Option pricing with a two-piece lognormal distribution

We derive a closed-form European option pricing model in a discrete-time utility-based setting where the underlying asset has a two-piece lognormal distribution. In our set-up, the market does not have to be dynamically complete, which makes our model applicable even in cases in which the underlying asset is illiquid. The two-piece lognormal distribution results from joining two opposite halves of two distinct lognormal distributions, where each half has a different scale parameter. Hence, it is possible to adjust the tails of the distributions to accommodate the implied volatility of in-the-money and out-of-the-money options. We show that our option pricing equation can generate several types of volatility patterns and therefore can be used to price a wide range of assets.