Distribution-based option pricing on lattice asset dynamics models

We provide an option pricing formula based on an arbitrarily given stock distribution, where the problem of optimally hedging the payoff on a European call option is considered through a self-financing trading strategy. An optimal hedging problem is solved on a trinomial lattice by assigning suitable probabilities on the lattice, where the underlying stock price distribution is derived directly from empirical stock price data which may possess heavy tails. We show that these probabilities are obtained from a network flow optimization. Numerical experiments illustrate that our formula generates the implied volatility smile, in contrast to the Black-Scholes formula.