Why Price and Implied-Distribution Calibrations Produce Different Q Models

# What is the meaning of the difference of Q measures calibrated to prices vs. implied probabillities?


# What is the meaning of the difference of Q measures calibrated to prices vs. implied probabillities?












What's the meaning of the differences between Q measures (and calibrated parameters of a model) fit to prices vs market implied probabilities?

Updated and Clarified Question: If I calibrate a Merton Jump Diffusion model to prices by minimizing the squared total dollar error $$\epsilon = (\vec{O} \cdot (\vec{P}_{est} - \vec{P}))^2$$ where $O$ is open interest, $P_{est}$ is the estimated prices from the MJD model and $P$ are the actual prices using drift equal to the risk free rate (30 day treasury yield), I'll find a calibrated set of volatility, mean jump amplitude, jump standard deviation, and jump intensity (number of jumps per year) under some $\mathbb{Q}$ measure. Using those values, I can generate an analytical PDF of the model's implied volatility is for an MJD path with those parameters.

I could, alternatively, find the implied probabilities, either by taking the second derivative of the options chain times $e^{rt}$ where r is the risk free rate, or I can find the probability distribution $f$ that minimize the above loss function when prices are $$ \vec{P}_{est} = e^{-rt}\mathbf{A} \times \vec{f}$$ where $\mathbf{A}$ is a matrix of payouts versus final outcome and strike. Either way finds basically the same $f$. That itself is a $\mathbb{Q}$ measure and not based on any model at all. I then can fit that $f$ to the MJD model's PDF to find a different set of calibrated set of volatility, mean jump amplitude, jump standard deviation, jump intensity, and drift that describes the MJD path that produces a distribution equal to the implied distribution.

If I do those two things, I end up with two different sets of parameters and two somewhat very different looking PDFs (see below). Those two PDFs are consistently very different looking: the direct prices-calibration has the mode at a lower value and higher variance than implied PDF-calibrated.

That brings us to the question: what is the meaning of that $\mathbb{Q}$ measure when calibrated to prices directly versus calibrated to the implied PDF? One seems to be "model parameters that explain prices under a $\mathbb{Q}$ measure" while the other seems to be "model parameters that would generate a path with an ultimate distribution that produces the discounted future values under a $\mathbb{Q}$ measure." Why aren't they the same distribution?

Original Question:

Background: Models commonly have SDEs that yield probability distributions of the underlying as a function of time. For example, the BS model is just a log normal; the Merton jump diffusion model is the infinite sum of weighted log normal distributions each weighted by the probability of a certain number of jumps in the time period.

Armed with this analytical distribution model, you can extract what the market "thinks" of an option chain by fitting the analytical distribution to the market implied probability (e.g., from the discounted second derivative of the option chain prices). Doing that will get you the model parameters that would evolve the SDE in to the implied probability.

You also can fit the model to the prices, directly. If you take the SDEs and apply portfolio hedge (or find a unique martingale, or other math tricks), you get all the traditional pricing models for call or puts. You can fit the current prices to those pricing models and also extract the same parameters. For BS, it's a delta hedge. For MJD, it's a hand-waiving "diversify portfolio so jumps wash out."

Example: Today's ES put chain for Oct 18 2024 contracts gets you the following sets of MJD parameters.

| Parameter | Implied Mkt Prob Calibration | Prices Calibration |
| Calibrated Volatility | 7.04% | 12.89% |
| Calibrated Jump Mean | -3.31% | -10.26% |
| Calibrated Jump Std | 5.24% | 15.99% |
| Calibrated intensity (jumps/year) | 3.79 | 0.44 |
| Calibrated drift | 4.32% | N/A |

Below are the market implied probabilities (black circles), the pdfs of the two calibrated models: one model calibrated to the price (green) and the other to the market implied probabilities (red).

Question: If you do both calibrations, you get two different sets of parameters. What is the meaning of the two different sets? I'm not asking "which one is correct," as they both are - they are asking different things, but I'm not sure what the intuition is between those two. One set fits prices well and probabilities poorly. One set fits probabilities well and prices poorly. Yet both were from the same underlying SDE and options chain.

Is this thinking correct? Does it have something to do with the fact that the MJD pricing model basically says "the jumps are riskless (which isn't necessarily true), and drift can be delta hedged away?" while the probability model doesn't assume anything about the market? Is it therefore a consequence of MJD not having a complete market, and the pricing model is wonky compared to the probability?

## Answer by Scott Howard (score 2)

https://quant.stackexchange.com/a/80582

Answering my own question after a bunch of thinking and playing with the data. Referring to Schoutens, et. al., "A Perfect Calibration ! Now What?", any stochastic process for $S_t$ that is complete, risk-neutral, and arbitrage-free has to yield the same implied probability distribution at time $T$ under a $\mathbb{Q}$ measure. Different processes have different intermediate distributions between $t$ and $T$, but since all those processes are martingales, the expected values at any intermediate time is the same for all processes. Why I was seeing two different distributions at time $T$ when calibrated to prices or implied probabilities comes from the fact that the weighting of errors are different and that I made a mistake of mixing $\mathbb{P}$ and $\mathbb{Q}$ measures.





In summary: the implied distribution from prices and from the calibrated $\mathbb{Q}$ process is supposed to be the same. Any differences are from choosing imperfect models that don't fit the data or uncertainty in the data.