Implied Binomial Trees with Cubic Spline Smoothing

Abstract

The binomial model is a workhorse of numerical option pricing. But the plain vanilla model produces a risk-neutral probability density for the stock price at expiration that becomes lognormal in the limit. This is consistent with Black–Scholes’ assumptions but not with actual option prices in the market. An alternative is an implied tree, which begins with the risk-neutral density extracted from the market prices for options with the desired maturity. The procedure constructs a lattice that is consistent with the desired distribution. An N-step tree has N + 1 terminal nodes, so fitting the tree to the market prices can be quite time consuming. The author proposes a simplification that greatly increases efficiency with negligible cost in accuracy. The trick is to calibrate only a subset of the terminal nodes to prices in the options market and fill in the other nodes by cubic spline interpolation.