Option P&L Fit Versus Likelihood for Return Distribution Models
Summary
The document considers how to fit a predicted distribution of future stock log returns using the realized profit and loss of European calls and puts at expiration. It asks how this objective translates mathematically and compares it with likelihood-based fitting, including a likelihood measure based on the probability mass within a return interval scaled to each stock’s volatility.
The responses distinguish aggregate P&L fit from likelihood. Optimizing aggregate option outcomes may suit a trader with a large, diversified sample, but may be unreliable with limited observations; likelihood focuses on avoiding surprising observations and may be more appropriate for smaller samples. A further caveat is that return distributions change over time, making direct distribution prediction difficult. Modeling distribution parameters or moments may be more practical. These are qualitative observations, not a demonstrated comparison across data sets, and the document gives no general proof that either objective will perform better.
Key ideas
- Fitting option P&L optimizes aggregate outcomes across the traded options.
- Aggregate P&L objectives may require large samples to produce reliable estimates.
- Likelihood fitting emphasizes how surprising individual observations are and may suit smaller samples.
- Return distributions are nonstationary, which complicates direct prediction.
- Predicting distribution parameters or moments may be more practical than predicting a full distribution.
Tags
Full text
# What best price fit by option P/L means in math terms? # What best price fit by option P/L means in math terms? I fit some algorithm predicting the distribution of stock future log returns, based on historical prices. The optimisation goal - best fit of European options premium. Done as backtesting on historical prices - start with 0 cash, and after selling millions of European Calls and Puts with various strikes, compared with actually realised outcomes from historical stock prices on expiration date - end up with 0 cash. I wonder what exactly I did, how "best option pricing fit" translates into math terms, and if there's a better, shorter, faster to compute way to express the goal. It feels as the fit is similar to optimising some momentum functions. P.S. The Likelihood after the option P/L fit is slightly worse than if fit done as best Likelihood. Note - I calculated Likelihood unusual way, as the actual probability of the ~1% spread, to be able to compare Likelihoods of stocks of various volatility (the Likelihood in its original form as the densities have different scale for different stocks and are not comparable) ``` spread = (cdf.quantile(0.95) - cdf.quantile(0.05)) / 100 likelihood = cdf.p(x+spread/2) - cdf.p(x-spread/2) ``` ## Answer by Alex Craft (score 0) https://quant.stackexchange.com/a/82273 It seems to me the difference is huge. With option P/L price fit the aggregate outcome optimised. For aggregate estimates to be correct - a large sample required, it doesn't work for small samples. Such goal may be good for huge trader who trades thousands of options on various stocks. It may be not good for small trader. With Likelihood the least surprise optimised. May not be the best for huge trader, but should be good for small trader. It should be correct even for the small sample. ## Answer by user93883 (score 0) https://quant.stackexchange.com/a/85799 Distributions are not stationary. Therefore, predicting them is hard. More practical is predicting model parameters or moments of the distribution.
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.