Historical Volatility Uncertainty and Black–Scholes Call Values
Summary
The document asks whether Black–Scholes call valuation should use a point estimate of variance from historical returns or average option values across the sampling distribution of that estimate. It describes treating the sample variance as uncertain, mapping variance draws through the pricing model, and comparing the resulting expected call value and interval with the value obtained from the point estimate. The example reports a higher expected value than the point-estimate price, while the probability that a draw produces a higher value is close to one half.
The responses raise separate issues rather than fully resolving the questions. One explains that under Black–Scholes assumptions, changing from the real-world to the risk-neutral measure changes drift but not volatility. Another argues that historical volatility is not the market-implied volatility used for practical option pricing. The answers do not reconcile estimation uncertainty with volatility risk premia, nor do they establish that a higher model value means the traded option is actually undervalued. The numerical illustration is model-based and depends on its assumptions and variance-estimation setup.
Key ideas
- A point estimate of variance and the expected option value over a variance distribution need not coincide.
- Uncertainty in historical variance estimates can produce a distribution of Black–Scholes call values.
- Within the stated Black–Scholes assumptions, the risk-neutral transformation changes drift while leaving volatility unchanged.
- The responses distinguish historical volatility estimates from market-implied volatility.
- A model value above a point-estimate value does not by itself establish that a market option is undervalued.
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# Expected value of Black-Scholes
# Expected value of Black-Scholes
(Apologies for any formatting mistakes)
Within the Black Scholes model, given that you are estimating the volatility from historical data - and all other parameters assumed exact - one usually substitutes the sample variance as a point estimate for the square of the volatility and evaluates the BScall using that point estimate.
However, why do we use a function of the point estimate instead of the expected value of the distribution of the estimate?
The sample variance follows a Chi-Squared distribution, so we now have a distribution of values of the Call Option based on the observed sample variance and degrees of freedom.
$$ D\sim BSCall \left( \frac{(n-1) \text{s}^2}{\chi_{n-1} ^2} \right) $$
The Expected Value of that distribution is rarely equal to the function of the point estimate.
Example, assume sample variance was .25 out of 52 weekly returns (so n=51 values used to estimate variance):
$$ S=100\\ K=95\\ r=0.10 \\ s^2=.25\\ T=0.25\\ $$
Yields the point estimate of
$$ BSCall(s^2)=13.6953 $$
But
$$ E[D]=13.8372 $$
with 95% confidence intervals of {12.2222, 15.9196}
In fact
$$ P[D>BSCall(s^2)]=0.525 $$
Question is two fold:
- For using historical data, why do we use a function of the point estimate instead of the expected value of the distribution of the estimate?
- If using the point estimate, does the above imply there is a 52% chance the call option is actually undervalued?
Thank you
## Answer by user25064 (score 4, accepted)
https://quant.stackexchange.com/a/9952
Two parts
- Real world vs risk neutral: Can we even estimate risk neutral volatility using historical data? There is a difference in distribution of the underlying stock price under the real world and risk neutral measures. Luckily, changing to the risk neutral measure does not affect volatility, only the drift. Thus, a real world measure of volatility will properly estimate the risk neutral volatility. In the BS framework, we assume that the stock price is an Ito drift diffusion process with constant coefficients. In equations; $$S_t = S_0 \exp\{(\mu - \sigma^2/2)t + \sigma W_t\} = S_0 \exp\{(r - \sigma^2/2)t + \sigma (W_t - \frac{\mu - r}{\sigma}t)\} \\ = S_0 \exp\{(r - \sigma^2/2)t + \sigma W_t^\star\} $$ see that volatility is the same when writing the equation for stock price in real world or risk neutral.
## Answer by Wilmer E. Henao (score 2)
https://quant.stackexchange.com/a/9946
Black-Scholes is just a model that tries to replicate what the market is doing. Unfortunately, any theoretical estimate of volatility (that is not the implied) that you come up with will be wrong.
In fact, you don't want to use historical volatilities at all.
The only correct volatility to use is the IMPLIED VOLATILITY. And the reason why it works is because it is designed to be the one that makes the model work. (sounds redundant, but this is why it's called implied... the one Implied by the model)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.