Maximum Entropy Calibration of Option-Implied Price Distributions

Summary

This question describes fitting a probability distribution for an asset’s terminal price to discrete option quotes. It proposes using a distribution inferred from market implied volatilities as a prior, then minimizing relative entropy from that prior alongside weighted squared errors between model-implied and observed call prices. The call payoffs act as constraints or calibration targets, while the weights control the cost of quote mismatches.

The central concept is relative entropy, also called Kullback–Leibler divergence: it measures how much a candidate distribution departs from the prior. Minimizing that divergence favors distributions close to the prior, while the price-error penalty allows adjustments to better match quotes. The document poses the rationale as a question and supplies no derivation, calibrated distribution, market data, or guidance for choosing penalty weights. Its formulation therefore outlines an approach but leaves important implementation and modeling choices unresolved.

Key ideas

• A prior terminal-price distribution can be inferred from market implied volatility quotes.
• Call option prices can be represented as expected payoffs under a candidate distribution.
• Relative entropy penalizes departures from the prior distribution.
• A weighted squared-error term trades closeness to market quotes against closeness to the prior.
• The proposed setup does not specify how to select penalty weights or validate the fitted distribution.

Tags

# Maximum entropy probability distribution for $S_T$ implied from discrete market quotes


# Maximum entropy probability distribution for $S_T$ implied from discrete market quotes












Consider a maturity $T$, for this maturity I have some implied volatility from market denoted $\sigma^{0}_{i}$. I want to interpolate these volatility using Entropy approach, by using $\sigma^{0}_{i}$ as prior.

From $\sigma^0_i$, market prices of options $C_i$ can be obtained, out of which a probability distribution function $\mathbb{P}_0$ for $S_T$ is derived.

I want to find a distribution $\mathbb{P}$ for $S_T$ such that

- The entropy is maximized

- $\mathbb{P}_0$ is used as prior distribution

- The distribution $\mathbb{P}$ yields prices as near from market prices as possible.

I know the probability distribution $\mathbb{P}$ is given by the ollowing minimization problem:

$\mathbb{E}^{\mathbb{P}}\left[\ln\left(\frac{d\mathbb{P}}{d\mathbb{P}^0}\right)-1\right]+\sum_{i}\omega_i\left(\mathbb{E}^{\mathbb{P}}\left[f_i(S)\right]-C_i\right)^2$

$\mathbb{P}_0$ is the market implied probability distribution of $S_T$ linked to the $\sigma^{0}_{i}$.

$f_i(S)=max(S_T-K,0)$, i.e. the payoff of market call options, with strike $K$

$C_i$ are market prices of call options

$\omega_i$ are weights

Then I understand that the term $\sum_{i}\omega_i\left(\mathbb{E}^{\mathbb{P}}\left[f_i(S)\right]-C_i\right)^2$ imposes a penalty to deviations to market prices.

But I do not understand the term

$\mathbb{E}^{\mathbb{P}}\left[\ln\left(\frac{d\mathbb{P}}{d\mathbb{P}^0}\right)-1\right]$

What is the rationale of this term? Why does it maximize entropy?

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.