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In this chapter, we have developed a new, more flexible and powerful method to construct risk-neutral, arbitrage-free, semi-recombining implied binomial trees that are consistent with given market prices of liquid-traded options. The advantage of our method for constructing implied binomial trees is that no interpolation or extrapolation steps are necessary and no prior guess about the benchmark distribution is required. This is achieved by using a’ smoothness criterion’ to recover the implied risk-neutral probability distribution. Additionally, we have to solve a quadratic programming optimization problem with linear inequality constraints, which can be easily solved with standard software. Furthermore, our method uses all the available information on market prices to estimate the IRNPD, since the IRNPD of each maturity date incorporates the IRNPDs of all previous maturity dates. Under the additional assumption that a volatility function exists, the method can be used to construct arbitrage-free, risk-neutral, recombining implied multinomial trees. As a result, we are able to price and hedge many plain-vanilla and exotic options in accordance with given market prices.

Further research should examine the empirical performance of the method and compare it to existing approaches in a more extensive test. Here, it is of special interest which method performs better — constructing implied binomial trees or constructing implied multinomial trees. This is equivalent to the question of whether the assumption of equal path probabilities in each sub-tree or the assumption of the existence of a volatility function leads to better empirical results.

In this chapter, we have developed a new, more flexible and powerful method to construct risk-neutral, arbitrage-free, semi-recombining implied binomial trees that are consistent with given market prices of liquid-traded options. The advantage of our method for constructing implied binomial trees is that no interpolation or extrapolation steps are necessary and no prior guess about the benchmark distribution is required. This is achieved by using a’ smoothness criterion’ to recover the implied risk-neutral probability distribution. Additionally, we have to solve a quadratic programming optimization problem with linear inequality constraints, which can be easily solved with standard software. Furthermore, our method uses all the available information on market prices to estimate the IRNPD, since the IRNPD of each maturity date incorporates the IRNPDs of all previous maturity dates. Under the additional assumption that a volatility function exists, the method can be used to construct arbitrage-free, risk-neutral, recombining implied multinomial trees. As a result, we are able to price and hedge many plain-vanilla and exotic options in accordance with given market prices.

Further research should examine the empirical performance of the method and compare it to existing approaches in a more extensive test. Here, it is of special interest which method performs better — constructing implied binomial trees or constructing implied multinomial trees. This is equivalent to the question of whether the assumption of equal path probabilities in each sub-tree or the assumption of the existence of a volatility function leads to better empirical results.