Pricing Binary Options from Empirical Returns with Gaussian KDE

Summary

The document proposes estimating cash-or-nothing call values from empirical log-return distributions. For each selected time horizon, it smooths observed returns with a Gaussian kernel density estimate, then uses the fitted distribution’s cumulative probability at the log strike-to-spot threshold as the option value. The approach treats the probability of finishing above the strike as the payoff value, assuming the asset pays no dividends.

The proposal is a modeling idea rather than a validated pricing framework: it provides no empirical tests or comparisons with market prices. Its formula omits discounting and does not establish that historical or empirical return probabilities are risk-neutral probabilities suitable for arbitrage-consistent pricing. Results may also depend on the sample period, bandwidth choice, horizon matching, and changes in the return distribution. KDE smoothness cannot by itself resolve these assumptions or risks.

Key ideas

• The method estimates each horizon’s log-return distribution from observed asset prices.
• Gaussian kernel density estimation smooths the discrete empirical return sample.
• The suggested call value is the fitted probability that the terminal price exceeds the strike.
• Pricing from empirical probabilities requires assumptions beyond the distribution fit, including an appropriate pricing measure.

Tags

# Pricing binary options with kernel density estimation


# Pricing binary options with kernel density estimation












Suppose I have a large enough set of prices of an asset, from which I can extract the following function: $f:T\to\mathcal{D}$, where $T$ is a fixed finite set of time intervals (say, 1 minute, 2 minutes, ..., $N$ minutes), and $\mathcal{D}$ is a set of finite discrete distributions of $\mathbb{R}$, such that $f(t)=D$ means the following: the (empirical) distribution of log-returns of a price in a time interval of length $t$ is $D$.

Since I assume most distributions $D$ will appear unimodal, if I tend to see them as random samples of an unknown underlying continuous unimodal distribution, a Gaussian kernel distribution estimation might be useful to model $D$ without over-smoothing. So, let $g:T\to\mathcal{D}'$ be a function which given a time interval $t\in T$ returns the Gaussian KDE of $f(t)$.

I'll now describe a method for pricing binary options (cash-or-nothing call): suppose the current price is $s$, the strike price is $k$, the time to maturity is $t$ (and assume $t\in T$), and assume there are no dividends. Let $\varphi$ be the cumulative distribution function of $g(t)$. Then, the price $c$ of the option is simply

$$c = \varphi\left(\ln\left(\frac{k}{s}\right)\right)$$

Do you think this is an appropriate pricing model? Do you see any expected pitfalls?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.