Numerics of Implied Binomial Trees

For about 20 years now, discrepancies between market option prices and Black and Scholes (BS) prices have widened. The observed market option price showed that the BS implied volatility, computed from the market option price by inverting the BS formula varies with strike price and time to expiration. These variations are known as “the volatility smile (skew)” and volatility term structure, respectively.

In this chapter, we describe the numerical construction of the IBT and compare the predicted implied price distributions. In Section 10.1, a detailed construction of the IBT algorithm for European options is presented. First, we introduce the Derman and Kani (1994) (DK) algorithm and show its possible drawbacks. Afterwards, we follow an alternative IBT algorithm by Barle and Cakici (1998) (BC), which modifies the DK method by a normalisation of the central nodes according to the forward price in order to increase its stability in the presence of high interest rates. In Section 10.2 we compare the SPD estimations with simulated conditional density from a diffusion process with a non-constant volatility. In the last section, we apply the IBT to a real data set containing underlying asset price, strike price, time to maturity, interest rate, and call/put option price from EUREX (Deutsche Börse Database). We compare the SPD estimated by real market data with those predicted by the IBT.