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?
