# 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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If you price options off the real-world probabilities like that, you are going to lose your shirt. –  Brian B May 12 at 20:25
Could you explain why you believe so? –  Bach May 13 at 6:23
Hi. I like the approach you are trying to take. But a couple questions, how come you didn't incorporate the risk free rate? Also you lost me at the end, what's the intuition behind taking the ln of (k/s) and then the cumulative probability of that value? –  Algo4Life Jun 17 at 17:20