option:
"The drift term of the process has no impact on the price of a call option, since we know that under the correct pricing measure we need the discounted stock price to have zero drift. This is achieved by changing the drift of the original process, rendering any initial drift term irrelevant"
- Mark Joshi, Quant Job Interview Questions and Answers.
So I guess my ultimate question is, if the stock price follows the following process:
$$ dS_t=\alpha(\mu-S_t)dt+\sigma S_tdZ$$
Is the price of an option on the stock just equal to the BSM price where $\sigma_{BSM} = \sigma$?
It would make sense to me that there is no effect, because the replicating portfolio argument still works, and we end up with the same PDE and boundary conditions, which would give the same price.
## Answer by Antoine Conze (score 6, accepted)
https://quant.stackexchange.com/a/45980
Joshi is correct. The no arbitrage argument implies that the stock price instantaneous return under the risk neutral measure is equal to the short rate, and the girsanov theorem implies that the instantaneous volatility $\sigma$ is the same under the historical measure and under the risk neutral measure, so under the risk neutral measure the stock price is a GBM with drift $r$ and volatility $\sigma$ and the option pricing formula is BS with volatility $\sigma$ even when the stock process is mean reverting under the historical measure.
However if the stock process is mean reverting under the historical measure, then an historical estimation of $\sigma$ must take into account the mean reversion and cannot rely on simply computing the standard deviation of $\delta t$ returns. For instance with a 100% mean reversion a 20% historical annual standard deviation would translate into approximately 30% instantaneous volatility. This is in essence the topic of the Lo and Wang paper, computing options prices with mean reversion under the historical measure assuming the historical standard deviation is known.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.