Dickson, Samuel Barnaby; (2002) Pricing and hedging European options in the presence of taxes. Doctoral thesis (Ph.D), University of London.

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Abstract

Following the work of Milevsky and Prisman (Milevsky (1997a)) we develop a tax-adjusted hedging algorithm that finds the tax-adjusted hedging price of a European option. We use a non-recombining binomial tree framework because the tax liability on the stock transactions is path dependent. In general, we find that in a Cox Ross Rubinstein environment (Cox(1979)) the hedging portfolio (of stock and bond) does not equal the option payoff, on a post-tax basis; there is a tax-mismatch. The algorithm uses an iterative procedure to force the tax-mismatch to zero across all the final nodes on the tree, thereby ensuring that the option writer is fully hedged on an after-tax basis. We can consider the non-recombining binomial tree framework as being one in which we have an equal number of unknowns (the deltas and bond amounts at each node on the tree) as linearly independent equations. We can find the general forms for the deltas and bonds by partially solving the system of equations. The simultaneous equation algorithm uses these general forms to find the tax-adjusted price of the option. This algorithm is less demanding on memory and computationally faster than the tax-adjusted hedging algorithm. The simultaneous equation approach allows us to derive an analytic formula for the tax-adjusted hedging price of the option, and in the one-period case we can use this to prove some of the empirical results found using the tax-adjusted hedging algorithm. We relax one of the assumptions made in the original framework - the tax year-end coincides with the option's maturity - and allow a tax year-end to occur during the life of the option. This requires us to consider two tax charges: one paid during the life of the option at the first tax year-end, and one paid after the option expires at the second tax year-end. The simultaneous equation approach is used again and we develop the tax year-adjusted simultaneous equation algorithm that finds the tax-adjusted price of the option when the tax year-end can occur during the option's life. Scholes has derived a modification to Black-Scholes, termed the tax-adjusted Black-Scholes equation (Scholes (1976)). We form a tax-adjusted risk neutral probability in the Cox Ross Rubinstein environment and use this to form the tax-adjusted binomial option pricing model. This is shown to be the discretetime precursor to the tax-adjusted Black-Scholes equation. The tax-adjusted Black-Scholes equation is generalised to relax the assumption in the original derivation that the derivative is taxed as income. A martingale derivation is given for this equation, as for Milevsky and Prisman's tax-adjusted Black- Scholes equation with dividends (Milevksy (1997a)).

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