The waterline tree for separable local-volatility models

The implied volatility is the unique volatility value that makes the celebrated Black-Scholes formula yields a traded option's price. Implied volatilities at varying strike prices and maturities form the implied volatility surface. Empirically, this surface is never flat. The local-volatility (LV) model is an option model that attempts to fit the implied volatility surface. It is popular because the preference freedom of the Black-Scholes model is retained. A tree consistent with the implied volatility surface generated by an LV model is known as implied tree. Past attempts to construct the implied tree, however, are prone to having invalid transition probabilities. In fact, this problem occurs even when the volatility surface is flat as in the Black-Scholes model. This paper unearths a potentially fundamental reason for that failure: the trees contain repelling fixed points. As efficient and valid trees for general LV models remain elusive despite decades of research, this paper turns to separable LV models. An efficient and valid binomial tree is then built for such models. Our novel tree is named the waterline tree because its upper part (the part that is above the water, so to speak) matches the moments of the price, whereas the lower part matches the moments of its logarithmic price. This break from traditional trees ensures that only attracting fixed points remain. Numerical results confirm the excellent performance of the waterline tree.